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A quantitative exploration of self-enforcing dynamic contract theory Sigouin, Christian 1998

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A QUANTITATIVE EXPLORATION OF SELF-ENFORCING DYNAMIC CONTRACT THEORY by CHRISTIAN SIGOUIN B.A.A., Ecole des Hautes Etudes Commerciales, 1992 M.sc., Ecole des Hautes Etudes Commerciales, 1993 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of economics) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November 1998 © Christian Sigouin, 1998 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e it f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s m a y b e g r a n t e d b y t h e h e a d o f m y d e p a r t m e n t o r b y h i s o r h e r r e p r e s e n t a t i v e s . It is u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t m y w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f ECQi\jQ N \ J C 5 T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r , C a n a d a D a t e AlwcM8£K gp 4 U ] t^fi D E - 6 (2788) Abstract This dissertation studies three different aspects linked to the literature on self-enforcing dynamic contracts. Namely, this dissertation examines how a solution to this type of eco-nomic models may be obtained numerically, how important enforcement issues might be for a common question in economics, and how the presence of self-enforcing constraints may be investigated empirically. It is composed of three essays. The first essay develops a numerical method designed to approximate the solution of models with self-enforcing constraints using a dynamic programming approach. This method may also be used to approximate the solution of general dynamic models with occasionally binding inequality constraints. It complements standard value function iteration algorithm with an interpo-lation scheme which preserves the concavity and the monotonicity of the value function. It has the advantage over usual value function iteration algorithms of procuring a reasonable degree of accuracy at a relatively lower computational cost. The second essay uses dynamic contract theory to analyze the joint behavior of in-vestment decisions and financial flows when contracts between lenders and borrowers are subject to enforcement constraints. In contrast to the usual belief that financing con-straints lead firms to underinvest, this essay shows that firms are likely to overinvest. While overinvestment is shown to be consistent with the empirical finding that investment spending is excessively sensitive to variations in internal funds' abundance, it does not give rise to a financial accelerator. The key feature of this model is that firms' production and financial capacities are simultaneously determined. Firms overinvest when external funds are relatively inexpensive if they apprehend the possibility of becoming financially constrained in the future. By increasing their production capacity in such a way, firms alleviate eventual shortages of funds arising from the fact that external finance has become limited. Finally, the third essay studies how a common implication arising from the litera-ture on self-enforcing contracts may be tested empirically. A key feature of a long-term self-enforcing contract is that the quantity subject to its terms evolves over time accord-ing to a simple updating rule; it is set to its full-enforcement level whenever doing so does not induce one of the agents to renege. Otherwise, it is set to a self-enforcing level. Using the example of Thomas and Worrall's (1988) labor contract model (to which pro-ductivity growth is added), it is shown that this updating rule may be expressed as an endogenous switching-regression model. Panel data may be used to estimate this model. ii When there are measurement errors, Monte-Carlo experiments show that the switching-regression model usually has a poor goodness of fit in small data sets. However, despite this finding, tests of the null hypothesis that conventional contract models generate the data under scrutiny still have a high power against the alternative hypothesis that this data is characterized by the presence of enforcement constraints. iii Contents Abstract ii Contents iv List of tables vi List of figures vii Acknowledgement viii Chapter 1 Introduction 1 Chapter 2 Shape-Preserving Approximations of the Value Functions arising from Dynamic Models with Inequality Constraints 2.1 Introduction 7 2.2 Description of the algorithm 11 2.3 Concave and monotone interpolation 18 2.4 Choosing the state space partition 29 2.5 Computation of optimal policies 30 2.6 Accuracy of the algorithm 33 2.7 Conclusion 35 Appendix 2.1 36 Appendix 2.2 39 Appendix 2.3 39 Appendix 2.4 40 Chapter 3 Investment Decisions, Financial Flows, and Self-Enforcing Contracts 3.1 Introduction 46 3.2 The model 51 3.3 Outside opportunities 55 3.4 The self-enforcing financial contract 57 3.5 Equilibrium 60 3.6 Financial flows 62 3.7 Investment decisions 65 3.8 Numerical results 70 3.9 Conclusion 75 iv Appendix 3.1 77 Appendix 3.2 84 Chapter 4 A Procedure for Investigating the Presence of Self-Enforcing Contraints Empirically 4.1 Introduction 93 4.2 A self-enforcing wage contract 96 4.3 Empirical implementation 102 4.4 Small sample properties 106 4.5 Conclusion 115 Appendix 4.1 116 Appendix 4.2 117 References 126 v List of tables N Table 2.1 Approximation errors 43 Table 3.1 Value of parameters 86 Table 3.2 Investment and internal funds' abundance 87 Table 3.3 Overinvestment statistics 88 Table 3.4 Summary statistics 89 Table 3.5 Moving-average representation of output 90 Table 4.1 Value of the parameters used for the simulation and the estimation exercises 119 Table 4.2 Percentage of periods where a self-enforcing bound is attained 120 Table 4.3 Estimation statistics 120 Table 4.4 Estimation results 121 Table 4.5 Root mean squared errors 124 Table 4.6 Accuracy in identifying actual realized regimes 124 Table 4.7 Proportion of rejected null hypotheses Ho in favor of the alternative hypothesis Hi : partial enforcement 125 . vi List of figures Figure 2.1 Implementation of concavity on a single square 44 Figure 2.2 Implementation of concavity on a mesh of points 45 Figure 3.1 Impulse responses - capital stock 91 Figure 3.1 Impulse responses - financial flows 92 vii Acknowledgement I would like to thank my thesis supervisor Paul Beaudry for his helpful comments, his encouragements, and his constant support. I am also indebted to Mick Devereux, John Cragg, Marc Duhamel, Roger Farmer, Francisco Gonzalez, Hashmat Khan, Cherie Metcalf, and Jim Nason for their helpful comments. The financial support of the Social Sciences and Humanities Research Council of Canada is gratefully acknowledged. viii Chapter 1 Introduction Issues related to the difficulty in enforcing contracts have been examined by a growing number of researchers in recent years. In particular, many have focused their attention on the dynamic aspects of commitment issues in long-term relationships. Concerns about agents' capacity to credibly commit themselves to fulfill their contractual engagements arise when these agents have other opportunities at their disposal. If contracts1 are not fully enforceable (through the actions of a third party, for example), agents may abandon any on-going relationship to take advantage of these opportunities. Long-term relationships prove to be useful in this context; incentives to renege may be mitigated in long-term relationships by counterbalancing any immediate gains from reneging with the promise of greater future benefits. Long-term contracts may thus support equilibria that would not be sustainable otherwise. The possibility that agents renege on their contractual engagements confers a crucial role to the opportunities available to them if they do so. The value of these opportunities imposes lower bounds on the value that any contract may offer to an agent. A contract will not be breached provided that it offers to each agent, in any period and any state of nature, at least as much value as the one offered by any other opportunity at their disposal. Otherwise, agents will eventually renege. A contract meeting this requirement is self-enforcing. Its terms are such that it is always in the agents' self-interest not to renege on their engagements; by design, behaving opportunistically would not procure them with more value.2 The impact of limited enforceability for long-term contracts has been analyzed in various contexts. For example, Thomas and Worrall (1988) consider a self-enforcing wage contract. Worrall (1990) examines a similar contract in the context of international lending. Marcet and Marimon (1992) study the impact of enforcement issues on growth in an international setting. Thomas and Worrall (1994) look at the relationship between investment and the 1 Throughout, the terms "contract" and "contractual relationship" are used to refer to explicit as well as implicit contracts. Models where the dynamic consistency of plans is required are closely related to models with self-enforcing contracts. See for example Chari and Kehoe (1993), and Kehoe and Levine (1993). 1 lack of enforceability in the context of foreign direct investment. Kletzer and Wright (1995) examine game-theoretical aspects of self-enforcing contracts. Kocherlakota (1996) derives empirical predictions from a self-enforcing risk-sharing arrangement. Albuquerque and Hopenhayn (1997) analyze the effect of partial enforcement on optimal lending contracts. Finally, Gauthier, Poitevin, and Gonzalez (1997) study how self-enforcing contracts are affected when agents have the possibility to make payments at the outset of a period. While the focus of these papers differs, a common conclusion emerges: Enforcement issues must not be neglected; they may influence the outcome of long-term contractual relationships in a significant way. Yet, the assumption that contracts are fully enforceable is common. This is usually meant as a simplifying assumption. Explicitly allowing for the fact that agents may breach a long-term contract requires the introduction of dynamic participation constraints which may bind only on occasion. These participation constraints (usually called self-enforcing constraints) impose the requirement that the contract in question be self-enforcing. That is, they compare the value offered to any agent by the contract with the one offered by alternative opportunities. As the value offered by the contract varies with optimal policies, and vice versa, the level of complexity involved in determining these policies may be substantial. This may explain in part why enforcement issues are often disregarded. Enforcement issues are also disregarded on the basis that courts of law may in prin-ciple be called upon to compel agents to fulfill their contractual engagements. For this reason, enforcement issues have typically been examined in international settings,3 since it is potentially difficult to revert to courts of law to enforce contracts across national boundaries. However, arguing that these issues may be relevant for all types of contracts, not only the ones developed in international contexts, does not deny the existence of a legal system. It has been argued by Hart and Holmstrom (1987) that it may be too costly to revert to courts to enforce the day-to-day governance of contracts. Moreover, it may be difficult for a third party to verify whether or not agents have effectively fulfilled their con-tractual engagements. The desirability or even the possibility to rely on the legal system to enforce contracts may therefore be limited. Hence, even if courts may be used to enforce contracts, there may be a need to design contracts in a manner which avoids this situation as much as possible. Enforcement issues may thus nevertheless play an important role in the design of contractual relationships. 3 See for example Worrall (1990), Marcet and Marimon (1992), and Thomas and Worrall (1994). 2 However, while considering enforcement issues seems rather important on theoretical grounds, there is no direct empirical evidence indicating of their relevance for the gover-nance of contractual relationships in practice. There are only a few papers attempting to investigate empirically whether there is evidence that enforcement issues matter for con-tractual relationships. Beaudry and Dinardo (1991, 1995) find evidence consistent with the fact that enforcement concerns may influence contractual relationships in the labor market. However, their findings are only indirect in the sense that they do not explicitly test the specific restrictions imposed on the data by self-enforcing constraints. Nevertheless, their results suggest that further empirical investigations may prove worthwhile. This dissertation extends the current literature on self-enforcing dynamic contracts in several directions: First, it proposes a numerical method allowing one to approximate the solution of dynamic contracts with self-enforcing constraints. Exact solutions for this type of models are not obtainable in general. Moreover, characterizing these solutions is usually quite complicated in view of the fact that self-enforcing constraints are constraints which may bind only on occasion. The numerical algorithm proposed in this dissertation gives one the possibility to investigate further the implications of enforcement issues in quite sophisticated models. Moreover, it may be used to evaluate the significance of theoretical results in a given setting. Second, this dissertation attempts to provide a better understanding of the possible macroeconomic implications of enforcement issues. In particular, it explores how enforce-ment constraints may influence the behavior of real variables. The focus of this endeavor is however quite different from the one in the current literature; in the spirit of Hart and Holmstrom (1987), enforcement issues are not examined within an international framework. Moreover, numerical results are presented in order to provide a quantitative assessment of the possible economic significance of these issues. Third, this dissertation proposes an em-pirical strategy making possible to proceed to a direct appraisal of the restrictions imposed on the data by self-enforcing constraints. Using this method may allow one to ascertain whether there is a real need to consider enforcement issues when examining economic problems. This dissertation is composed of three essays. The first essay develops a numerical dynamic programming algorithm. This algorithm is specifically designed to approximate the solution of dynamic economic models characterized by the presence of inequality con-straints which bind (hold with equality) only on occasion. Self-enforcing dynamic contracts fall within this category of models. In a self-enforcing contract, the value offered by any 3 alternative opportunities to this contract imposes lower bounds on the value that it may offer to each agent. These lower bounds may or may not be attained in a particular period, that is, the self-enforcing constraints may bind only on specific occasions.4 The algorithm proposed in this essay complements the standard value function iteration algorithm with a shape-preserving approximation scheme which allows one to compute optimal policies by solving a system of nonlinear equations numerically. Standard numerical algorithm find a value function which solves the Bellman equation, iteratively, by successive approximations. They proceed by replacing the continuous state space of the value function with a finite partition of points. With this restricted state space, the value function only needs to be evaluated at a finite number of points. Therefore, iterations on the Bellman equation to find its solution only involve finite matrices. This approach may alternatively be seen as approximating the value function with a collection of step functions. In view of the poor properties of this approximation scheme, the first essay proposes to use a smoother approximation of the value function. This approximation is required to preserve the intrinsic curvature properties of the value function, namely, concavity and monotonicity (if necessary). Failing to enforce concavity on the function approximating the value function would render the approach invalid: Each iteration on the Bellman equation involves the compu-tation of optimal policies, conditional on the current approximation of the value function. Iterating on the Bellman equation using approximations which may not be concave ev-erywhere would lead to erroneous results. This would occur because optimal policies cannot be determined uniquely if the approximate value function used is not concave. A shape-preserving approximation scheme avoids this situation. This algorithm has several advantages. It allows one to obtain a solution to the Bellman equation, an approximate value function, with a reasonable degree of accuracy at a fairly lower computational bur-den than standard methods. Moreover, it is especially well-suited for dynamic contract models with self-enforcing constraints. Self-enforcing constraints sometimes involve a lower bound on the value function itself. Thus, the precision with which optimal policies are determined in this case, directly depends on the precision with which the value function is approximated. The fact that a self-enforcing constraint is binding or that it may bind in the near future affects decision rules in a significant way. In general, policy functions do not vary smoothly with the state of the economic system. They usually have "kinks" at the points of the state space where an inequality constraint begins to hold with equality. 4 The second essay examines how firms' investment decisions are affected when they are funded through self-enforcing financial contracts. In this essay, firms finance investment spending by entering infinitely-lived contractual relationships with lenders. Investment decisions, financial flows, and financing constraints are simultaneously determined. The capital stock play a dual role in this economy; it is a factor of production and an asset which may be pledged as collateral to secure borrowings. Increasing investment expenditures does not only increase firms' production capacity but also improves their future financial capacity. Firms and lenders cannot commit themselves to respect the terms of financial arrangements; firms may default, lenders may liquidate the collateral capital stock. A feasible financial contract in this context is one that offers to the firms and the lenders as least as much value as these options do. Lenders and firms adhere to such contract because it is in their self-interest to do so; they do not gain anything by reneging. The dynamic interaction between financial flows and firms' investment decisions is investigated in this context. Several properties of the model are explored both analyti-cally and numerically. It is shown that the inability of borrowers and lenders to commit themselves to a contract can help explain the empirical finding that investment spending is excessively sensitive to firms' abundance of internal funds. This excess sensitivity of in-vestment is often taken as evidence that firms faced with financing constraints underinvest; an internal funds' windfall allows them to increase investment spending further, hence the positive relationship between internal funds and investment. However, the model studied here suggests quite a different interpretation. Firms typically overinvest at the end of an expansion, in anticipation of bad times, when internal funds are scarce. Increasing investment above the one dictated by opportuni-ties augments firms' capacity to raise external finance later on because an increased capital stock allows them to collateralize more borrowings in the future. This finding stresses the importance of examining firms' investment decisions in a fully dynamic framework in order to take into account of the potential effects that the endogenous nature of collateral assets may have. In addition, it is shown in this essay that in spite of the fact that investment spending is excessively sensitive to the availability of internal funds in this model, output levels do not exhibit excess volatility nor much propagation. These findings contrast with the conclusions usually drawn from the empirical evidence of excess sensitivity. Finally, the third essay studies how a common implication arising from the literature on self-enforcing contracts may be tested empirically. This is done using the example of Thomas and Worrall's (1988) labor contract model, to which productivity growth is added. 5 This essay demonstrates a key feature of long-term self-enforcing contracts. That is, the quantity subject to its terms (the wage level in this precise example) evolves over time according to a simple updating rule; it is set to its full-enforcement level whenever doing so does not induce one of the agents to renege. Otherwise, it is set to a self-enforcing level. The third essay shows that this updating rule may be expressed as a two-sided endogenous switching-regression model. Switches between regimes are triggered by the relative position of the full-enforcement wage level with respect to a compact interval of self-enforcing wage levels. In any period, the optimal wage differs from the full-enforcement level by the least amount possible such that it lies within this interval. Estimating the switching-regression model amounts to estimating the value taken by the upper end and the lower end of this interval in any period. It is shown for this specific example that these values depend on the strength of labor demand in any period and the level of skills attained by an individual worker. Parameterizing the value of each interval's ends as function of these variables allows one to estimate the switching-regression model. Note that as the value of the lower end and upper end of the interval correspond to the limits imposed on the wage level by the self-enforcing constraints, estimating this model therefore constitutes a direct appraisal of the presence of self-enforcing constraints. It is shown that the switching-regression model may be estimated using panel data. Monte-carlo experiments are used to evaluate the empirical strategy's capacity to detect the presence of self-enforcing constraints. In particular, the performance of the approach proposed in this essay is examined in the case where there is measurement errors. This situation is likely to occur since precise data on individual contractual relationship may be hard to obtain. In the presence of measurement errors, the experiments undertaken show that the switching-regression model usually has a poor goodness of fit in small data sets. However, tests of the null hypothesis that conventional contract models generate the data under scrutiny still have a high power against the alternative hypothesis that this data is characterized by the presence of enforcement constraints. In all the experiments considered, it is in general possible to detect the presence of self-enforcing constraints most of the time despite the fact that parameters estimates are quite imprecise. 6 Chapter 2 Shape-Preserving Approximations of the Value Functions arising from Dynamic Models with Inequality Constraints 2.1 I n t r o d u c t i o n . In general, it is not possible to obtain an explicit solution for most dynamic economic models. Their solutions usually have to be approximated numerically. Obtaining such ap-proximate solutions becomes problematic when dynamic models with occasionally binding inequality constraints are considered. Dynamic models featuring this type of constraints are common in the recent literature. For instance, models with liquidity or credit con-straints fall into this category.1 It is also the case for models of the dynamic contracting literature.2 Occasionally binding inequality constraints affect optimal decisions in a non-trivial way. On the one hand, in any period, binding inequality constraints limit the set of feasible policies. On the other hand, decisions made in any period affect the likelihood of whether or not these constraints will bind in the future. Explicitly taking this dynamic interaction into account is not an easy task. This essay proposes a numerical method designed to approximate the solution of a dynamic programming problem involving occasionally binding inequality constraints. Ob-taining the solution of a dynamic programming problem entails that the underlying value function be found. That is, one has to find a function v which solves a functional equation of the form v = T(v). This functional equation is commonly known as Bellman's equa-tion. In principle, the function solving the Bellman's equation may be found by successive approximations, starting from an initial arbitrary function u°, by simply iterating on the recursion vn = T(vn~l) until a fixed point v is attained.3 Numerical dynamic programming amounts to finding an approximate numerical representation vn for each function vn. In fact, this is equivalent to confining the functions obtained from iterating on T( •) to a finite-dimensional subspace of the space containing the actual value function.4 A familiar approach used to solve dynamic programming problems numerically is to 1 See for example, Deaton (1991), Huggett (1993), Hansen and Imrohoroglu (1992). 2 See for example, Gertler (1992), Marcet and Marimon (1992), and Thomas and Worrall (1988, 1994). See Stokey and Lucas (1989) and Rust (1996) for a statement of the conditions under which this is possible. 4 One may also see this procedure as iterating on the parameters vector characterizing v. 7 discretize the value function's domain (the state space) into a finite partition of points.5 As pointed out by Bertsekas (1976), when the points of this partition are considered as interpolation nodes rather than as the state space itself, this procedure is equivalent to approximating each function vn with piecewise constant functions (i.e. step functions). In principle, nothing prevents one from using approximation schemes which have better con-vergence properties. For example, Bellman, Kalaba and Kotkin (1963) and Hartley (1995) propose the use of polynomials. Judd (1990) uses Chebyshev polynomials. The idea is that smooth interpolation schemes may provide a better approximation of each vn than a piecewise constant interpolation scheme, using a relatively smaller number of interpolation nodes.6 This may lead to a substantial reduction of the space and computing time required to obtain a solution.7 However, a fundamental difficulty arises if relatively smooth approximation schemes are used. In most economic applications, the value function is concave (convex) and sometimes, monotonically increasing or decreasing in its arguments. In general, smooth interpolation schemes do not preserve the shape of the function being approximated; they tend to exhibit unwarranted wrinkles or erratic oscillations between interpolation nodes, especially when a drastic change in slopes is required. Thus, the numerical representation of the actual value function will not have the appropriate shape in general; it may be convex on a portion of its domain while the actual value function is concave. This is troublesome. For example, computing T(vn) at one point of the state space requires that a static maximization problem be solved. The solution of this maximization problem is unique provided that vn is concave. Hence, if the function vn is approximated with a function vn that is not concave everywhere on its domain, T(vn) may substantially differ from T(vn). Iterations on the operator T( •) may thereby become unstable. Therefore, gains in efficiency may be seriously limited. The use of smooth interpolation schemes is only valid to the extent that it yields approximations which have the same intrinsic curvature properties as the actual value function; namely, concavity and monotonicity. This essay proposes a shape-preserving interpolation scheme which ensures that each iteration using the operator T( •) yields approximate functions vn that are concave and possibly monotone. Concavity and monotonicity are enforced on the interpolants, when 5 See for example Christiano (1990) and Tauchen (1990). In general, an extremely large number of interpolation nodes is required to achieve a reasonable level of accuracy with piecewise constant interpolants. Another advantage of relatively smooth approximation schemes is that allows one to use first-order conditions in order to compute optimal policies at each iteration. 8 necessary, using a procedure similar to the ones encountered in the shape-preserving inter-polation literature (see for example Carlson and Fritsch (1985, 1989), Fontanella (1987), and Dougherty, Edelman and Hyman (1989)). That is, intervals of admissible values are found for the parameters characterizing the interpolant. If one parameter's value lies out-side these intervals, the interpolant does not have the proper shape; this parameter's value has to be adjusted accordingly. Judd (1996) also suggests a similar approach. He proposes the use of a shape-preserving interpolation scheme, due to Schumaker (1983), based on piecewise quadratic Hermite polynomials. The interpolation scheme set forth in this essay is based on piecewise cubic Her-mite polynomials. Approximations based on these polynomials have better convergence properties than those based on piecewise quadratic Hermite polynomials.8 This type of interpolation scheme is chosen for several reasons. First, a piecewise cubic Hermite ap-proximation is built from a collection of cubic polynomials pieced together. At any given point of its domain, the shape of the resulting approximation is uniquely determined by the data provided at nearby interpolation nodes. It may thus be adjusted locally without affecting concavity and/or monotonicity between other interpolation nodes. This consid-erably simplifies the task of enforcing concavity and monotonicity since only a small set of parameters need to be adjusted. Second, as opposed to smoother interpolation schemes, piecewise cubic Hermite approximations allow for rapid change in curvature.9 This property is desirable when approximating value functions arising from models with occasionally binding inequality constraints. Such value functions generally exhibit sudden changes in their slopes; as inequality constraints begin or cease to bind, kinks usually appear in the value functions' first derivatives, thereby leading to discontinuous second derivatives. By nature, relatively smooth interpolation schemes cannot generate this kind of functions. Using such schemes would only result in approximations displaying an erratic behavior near the nodes where the value functions' slopes are required to change dramatically. Finally, an Hermite interpolation scheme requires that the interpolant's first derivatives be provided at each interpolation node. For a value function, these may be obtained from the so-called envelope conditions. Therefore, piecewise cubic Hermite interpolants should provide approximations at each interpolation node which are consistent with the fact that a value function is being interpolated (i.e. the first derivatives near each 8 • * • 1 Piecewise cubic Hermite polynomials yield approximations that are C everywhere and possibly, but not 1 necessarily, C at interpolation nodes. Piecewise quadratic Hermite polynomials are only C at interpolation nodes. 9 But possibly smoother than the one produced by piecewise quadratic Hermite polynomials. 9 interpolation node agree with the envelope conditions). This is likely to improve the accuracy of the value functions approximations. Christiano and Fischer (1994) describe and compare several algorithms which may be used to solve dynamic models with occasionally binding inequality constraints. These al-gorithms are essentially methods for finding sequences of optimal policies and sequences of Kuhn-Tucker multipliers which simultaneously solve the stochastic Euler equations10 and the Kuhn-Tucker conditions characterizing an optimal solution. Numerical methods seek-ing the solution to stochastic Euler equations are usually thought as being more efficient than methods solving dynamic programming problems.11 Their use is however limited to a specific class of models. The dynamic programming approach allows one to seek the solutions to a broader class of dynamic models. For example, in a model where the decision-maker is allowed to choose a nature-contingent menu of values for an endogenous state variable rather than a single value for any future realization of nature, equilibrium conditions cannot be summarized with a single stochastic Euler equation. In this case, one Euler equation (one first-order condition) is required for each possible future realization of nature12 as the endogenous state variables in future periods take on a different value in each state of nature. This is a feature found in many models of the dynamic contracting literature because dynamic contracts often allow for state-contingent intertemporal trades between agents.13 Moreover, in some models, the value attached to future actions explicitly enters the constraints set.14 This occurs for instance if dynamic consistency constraints are introduced in a model. The computation of the value of any action undertaken is in general required by this type of constraints. These quantities are usually readily available using dynamic programming. Section 2.2 describes a general class of dynamic programming problems for which a so-The terms "stochastic Euler equation" refer to the arbitrage relationship relating the current marginal cost of undertaking an action and its future expected marginal benefits (see Stokey and Lucas 1989, p. 280). 1 1 Numerical dynamic programming algorithms usually suffer from the well-known curse of dimensionality which states that the time required to compute the solution of a dynamic programming problem rises exponentially with the number of state variables. 12 • In this type of models, states of nature are usually drawn from a discrete set of possible values. 1 3 See for example, Thomas and Worrall (1988, 1990, 1994), Kletzer and Wright (1995), and Kocher-lakota (1996) 1 4 See for example Thomas and Worrall (1988, 1994) and Marcet and Marimon (1992) where optimal policies must at least deliver a specified minimum amount of discounted expected utility to an agent in any period. 10 lution may be obtained using the algorithm proposed in this essay. Attention is limited to dynamic programmes where uncertainty is described by a finite set of states of nature. In addition, only cases with at most two endogenous state variables are considered. Allowing for more general problems is not impossible but increasingly difficult. Nonetheless, a wide variety of economic models may be examined within this framework. The basic strategy followed to approximate numerically the solution of dynamic programming problems is also introduced in this section. Specific elements of the algorithm are discussed in subsequent sections. The shape-preserving interpolation scheme used to approximate value functions is set forth in section 2.3. Conditions guaranteeing the concavity and the monotonicity of piecewise cubic Hermite interpolants are presented. Section 2.5 describes how optimal policies are computed using Kuhn-Tucker conditions. The system of nonlinear equations involving occasionally binding inequality constraints is transformed into a system of equa-tions which does not involve any. This enables the use of a standard Newton's algorithm to solve for optimal policies. Section 2.6 provides some statistics allowing one to evaluate the algorithm's accuracy. Finally, section 2.7 concludes with some observations. 2.2 Description of the algorithm. This section begins with a description of the class of dynamic economic problems for which a solution may be approximated numerically using the method proposed in this essay. For the sake of exposition, it is assumed that the underlying economic models have infinite time horizons. Extension of the numerical method to solve economic models with finite horizons is trivial. 1 5 This essay considers dynamic economic problems which may be written as dynamic programming problems of a given general form. That is, the state space, the transition probabilities for the exogenous state variables, the constraints set, and the functional equation characterizing a general dynamic programming problem is described. Approximate solutions to economic models which may be characterized within this framework may be obtained using the numerical method explained thereon. This method is restricted to the class of dynamic programming problems for which the concavity of the value function may be established theoretically. Let S denote the state space of some economic model. This state space is composed of mx endogenous state variables and of TXIQ exogenous state variables. Let X be the set of possible values taken by the endogenous state variables and let 0 be the set of possible 1 5 At the limit, the strategy proposed in section 2.5 to compute optimal policies may be used to solve static maximization problems subject to occasionally binding inequality constraints. 11 values taken by the exogenous state variables. The set of possible states of the economic model is thus given by S — X x Q. For simplicity, it is assumed that 0 is a finite set containing N elements, that is 0 = {9i,82,... , # A T } . The probability of drawing state 0j from 0 conditional on state 6j having been drawn previously (i.e. Pr(0' = 9{ 16 = 9j)) is denoted by 7 ^ . It is assumed that endogenous state variables may take on different values depending on the particular realization of the exogenous state variables. That is, an element of S is denoted by a pair (x(0),0) where x(9) EX and 0G 0 . 1 6 For simplicity, x(6a) is written as x3 for 03G@ and a nature-contingent menu of values taken by the endogenous state variables (xi, x2,..., #jv) as x. Primes are used to distinguish between the values taken by a state variable in two successive periods. In any given period, x' denotes a nature-contingent menu of endogenous state variables for next period. Let g(x', c, xs, 93) be a mg-dimensional vector function where c is a mc-dimensional vector of control variables. It is assumed that the set of feasibility constraints, when the current state is (x3,93), may be written as mg inequalities g(x', c, x3, Ga) < 0. These in-equality constraints may bind at all times (i.e. become equality constraints in equilibrium) or bind only on some occasions. The set of feasible values for the control variables is thus given by the correspondence17 This essay considers dynamic economic models which, given any state vector (x3,63) ES, may be written as the following dynamic programming problem where r(c, x3,Qa) is a real-valued instantaneous return function and where 8 € (0,1) is a discount factor. The function v is the value function. The functional equation (1) is commonly referred to as Bellman's equation.18 It is assumed that the constraints corre-spondence and the instantaneous return function are such that it is possible to establish that the value function is concave. This general formulation allows one to handle models of the dynamic contracting literature where state-contingent intertemporal trades are allowed between the agents. 17 The reason for indexing the correspondence Q with the set X will become clear later on. 1 8 See Bellman (1957). nx(x9,6a) = {(x',c) | {x',c)eXN xnm% g(x',c,x3,93) <0}. N (1) s.t. (x',c)e£lx{x3,03) 12 Finding the solution to the dynamic programming problem (1) entails that the under-lying value function v be found. A value function iteration algorithm finds the value func-tion solving (1) iteratively, by successive approximations. The argument is the following: Define the right-hand side of the functional equation (1) as an operator T: C(S) —> C(S) mapping some appropriate functional space on S into itself. That is, for some func-tion uGC(S), let N T(u)(xs,9a) = max{r(c,x9,93) + py^uix'^e^TTsi} s.t. (x',c)eQ,x(x3,9a;v) for (x„,03) £ S, s = 1,2,... , i V . 1 9 Since v = T(i>), by definition, finding a solution to the functional equation (1) amounts to finding a fixed point of the operator T. Under appropri-ate conditions,20 the sequence of functions {vn} obtained from the recursion vn — T(vn~l) (given some initial function v°) eventually converges to v as n increases. Hence, the value function solving the functional equation (1) may be found, in principle, by iterating suc-cessively on the mapping T until a fixed point is reached. However, in practice, this task is not so simple; a functional space is an object too intangible to be handled easily in practice. The conventional approach taken to address this difficulty is to discretize the state space S. Let Z be a compact subset of X and let ZQ C Z be a discrete and finite parti-tion of Z with nz elements. Then, the set ZQ X 0 is a discrete partition of Z x 0. As the set Zo x 0 is finite, the image of a function v£C(Zo X 0) may be represented by a finite matrix. Therefore, defining the operator T on C(ZQ X 0) rather than defining it on C(S) reduces the iterative procedure described above to a problem involving finite matrices uniquely. For example, this approach is described in Christiano (1990) and in Tauchen (1990). However, while finding an approximate solution to equation (1) by suc-cessive approximations becomes a tractable problem under this approach, it is an highly inefficient way to do so. This is commonly referred to as the "curse of dimensionality". A very fine partition of Z is often required to achieve any reasonable degree of accuracy because the policy correspondence Q x ( r E s , # 3 ) is replaced by Clz0(xs^a) in this case. The 1 9 Note that the policy correspondence fi has been made conditional on the function v in the definition of T. In models of the dynamic contracting literature, the value function itself often appears in the constraints set. The reference to the specific function v used to compute optimal policies is thus essential in this context. 2 0 See Stokey and Lucas (1989) or Thomas and Worrall (1994) for the case of dynamic contracts. It is assumed that the dynamic model behind functional equation (1) meets the appropriate conditions. 13 policies chosen when the operator T is restricted to C[ZQ X 0) are optimal conditionally on the fact that the values of x' are taken from a discrete set ZQ . However, endogenous state variables are usually continuous quantities in most economic applications. Hence, optimal policies chosen from £lz0(') m a y be a poor approximation for those chosen from •) unless ZQ is a sufficiently fine partition. Obtaining a solution using a relatively large number of partition points may prove itself highly time consuming, especially as the number of state variables increases. In view of this difficulty, an alternative approach has been proposed by Bellman, Kalaba and Kotkin (1963). Rather than approximating the state space with a discrete one, they suggest to approximate the value function with an interpolant, that is, a continuous function characterized by a finite set of parameters.21 Consider a family of functions on Z x 0 for which each individual element may be identified uniquely with the value of a parameter vector a. An individual function of this family is denoted by v(x, 9; a). The function v is said an interpolant of v if for a given vector of parameters a. In the case of Hermite interpolation, the derivatives of v also agree with those of v at each point of ZQ X 0 (see section 2.3 for more details). Informally, the argument is that one could seek an approximation of the value function v by iterating on the vector of parameters a rather than on a set of function's values, as it is the case with the previous approach (e.g. see Judd 1990 and Hartley 1996). The advantage of this approach with respect to the preceding one is that the values of x' no longer have to be confined to the finite set ZQ , only to Z N . Then, if v is a good approximation of i / , this may allow to reduce substantially the number of points in the partition ZQ needed to achieve a reasonable degree of accuracy because these points now only serve as support abscissas for the interpolation scheme and not as the state space itself. More formally, let the interpolant of the real-valued function u on C(Z x 0), v( •; a), belong to a family of interpolants tpa for which a = {u(x,9) G 7£ | V (x, 9) G ZQ X 0}. In this case, a belongs to A = lZn2xN. Define an operator L from T on C(ZQ X 0) by L(a) = T(v( •; a)). That is, let v(x3, 6S; a) — v(x3, 9S) for each (x3,93) G ZQ X 0, N L(a)(xa,93) = max (x',c) {r(c, x3,93) + S^v^O^a)^} (2) s.t. (x',c)en,z(xa,93;a) 21 This approach is used in Judd (1990) and Hartley (1996) for example 14 for (xa,9a) € ZQ x © and vE<pa- Clearly, L(a) also belongs to A. Therefore, L(a) maps A onto itself. The idea is to generate a sequence {a n} from the recursion an = L(an~l). Hence, one may seek a fixed point in some finite-dimensional parameters' space A rather than in an infinite-dimensional functional space. The interpolant v formed from each element of the sequence {a n} provides an approximation for each function of the sequence {vn} that would be obtained if the operator T was applied on C(Z x 0). In effect, this procedure amounts to restricting the operator T to a closed subspace of C(Z x ©) . That is, an iterative procedure based on L produces a sequence of interpolants {{>(•; a")} fully characterized by a finite-dimensional vector instead of a sequence of functions {vn} from an infinite-dimensional space. The only hope is that both sequences converge to the same limit. This will usually depend on the convergence properties of the type of interpolants chosen for v and on the fineness of the partition ZQ. Bertsekas (1976) points out that the conventional approach described previously (i.e. using a discrete state space) is equivalent to approximating the value function with piece-wise constant interpolants. Notice that this type of interpolants belongs to ipa. Thus, this approach is a special case of the one based on equation (2). Convergence to a fixed point is guaranteed in this case. However, unless an extremely fine partition of the domain is used, piecewise-constant interpolants produce approximations of very poor quality. Hence, the fixed point found using this approach may differ substantially from the actual limit of the sequence {vn}. One could in principle use a class of interpolant in (fa that produces approximations of better quality with a coarser partition of the domain (i.e. a reduced number of interpolation nodes). Convergence (pointwise) to a fixed point should also oc-cur in this case since this approach is essentially the same as the preceeding one; only the degree of accuracy of the value function's approximation between interpolation nodes is increased. Approximating the value function v with a relatively accurate interpolant rather than approximating its domain with a finite partition should allow one to benefit from the advantages of solving a dynamic economic model using a dynamic programming method at a lower computing cost. However, while certainly appealing, as such, this approach has a serious shortcoming in practice: The difficulty with interpolants having good convergence properties (i.e. interpolants producing relatively smooth surfaces), in practice, is that they tend to produce approximations displaying wrinkles or oscillations between support abscissas (between points of the partition ZQ): Even if in theory, an interpolant converges to some function, it might do so by oscillating around it. This is quite undesirable in the 15 present context. Well-behaved value functions generally have properties like concavity and monotonic-ity. A simple interpolation scheme may thus produce an approximation of the value func-tion exhibiting a spurious behavior between support abscissas (interpolation nodes). Note that each iteration using the operator L entails that optimal policies be computed. As the concavity of the value function is a crucial property permitting to adequately deter-mine these policies22, using an approximation of the value function which does not have the proper shape to compute optimal policies will lead to erroneous results. Hence, each application of the operator L will compound errors due to the wrong choice of optimal policies. This may prevent the sequence {u( •; an)} to converge to the same limit as the se-quence {vn} does; if not at all. Therefore, while appealing in theory, the approach proposed by Bellman, Kalaba and Kotkin (1963) and advocated by Judd (1990) and Hartley (1996) may fail in practice. This essay specifically addresses this problem. The interpolation scheme considered in this essay delivers an approximation of the value function which has the same shape as the one possessed in theory by the value function v solving the functional equation (1). Given the nature of most economic applications, any sequence {v( •; an)} originating from successive applications of the operator L is required to be composed of concave and possibly, but not necessarily, monotone functions.23 This guarantees that optimal policies will be determined adequately every time the operator L is applied. Judd (1996) advocates the use of a similar approach. The strategy followed to enforce concavity and/or monotonicity in this essay is similar to the one used in the shape-preserving interpolation literature. Shape-preserving interpolation schemes have been devised, for example, by Carlson and Fritsch (1985), Fontanella (1987), and Dougherty, Edelman and Hyman (1989). Such schemes are simple in essence; in general, shape-preserving constraints like monotonicity and concavity, are translated into bounds on the parameters characterizing a certain type of interpolants. If some parameters fail to fall within those bounds, they are adjusted accordingly. The shape-preserving interpolation scheme proposed in this essay uses piecewise cu-bic Hermite interpolants. This class of interpolants is chosen for four reasons. First, the Simply observe that the value function is part of the objective function in the maximization exercise in the definition of L. 23 . » Of course, this approach is appropriate only to the extent that the value function v itself has these properties. 16 parameters used to characterize an interpolant of this type only have local impacts on its overall shape. This greatly simplifies the task of imposing concavity and monotonicity because some parameters can be varied to adjust the shape of the interpolant where it fails to be concave or monotone without affecting the rest of its shape. Second, piecewise cubic Hermite interpolants produce approximating functions that are continuously differ-entiable once and possibly but not necessarily twice. This is an advantage when dealing with occasionally binding constraints. The slope of the value function usually changes dramatically as a constraint ceases or begins to bind, therefore creating non-differentiable first partial derivatives and "kinks" in the value function. An interpolant which produces approximating functions with relatively high degrees of smoothness are not generally able to capture appropriately these sudden changes in the slope of the value function. Third, a piecewise cubic Hermite scheme requires that the interpolant's first deriva-tives agree with the ones of the function being approximated at each point of the par-tition ZQ x ©. As the first partial derivatives of the value function at each point of the partition ZQ X 0 may be computed from the so-called envelope conditions, the interpolant's first derivatives at each of these points will have a value consistent with the fact that v indeed approximates a value function. Finally, given the value of the partial derivatives of 0( •; a) at each point of the partition ZQ X 0, the parameters vector characterizing this type of interpolant is given by a = {v(xa, 8a) \ (xa,9a) £ ZQ X 0}. That is, the vector a cor-responds to the values taken by u( •) at each point of ZQ X 0. The sequence of parameters vectors {an} generated from the recursion v(-;an) = L(an~l)(-) is intimately related to the sequence of functions {vn} obtained from the algorithm where the continuous state space is replaced with a discrete partition. Simply observe that an = L(an~l)(x, 9) for any (x,9) £ ZQ X 0. Therefore, the algorithm proposed in this essay retains the spirit of the standard algorithms used to solve dynamic programming problems. The algorithm proceeds as the standard value function iteration algorithm: Begin-ning with an initial parameters vector a° such that v( •; a°) is concave, L(a°) is computed. That is, optimal policies are solved for using the method described in section 2.5. They are then used to calculate L(a°). Setting v( •; a1) equal to L(a°) provides a new value for the parameters vector a 1. Its value is then verified against the concavity and monotonicity constraints set forth in section 2.3. The value of a1 is adjusted accordingly if one of these constraints is violated. Since a1 is such that the concavity of v( •; a1) is guaranteed, L(al) may be computed without any problem. Hence, the whole procedure may be repeated to obtain the vector a 2 and so on, until the convergence of the sequence of vectors an so 17 generated is attained. A complete step-by-step description of the algorithm is presented in the appendix 2.3. Details of the algorithm, namely, the interpolation scheme and the com-putation of optimal policies in the presence of occasionally binding inequality constraints, are discussed in the following sections. 2.3 Concave and monotone interpolation. A shape-preserving piecewise cubic Hermite interpolation scheme is developed in this sec-tion. Note that the explicit dependence of the value function on the exogenous state of nature 9 is omitted in this section. As uncertainty is described by a discrete set, the effect of 9 on the approximation of the value function v may be embedded in the parameter vector by having one such vector for each state of nature in 0 (that is, v(xa,9a; o) may be written as v(x3;a(0a))). The interpolation problem considered in this essay is defined in an unconventional way. Usually, one has at its disposal a set of ordinates describing the function being interpolated at each point of a given partition of its domain (i.e. ZQ in the present essay). Considering that a piecewise cubic Hermite interpolant is uniquely deter-mined by the value taken by this function and its derivatives at each one of these points, an algorithm for constructing such an interpolant is basically a procedure for computing a set of derivatives values. For the problem at hand, the set of value function's ordinates are not readily available but its derivatives may be obtained from the so-called envelope conditions arising from the underlying dynamic programme. Therefore, the interpolation problem may be defined the other way around. That is, its aim will be to determine an approximation of the value function, given a set of derivatives values. Hence, the approach taken in this essay towards shape-preserving interpolation will differ from the standard literature (e.g. Carlson and Fritsch (1985), Fontanella (1987), and Dougherty, Edelman and Hyman (1989)) in one crucial aspect: Usually, shape-preserving constraints on a piecewise cubic interpolant are translated into bounds of admissible values for its derivatives at each support abscissa. This comes from the fact that these are in some sense the free parameters of the interpolation exercise. For the present application, it is more sensible to define those shape-preserving conditions on the values taken by the interpolant at each of its support abscissas rather than on the value of its derivatives since the value function is unknown a priori but its derivatives are. Univariate results are presented first. Bivariate results will follow. Let the inter-val Z = [z, z] be the (restricted) domain of the value function (holding constant the 18 state of nature) and let the set {zi}^ be a partition of this interval (i.e. Zo) such that z = zi < Z2 < . . . < znt = z. This partition forms the set of support abscissas for the interpolation. Let v be a concave and possibly monotone function with ordinates {v(zi)Yllx on ZQ and suppose that the value of its first partial derivatives at each point of ZQ are known and given by the set {^(^i)}"^ • By definition, the elements of this set are arranged in a descending order since v is concave and are all of the same sign if v is monotone. The goal is to construct an interpolant v of v on [z,z] which is concave and monotone (only if v is). A piecewise cubic Hermite interpolant is used. That is, a cubic polynomial is defined on each subinterval of [z,z]. These polynomials are pieced together to form the interpolant v on [z,z]. In order to do so, a cubic Hermite interpolant p is defined on a standard interval [0,1]. From this interpolant, the function v may be formed on each subinterval [zi, Zi+i] by using a simple transformation of variables. Let the interpolant p be given by the following expression: p{x) =p{0)H0(x) +p{\)Hl{x) +Px(0)KQ(X)+Px(1)K1(X), (3) for xG[0,1], where the Hi and K{ are cubic Hermite basis functions for [0,1] (see Ap-pendix 2.2). Take any zE[z,z]. Necessarily, z is enclosed in some subinterval [zi,Zi+i] of [z,z\. Thus, if we let Az* = Zi+i — Zi, then (z — Zi)/Az{ maps z into [0,1]. In addition, if in equation (3),p(0) = P(l) = v(zi+1), px(0) = AziVz(zi), and px(l) = AziVz(zi+i); Then, v(z; a) may be defined on each subinterval [z{, Zi+i] by v(z\a) =p({z - Zi)/Azi) = v(zi)Ho{(z - Zi)/AZi) + vizi+^Hidz - Zi)/Az{) (4) + AziVz(zi)Ko((z - Zi)/Az{) + AziVz(zi+i)Ki((z - Zi)/Azi). From the expression above, it is clear that v is completely characterized on [z, z] by a parameter vector a composed of {u(^)}" :i1 and {v^Zij}^. The set of derivatives values {vz(z{)YlL\ is readily available from the dynamic pro-gramming problem's envelope conditions. The value function ordinates are only available after the mapping L is applied. Hence, the shape-preserving interpolation scheme pro-posed in this essay will treat the value function ordinates as free parameters: Given the set of derivatives values {^(.Zi)}"^, the set of ordinates {vizi)}^ obtained once the opera-tor L has been applied will be adjusted so as to render v concave (and possibly monotone) if it fails to be. Note that using the same transformation of variables, it follows from 19 equation (4) that vz(z ; a) = Azi~lpx({z - zfi/Azi), (5) vZz(z ; a) = Azi pxx((z - Zi)/Azi). Therefore, on each subinterval of [z,z], v inherits all the curvature properties possessed by the interpolant p. For simplicity, all the following results are derived using p rather than v. Extension of these results to v are straightforward. Lemma 1. The interpolant p defined above by equation (3) is concave on [0,1] if and only if p(0) + §px(0) + §p x ( l ) < p(l) < p(0) + |p x(0) + |p x (l) and px(0) > p x(l). Moreover, p(x) is monotone on [0,1] if pz(0) and p x(l) are of the same sign. Proof. See appendix 2.1. || As it should be clear from lemma 1, monotonicity is enforced only as a by-product of concavity. The primary goal is to approximate concave value functions which may or may not be monotone. Monotonicity is not enforced on its own without concavity being imposed, but the converse remains possible. Theorem 1 below extend lemma 1 to 0 on [z, z]. Theorem 1. Let {vz(zi)Yl^l D e a g i y e n s e ^ of derivative values such that for j > k, Vz(zk) > VZ(ZJ). Then, if the constraint v(zi) + \AziVz(zi) + lAziVz(zi+i) < v(zi+i) < v(z{) + \AziVz(zi) + \AziVz(zi+i) is enforced successively on v(zi+i), starting from i = 1 and ending with i = nz — 1, the interpolant v defined by equation (4) is concave on [z,z]. Moreover, if sign(v z(z{)) = sign(v,(zi)) for i = 2 ,3 , . . . , nz, then v is also monotone. Proof. See appendix 2.1. || The procedure proposed in theorem 1 for implementing monotonicity and concavity in the univariate case is simple. Beginning with i = 1, the constraints on v(zi+i) stated in theorem 1 are checked sequentially. That is, if the value of v(z2) fails to fall within the bounds computed using the values of v(z\), vz(z\) and vz(z2), it is adjusted accordingly. Then, the value of v(zz) is checked against a similar pair of constraints and is adjusted if necessary, and so on until i = nz — 1. This procedure produces a curve v on [z,z] which is concave everywhere. In addition, v is also monotone if the derivative values {vzi^i)}^ are 20 all of the same sign. Enforcing the concavity constraints of theorem 1 on the approximate value function v, after the operator L has been applied, ensures that the optimal policies may be determined adequately. The concavity of v guarantees uniqueness of the optimal policy vector found when L is computed. Value functions characterized by two endogenous state variables are approximated with a piecewise bicubic Hermite interpolant. Let Z — [z,z] x [w, w] be the (restricted) domain of the value function v for a given 6 G 0. Let { .Zj}"^ be a partition of the in-terval [z, z] such that z = z\ < Z2 < ... < znz = z and similarly, let {wi}™^ be a partition of the interval [w, w] such that w = Wi<w2<---< wHw = w. Together, these two parti-tions form ZQ. Moreover, let v be a concave function with derivatives values given by the sets {vz(zi,Wj)}, {vw(zi,Wj)}, and {vzw(zi,Wj)} for i = 1, 2 , . . . , nz and j = l,2,...,nw. As in the univariate case, if v is monotone in z, the first derivatives {vz(zi, Wj)} are all of the same sign and similarly for monotonicity in w. Again, the goal is to make sure that v is a concave interpolant, given the sets of derivatives values. Note however that in this case, a set of cross partial derivatives {vzw(zi, Wj)} need to be supplied in addition to the first partial derivatives. They are more difficult to obtain from the underlying dynamic programming problem than the first partial derivatives are. They may be approximated with difference quotients based on the first partial derivatives. It will be seen however that the values of the cross partial derivatives may have to be adjusted to maintain the concavity of the interpolant v. Proceeding similarly as in the univariate bicubic Hermite interpolant p is defined on a standard area [0,1] x [0,1]. For (re, y) G [0,1] x [0,1], let p(x, y) be given by l l i i P(x, V) = X1SI^ C' b)Hc{x)Hb{y) + Y,P*(C' b)Kc(x)Hb(y) c=0 6=0 c=0 6=0 / „ x 1 1 1 1 ( 6 ) + X) Y^Py^ b)Hc(x)Kb(y) + Y,Y,P*y(c' b)Kc{x)Kb(y) c=0 6=0 c=0 6=0 where the functions Hi and Ki are the same cubic Hermite basis functions defined previ-ously. The interpolant v on [z, z] x [w, w] may be defined from p on [0,1] x [0,1] by using a similar transformation of variables as in the univariate case. Take any pair (z, w) G [z, z] x [w, w] and consider the smaller area [zi, Zi+i] x [WJ, Wj+i] in which the pair (z,w) is contained. Let A-Zj = Zi+i — Zi and AWJ = — Wj. Then, setting p(c,b) = v(zi+c,wj+b), px(c,b) = AziVz(zi+c,wj+b), py{c,b) = AwjVw(zi+c,wj+b), 21 and pXy(c, b) = AziAvjjVzw(zi+c, Wj+b) for c = 0,1 and b — 0,1 in equation (6), v(z, w ; a) may be obtained from p using v(z,w;a) = p((z - Zi)/Azi, [w - Wj)/AWJ) (7) for any (z,w)E[zi,Zi+i] x [WJ, Wj+i], Using the same argument as in the univariate case, the interpolants v and p share the same curvature properties under this transformation of variables. Note from expressions (6) and (7) that v(z, w;a) is fully characterized on \z,z] x [w,w] by a parameter vector a composed of the sets {v(zi,Wj)}, {vz(zi,Wj)}, {vvfawj)}, and {v^z^Wj)}. Concavity is more tricky to enforce on v in the bivariate case than it is for the univariate case. Formally, v is concave if the inequalities vzz(z,w) < 0, vww(z,w) < 0, and vzz(z,w)vwui(z,w) — vzw(z,w)2 > 0 are respected everywhere on [z,z] x [w, w]. The last inequality is especially troublesome. Conditions such that this inequality is respected involve a nontrivial set of bounds on the free parameters characterizing v. Any algorithm designed to enforce these bounds would be rather intractable. Therefore, only the first two inequalities are considered. This amounts to imposing concavity along one axis, holding the other variable constant. This definition of concavity along axes is the one usually considered in the shape-preserving literature (e.g. see Fontanella (1987)). Enforcing this type of concavity is likely to eliminate much of the unwarranted oscillations in the shape of v. However, this approximate value function may no longer have the properties required for the maximization exercise to yield adequate optimal policy vectors. Nevertheless, there are some possible solutions to this problem. First, the concavity of v at each point of the partition ZQ may be imposed by varying the cross partial derivatives values {vzw(zi,Wj)} so as the joint concavity condition vzz(z, w)vww(z, w) — vZVJ(z, w)2 > 0 is met everywhere on ZQ. Still, this would not guarantee joint concavity everywhere on Z. However, by proceeding this way, the problem may become less severe if the partition ZQ is sufficiently fine. Second, when optimal policies are sought along the lines set forth in section 2,5, the value of vZVJ(z, w) may be corrected in a manner analogous to the quadratic hill-climbing method of Goldfeld, Quandt, and Trotter (1966) employed in numerical opti-mization algorithms in order to deal with a similar problem. Yet, although a weaker form of concavity is enforced on v, this does not mean that the ultimate solution obtained with this algorithm will not be concave. As v gets closer to the "true" solution v, one would expect v to become concave without having to enforce it. Imposing concavity should only be necessary at early stages of the iterations on the operator L. 22 Lemma 4.1 and lemma 4.2 below present sufficient conditions ensuring the concavity of p along axes in [0,1] x [0,1]. These results are extended to v thereafter. Lemma 3 below will be helpful when proving subsequent results. Lemma 3. Let p(x) for x € [0,1] be the interpolant defined by equation (3) above, where p(0) and p(l) are of the same sign. Then p(x) does not change sign on [0,1] if i) p(0) > -Jpx(0) and p(l) > i p x ( l ) when p(0) > 0 and p(l) > 0, ii) p(0) < -|px(0) and p(l) < §p x(l) when p(0) < 0 and p(l) < 0. Proof, see Carlson and Fristch (1985), lemma 2. || Lemma 4.1 Let p(x,y) be the interpolant defined on [0,1] x [0,1] by equation (6) and let Ax(a) = Py(l,0) - Py{0,0) - apxy(0,0) - (1 - a)pxy{l, 0), M&) = Pj/(1,1) - Py(0,1) - aPxy{0,1) - (1 - a)pxy(l, 1). Then Pxx(x, y) < 0 for any pair (x, y) £ [0,1] x [0,1] if the following inequalities i) p(l ,0) >p(0,0) + |px(0,0) + |p*(l,0) + m a x { 0 , - l ^ d ) } , ii) p(l,0) < p(0,0) + |px(0,0) + ip x ( l , 0 ) - r - m i n j O , - ^ ^ ! ) } , iii) p ( l , 1) > p(0,1) + ip x (0,1) + §p x(l, 1) + max{0,\A2{\)}, iv) p ( l , 1) < p(0,1) + |px(0,1) + J p x ( l , 1) + min{0,\A2{\)}, (8) and are satisfied. Px(0,0) - m a x j C A x d ) } - m a x { 0 ,-^(i)} >p*(l,0), px(0,1) - max{0, - A 2 ( | ) } - max{0, A 2 ( | ) } > p x ( l , 1), (9) Lemma 4.2 Let p(x, y) be the interpolant defined on [0,1] x [0,1] by equation (6) and let Bi(a) = px(0,1) - px(0,0) - apxy(0,0) - (1 - a)Pxy{0,1), B2(a) = p x ( l , 1) - p x ( l , 0) - apxy(\, 0) - (1 - a)p x y ( l , 1). Then Pyy(x, y) < 0 for any pair (x, y) E [0,1] x [0,1] if the following inequalities i) p(0,1) > p(0,0) + ipy(0,0) + |py(0,1) + max{0, - | B ! ( i ) } , ii) p(0,l) <p(0,0) + fpy(0,0) + ip v(0, l ) + m i n { 0 , - i J B 1 ( | ) } , iii) p(l , 1) > p(l , 0) + | p y ( l , 0) + | p y ( l , 1) + max{0, | B 2 ( i ) } , iv) p(l , 1) < p(l,0) + |p y(l ,0) + ip„(l, 1) +min{0, i S 2 ( | ) } , 23 and p J /(0,0)-max{0,S 1(|)} -max{0,-/3i(i)} >p y(0,l), py(l, 0) - max{0, -B2(l)} - max{0, B2(\)} > py(l, 1), (11) are satisfied. Proof. See appendix 2.1. || Note that lemma 4.1 and lemma 4.2 have been stated separately for one particular reason: The last two inequalities of conditions (8) and (10) impose bounds on the value that p(l, 1) is allowed to take. However, nothing guarantees a priori that the bounds on p(l, 1) in (8) and those in (10) have a nonempty intersection. Lemma 5 provides a consistency condition which renders lemma 4.1 and lemma 4.2 compatible with each other. Lemma 5. Let A2( •) and B2( •) be the quantities previously defined in lemma 4.1 and lemma 4.2. Then, the bounds on p(l, 1) defined by part iii) and part iv) of condition (8) in lemma 4.1 are consistent with those defined by part iii) and part iv) of condition (10) in lemma 4.2 if the value of p(0,1) satisfies Proof. See appendix 2.1. || Lemma 4.1, Lemma 4.2, and lemma 5 allow the enforcement of concavity along the x and the y axes on the interpolant p(x, y). In view of the consistency condition in lemma 5 however, it has to be done in a precise order: The procedure is illustrated in figure 2.1. The value of p(x, y) at the point (0, 0) is always taken as given. The procedure amounts to ad-justing p(l, 0), p(0,1), and p(l, 1) with respect to the conditions enumerated in lemma 4.1, lemma 4.2 and lemma 5. First, parts i) and ii) of condition (8) are enforced (figure 2.1, panel A): p(l,0) is adjusted in relation with p(0,0). Second, the value of p(x,y) at the point (0,1) is adjusted with respect to p(0,0) and p(l, 0), that is, parts i) and ii) of condi-tion (10) and condition (12) are enforced (panel B). Finally, this permits to impose parts iii) and iv) of conditions (8) and (10) simultaneously by adjusting p(l, 1) accordingly (panel C). The resulting interpolant is concave along both axes. Note that conditions (9) and (11) p(0,1) > p(l,0) + ip y ( l , 0) + §p„(l, 1) + max{0, \B2{\)} - [§p x (0 , l ) + |p x ( l , l ) +min{0, |A 2 ( |)}] , p(0,1) < p(l, 0) + |p y ( l , 0) + |p y ( l , 1) + min{0, JB 2(|)} - [±p x (0,1) + |p x ( l , 1) + max{0,\A2{\)}}. (12) 24 respectively ensure that the intervals defined by conditions (8) and (10) are nonempty. These conditions actually define bounds of admissible values for the set of cross partial derivatives as a function of the first partial derivatives. Imposing monotonicity on p(x, y) also involves constraints on the value taken by cross partial derivatives. These are stated in lemma 6 below. Again, monotonicity is not enforced on its own; the conditions in lemma 6 below only apply to a concave interpolant. Lemma 6. Let p(x, y) be an interpolant defined by (6) for which the values of p(0,0), p(l ,0), p(0,1), and p ( l , l ) are such that the conditions stated in lemma 4.1, lemma 4.2, and lemma 5 are satisfied. Moreover, let fx = 1 if p x(0,0) > 0, £ x = — 1 if p x(0,0) < 0 and define £ y similarly from p v(0, 0). a) If Px(0,0), p x ( l , 0), Px(0,1), and p x ( l , 1) are all of the same sign, and if in addition, it is the case that f*P* y(0,0) > -3fxPx(0,0) , £cPxy ( l , 0 ) > -3 fxPx ( l , 0 ) , 6 ^ ( 0 , 1 ) < 3fxPx(0 , l ) , & P * y ( l , l ) < 3 f x P * ( l , l ) , then p(x, y) is monotone in x on [0,1] x [0,1]. b) If Py(0, 0), p y ( l , 0), p y(0,1), and p y ( l , 1) are all of the same sign, and if in addition, it is the case that £ y P x y ( 0 , 0 ) > - 3 f y p y ( 0 , 0 ) , & P x y ( 0 , l ) > - 3 ^ , ( 0 , 1 ) , (14) tyPXy(hO)<3LzyPy(l,0), 4PX 3 / (1 ,1) < 3 f y p y ( l , l ) , then p(x, y) is monotone in y on [0,1] x [0,1]. Proof. See appendix 2.1. || Much like as in the univariate case, lemma 4.1, lemma 4.2, and lemma 5 may be translated into concavity constraints applying to the interpolant v. Recall that on each subrectangle [,z;,.Zj+i] x [WJ, tfy+i] of [z, z] x [w,«J], this interpolant can be uncovered from p(x, y) using a simple transformation of variables. Therefore, the procedure for enforcing concavity on every subrectangle [zi, Zi+i] x [WJ, Wj+i] may be imagined as being a mere extension of the results previously obtained. This would however neglect an important fact; these subrectangles are pieced together, side by side, to form the entire domain of v. 25 Simply adjusting the function values {v(zi,Wj)} according to the conditions stated in the preceding lemmata, independently on each subrectangle, would not necessarily produce a surface with the required properties. Adjusting a function value v(z{,Wj) so that t) is concave on some subrectangle of the domain is likely to affect the curvature properties of v on all other adjacent subrectangles. For example, the left side of a subrectangle is the right side of the one to its right. Therefore, enforcing concavity on the first subrectangle may affect the shape of the interpolant on the subrectangle to its right. This possibility is not taken into account by the procedure illustrated in figure 2.1. Observe that the value of p(l,0) in lemma 4.1 is subject to different concavity conditions than the value of p( l , 1) in lemma 4.2. However, as far as enforcing the concavity of v is con-cerned, these two sets of conditions should apply to the same point. The value of v(zi, Wj) corresponding to p(l , 1) in one subrectangle of [z,z] x [w,w] also corresponds to p(l,0) in the next subrectangle to its right. A procedure designed to enforce concavity on v must take this fact into account. Moreover, this procedure must also be such that adjusting one function value v(zi,Wj) to enforce concavity on a particular subrectangle of [z,z] x [w,w] does not affect the curvature of v on another region of the domain where concavity has already been ascertained. Theorem 2 below states sufficient conditions ensuring the con-cavity of v. These conditions take the form of bounds of admissible values for v(zi,Wj). Implicit to theorem 2 is a specific sequence according to which these bounds have to be enforced. This specific sequence guarantees that once the interpolant has been deemed concave on some subregion of the domain, its shape on this subregion is not affected if concavity adjustments are made in other regions of the domain. Basically, the procedure consists in adjusting all the function values {v(zi,Wj)} from left to right by proceeding first down the z-axis, holding w constant, and then move gradually along the tu-axis until the entire partition of points has been covered. This procedure is illustrated in figure 2.2 for the case where [z,z] x [w,w] is parti-tioned into a 4 by 3 mesh of points (i.e. nz = 4, nw = 3). First, the function values on the leftmost side of the partition are adjusted (panel A). This is equivalent to what would be done in the univariate case. Then, the consistency condition is enforced on the top-left rectangle, just as it was done for p(x,y) (panel B). However, when the value of v(z2,w2) is adjusted (panel C), it has to be done in a manner which not only preserve concavity of v on [/ZI,M>I] x [^2,^2] but also takes into account the concavity bounds along the left side of the rectangular region [-Zi,uv2] x [22)^3] and the consistency condition needed for enforcing concavity on [z2, wi] x [z3, w2]. This last adjustment is repeated while descending 26 down the z-axis (panel D) until the bottom of the partition is reached (panel E). Then, the situation which prevails (panel F) when the point (z\, W3) is reached, is in every aspect identical to the one faced in panel B at the point (-21,102)- Thus, the procedure described above may be repeated until the rightmost side of the partition is attained. By proceeding along these lines, from left to right, every rectangular region of the partition in which con-cavity has already been enforced (if necessary) is left unaffected when further adjustments are made on other regions. Theorem 2 and theorem 3 are stated below. Theorem 3 states the conditions required for monotonicity. It is an extension of lemma 6 by using the same argument as the preceding one for concavity. Theorem 2. Let v(z, w ; a) be the interpolant defined on [z, z] x [w, w] by equation (7), let AZQ = AzU:t = 0, AWQ = AwUw = 0, and let Ai:j(a) = v w ( z i + l ,w j + x ) - vw(zi,wj+i) - aAziVzw(zi,wj+i) - (1 - a)AziVzw(zi+1,wj+i), Bij(a) = vz(zi+i,wj+i) - vz(zi+i,Wj) - aAwiVzw(zi+i,Wj) - (1 - a)AwiVzw(zi+u wj+i). If, given sets of derivatives values {vz(zi, Wj)}, {vw(zi, Wj)}, and {vzw(zi, Wj)} satisfying vz(zi,Wj+i) - Azi'1 max.{-AwjAij(l), Awj+iAij(l)} - Azi~l max{AwjAij(l), -AtWj+i^d )} > vz(zi+1, wj+l), vw(zi+i,Wj) - Awf1 m&x{-AziBij(l), Azi+lBij(l)} - Awj~1vaax{AziBij(\),-Azi+iBij(i)} > ^ (zi+i,w,+1), the constraints v(zi+i,wj+i) > v(zi,wj+i) + \AziVz(zi,Wj+{) + lAziVz(zi+i,wj+l) + Imax{AwjAijd), -Awj+iAi:i(l)}, V(zi+i, WJ+L) <v(zU Wj+i) + \AziVz(Zi, WJ+1) + \AZiVz(zi+X, Wj+i) + |min{AwjAij(§), -Awj+iAi:j(l)}, for i = 1,2,..., nz - 1, j = 0,1 , . . . , nw - 1, v(zi+i,Wj+i) > v(zi+i,Wj) + lAWjV^Zi+uWj) + lAWjV^Zi+uWj+x) + I max{ Azi B{j ( i) , - A z i + l B{j (±)}, v(zi+i,WJ+1) < v(zi+i,WJ) + lAWjVw{zi+uWj) + \AWjVw(Zi+x,WJ+1) + I min{ AziBij (|), - A z i + i % ( | ) } , 27 (15) (16) (17) for i — 0 ,1 , . . . , nz — 1, j = 1,2,..., nw — 1, and v(zi+i,Wj+i) > v(zi+2,Wj) + lAWjVuiZi^Wj) + lAwjVw(zi+i,wj+1) + i m a x { A z i + i S ( i + 1 ) j ( i ) , - A z i + 2 B ( i + 1 ) j ( i ) } - [lAzi+iVz(zi+i,Wj+i) + \ AZi+iVz(zi+2,Wj+i) (18) V(ZJ+I ,« ;J + I ) < v(zi+2,Wj) + lAwjVw(zi+2,Wj) + \AwjVw(zi+i,wj+1) + lmin{Azi+1B{i+l)j(l), - Azi+2B{i+l)j(\)} - [lAzi+ivz(zi+1,wj+i) + lAzi+ivz(zi+2,wj+l) + | max { AwjA^jd),-Awj+1A(i+l)j(l)}], for i = 0,1 , . . . , nz — 2, j = 1,2,..., nw — 1, are enforced successively on the set of function values {v(z{+i,Wj+i)}, then the interpolant v(z, w;a) is concave along the z and the w axes on [z, z] x [w, w]. Proof. See appendix 2.1. || Ensuring that v is a concave interpolant of v is quite simple computationally: Once a sequence of values {v(z{, Wj)} is obtained after an iteration on the operator L, the concavity constraints (16), (17), and (18) stated in theorem 2 are verified successively following the procedure illustrated in figure 2.2. If a concavity constraint is not respected at any point, the corresponding value of v(z{, Wj) is adjusted accordingly, and the constraint verification is resumed. This procedure guarantees the concavity of v(z, w ; a) along both z and w axes on [z,z] x [w, w] if the derivatives values satisfy condition (15). Condition (15) may be verified by adjusting the set of cross partial derivatives, following a similar procedure as the one used to adjust the function ordinates. Appendix 2.4 presents the intervals that are implied by condition (15) for the values taken by the cross partial derivatives. Theorem 3 below shows that the monotonicity of v may also verified by varying the cross partial derivatives' values on ZQ. Theorem 3. Let v(z,w;a) be an interpolant for which the set of function val-ues {v(zi, Wj)} is such the conditions in theorem 2 are satisfied. Let ^  = 1 if vz(z\, wi) > 0, £z — — 1 if ve(zi,wi) < 0 and define £ w similarly from vw(z\,wi). In addition, set A^o = Aznz = 0 and Aw0 = Awnw = 0 in all the following expressions. i) If the derivatives values {vz(zi, Wj)} are all of the same sign, and if in addition, it is 28 the case that 3£gvt(zi+l,wj+l) . -3ZzVg(zi+i,wj+l) . . r — > £zVzw(Zi+UWj+l) > J- (19) for i = 0 ,1 , . . . , nz — 1, j = 0 ,1 , . . . , nw — 1, then v(z, w;a) is monotone in z on all its domain [z, z] x [w, w}. ii) If the derivatives values {vw(zi,Wj)} are all of the same sign, and if in addition, it is the case that 3£_wvw{zi+i,wj+i) c f v -3£wvw{zi+i,wj+1) T > SwVzw(Zi+l,Wj+i) > (2uj for i = 0,1,...,n z — 1, j = 0,1,.. . ,n w — 1, then v(z,w;a) is monotone in w on all its domain [z, z] x tw,uJ]. Proof. See appendix 2.1. || 2.4 Choosing the state space partition. The procedure proposed in section 2.3 in theorems 1, 2, and 3 to enforce concavity and monotonicity on the numerical representation of the actual value function works with a given partition ZQ of the state space Z. It ensures that the value function's approximation have the proper shape at each iteration. Ideally, however, one does not want to have to enforce concavity or monotonicity at every iteration on T( •). Enforcing concavity should be more frequent in the first few iterations when the value function's approximations are fairly inaccurate; it should not be needed subsequently. Choosing the partition ZQ appropriately may facilitate this. It may improve the approximations' accuracy and may reduce the need for enforcing concavity. In the case of piecewise Hermite interpolation, there is no "optimal way" to choose the points forming ZQ. This is the case for other types of polynomials (Chebychev polynomials, for example). Note, however, that using a uniformly-spaced grid of points for ZQ is not always a good idea. This may result in using too few interpolation nodes in regions of the domain where the function being approximated has a relatively high curvature, and superfluous interpolation nodes elsewhere. Gains in accuracy may be realized by using a fine interpolation grid in regions with high curvature. Section 2.6 provides an example of this. In addition, the presence of occasionally binding inequality constraints usually leads to kinks in the first derivatives. Gains in accuracy may be realized if there is an 29 interpolation node corresponding to the point where a kink occurs.24 Overall, increasing the accuracy with which the value function is approximated should reduce the need for enforcing concavity as more and more iterations on T( •) are performed. 2.5 Computation of optimal policies. Given the value function v and a point (xa,9a) of the state space, the right-hand side of the functional equation (1) may be seen as a static maximization problem: N ,, aAr(c>x»d')+py2v(xi,0i)*si}- (21) {x',c)eiix{x.,e.) r~r Under various curvature restrictions on the functions v, r, and g (see Mangasarian (1969), Theorem 10), first-order conditions are sufficient to determine the optimal policy vec-tor (x'*,c*). Forming the Lagrangian and setting the first derivatives to zero yields the first-order conditions25 Vcr{c*, xa,9a)- u*JVcg(x", c*,xa, 6S) = 0, (22) /3%lv{x[*,9i)Trai-u*T\xlig(x'\c\xa,Ga) = 0 for i = 1, 2,...,N, (23) Vj9j{x'*,c*,xa,9a) =0 for j = 1,2,..., mg, (24) gj{x'\c\xa,9a) < 0 for j = l , 2 , . . . , m s , (25) UJ*>0 for j = l , 2 , . . . ,m f l , (26) where u* is a ms-dimensional vector of Kuhn-Tucker multipliers, u>j its jth element, and <7j( •) the jth function in g( •). Note that contrary to the common practice, there is no attempt to eliminate the derivatives of the value function from these first-order conditions by using the envelope conditions arising from the functional equation (1). By proceeding this way, there is no need to explicitly consider transversality conditions. However, if the value function's derivatives are eliminated from equation (23) by using the envelope conditions, these first-order conditions are no longer sufficient to identify an optimal policy vector (x'*, c*), unless 2 4 Such a point may be endogenous (i.e. vary with each iteration). However, the fact that a piece-wise interpolation scheme is used allows one to move some interpolation nodes after an application of the mapping T{ •). 2 5 Hereafter, \^f(q, r) denotes the partial derivative of a vector function / with respect to the vector q of arguments 30 the transversality conditions are verified (see Stokey and Lucas (1992), pp. 97-100 for a complete argument). These transversality conditions are usually omitted in algorithm seek-ing the solution of dynamic models by solving stochastic Euler equations iteratively (See Christiano and Fischer 1994). However, if these algorithm are initialized with relatively bad starting values, they may converge to solutions that do not satisfy the transversality conditions. This cannot occur with the procedure described in this section as long as v is concave. Observe that equations (22), (23), and (24) form a system of (mc + mg + mxmeN) nonlinear equations in as many unknowns (x1*,c*,ui*). Equations (25) and (26) are only acting to restrict the set of values from which these unknowns may be chosen from. This observation will serve as a basis for the strategy followed thereon. Looking at the problem from this angle, finding the optimal policy vector amounts to solving the nonlinear system of equations (22)-(24), taking into account the inequality constraints (25) and (26). This may be handled using numerical methods. The task is however complicated further if the constraint set characterized by the function g includes inequality constraints that bind oc-casionally. The difficulty with this type of constraints is that they act in a way which limits the admissible values of some maximizers to bounded intervals. That the multipliers u* have to be included into bounded intervals is certainly clear from the constraints in (26), but it may also be required of some control variable by a constraint in (25). Unfortunately, numerical algorithms designed to solve nonlinear system of equations of the form F(z) = 0 iteratively, usually work with each element of the vector z being taken from the real line. Therefore, in their "crude" form, these algorithm are not well-suited to the problem at hand because they could be led astray if a vector (x1*, c*,u>*) for which the constraints (25) and (26) are not respected, is generated at one iteration. This matter is addressed by using a transformation of variables which allows one to eliminate constraints (25) and (26) from the system of equations (22) — (26). Without this set of constraints, the modified system of equations may be solved by using a standard Newton algorithm. The idea is to find from constraints (25) and (26), the lower bound and the upper bound within which each maximizer in (x'*,c*,u*) has to be contained, plus or minus infinity being replaced by arbitrary large constants. For example, suppose that the jth constraint gj(-) is represented by a function h(u,xa,9a) which is convex and in-creasing in the control variable u. Define u as the value of u for which h(u, x9,0a) = 0 (this value may have to be found numerically). Then, the requirement that u satisfies the con-straint h(u, xa,9a) < 0 may be replaced by u < u. Another constraint could require that u 31 exceeds some lower bound u (a non-negativity constraint say), in which case the admissible values for the control variable u would be described by the compact interval u > u > 0. Let (x,c,uJ) and (x,c,u) be vectors of upper and lower bounds for (x'*,c*,u>*) de-fined from the set of occasionally binding inequality constraints. Each maximizer may be expressed as a convex combination of the bounds within which it should be included. Con-sider the (m c -+- mg + mxm0 A^-dimensional vector function p(z) for which each individual element given by 0.5[1 + COS(ZJ)]. Each individual element of p(z) maps each individual real element of the vector z, Zj, into [0,1]. From the function p(z), define the vector function (x'(z), c(z), w(z)) by {x'(z),c{z),u{z)) = (l-p(z)) (x,c,u)+p(z) (x,c,uJ). This function gives any vector (x', c, CJ) as a convex combination of the bounds defined previously (each element of p(z) is in [0,1]). Thus, there exists a vector z* G7£mc+ms+m*m»- /v such that (X'\C*,LJ*) = (X'(Z*),C(Z*)MZ*)). Moreover, by definition, (x'(z),c(z),u(z)) is such that the constraints (25) and (26) are always satisfied for any z. Therefore, system of equations (22)-(24) may be rewritten to form a system of non-linear equations of the form F(z*) = 0: Vcr(c(z*),xa,es) - u(z*)TVcg(x'(^),c(z*),xs,63) = 0, (27) / ^ ( ^ ( z * ) , ^ - u{z*fVxlig(x'(z*),c(z*),x3,e3) = 0 for i = 1,2,...,N, (28) ^ • ( / ^ • ( i V l . ^ V J ^ O for j = l , 2 , . . . , m f l . (29) The vector z* may be found numerically using a standard Newton algorithm. That is, by iterating on the recursive expression zn+1 = zn — VzF(zn)~lF(zn), beginning with some initial vector z°, until a vector z* such that H-F^z*)!^ < e is found.26 Any vector z n + 1 obtained from this recursion is such that the constraints (25) and (26) are respected. Finding optimal policies following the procedure developed in this section is of course only valid if the value function v is known a priori. However, the same argument is still valid if it is built from the operator L rather than from the operator T, as long as the 2 6 ||Ft**)||oo=maxi{F i(**)}. 32 interpolant v has the same curvature properties as the "true" value function v. This is guar-anteed by the shape-preserving interpolation scheme devised in section 2.3.2 7 Therefore, optimal policies solving L may also be found by solving the system of equations (27)-(29). Note that it is a good idea to include in the constraint set, inequality constraints which force the optimal value of x'* to be enclosed within Z, the domain of the interpolant v, rather than within X, the domain of the "true" value function. Failing to do so may render invalid the use of v as nothing would prevent this interpolant from being evaluated at a point outside its domain when the system of equations (27)-(29) is solved. 2.6 Accuracy of the algorithm. This section presents statistics giving an indication of the degree of accuracy which may be attained when a dynamic programming problem is solved using the numerical method described in this essay. In order to do so, the solution of a dynamic programming problem is approximated using interpolation grids of various sizes. Each one of these approximate solutions is compared to a benchmark case. Consider the following dynamic programming problem: N V(k, Qj) = max{u(c) + 3 £ V(k', 0,-)**} i = 1 (30) s.t. c = 9jBkQ+ {l-6)k-k', k'- (l-6)k > 0, for k > 0 and 0,- G 0. This dynamic programming problem corresponds to the standard stochastic growth model to which a constraint requiring non-negative investment is added. It is assumed that u(c) = c 1 _ < 7 / ( l — a). The solution of this problem is approximated for a share of capital a equal to 0.36, for a coefficient of relative risk aversion a equal to 0.5, and for a depreciation rate 8 of 0.025. The value of B is set to 0.097. In addition, it is assumed that the state of nature 9 takes on one of the two possible values {0.7,1.3} with probability 0.5. Approximate solutions are obtained over the range k e [0.05,2.25], using uniformly-spaced grids of interpolation points of size n £ {21,41,61,81,101}. These solutions are then Note that even if "joint" concavity cannot be entirely enforced in the bivariate case, this may only pose a problem for the first few iterations on L. As the approximations should converge to the true value function, the need for enforcing concavity will usually be present for the first few iterations. 33 compared to a benchmark solution obtained using a grid containing 4001 evenly-spaced interpolation points.28 An exact solution is not available for this dynamic programming problem. Thus, it is not possible to determine how close these approximate solutions are from the actual solution. Examining how close they are from the benchmark solution, however, allows one to evaluate how solutions using relatively coarse interpolation grids perform with respect to one using a large number of interpolation points. Nevertheless, since in principle, the benchmark case approximates the actual solution with a fair amount of precision, the statistics presented thereon may still be considered as evaluating the algorithm's accuracy. The accuracy in determining optimal policies and in approximating the value function are examined in table 2.1. The precision of the shape-preserving interpolation scheme is measured using the absolute percentage deviation of the approximate value function from the benchmark case. Let Vn(k,9) be the value function approximation at (k,9) obtained with a grid of n interpolation points. Then, the absolute percentage deviation, %AVn(k, 9), is given by %AVn(k,9) = 100 x \Vn(k,G) - W M ) | / W M ) . (31) Similarly, the errors in the computed optimal policies 9) for each grid size n, are given by %Ak*n(k,9) = 100 x \k*n(k,9) - Kooi(k,0)\/K00l(k,9). (32) For each n and each 9 G 0, the value of %AVn(k, 9) and of %Afc* (k, 9) are computed at each point of an evenly-spaced partition of [0.05,2.25].29 The grid sizes considered to evaluate the algorithm's accuracy are reported in the first column of table 2.1. The second column reports the implied distance between each interpolation nodes. The third column reports the maximal value of %AVn(k,9) while the fourth column presents the maximal value of %Afc*(fc, 9). Table 2.1 has three panels. The first panel presents the value of these statistics over the interval [0.05,2.05]. The values in panel B are computed over the interval [0.25,2.25] only. The results in table 2.1 show that the algorithm begins to perform relatively well when n exceeds 60. By comparing panel A to panel B, it is readily seen that large approximation errors are concentrated below k = 0.25. For example, the maximal error in optimal policies is reduced by a factor of 70 in the case where n = 20. Large approximations errors are concentrated at the 2 8 The distance between each interpolation points is 5.0 X 10~4. 29 4001 partition points are used. 34 bottom of the interval [0.05, 2.05] because the marginal productivity of capital is very high for small values of the capital stock. Since the slope of the value function corresponds to the marginal productivity of capital, the value function itself has a high curvature over this region. It appears that the interpolation grids used in the exercise have too few points in the lower portion of [0.05,2.05] to be able to approximate a function with such a high curvature. However, choosing interpolation nodes more adequately may resolve this problem. This is shown in panel C of table 2.1. This panel reports results obtained with an interpolation grid of 61 points; thirty-one of them evenly spaced between 0.05 and 0.25 and thirty between 0.25 and 2.05. The performance of the algorithm improves substantially with this new grid. The results reported in panel C outperform the ones computed in panel A for n = 100. Errors in the computed optimal policies are less than three thousandth of one per cent. Overall, complementing standard value function iteration algorithms with a shape-preserving interpolation scheme appears to yield significant benefits. The results presented in table 2.1 show that using a relatively coarse partition of the state space, one may obtain approximate optimal policies with a reasonable degree of accuracy. 2.7 Conclusion. An algorithm designed to approximate numerically the solution of dynamic models with occasionally binding constraints has been described. While retaining the substance of standard value function iteration algorithms, this algorithm may provide a better approxi-mation of the value function, of its derivatives, and of the multipliers associated with each constraint at a lower computing cost. Moreover, it allows one to handle quite easily models with state-contingent endogenous state variables and models where the value function itself is part of the constraint set. Hence, the chief advantage of this algorithm is that it allows one to enjoy the benefits of solving a dynamic model as a dynamic programming problem at a smaller computational cost than standard numerical methods. The results presented in section 2.6 show that this may be done with a relatively high degree of accuracy. 35 Appendix 2 .1 Proofs of the lemmata and theorems found i n the text are provided i n this appendix: Proof of lemma 1. Suffiency: F i rs t note that p x (0 ) > px(l) guarantees that the upper and lower bounds on the value of p ( l ) form a nonempty interval. In addi t ion, since Pxx(O) = 6[p(l) - p(0) - lPx(0) - lPx(l) and Pxx( l ) = -6 [p ( l ) - P ( 0 ) - !P*(0) - § p B ( l ) ] , the second derivative of p is negative at both ends of [0,1] i f p ( l ) falls w i th in those bounds. Hence, s incep x x is l inear on [0,1] (p(x) is a cubic polynomial ) , it is also negative everywhere in [0,1]: The concavity of p follows. Necessity: Observe that the concavity of p(x) implies that p x (0 ) > px(x) > p x ( l ) . Expand ing this expression yields the required inequalit ies. Monoton ic i ty follows f rom the concavity of p: Concav i ty implies that px(x) is decreas-ing on [0,1] and thus necessarily contained between px(0) and p x ( l ) since px(0) > p x ( l ) . Therefore, the sign of px(x) on (0,1) w i l l be the same as the one of p x (0) and p x ( l ) . || Proof of theorem 1. Theorem 1 follows direct ly from lemma 1 by using p(0) = v(z{), p ( l ) = v(z{+i), px(0) = AziVz(z{), p x ( l ) = AziVz(zi+i) and the expressions found i n (5) i n the proof of lemma 1. Enforc ing the constraints sequentially on each segment of [z, z] allows v to remain concave on the subintervals already considered. || Proof of lemmata 1 and 2. Lemma 4.1 is proven. The proof of lemma 4.2 is similar. F i rs t note that equation (6) may be rewri t ten as p{x,y) =p{0,y)H0(x)+p{l,y)Hi{x) + px{0,y)KQ{x) + p x ( l , y ) f f 1 ( x ) , wherep x (0 , y) a n d p x ( l , y) are obtained by differentiating (6) w i th respect to x and setting x to 0 and 1 thereafter. Hold ing y constant, p(x , y) is an univariate cubic Hermite interpolant in x . Take any y £ [0,1]. Then , using lemma 1 we have that p x x ( x , y) < 0 for a l l x G [0,1] if and only i f a) Px(0,y) > px(l,y), and b) p(0,j/) + iPx (0 ,y ) + | p x ( l , y ) < p ( l , y ) <p(0 , jf) + |p x (0 , j / ) + | p x ( l , y ) . Wr i t i ng px(0,y) and p x ( l , y ) expl ic i t ly and working the algebra, we have that Px(0,y) - P x ( l , y ) = [Px(0,0) - P x ( l , 0 ) ] t f 0 ( y ) + [Px(0,l) - p x ( l , l ) ] f f i ( y ) + \pxy(0,0) - p x y ( l , 0)]K0(y) + \pxy(0,1) - pxy{l, l)]Ki(y). 36 The left-hand side of the expression above is nothing else than an univariate cubic Hermite interpolant in y. Remark that Condition (9) of lemma 4.1 ensures that the coefficients in front of Ho(y) and H\(y) are positive. In addition, from the conditions in (9), we have that P*(0,0)-p x(l,0) >max{0,A 1 ( |)} +max{0 , -Ai(i)} > -\[pxy{Q,0) -p x y (l ,0)], P*(0, l)-p x ( l , l ) > m a x { 0 , - A 2 ( f ) } + m a x { 0 , A 2 ( ± ) } > ± [ p i y ( 0 , 1 ) - p X ! / ( l , 1)]. Hence, using part i) of lemma 3, px{0,y) — Px(l ,y) > 0. That is, condition a) above is satisfied for any y € [0,1]. Condition b) is also satisfied: Remark that p(l,y) -p(0,y) - |px(0,y) - |p*(l,y) =[p(l,0) -p(0,0) - |p*(0,0) - |px(l,0)]H 0(y) [p(l, 1) - P(0, 1) - |px(0,1) - |p x ( l , l)]JTi(y) + Al(l)K0(y) + A2(l)Kl(y)-Condition (8), part i) and part iii), ensures that the coefficients in front of Ho(y) and H\(y) are both positive. Moreover, they also imply that p(l ,0) -p(0,0) - !PX(0,0) - |px(l,0) > -\AX{\) p ( l , 1) - p(0,1) - \px{0,1) - |px(l, 1) > | A 2 ( | ) . Hence, from part i) of lemma 3, we have that p( l ,y) — p(0, y) — |px(0, y) — |p x ( l , y ) > 0 for any y € [0,1]. That is, the first part of condition b) is satisfied. In a similar manner, part ii) and part iv) of condition (8) ensure that the second part of condition b) is met. Therefore, pxx(x, y) is negative for any y€ [0,1] since conditions a) and b) above are both fulfilled. As the choice of y was arbitrary, the conditions enumerated in lemma 4.1 are sufficient for having Pxx(x, y) < 0 for any (x, y) £ [0,1] x [0,1]. || Proof of lemma 5. Write the intervals defined by parts iii) and iv) of condition (8) and parts iii) and iv) of condition (10) respectively as & > P ( 1 . 1 ) > 6 and & > p ( M ) > These two intervals have a nonempty intersection if f i > f4 and £ 3 > f2- Writing explicitly these two inequalities, we find the ones expressed in condition (12) of lemma 5. || Proof of lemma 6. We prove monotonicity of p(x, y) in x as the case for monotonicity in y can be proven following the same way. The proof parallels the one for the univariate case. Given that all derivatives with respect to x are of the same sign and since lemma 4.1 37 already ensures that px(0, y) > p x ( l ,y) for any y G [0,1], it remains to show that px(0, y) and px(l,y) do not change sign as y is varied. This way, the concavity in x of p(x,y) will guarantee that px(x,y) does not change sign as x is varied. Upon differentiating equation (6) with respect to x and setting x — 0 in the result, one obtains P*(0,y) =Px(0,0)# o(y) +Px(0,l)H1(y) + pxy(0,0)K0(y) +Pxy{0, l ) # i ( y ) . This is an univariate interpolant with coefficients in front of Ho(y) and H\(y) having both the same sign. Therefore, lemma 3 implies that px(0, y) does not change sign on [0,1] if e»Pxi,(0,0) > -3£ xp x(0,0), €«P*i,(0,l) <3£xPx(0 , l ) , where £ x was defined in lemma 6. Similarly, p x ( l , y) can be written as Px( l ,y) =Px(l,0)£T 0(y) + P x ( M ) f f i ( y ) +Pxy(l,0)KQ(y) +pxy(l,l)K1(y). Again, using lemma 3, p x ( l , y) does not change sign on [0,1] if £tP* y(l,0) > -3£ x p x ( l ,0) , €rPxy ( l , l ) <3exPx(l,l). The four conditions above on the cross partial derivatives are the same as the ones stated in lemma 6, part a). Therefore, lemma 6 provides sufficient conditions for the monotonicity of p(x,y). || Proof of theorem 2. Theorem 2 follows directly from the definition of v from p, lemma 4.1, lemma 4.2 and lemma 5 when the discussion before theorem 2 concerning adjacent subrectangles of [z,z] x [w, w] is taken into account. Condition (16) above corre-sponds to condition (8) of lemma 4.1, condition (17) to condition (10) of lemma 4.2, and condition (18) to the one stated in lemma 5. Condition (15) corresponds to the constraints found in (9) and (11). || Proof of theorem 3. Theorem 3 follows directly from lemma 6 when again, the conditions in lemma 6 are adjusted to take into account the ones on adjacent subrectangles of Z. || 38 Appendix 2.2 The cubic Hermite basis functions on [0,1] are given by HQ{x) = 2x3 - Sx2 + 1, Hi(x) = -2xz + 3x2, KQ{X) = x3 - 2x2 + x, Ki(x) = x3 - x2, for x € [0,1]. Appendix 2.3 Description of the algorithm. Step 0.0: Initialization 0.1 Restrict the endogenous state space to a compact set Z C. X and choose a finite partition ZQ of Z. 0.2 Choose an initial guess for the approximate value function. That is, define an inter-polant v(xs, 93; a°) on ZQ X 0 by choosing a vector a° = {(v°{x3,93), Vxv°{x3,93)) \ {x3,93) <EZQ X 0} such that t>( •; a°) is concave. 0.3 Set n = 0, and choose the tolerance levels e\ and e%. Step 1.0: Determination of optimal policies 1.1 Form the interpolant v( •; an) from the vector of parameters a" 1.2 Find a policy vector (x'n(x3,93),cn(x3,9a)) which solves N (x'n(x3,93), cn{x3,9a)) = argmax{r(c, x3,93) + B t)(asj, 9{; an)7ri3} i=l s.t. (x',c) G Qz{x3,93;an) for each (X3,93)EZQ X 0, by following the procedure described in section 4. 39 Step 2.0: Update of the value function's approximation 2.1 Compute the set of values {vn+1(xa, 9a)} for each (xa,9a)£Zo x 9 from N vn+1{xa, 9a) = r(cn(xa, 0a), xa, 9a) + 0 £ v(x?(x„ 6a), 9{; an)nia i=i 2.2 Compute the set of derivatives values {^7xvn+1 (xa, 9a)} for each (xa, 93)EZQ x 0. That is, compute the first derivatives from the dynamic programming problem's envelope conditions using (x'n(xa, 0a), cn(xa, 9g)) and from these first derivatives, compute the cross partial derivatives (if necessary) with a difference quotient. 2.3 In the bivariate case, adjust the cross partial derivatives so that condition (15) in the-orem 2 is satisfied and enforce the conditions of theorem 3 if monotonicity is required. 2.4 Form an+1 = {(vnU(xa,0a),Vxvn+1(xs,9a)) \ (xa,0a)eZ0 x 0}, the updated vector of parameters, from these derivatives values and the new function's ordinates. 2.5 Verify that the vector a n + 1 is such that i>( •; an+l) is concave and adjust it, if necessary, by following the procedure described in theorem 1 or in theorem 2. Step 3.0: Verification of convergence 3.1 End the algorithm if \\(x'n+\cn+1) - (x'n,cn)\\00 < e i and \\vn+1 - < e2. 3.2 Increment n by one and go back to step 1.0 Appendix 2.4 The condition vt(zi,wj+i) - Azr1 max{-AwjAijd), Awj+1Ai:J(l)} - Azf1 m&xlAwjAijd), -Awj+lAij{l)} > vz(zi+1,wj+l) in condition (15) of theorem 2 leads to the following conditions on the cross partial deriva-tives values {vzw(zi+i,Wj+i)}: If A^l) > 0, A{j(l) < 0 and j < nw - 1 vzw(zi+i,wjU) < vzw(zi,wj+i) - ZAwjlx[vz(zi+i,wj+i) - vz(zi,wj+i)] If Aij{\) < 0, Aij(l) > 0 and j > 0 Vzw(zi+i,Wj+i) > Vzw(Zi,Wj+i) + ZAw^[vz(zi+l,Wj+X) - Vz(Zi,Wj+i)} 40 If A i ( f ) > 0 , A y ( f ) > 0 Vzw{Zi+l,WJ+1) > - y - —=— Vzw(Zi,WJ+1) ±AiVj+x + | AWJ A w j + i + AWJ v w ( z i + i , w j + 1 ) - vw(zj,wj+i) l A w j + i + l A w j Azi + v z ( z i + 1 , w j + i ) - Vz(Zi,Wj+i) l A w j + i + \ A v j j l f ^ ( | ) < 0 , ^ ( | ) < 0 \ A w j + i + lAwj Vzw(zi+l,Wj+i) < - y - — f — Vzw{Zi,WJ+L) § A Wj+i + %AWJ A w j + i + AWJ v w ( z i + 1 , w j + i ) - vw(zi,wj+1) ^AWJ+I + \AWJ Azi _ Vz(zi+i,Wj+i) - Vz(Zi,WJ+1) l A w j + 1 + ±AWJ The condition vw(zi+i,Wj) - Awf1 m&xl-AziBijd), Azi+iBijd)} - Awf1 maxlAziBijd), -Azi+iBij^)} > vVJ{zi+1,wj+l) in condition (15) of theorem 2 leads to the following conditions on the cross partial deriva-tives values {vzw(zi+i,Wj+i)}: If Bijd) > 0, B ^ D < 0 and i < n z - 1 Vzw(Zi+i,Wj+X) < Vzw(Zi+i,Wj) - 3Az^+\[vw(zi+i, Wj+i) - Vw(zi+i,Wj)] If Bij(\) < 0, Bij(l) > 0 and i > 0 vew(zi+i,wj+i) > vzw(zi+i,Wj) + 3Az7l[vw{zi+i,Wj+{) - vw(zi+i,Wj)] H B y ( | ) > 0 1 B y ( | ) > 0 l A z i + i + \Azi Vzw(ZI+1,WJ+I) > - — » Vzw(Zi+i,Wj) 3 AZi+i + 3 AZi A z i + i + AZJ v z ( z i + i , Wj+i) - v z ( z i + 1 , Wj) \ A z i + i + \Azi AWJ + 3' Vw(Zj+i, Wj+i) - ^ ( g j + i , Wj) ^AZi+l + 41 ffBy(|)<0, S y ( | ) < 0 gAzi+i + \Az{ + Azi+i + AZJ vg(zi+i,Wj+i) - vz(zi+i,Wj) \Azi+i + hAzi Aw; 3 « — T " 3 i - i ' & i i_»u/j l A z i + i + i A ^ i 42 Table 2.1 Approximation errors A. A; 6 [0.05,2.25] g r i d s i z e ( n ) g r i d s p a c i n g m a x M { % A V ; ( M ) } m a x M { % A A £ ( M ) } 21 0.100 5.25xl0-2 7.64 x l O - 1 41 0.050 4.21 x l O - 3 1.10 xlO" 1 61 0.033 1.08 xlO" 3 4.17 xlO" 2 81 0.025 3.43 xlO" 4 1.98 xlO" 2 101 0.020 1.17 xlO" 4 1.00 x l O - 2 B. JfcG [0.25,2.25] g r i d s i z e ( n ) g r i d s p a c i n g m a x M { % A K ( M ) } m a x M { % A A £ ( M ) } 21 0.100 9.61 xlO" 4 1.13 xl0~ 2 41 0.050 5.76 xlO" 5 1.41 x l O - 3 61 0.033 1.38 xlO" 5 5.19 xl0~ 4 81 0.025 4.39 x IO - 6 1.93 xlO" 4 101 0.020 2.51 x 10~6 1.01 xlO" 4 C. k €[0.05,2.25] g r i d s i z e ( n ) g r i d s p a c i n g m a x M { % A V ; ( M ) } m a x M { % A ^ ( M ) } 61 0.006/0.06 1.01 x l O - 4 2.45 x l O - 3 43 44 45 Chapter 3 Investment Decisions, Financial Flows, and Self-Enforcing Contracts 3.1 Introduction. The assumption that capital markets are perfect allows one to study firms' investment decisions independently of their financial situation. However, following the work of Fazzari, Hubbard, and Peterson (1988), a substantial body of empirical evidence has raised doubts as to the validity of this assumption.1 In particular, it has been shown that investment expenditures of firms identified a priori as likely to be faced with financing constraints are in general excessively sensitive to measures of internal funds' abundance. Combined with the finding of a weaker link for firms in a better financial situation,2 this finding has led many to conclude that investment expenditures are indeed affected by financing considerations. This conclusion is reached on the basis that firms with limited external funding may increase investment expenditures further (to attain investment levels which allow firms to take advantage of all opportunities) only through internal finance. Hence, increases in investment spending of constrained firms should coincide with internal funds' windfalls, holding constant investment opportunities. The existence of financing constraints is often motivated on the grounds that borrow-ers and lenders may not have the same information as to actual returns on investment projects or as to their intrinsic quality. Financial contracts in this context usually in-volve agency costs (deadweight losses) with respect to a situation where information is symmetric. Agency costs increase the cost firms have to pay in order to raise external finance compared with the one of using internal funds. Rising agency costs compel them to self-finance a significant portion of their investment expenditures. That is, the presence of imperfect information typically constrains firms' access to external finance by making its cost prohibitive. For example, financing constraints arise from imperfect information in Bernanke and Gertler (1989, 1990), Calomiris and Hubbard (1990), Gertler (1992) and Gertler and Gilchrist (1994). 1 See for example Blundell, Bond, Devereux and Schiantarelli (1992), Calem and Rizzo (1995), Chirinko and Schaller (1995), Fazzari and Peterson (1993), Hubbard, Kashyap, and Whited (1995), Schaller (1993). The two classes of firms have been distinguished, for example, according to their size, their maturity, their type of ownership, their membership to a group, and their dividend payment practice. 46 Another approach links the existence of financing constraints to the inability of bor-rowers and lenders to credibly commit themselves to respect the terms of financial con-tracts. A financial contract may be beneficial to its parties, ex ante, when it is signed. However, ex post, as the economic environment evolves and as uncertainty unravels, ei-ther the borrower or the lender may be worse off if it abides by the contract's terms. For instance, it could be in one agent's interest to abandon an ongoing financial contract so as to benefit from some other opportunity at its disposal. If there is no mechanism prevent-ing him to do so, that is, if the contract is not enforceable, this agent will renege on his engagements. As in the imperfect information case, this possibility may seriously limit the set of feasible financial flows between the borrowers and the lenders. Financing constraints arising from the lack of credible commitment are found in Hart and Moore (1994), Kehoe and Levine (1993), and Kiyotaki and Moore (1997), Marcet and Marimon (1992). Both types of markets' imperfections generally lead to predictions that are consistent with Fazzari, Hubbard, and Peterson's (1988) finding. That is, firms typically underinvest because of their limited financial capacity; windfalls in internal funds allows them to finance greater investment expenditures; hence the positive relationship between firms' investment expenditures and measures of internal funds' abundance. Several authors have emphasized the potential implications of this finding at the macroeconomics level. For example, it has been argued that financial markets imperfections may lead to the formation of endogenous cycles (Greenwald and Stiglitz 1993, Kiyotaki and Moore 1997), may provide a channel through which the impact of business fluctuations is amplified and propagated over time (Bernanke and Gertler 1989, Gertler 1992, Greenwald and Stiglitz 1993), or may induce asymmetric dynamics (Bernanke and Gerter 1989). Marcet and Marimon (1992) compare the effects of both types of markets' imperfections on investment spending. They find that the impossibility to enforce contracts may have more pervasive effects on investment spending than the presence of imperfect information. The goal of this essay is to examine further how firms' investment decisions are affected by the impossibility to fully enforce financial contracts. This is done in an infinite horizon framework where firms finance capital accumulation by entering long-term contractual re-lationships with lenders. This issue has been examined in Marcet and Marimon (1992) and Thomas and Worrall (1994) following a similar approach. However, these two papers study international financial relationships where borrowers may take possession of the cap-ital stock if the contract is breached. In contrast, the focus of this essay is on understanding domestic financial relationships rather than international ones. The key distinction with 47 Marcet and Marimon (1992) or Thomas and Worrall (1994) is that borrowers pledge their capital stock as collateral when they borrow. Lenders may force them into bankruptcy by refusing to reschedule or renew their debts; in which case, they take possession of the capital stock. Hart and Moore (1994) and Albuquerque and Hopenhayn (1997) study en-forcement problems in domestic settings where investment spending takes place uniquely in the initial period. Investment spending occurs every period in this essay; not only in the initial one. This allows us to observe the dynamic interaction between investment expenditures and financial flows. Several issues are investigated. First, it is assessed whether financial constraints aris-ing from the impossibility to enforce contracts influence investment spending in a manner consistent with Fazzari, Hubbard, and Peterson's (1988) empirical finding. Second, many implications usually linked to the existence of financing constraints are studied. In particu-lar, the dynamic properties of output are examined; the presence of a financial accelerator is ascertained by examining if financing constraints contribute to the amplification and the propagation of business fluctuations. The presence of asymmetric output dynamics is also investigated. Finally, the dynamic interaction between financial variables and investment decisions is scrutinized in order to provide a better understanding of the mechanism at work when contracts cannot be enforced. The essay proceeds as follows: A standard infinite horizon stochastic accumulation model is used as a starting point.3 It is assumed that a risk-averse entrepreneur has access to a capital intensive production technology which is subject to random fluctuations. This entrepreneur is assumed risk-averse in order to capture the behavior of a firm with limited access to financial markets. The entrepreneur may borrow from a risk-neutral lender to finance investment by entering a multiperiod contractual relationship allowing for state-contingent intertemporal trades.4 This contract allocates the property rights over the existing capital stock to the lender. There is no other structure imposed a priori on the form taken by the financial contract.5 Its form is solely molded by the fact that the J Note that this framework allows one to consider the effect of shocks that are rationally anticipated. The models of Gertler (1992) and Kiyotaki and Moore (1997), for example, are essentially deterministic in the sense that the shocks considered in these papers are zero probability events. 4 Given agents' preferences, gains from trade may also stem from an insurance motive. 5 This is a desirable property considering the fact that several results in the literature examining financial relationships when capital markets are imperfect, have been shown to rest exclusively on the type of financial structure postulated a priori. See for instance de Mezza and Webb (1987). Moreover, in view of the rapid emergence of financial innovations in the 80's, several authors, notably Gertler (1988), have stressed the 48 contract's terms are not enforceable; only property rights over the accumulated capital stock are enforceable. As in Hart and Moore (1994) and Kiyotaki and Moore (1997), it is assumed that the entrepreneur's human capital is inalienable; he always has the freedom to withdraw from the contractual relationship by defaulting. If he defaults, the entrepreneur may take advantage of some alternative opportunity at his disposal but he is barred from ever contracting again in the financial markets. The lender also has the possibility to withdraw from the financial contract at any time; in which case, he takes possession of the firm's accumulated capital stock.6 As neither the lender nor the entrepreneur are able commit themselves to respect the terms of a contract, a feasible contract in this context is one which offers to both of them at least as much value as the one offered by their alternative opportunities, at any point in time; otherwise, the contract will be breached eventually. That is, as in Thomas and Worrall (1988, 1994), the financial contract must be self-enforcing.7 A self-enforcing contract is never breached by its parties. They have no interest in doing so considering that the contract offers them no less value than other alternatives at their disposal; remaining part of the financial relationship is always in their self-interest. Financial markets are developed in Kehoe and Levine (1993) following a similar ratio-nale. However, their model does not allow for an intertemporal production technology. One important aspect of the financial contract considered in this essay—absent for example in Kehoe and Levine (1993) or in the two-periods models based on imperfect information8—is the endogenous nature of the collateral. The capital stock has a dual role in this model: it is a factor of production and an asset which may be pledged as collateral. Hence, in-creasing investment spending does not only increase the firm's production capacity and thereby the abundance of internal funds but also the firm's capacity to obtain external finance. It is shown that this interaction between the firm's investment spending and its importance of letting financial markets evolve endogenously with real variables; allowing for state-contingent trade permits to address this concern. In Hart and Moore (1994) and Kiyotaki and Moore (1997), the lender may only seize the borrower's assets when the latter repudiates the debt. Here, the lender may seize the collateral assets regardless of whether or not the borrower has repudiated the debt. In Marcet and Marimon (1992), the entrepreneur retains the property rights over the capital stock; The lender may never seized his assets. See also Kletzer and Wright (1995) and Kocherlakota (1996) for similar contracts. Self-enforcing constraints on the contract take the form of dynamic participation constraints. Q For example, Calomiris and Hubbard (1990) consider shocks to the entrepreneurs' initial wealth but cannot account for the endogenous accumulation of wealth. 49 financial capacity leads to investment movements which are consistent with the findings of Fazzari, Hubbard, and Peterson (1988) and others. Namely, that investment expenditures are excessively sensitive to internal funds' abundance. However, as it will be shown, the interpretation of this result is quite different in an environment with partial enforcement. Rather than being the consequence of chronic underinvestment, the positive relationship between investment expenditures and internal funds' abundance arises from overinvest-ment. In fact, the entrepreneur takes advantage of good times to obtain external funds at a relatively low cost in order to increase investment spending. This raises the firm's fu-ture production capacity and thereby provides it with additional internal funds when bad times occurs.9 However, while investment appears excessively sensitive to internal funds' abundance, the output level does not exhibit excess volatility nor much propagation. Over-investment actually reduces the volatility of output; increasing investment expenditures above the full-enforcement level in good times helps to dampen future reductions in output when bad times begin. The prime effect of the difficulty to enforce contracts is to induce asymmetric dynamics. It is shown that the entrepreneur's decisions differ based on whether a positive or a negative shock to the availability of internal funds occurs. Variations in the likelihood that a financing constraint will bind in the future are responsible for this asymmetric behavior. The essay is organized as follows: Section 3.2 describes the economy under consider-ation and the basis for multiperiod contracting in this economy. The set of opportunities available to borrowers and lenders are described in the third section. A self-enforcing financial contract is designed in section 3.4. The fifth section sets forth equilibrium con-ditions describing the optimal contract. The behavior of financial flows is examined in section 3.6. Investment decisions are studied in section 3.7. Section 3.8 presents numerical results which allow one to assess and quantify the importance of the properties identified in section 3.7 and 3.8. The model's dynamic properties are also investigated. Finally, concluding remarks are offered in section 3.9. a The possibility that financial market imperfections lead to overinvestment is not a novelty. Apple-baum and Harris (1978) show that both underinvestment and overinvestment may occur when borrowing is precluded and all retained earnings are distributed. Using a similar model, Schworm (1980) shows that overinvestment does not occur if firms may save a portion of their retained earnings. However, the results presented in this essay may not be as sensitive as the ones presented in Applebaum and Harris (1978) in view of the fact that the financial contract considered here has a more general form. 50 3.2 The economy. Consider an economy populated by two infinitely-lived agents; an entrepreneur and a lender. The entrepreneur is a risk-averse agent. He owns a state-contingent production technology which, each period t = 1,2,..., oo, allows him to transform k units of the capital good into g(k,9) units of goods which may either be consumed or invested.10 The state of nature 9 is drawn from a discrete set of possible values 0 = {9X, 92,..., #„}, where n > 2. Both agents know the realization of 9. Let hl = {9\, 92,. . . , 9t} denote the history of states of nature from the first period up to time t, with h° being the empty history {0}. History {/it-1,#s} is denoted by hi and history { / i t - 1 , b y h),1. Histories are generated by a stationary first-order Markov process with transition matrix n = [irij], where 7Ty = Pr(0t = 9j \ 9t-i = The entrepreneur may accumulate capital. The capital stock at the end of history W is given by k1 = (1 — S)^"1 + il, where il denotes investment spending and 8 is the depreciation rate. Investment expenditures are irreversible. That is, the constraint il > 0 is imposed on investment spending.11 The entrepreneur is initially endowed with 77 units of the capital good. The entrepreneur's preferences over consumption plans {o7'}^ in history h* are de-scribed by the time-separable expected utility function u(<*) + Et /^"VcO, (1) j=t+i where j3 is the time discount rate and Et the expectation operator conditional on history h1. The momentary utility function u(-) is assumed increasing and strictly concave. The entrepreneur has access to a financial market. The lender represents the financial market as a whole; he is a risk-neutral agent with a large amount of funds X at his disposal. In any history, he may offer a long-term financial contract to the entrepreneur. Let r > 1 be the first period for which a financial contract becomes effective between the two agents: This contract is negotiated at the outset of history hT. At that time, the lender and the entrepreneur agree on a sequence of transfers {acJ'} .^T accounting for all subsequent trades between them.12 When the value of xl is negative, the lender transfers funds to the 1 0 The production function g(k, 6) is assumed concave, increasing with the capital stock, and decreasing with investment. 1 1 The effects of relaxing this constraint are explored subsequently. To avoid equilibria with rationing, it is assumed that the entrepreneur's financial needs are insignificant with respect to the size of the financial market. Hence, X is large enough for a;' > —X to hold in every history. 51 entrepreneur , tha t is , there is a net increase i n bor rowings ; a pos i t ive va lue is a n i n d i c a t i o n of the converse. T h e q u a n t i t y —x* is thus the net f inanc ia l flow rece ived b y the ent repreneur i n h is -to ry h*. T h e r e f o r e , the entrepreneur 's choices over c o n s u m p t i o n a n d investment have to o b e y the resources constra in t ct + t*=g(kt-\&t)-xt (2) for t = 1,2,... , o o . 1 3 A p a r t i c u l a r sequence o f net flows or transfers { x J } ^ t , is eva lua ted b y the lender i n h is tory hl a c c o r d i n g t o 1 4 oo xt + Et'E p-*xi. (3) j=t+i A s al l i n f o r m a t i o n is p u b l i c a n d c o m m o n knowledge i n this economy, financial flows m a y b e t ied t o each possib le rea l i za t ion o f nature . T h e r e f o r e , a f inanc ia l cont rac t prescr ibes al l poss ib le t ransac t ions between the agents for al l poss ib le fu ture cont ingencies . T h a t is, a cont rac t does not o n l y speci fy a un ique t ransfer for each p e r i o d regardless o f the rea l i za t ion of na tu re , b u t a state-cont ingent m e n u of t ransfers. G a i n s from t r a d e be tween the entrepreneur a n d the lender arise n a t u r a l l y i n th is set t ing. T h a t is , the cont ract is not i n general the t r i v ia l one where al l t ransfers x* are set equa l to zero i n a n y p e r i o d . F i r s t , there is a g r o w t h m o t i v e for t rade o r i g i n a t i n g f r o m the fact tha t the entrepreneur is not indifferent to the t i m i n g o f his c o n s u m p t i o n whi le the lender is: In the in i t ia l p e r i o d , the a m o u n t of funds avai lable to the entrepreneur m a y fal l shor t o f the a m o u n t of funds requ i red to a t ta in des i red investment levels. In these c o n d i t i o n s , the lender m a y be wi l l ing to p r o v i d e the ent repreneur w i t h the requ i red a m o u n t o f funds to i m m e d i a t e l y fill this g a p i n r e t u r n for a sequence of fu ture repayments . S e c o n d , there is a n insurance m o t i v e for t rade emerg ing f r o m the agents ' a t t i tude t o w a r d r isk . G i v e n his avers ion to r isk, the entrepreneur w o u l d ra ther have a s m o o t h c o n s u m p t i o n p a t t e r n t h a n a volat i le one. T h e lender is indifferent to r isk. T h e agents m a y thus enter a r i s k - s h a r i n g agreement where the lender absorb a p o r t i o n of the r isk inherent to the p r o d u c t i o n technology. If cont rac ts ' te rms are enforceable , b o t h m o t i v e s 1 5 l ead to a n 1 o if T > 1, the contract does not become effective during the model's first period. For any period j < r, this resources constraint is still valid since x1 — 0 when no contract exists. 1 4 Note that it is assumed that the entrepreneur's and the lender's discount rates are the same in order to avoid that our results be solely driven by the fact that one agent is more impatient than the other. 1 5 A third motive would arise if the agents' discount factor were different. For simplicity, it is assumed that they are the same. Otherwise, one agent would become more impatient than the other. 52 optimal contract where transfers are such that the entrepreneur's consumption is constant over time and investment is set to its socially efficient level. Hereafter, this contract is referred to as the full-enforcement contract. It is assumed that the contract's terms between the lender and the entrepreneur are not enforceable. The full-enforcement contract may not be feasible in this context. Even though this contract is in principle beneficial to the entrepreneur and the lender, ex ante, in history hT, this may not be the case ex post, as the situation of each agents evolves and as uncertainty unravels. One of the agents may be in a situation where following the contract's terms is in fact less attractive than taking some other course of action. For example, the lender may have other lending opportunities. If the contract specifies a sequence of transfers that would render its discounted expected return inferior to the return on another lending opportunity; the lender will terminate the contract, and advantage of this opportunity. Similarly, it may become in the entrepreneur's interest to default, if by abandoning the contractual relationship, he also gain access to other opportunities. Let V(kt~1,9t) be a function giving the value attached by the entrepreneur in history hl to some opportunity at his disposal outside the contractual relationship when the capital stock installed is fct_1 and the realization of nature is 9t. Define W(kt~l,9t) in a same way for the lender. These functions are value functions arising from dynamic optimization problems solved independently from the long-term contract. They provide an exhaustive description of both agents' set of opportunities outside the contractual relationship.16 As both agents cannot be forced to respect their contractual engagements, a feasible contract must offer to both of them, in every period, at least as much value as the one offered by the outside opportunities at their disposal; otherwise, one agent will eventually breach the contract. That is, the incapacity to enforce contracts implies that a contract is feasible only to the extent that its parties do not have any interest in breaching it. A contract which is enforced by its parties own volition, because it is in their self-interest to respect their engagements, is labeled self-enforcing (see Thomas and Worrall 1988, 1990, 1994). A self-enforcing contract mitigates its parties' incentives to behave opportunistically by offsetting any short-term gains obtained after reneging with long-term benefits. This is made possible by the assumption that once an agent leaves the contractual relationship, 16 They are referred to as outside opportunities. 53 he cannot re-enter it at a later date17—the relationship is broken forever after.18 In this context, to the extent that long-term gains from bilateral contracting exist, opportunities outside the contract constitute the punishment for deviating from the contract's terms. A long-term contract is thus enforced by threatening agents with these "punishments" . 1 9 In view of this threat, they only remain in the relationship because it is in their self-interest to do so. Financial contracts are usually assumed enforceable on the basis that courts could in principle be called upon to compel borrowers and lenders to respect their contrac-tual engagements. Except maybe for Albuquerque and Hopenhayn (1997), Hart and Moore (1994), Kehoe and Levine (1993), and Kiyotaki and Moore (1997), the implications of partial enforcement for financial contracts have been typically examined in international settings because of the inherent difficulty to use courts to enforce contracts across national boundaries.20 Arguing that financial contracts are also difficult to enforce in the context of domestic financial markets does not however deny the existence of a legal system. Hart and Holmstrom (1987) argue that because of transaction costs,21 it would be too costly to enforce contracts on a day-to-day basis by reverting to courts; contracts are rather enforced by custom, good faith, and reputation, that is, through the self-discipline of their parties. Even if using courts to enforce the letter of a contract may be too costly, courts nevertheless play a crucial role in a domestic setting. The key distinction between a financial contract in an domestic context and one in an international context lies in the distribution of the property rights over the borrower's assets if the financial arrangement is terminated. In a domestic context, borrowings may be secured by collateral assets since it is possible for courts to enforce lenders' property rights over these assets; in an There is only one lender in this model. Strictly speaking, this implies that the borrower cannot leave one financial relationship for another one. This may seem to be a strong assumption. However, Kletzer and Wright (1995) provide arguments suggesting that self-enforcing contracts may be immune to the addition of other lenders. This is not a strong assumption in the absence of private information. As everything is common knowledge, if breaching the contract and re-entering it later on was optimal, this could be "replicated" inside the contractual relationship. 1 9 For a complete game-theoretic treatment of self-enforcing contracts, see Kletzer and Wright (1995) and Kocherlakota (1996). Subgame perfect equilibria of the model which are strongly renegotiation-proof are selected. See for example Bulow and Rogoff (1989), Thomas and Worall (1994), and Marcet and Marimon (1992). n i Their argument is based on the idea that it may be costly for a third party to verify if the contract's terms are being followed to the letter by its parties. In a sense, there is an informational asymmetry between the courts and the parties to the financial contract. 54 international context, this is not possible. One could see the prime role of courts as one of enforcing property rights over collateral assets rather than one of enforcing the letter of each financial contract. This is the approach taken in this essay. Hence, while on the one hand it is assumed that contracts cannot be enforced, it is assumed on the other hand that property rights are enforceable. This confers a dual role to the capital stock; it is not only a factor of production but also an asset which may be pledged as collateral by the entrepreneur to secure its borrowings. Property rights over the capital stocks are implicitly embedded in the value func-tions V( •) and W(-); they specify how the capital stock is distributed between the en-trepreneur and the lender if the contract is breached. Thus, these functions give the value each agent may obtain from his respective share of the capital stock outside the contrac-tual relationship. In Marcet and Marimon (1992) for instance, the entrepreneur retains the totality of the accumulated capital stock if the contract is breached. A similar assumption is made in Thomas and Worrall (1994). They reflect property rights in an international context. In a domestic context, the lender may be able to alienate most of the installed capital stock. This fact is likely to influence in a significant way the form taken by the op-timal financial arrangement with respect to the ones found in Thomas and Worrall (1994) or Marcet and Marimon (1992). 3.3 Outside opportunities. At the beginning of each period t, the amount of goods available for investment spending, consumption, and debt repayments is g(kt~1,6t) + (1 —£)fc t -1 (i.e. cash plus fixed assets). This is the quantity of internal funds available in history / i * . It is assumed that the firm is liquidated if the contract is breached. Hence, g(kt~1,dt) + (1 — 5)fc t _ 1 is also the amount of goods over which both agents may exercise property rights upon termination of the contractual arrangement. The set of outside opportunities must specify each agents' respective share of these goods if the firm is liquidated and their value in alternative utilization. Outside opportunities also determine if it is initially in both agents' interest to enter a financial agreement. Property rights are described first. A description of the set of alternative opportunities to the financial contract follows. It is assumed that the entrepreneur's initial endowment of the capital good, 77, cannot be alienated by the lender under any circumstances. In the terminology of Kehoe and Levine (1993), the quantity r\ is a "private endowment"; an asset which cannot be taken away from its proprietor. This portion 77 of the capital stock may be thought of as an asset 55 which has value to no other agents than its owner because its utilization requires owner-specific knowledge. Alternatively, 77 may be considered as human capital; in which case, the model may be interpreted along the lines of Hart and Moore (1994). Any amount of capital exceeding 77 may be pledged as collateral in a contractual relationship.22 In addition, it is assumed that the lender has no claim on current output. Therefore, if a financial contract is terminated in period t—either because the lender refuses to renew the financial arrangement or because the entrepreneur defaults—the amount of goods which may be attached by the lender for payment of the outstanding debt is limited to (1—6)(kl~l — 77). At the same time, g(kt~1,9t) — g(r),9t) goods are supposed wasted in legal fees or other expenses incurred at the time of the firm's liquidation.23 The entrepreneur is thus left with g(n,9t) + (1 _^)»7 goods. Both agents may use their respective shares of the goods available after the firm's liquidation to take advantage of their alternative opportunities. Rather than taking part in a financial contract with the lender, the entrepreneur may finance investment through retained earnings exclusively. The value of this option in history when kt-i is the capital stock installed,24 denoted by v(kt-i,9j), may be obtained by solving the dynamic programming problem n v(kt-i,9j) = max{tx(ct) s.t. ct = g(kt-i,9j) + (l-6)kt-i ~ h, kt-(l-6)kt-i > 0 . Since k° = rj, the entrepreneur will agree to a financial contract with the lender in history hT provided that the terms of the contract guarantee him an expected utility level no less than v(n,9r). Moreover, even when he is part of an ongoing contractual relationship, the entrepreneur may always revert to this initial option. This assumption is akin to the one in Hart and Moore (1994); the entrepreneur cannot be compelled to work. He may always renege out of a financial contract by repudiating the debt. If he does, however, he is barred from entering any other financial relationship in the future. Thus, in view of the division of 22 Allowing the entrepreneur to keep a portion of its assets is a plausible assumption. Kehoe and Levine (1993) point out that "modern bankruptcy laws make it possible for agents to preserve some as-sets even in the face of bankruptcy" (p. 869). 23 * This assumption is not necessary. It is only made for convenience. An alternative assumption would be that the firm's liquidation occurs before production takes place, and that the entrepreneur may resume production right away using his remaining share of the capital stock. Note that fct_i is used instead of fct_1. Making explicitly reference to history is useless in this context because this option excludes any contractual arrangement with another party. 56 assets when the contract is breached, the entrepreneur remains in the financial relationship in history hl if it offers him an expected utility no smaller than v(n,9t).25 Therefore, the entrepreneur's valuation of his outside opportunity, V(fc t _ 1,9 t), is equal to v(n,9t). The lender has the possibility to invest funds at a constant interest rate r = 1/8 — 1 in addition of taking part in a financial contract with the entrepreneur. Therefore, in history hT, he enters a contractual relationship with the entrepreneur if the expected discounted sum of transfers is non-negative; that is, if the contract has a positive valuation. Once he lends to the entrepreneur, the lender always has the option to liquidate the firm. In which case, he may exercise his property rights over the outstanding capital stock. As he may lend at the constant interest rate r, the value of this option is (1 —6)(A; t_1 — rj). Hence, the lender resumes an ongoing relationship with the entrepreneur in any history hl if the contract's value is no less than (1 — 6)(kt^1 — rf). The lender's outside opportunity valuation, W(Jfc1_1,9 t), is thus equal to (l-6)(kt-1 - ry).26 3.4 The self-enforcing financial contract. A long-term contract, beginning at date r with an accumulated capital stock fcr-1, is completely described by an history-contingent sequence of transfers, consumption, and in-vestment {xi, o3, V}'jLT satisfying the resources constraint (2) from this date onwards. This contract is never breached if it offers to both agents no less value than outside opportunities do. Hence, the contract is never breached by the entrepreneur if oo uic1) + EtY, PM*) > v(V, 9t) (4) j=t+i is satisfied for each period t = T , T + 1,..., co, in every history. In a similar manner, the lender never terminates the contract if oo x* + Et ^2 ^ ( l - t f X * * - 1 - V) (5) j=t+i is satisfied for each period t = r, r + 1,..., oo, in every history. A long-term contract is self-enforcing if both self-enforcing constraints (4) and (5) are satisfied for each period and When he reneges, the entrepreneur can keep g(r], 6t) + (1 — 6)77 goods and self-finance investment from then on. The value of doing so at time t is t>(?7, 9t). 2 6 Note that W(kl~l, 6t) is formally equal to wt~l(l—S)(kt~l — rf) where wl~l is equal to one if a contract was in effect at the end of history hl~^ and zero otherwise. In addition, notice that fc'-1 > 77 is also implicitly assumed. Otherwise, (1 —6)(fct_1 — 77) could be negative, prompting the lender to renege. This constraint is however always satisfied for small values of 77. 57 in every history. Note that in view of the self-enforcing constraints (4) and (5), the agents never opt for outside opportunities; they would not gain anything by doing so because the contract's terms always offer them at least as much value as outside opportunities do. Note however that even though the termination of the contract is never an outcome of this model, the contract itself is essentially shaped by this very possibility. For instance, even if the entrepreneur may abandon the financial contract by repudiating the debt, he will never do; the debt is always rescheduled so as to avoid this possibility. The sequences of constraints (2), (4), (5), and i* > 0 determine the set of feasible allocations sustainable by a self-enforcing long-term contract from date r onwards. The goal is to characterize the set of Pareto efficient allocations.27 This is done by choosing among the set of feasible allocations, the ones which maximize the lender's expected present value of transfers (3) for all feasible28 level UT of the entrepreneur's expected utility given by ( l ) . 2 9 For a given expected utility level UT, this maximum expected discounted sum of transfers corresponds to a point on the Pareto frontier (or the contract curve). The whole Pareto frontier may be traced by simply varying UT. Define f(Ul,kT~l,6s) as the value of the Pareto frontier in history hTs when the entrepreneur is given the expected utility level C/J and the capital stock installed at that date is kT~l > 0. Given the above discussion, this value is found as a solution to oo oo f{UTa,kT-\ea) = max{xr + ET ^ ^ ' " V | u(cT) + ET ^ P^utf) > UTS) j=T+l j=T+l by choosing a sequence {xj, cj, V}f=T satisfying the sequences of constraints (2), (4), (5), and i* > 0 from period r onward. Following the argument in Thomas and Worrall (1988, 1990, 1994), the Pareto frontier may be expressed recursively as a solution to a dynamic programming problem where the entrepreneur's expected utility level is treated as a state variable. That is, the Pareto frontier defined above may be shown to be the fixed point of a dynamic programming 27 Formally, the sequences of constraints (2), (4), and (5) allows one to identify the set of all subgame-perfect equilibria. Renegotiation-proof subgame-perfect equilibria are selected using Rubinstein's strong perfection as refinement criterion. This consists in choosing equilibria that are immune to renegotiation when possible deviations from an ongoing equilibrium path may be any other subgame-perfect equilibrium. The utility level If is feasible in the sense that it satisfies (4) in history hT. 29 The set of Pareto efficient allocations could also be characterized by defining the problem the other way around: Maximizing the entrepreneur's utility rather than the lender's payoff subject to a similar participation constraint would not change the results presented thereon. 58 problem where each period t = r, r + 1,..., oo, for every state of nature s — 1,2,..., n, a policy {k^c^iU**1}^} is chosen to solve30 n «'=i s.t. x\ = gik1'1,68) + (1 - 6)fct-1 - k\ - cl n <<?.)+P^Up^ZUi (6) i = l Ul^^v^Oi) for i = l ,2 , . . . ,n , **, 00 > (1 - <5)(fc< - i , ) for • = 1,2,.... n, Jfc* - (1 - tf)**-1. > 0. This formulation demonstrates that a self-enforcing long-term contract may be imple-mented via an infinite sequence of one-period contracts. Here, the entrepreneur's expected utility level plays a role akin to the one played by a credit record; it links the series of short-term contracts together. Past history does not have to be explicitly "recalled" throughout the implementation of the contract as long as the entrepreneur's expected utility level is; the entrepreneur's expected utility level serves as a sufficient statistic summarizing past history.31 This way, as 6 is a Markov process, only its current realization matters (directly) for the future and not its past realizations. The set {t/^+1}f=1 is a state-contingent menu of future utility levels promised to the entrepreneur by the lender in period t. The idea is the following: The lender enters each period t having previously promised to the entrepreneur a certain utility level for each possible realization of nature. At the beginning of period t, both agents learn the current realization of nature 8t. This determines the amount of expected utility received by the entrepreneur in period t. The second constraint of the dynamic programme (6) ensures that the entrepreneur obtains from the contract at least the amount of expected utility level he was promised previously. Taking this utility level as given, the lender maximizes the expected discounted value of the series of current and future transfers by choosing a consumption level, an investment level, and a contingent menu of future expected utility levels. The third and fourth constraints guarantee that these utility levels are such that neither the entrepreneur nor the lender will have an incentive to breach the contract in period t + 1. See appendix 3.1. for a complete exposition. References to history in the dynamic programme (6) could be removed without loss of generality, they are not suppressed to keep continuity in the notation. 59 When the state of nature is revealed in period t + 1, a utility level is chosen from this menu. The lender is then faced with a similar problem at that time, and so on for all the following periods. Define r > 1 as the first period for which a self-enforcing contract is feasible. That is, let r be the first period where the entrepreneur's expected utility level UT is such that the inequalities UT > v(n, 0T) and f(UT, kT~1,6T) > (l—6)(kT~l—n) are satisfied (given a capital stock fcT_1). Then, the long-term contract generated by solving (6) is self-enforcing from period r onward because the forward-looking self-enforcing constraints ensure that any subsequent expected utility level granted to the entrepreneur automatically satisfies a similar requirement. Note that the functional equation (6) does not constitute a standard dynamic prob-lem. Standard contraction mapping arguments may not be applied to characterize the Pareto frontier because of the presence of the value function itself in the constraint set. Nevertheless, it may be shown that the Pareto frontier is the limit of a sequence of func-tions obtained from an iterative scheme initialized with the first-best Pareto frontier. It may also be shown that the Pareto frontier, /(•)> is a continuously differentiable (once) and concave function. However, this latter property depends on the specific form taken by outside opportunities. Detailed proofs of these claims are presented in appendix 3.1 in lemmata 1, 2, and 3. The argument followed is similar to the one used in Thomas and Worrall (1994). 3.5 Equilibrium. Let A*, {/37r3j$is^}i==i' a n a {P'K3i'ipl^1}'i=i be the multipliers respectively associated with the second, the third and the fourth constraint of the dynamic programme (6). Setting the first-order conditions to zero, yields the following expressions: -1 + X'Mcl) = 0, (7) 7 i n /^E^toW1'**-'*) " t 1 " 6 ) ] ^ + 0 £ / * ( C ' £ ' 1 ' - 1 + A*S = 0, (8) i=l i=l (1 + ^X)fu(Ult\klBi) + C 1 + A*. = 0, for i = 1,2,...,n. (9) Together with the Kuhn-Tucker conditions kt3-(l-8)kt-l>0, ri>o, [ ^ - ( l - ^ ^ V ^ O , (10) u\t x > v(v, ox, > o, ml - v(rj, eMt 1 = o, (11) 60 for every i = 1,2,..., n; the equalities3 2 u(ca)+pJ2utis+\3i = Uta, (13) i=l **. = 0 (* T _ 1 , * . ) + ( l - * ) * 1 " 1 - h\ - cl (14) and the envelope conditions fu(yl,kt-\ea) = -\\, (is) MUl,^-1^) = 1 -S + gdk'-W - (l-8)nl, (16) these expressions characterize the solution of the dynamic programme (6) when the state in period t is described by the triple (Ul, kl~l, 0a). Suppose that the self-enforcing constraints were ignored. Without the equilibrium conditions (11) and (12), the multipliers I/J^1 and <f>\^1 are equal to zero, and the full-enforcement equilibrium is attained. That is, the entrepreneur expected utility level is equated across all realizations of nature and his consumption level remains constant, both over time and across states of nature; the lender assumes all the risk inherent to the production technology. Investment expenditures are determined by equating the marginal cost of installing one more unit of capital to the expected marginal increase in future revenues brought by this additional unit of capital. This specific contract may be seen as one where the lender rents the entrepreneur's production technology by paying him a fixed payment c 1 every period. This full-enforcement outcome may not be attainable when both agents have access to alternative opportunities. The self-enforcing constraints (the equilibrium conditions (11) and (12)) impose bounds on the amount of future expected utility levels that may be promised to the entrepreneur in period t. For any capital stock level k, define IJi(k) and XJi(k) as functions giving the expected utility levels of the entrepreneur for which XJi(k)=v(k,9i), m(k),k,9i) = (l-6)(k-r1), hold for each i = 1,2,... ,n. Clearly, Ujs+1 satisfies the entrepreneur's self-enforcing con-straint provided that Ufs+l > UJ(J7). The value of U^ry) is the lowest expected utility level 32 The first one of these equalities may be shown to hold whenever U1 > v(rj, 0S). 61 that the entrepreneur may obtain from the contract. Any expected utility level below U^ry) would lead the entrepreneur to breach the contract as it would offer him less than v(n, 9{). Similarly, observing that the envelope condition (15) implies that / (•) is decreasing in the level of utility given to the entrepreneur,33 it is seen that Uf*1 satisfies the lender's self-enforcing constraint if Ui(fc) > Uj^1. The quantity Ui(fc) is the highest expected util-ity level that may be granted to the entrepreneur when enforcement problems exist. Any expected utility level above tji(A;) would lead the lender to renege because in this case, the contract would offer him less value than what other opportunities at his disposal have to offer. The financial contract's continuation after history h\ is self-enforcing provided that for each i, the constraint U^ry) < Uj+l < Uj(A;*) is satisfied. In this case, neither the lender nor the entrepreneur have an incentive to renege in history h^1. The full-enforcement contract is only feasible in this context if there exists an expected utility level U* such that for the full-enforcement capital stock k*, the constraints \Ji(r)) < U* < \Ji(k*) are satisfied for all i. Such an expected utility level does not exist in general. Therefore, the financial contract cannot in general be assimilated to a lease. Financial flows have a non-trivial behavior in this context. The fact that the maximum amount of expected utility that may be promised to the entrepreneur depends on the capital stock becomes a crucial aspect. It introduces a dynamic interaction between the entrepreneur's financial situation and his investment decisions. 3.6 Financial flows. The equilibrium conditions (7) — (14) determine the optimal investment level and a menu of state-contingent future expected utility levels in history hls, given the value of Ul, fct_1, and 9B. The contract's properties may thus be examined by solving the dynamic pro-gramme (6) sequentially, beginning with k° — n and some expected utility level Ul. The value of Ul is arbitrary. It reflects each agents' initial bargaining power. The higher Ul is, the higher is the entrepreneur's bargaining power when the contract is signed. Assumption. U1 is such that f(Ul,ri,9\) — 0. That is, it is assumed for the remaining of the essay that the entrepreneur has all the bargaining power when the contract is signed. The lender is just indifferent between signing the contract and taking advantage of his alternative opportunity to the contract. 3 3 This is true if in (15), V1, > TJ,. 62 This assumption is akin to assuming that the financial market is composed of numerous risk-neutral lenders; competition among lenders drives the contract's net expected value to zero. Financial flows are studied under this assumption: Proposition 1. Define Cj(fc) and Ci(k) as functions satisfying the relations Then in equilibrium, for t > 2, in any history h\, the entrepreneur's consumption level is given by and by c\ — c~i(rj) in history h\. Proof. See appendix 3.1. || Beginning from c1, the optimal consumption level in any subsequent history has a behavior similar to the one observed in Thomas and Worrall's (1988) model. It follows a ratchet-like behavior; it moves by the least amount feasible to keep the agents from breaching the contract. That is, the consumption level is kept constant from one period to the other whenever doing so is feasible (self-enforcing), and is set to a new level when-ever keeping it constant would induce one of the agents to renege. However, Thomas and Worrall (1988) do not allow for intertemporal production. Here, an additional dimen-sion is present; investment expenditures in history hl~l affect the size of the each inter-val [cj(fc), Ci(k)] in history h\ and thereby influence the degree of consumption smoothing attainable by the entrepreneur. This has a significant impact on the path followed by financial flows between the lender and the entrepreneur. Net financial flows in history h* are given by xt — yt — c* — il. As investment expen-ditures are irreversible (i.e. il > 0), they always constitute new borrowings; the remaining portion of x*, m* = y* — c4 amounts to the net payment made to the lender. These net payments evolve in the following manner: Proposition 2. Let rn^k) = g(k,6i) — c~i(k) and Tn~i(k) = g(k,6i) — c^k). Then, in any history hi, t > 2, the net payment m* made to the lender by the entrepreneur is given udc^k))-1 = - ^ ( 1 3 ^ ) ^ , 6 ^ , ' c ^ - 1 ) , if c*-1 > Ciik1'1), if Cj(A; t _ 1 ) > c*-1 > c^A;*-1), if c*"1 < c^fc*-1); 63 if m*-1 + (?/ - y*-1) > mi (A;*-1) y*"1), if m ^ - 1 ) > m*"1 + (yf - y*"1) > m^A;*-1) if m*"1 + (y* - y*-1) < m^A;*-1) when the capital stock installed is A;*-1 and where m 1 = y(A;0, 6\) — c 1. Proof. See appendix 3.1. || With full-enforcement, the optimal sequence of net payments is generated by the recursion m* = m t _ 1 + (y* — y t _ 1 ) , where m1 = y 1 — c 1. This precise sequence of payments may not be feasible however, if the financial contract is required to be self-enforcing. The quantity mi(A; t _ 1) is the largest payment the entrepreneur may credibly commit to make in history h\\ any payment exceeding this amount would prompt the entrepreneur to default as he would be better off opting for his alternative opportunity. The quantity m^A;*-1) is the smallest payment the lender is willing to accept from the entrepreneur in history h\; a smaller payment would make the contract less attractive to the lender than his outside opportunity. Thus, a payment inferior to m^A;*-1) would induce the lender to breach the contract. Proposition 2 states that with partial enforcement, the net payments are set to their full-enforcement levels as long as doing so does not induce one of the agents to renege. When making a full-enforcement payment would lead one of the agent to breach the contract, the sequence of payments is reset to a new level which prevents him to do so; full-enforcement payments are then resumed from this new level. Definition. The entrepreneur is said to be financially constrained in history h\ if his expected utility level Ul is equal to V3(kt~1). From the definition of m^A;* -1) in proposition 2 and the definition of c s(A; t _ 1) in proposition 1, it is clear that the net payment made by the entrepreneur in history ha, m*, is equal to m^A;* -1) if his expected utility level reaches U s (A; t _ 1 ) . This payment exceeds the full-enforcement payment by c t _ 1 — c a(A; t _ 1) units; that is, the entrepreneur is compelled to self-finance a portion of the firm's liquidity needs by reducing his consumption level. The quantity of funds obtained by the entrepreneur falls short of the quantity he would like to obtain. However, increasing the net payment to m a(A; t _ 1) keeps the lender from liquidating the firm by raising his valuation of the financial contract. Partial enforcement of the contract introduces financing constraints in an unusual way; they do not limit the stock of debt per se. Rather, they impose a lower bound on the value of the entrepreneur's debt. by mi m^Ar*-1), m^Ar*"1), m 64 3.7 Investment decisions. Optimal investment decisions are examined in this section. In order to simplify notation, it is assumed for the rest of the analysis that the entrepreneur's momentary utility function is logarithmic (i.e. u(cl) — ln(c1)) and that the function g(kt~1,9t) is a Cobb-Douglas production function of the form QtBk1'101.34 The shock 9t may be interpreted as a demand shock, that is, a shock to the relative price of output and capital (the price of capital is normalized to one). It may alternatively be interpreted as a productivity shock. The set @ and the associated transition matrix TI are chosen so as to approximate a first-order autoregressive process of the form ln(# t+i) = p ln(9t) + et+i, where et+i has a Normal distribution with mean zero and variance cr 2. 3 5 To get a better understanding of investment decisions, the irreversibility constraint on investment spending, il > 0, is initially relaxed. The effects of irreversibility are then discussed. Reversible investment The Pareto frontier has a relatively simple characterization when investment expendi-tures are reversible. In this case, decisions rules in history h\ do not depend on the amount of capital inherited from the previous period, only on the value of U\ and 9a. This implies that the Pareto frontier may be written as That is, the Pareto frontier is additively separable in capital and utility. Moreover, /(•) is concave,36 strictly decreasing in utility for U\ G [IJ9(A;t~1),tjs(A;t_1)] and strictly increasing in capital. These properties of the Pareto frontier allows one to unambiguously characterize the investment level solving equation (8). Proposition 3. Let i * be the full-enforcement investment expenditures in history ha. Then, the partial-enforcement investment expenditures in history ha are determined by = fc&rt1 + d^M\ + (i-8)d(ut,9a), (l?) d(Ul,93) = (1 + E ^ 1 ( » i / ^ » ) T « ) 1 / ( 1 " ° ) - 1, (18) i=l 3 4 These assumptions are not essential to show the following results. The procedure used to determine 0 and IT mimics the one found in Deaton (1991). It is described in appendix 3.2. 3 6 The value functions W( •) and V( • ) satisfy the conditions of lemma 2 in appendix 3.1. 65 where E39 = £?=19^ai and d{Ul,93) > 0. Corollary. ia > i*3. Proof. See appendix 3.1. || Proposition 3 states that investment spending may exceed the full-enforcement level; Rather than underinvesting, the entrepreneur overinvests. The full-enforcement invest-ment level, i * , reflects investment opportunities in history ha. It is the investment level which would be observed if financial concerns were disregarded. The function d(Ul,9a) reflects the entrepreneur's financial situation; the value of each Kuhn-Tucker multipli-ers tplf1 is different from zero if Ufal = Uj(A;*) in history h^1, that is, if a borrowing constraint is suffered in history h}^1. The value of d(Ul,93) is thus zero in history ha un-less experiencing a financing constraint in history ht+l is possible; in which case, its value is positive. Therefore, the temporal behavior of investment expenditures exhibits regime switches depending on the firm's financial situation: Investment expenditures correspond to investment opportunities if the entrepreneur does not apprehend a financing constraint next period; they exceed the full-enforcement level if experiencing a financing constraint is a likely possibility. The key aspect of equation (18) is that in the presence of partial enforcement, invest-ment expenditures are not only determined by expectations about their future profitability but also by past history, through the value taken by d(Ul,93). High expected utility lev-els increase the probability that a financing constraint will be binding in the future (see proposition 1 or proposition 2). This occurs because the lender's valuation of the financial contract decreases as the entrepreneur's expected utility increases, thereby making more likely the possibility that the lender's self-enforcing constraint will bind for low realization of 9. Hence, if the entrepreneur is granted relatively high expected utility levels when internal funds are abundant—a situation occurring when his outside opportunity becomes more attractive in good times—the likelihood of experiencing a financing constraint in the future, when internal funds become scarce, increases; prompting the entrepreneur to overinvest. Therefore, investment expenditures may become excessively sensitive to the abundance of internal funds. That is, they may behave in a manner consistent with Faz-zari, Hubbard, and Peterson's (1988) empirical findings. The mechanism leading to this result is however quite different than the one present when financing constraints originate from imperfect information. Proposition 4 below illustrates this fact: 66 Proposition 4. The value of f(U,k,9) — (1 — S)(k — n), the lender's net valuation of the financial contract, increases when the capital stock is increased. Corollary, i) The function Vi(k) is increasing and concave in k for any i = 1,2,..., n. ii) The amount of self-finance provided by the entrepreneur when he experiences a financing constraint, c— c~i(k), decreases with the amount of capital installed. Proof. See appendix 3.1. || Proposition 4 allows one to investigate why overinvestment occurs in equilibrium. Suppose that a financing constraint is likely to be binding in some history hitfi1. Then, the highest expected utility level which may be granted to the entrepreneur if history / i * * 1 occurs is \Jj(k); otherwise, the lender reneges. From proposition 1, it is clear that the entrepreneur's consumption level in history h*^1 has to fall with respect to the one in history ha. Suppose that the full-enforcement capital stock k* is installed in history ha. Then, the entrepreneur's consumption level and the menu of future expected utility lev-els {C^, + 1}" = 1 promised to him in history hi, are such that c\+l = ca for i ^ j, c**1 = Cj(A;*) (see proposition 1), and u(c*) + B $^"=1 U^ir^ = Uj is satisfied. Since the expected utility level Ul is given in history ha, the fact that U1^1 — U,-(A;*) is less than the full-enforcement level implies that the value of c* and of all the utility levels {^ s + 1}™ = 1, i ^ j, is higher than it would be otherwise. The lender gains in this situation if more capital than the full-enforcement level k* is installed. Proposition 4 states that the lender's self-enforcing constraints are relaxed as the capital stock is increased; that is, installing more capital raises the range of self-enforcing utility levels. Suppose the capital stock is increased to some level kla greater than k*a. Then, the entrepreneur's expected utility level in history h^1 increases from Uj(fc*) to \jj(kl). On the one hand, this augmentation of expected utility in history h*^1 increases the value of Cj(k*a) to c~j(ka). On the other hand, since this would give to the entrepreneur an expected utility level superior to the one he has been promised in history ha, the value of c* and, as a result, the value of each c*^ "1, i ^ j may be lowered until the entrepreneur is made just indifferent with respect to the situation where A;* was installed. That is, the entrepreneur obtains the same expected utility level Ul whether k\ or k* is installed. However, the state-contingent menu of future expected utility levels has been changed. The self-enforcing constraints impose a cost on the lender. With full-enforcement, the lender is able to equalize the entrepreneur's utility levels across all states of nature. 67 The self-enforcing constraints act so as to spread apart from each other the utility levels granted to the entrepreneur in each state of nature. This is costly because the lender's valuation of the contract, / ( • ) , is concave in utility; stretching the state-contingent menu of future utilities reduces the value of the contract's continuation. When the capital stock is increased from k* to k\, the entrepreneur's future expected utility level is augmented in the state of nature where a financing constraint is binding (history h^1) and is decreased in all other states of nature. That is, installing more capital when a financing constraint is anticipated reduces the difference between all the state-contingent future expected utility levels promised to the entrepreneur and raises the value of the contract's continuation. In addition, in equilibrium, overinvestment is partly financed by the reduction of the entrepreneur's current consumption level. Overall, overinvestment makes the lender better off. This is confirmed by proposition 5. Proposition 5. The expected marginal cost of installing one additional unit of capital in history h3 is 1 + P *£> WMk-lHi - P A:*"1, O^XJXkl)^ < 1 (18) i€l i€l where I = { t | xj^1 > 0, n > i > 1}. Proof. See appendix 3.1. || The first term in equation (18) is the direct cost of raising investment expenditures in history hs. If a financing constraint is not expected, the marginal cost is unity and investment spending attains its full-enforcement level. The second term represents the additional cost incurred by the entrepreneur for increasing the entrepreneur's utility level in states of nature where financing constraints are binding. The third term takes into account the savings realized by decreasing the expected utility levels in all other states of nature. Overall, this last effect dominates; the marginal cost of raising investment is less than unity if reaching at least one financing constraint is anticipated. The capital stock is therefore raised above the full-enforcement level. Note that this reduction of the marginal cost of providing external finance through an increase of investment expenditures translates itself into a reduction of the cost differential between external and internal funds. In short, investment spending in excess of the one solely reflecting investment oppor-tunities should be observed when the entrepreneur fears that he may experience a financing constraint in the near future, if the realization of nature is a relatively low one. He does so in order to reduce the amount of self-finance (reduction of his consumption level) he will 68 have to put up if such an eventuality occurs. If the model is not taken too literally, the entrepreneur's consumption level may be seen as representing various costs of sales (com-pensation of employees, advertising, overhead expenses, administrative expenses, perks, etc..) or specific engagements with suppliers which are to some extent fixed, at least in the short-run.37 The organization as a whole may be averse to abrupt variations of these expenses because they may be the result of a painful reorganization of the firm (i.e. layoffs, wages or administrative expenses cutbacks, etc.) . Whereas in good times (high realization of 9), sales and external funds are sufficient to maintain a certain level of expenses, they may be insufficient to sustain the same level of expenses in bad times (low realization of 9) if a financing constraint is experienced. In this context, a firm with high "quasi-fixed" ex-penses in proportion of its revenues runs a high risk of being forced to undergo a significant reorganization of its activities, the more variable are its sales revenues. Overinvestment allows a firm to insure itself to a certain extent against this risk by allowing it to dampen variations in its sales revenues. Irreversible investment. The irreversibility of investment expenditures may have a significant impact on the financial contract. Considering this additional constraint introduces the notion of produc-tion capacity. New capital installations cannot be taken down through divestment; once installed, new units of capital increases the firm's production capacity persistently. The production capacity may only be reduced through the depreciation of the capital stock, by postponing new installations. Whereas in the case of reversible investment, it is possible to revert to efficient production levels quite rapidly, if overinvestment occurred in the past, this is not possible when investment expenditures are irreversible. Irreversibility introduces a dynamic interaction between the production capacity and the financial capacity; it may contribute to propagate the effects of financing constraints over time. The overall effect of this additional constraint on investment spending is nontrivial. Fama (1990) emphasizes the importance of the fact that nearly 90% of the outflows of funds for U.S. nonfmancial corporations constitute payments on fixed-payoff contracts (which are cost of sales for a large part). 6 9 Using the envelope condition (16) in the equitibrium condition (8), one obtains n 0 + aBQ^r1- [ l - 8 ) ^ s i - 1 + t=i The term on the second line of equation (8') reflects the impact of financing constraints on investment in history h]*1. The impact of financing constraints on investment is am-biguous in this case. The incentives to overinvest38 are dampened or counterbalanced by the fact that an increased production capacity may prove to be costly later on, if it substantially exceeds desired production levels in the future. Underinvestment as well as overinvestment may both occur in equilibrium since the term on the second line of (8') is not necessarily positive in this case. Nevertheless, overinvestment is likely to dominate because underinvestment is possible only if some p^1 are positive. Moreover, note that the introduction of the irreversibility constraint itself narrows the constraint set and thereby reduces gains from trade between the lender and the en-trepreneur. This may increase the likelihood that a financing constraint becomes binding in any period. Overinvestment may thereby be observed more frequently, even if its mag-nitude might be reduced. Unfortunately, these are only speculations: The equivalent of proposition 3 cannot be readily proven in this case since it is not possible to obtain an analytic solution for the full-commitment investment level. Similarly, the equivalent of proposition 4 cannot be obtained because the Pareto frontier is no longer additively sepa-rable in utility and capital. The properties of investment expenditures in this context may nevertheless be explored further by reverting to numerical techniques. This is the object of section 3.8 3 . 8 Numerical results. This section examines the dynamic properties of the equilibrium financial contract between the lender and the entrepreneur for a specific parameterization of the model. The parame-ters' values39 used throughout this exercise are presented in table 3.1. The characterization oo Note that overinvestment refers to investment in excess of the one that would be observed under full-commitment and irreversibility. Obviously, there is almost always underinvestment with respect to the case where investment is reversible. An annual frequency is considered. The parameters a, (3, and 6 are set to standard values in the real 70 of the process 9 is also found in this table. Note that as this is not a general equilibrium model, there is no explicit attempt to match empirical moments; the goal of this exercise is to provide some insights concerning the dynamic properties inherent to a self-enforcing financial contract for a reasonable parameterization of the model. The results presented in table 3.2 to table 3.4 are obtained by simulating the optimal path followed by each en-dogenous state variable for 200 sample realizations of the shock 9. Each sample spans 20 periods. This choice allows one to compare these results (especially the ones in table 3.2) with the ones found in the empirical literature. The results in table 3.5 are computed from one sample realization of 9 of 4000 periods. Each table reports results for the reversible and the irreversible investment cases. The numerical solution of the dynamic programme (6) is obtained through a value iteration algorithm where the value function at each iteration is approximated using a shape-preserving interpolant. Details of this procedure are found in Sigouin (1997a).40 Beginning with Fazzari, Hubbard, and Peterson's (1988) work, several papers have shown that investment spending is excessively sensitive to measures of internal funds' abundance for firms identified a priori as being financially constrained.41 This result is interpreted as evidence that financial concerns indeed affect investment decisions. It is substantiated by the finding that this apparent excess sensitivity of investment does not seem to exist for firms identified as being unconstrained on a priori grounds. In particular, these results have been obtained by estimating equations of the following form: *t . Qt . it-i , ifundst - — = a0 + ai- \ - a 2 z — + a3 — + e t , (19) « t - i kt-i kt-2 kt-i where Qt and ifundst are variables measuring investment opportunities and the avail-ability of internal funds respectively. Investment opportunities are typically measured by business cycles literature. The technology parameter B is chosen so as that the mean capital stock is unity. Choosing an appropriate value for rj is troublesome as its empirical counterpart is not readily available. The parameter 77 is set to a low value which ensures a sufficient degree of difference in v(t]y0i) from one state of nature to the other in order to keep the problem interesting. All the computations are done with five possible states of nature (i.e. n = 5). 4 0 Note that one must revert to a value iteration algorithm to obtain a solution to (6) since that the lender's self-enforcing constraint (the fourth constraint) implies that an expression for the value function /( • ) is necessarily required. The algorithm in Sigouin (1997a) approximates /(•) with a concave and monotone function. 4 1 See for example Blundell, Bond, Devereux and Schiantarelli (1992), Calem and Rizzo (1995), Chirinko and Schaller (1995), Fazzari and Peterson (1993), Hubbard, Kashyap, and Whited (1995), Schaller (1993). Considering the size of the literature examining the relationship between investment and internal funds' abundance, this list is not exhaustive. 71 average Tobin's q and internal funds' abundance by cash flows or other measures of firms' available liquidity. If Qt measures investment opportunities properly and capital mar-kets are perfect, financial concerns are irrelevant and investment spending only reflects investment opportunities; the value of should therefore be zero. Fazzari, Hubbard, and Peterson (1988) show that the value of a$ is greater than zero for firms a priori identified as financially constrained while it is close to zero for the unconstrained ones. A similar exercise is performed using simulated data from the model with enforcement constraints. In equation (19), the full-enforcement investment level i* is used to measure investment opportunities (i.e. Qt). The availability of internal funds (ifundst) is measured with OtBk1"10 + (1—6)fct_1. This measure of internal funds' abundance is used rather than cash flows (the conventional measure in empirical studies) because the general nature of the financial contract does not allow one to compute this latter measure.42 By construction, the value of a3 is zero for firms which are never faced with financing constraints since i t/fc t_i = %1/kt-i for these firms. Equation (19) is estimated using simulated data on financially constrained firms. The estimation results are presented in table 3.2. The estimates of as are significantly different from zero. Therefore, financing constraints arising from the impossibility to enforce contracts influence investment spending in a manner consistent with Fazzari, Hubbard, and Peterson's (1988) empirical finding.43 Table 3.3 quantifies the relevance of overinvestment. In spite of the fact that a fi-nancing constraint is experienced only 12% of the time, overinvestment is observed more than half of the time in the case where investment expenditures are irreversible. The interaction between the irreversibility constraint and the financing constraints is clear. Overinvestment occurs twice as more often in this case than in the case where investment expenditures are reversible. Underinvestment almost never occurs. On the one hand, the constraint il > 0 reduces possible gains from trade and makes the occurrence of a binding financing constraint more likely. On the other hand, as it may be seen from the third panel of table 3.3, the fact that capital expenditures are irreversible reduces the size of overinvestment. Overinvestment is not only more frequent but also slightly more persistent when the constraint i* > 0 is introduced. Episodes of sustained overinvestment last 3.5 periods on average in this case compared to 2.2 in the reversible investment case. Half of the time, however, overinvestment is Firms' borrowing cost is unknown. Therefore, actual profits cannot be measured. The hypothesis that c*o = a2 = «3 is also rejected for both models. 72 observed for at most two consecutive periods. Overall, overinvestement appears significant even though financing constraints are suffered quite infrequently. Investment spending may deviate from the the full-enforcement level by as much as 80%. However, half of the time, deviations do not exceed 2%. Overinvestment does not only increase the average capital stock installed by the entrepreneur but also reduces its variability. Indeed, it is seen from table 3.4 that the capital stock is markedly less variable than the one observed in an economy with full enforcement. As a result, the output volatility is also reduced. This finding suggests that overinvestment occurs in periods where the capital stock declines; it dampens downturns rather than exacerbating upturns. This is consistent with the fact that experiencing a financing constraint becomes more likely as the value of 9 declines. Table 3.5 investigates if financing constraints give rise to endogenous dynamics. In the spirit of Cogley and Nason (1993), this is done by studying the moving average (MA) representation of output. The moving average representation of output expresses output as a function of all present and past innovations to the exogenous random variable 9. The moving average representation of output observed in an economy with full enforcement serves as a baseline case. Any difference from this baseline case may be attributed to the presence of financing constraints. The deviation of output from its mean is denoted by yt in the partial enforcement case and by y* in the full enforcement case. These moving average representations are obtained by estimating autoregressive-moving average (ARMA) repre-sentations. The order of each A R M A representation used is motivated by proposition 6 below:44 Proposition 6. The law of motion of the capital stock and the entrepreneur's ex-pected utility level may be expressed as Ar* =K(U\kt-\9t) U*+l = U(U*,kt-\6t,0t+l), for some functions JC( •) and U(-). If these functions are approximated by log-linear functions around the mean value of their respective arguments, then the logarithm of output has an approximate ARMA(3,2) representation in the partial enforcement case and an approximate ARMA(2,1) representation in the full enforcement case. Proof. See appendix 3.1. || 4 4 The parameters' value in table 3.5 are obtained by first estimating equations (A.8) and (A.9) of ap-pendix 3.1 and the law of motion of the random disturbance 0. The results hence obtained are used to compute the value of the parameters in equation (A. 10). 73 From the fact that the entrepreneur's expected utility level remains constant in the full enforcement case, it is clear that its behavior in the partial enforcement case is respon-sible for the additional output dynamics, that is, for the addition of one moving average coefficient and one autoregressive coefficient with respect to the full-enforcement A R M A representation of output. The moving average representations of output in table 3.5 are obtained by inverting these two A R M A representations. The first term of each moving average representation constitutes the impulse dynamics, that is, the dynamic behavior of output due to the behavior of the exogenous shock 9. The second term captures the dynamic behavior of the output series specific to each specification of the model. The last term of each moving average representation of yt accounts for the output dynamics inherent to partial enforcement. It is readily seen that the model specific component (second term) are quite similar between yt and y*. This confirms the fact that the third component is really arising from the presence of enforcement constraints. The numerator and the denominator in the component specific to the presence of financing constraints are virtually the same. They nearly cancel each other. Therefore, the dynamic behavior of both output series following an innovation in 9 appears rather similar. Moreover, the persistence of output is not increased in a significant way. The long-run impact of an innovation in 9 is quite similar for both output series. Overall, financing constraints arising from partial enforcement do not seem to be responsible for increasing the persistence of the output series' response to an innovation in 9. Their presence seems to alter short-term dynamics uniquely. Figure 3.1 and figure 3.2 present impulse responses following a positive and a negative innovation to 9. These impulse responses reflect the entrepreneur's decisions when he apprehends the possibility of experiencing a financing constraint. They were computed as follows: Three thousand realizations of 9 were first simulated. Each draw has a length of 20 periods. The model's responses to each sequence of shocks were computed beginning from the same initial value of the state variables. The exercise was initialized from a situation where the capital stock has been near its mean for a while, the value of 9 was at his mean, and the entrepreneur's expected utility level was such that the probability of experiencing a financing constraint was near one fourth. The conditional expectation at time t of each variable at time t -f- i , i — 0 ,1 , . . . , 20 is estimated by the average of these 3000 responses. The top portion of figure 3.1 presents the responses of capital after a positive and after a negative innovation to 9. The dashed line is the response of the capital stock if the contract was fully enforceable from then on. Clearly, the responses of capital are 74 asyinmetric. However, the bulk of this asymmetric behavior stems from the irreversibility constraint since the impulse responses are also asymmetric when there is full enforcement. The impact of overinvestment on the capital stock may be measured by the difference between the solid and the dashed lines. It is plotted on the bottom portion of figure 3.1 (solid line). The evolution of the probability, in any period, that a financing constraint will bind in the next period is also plotted (dashed line). Overinvestment occurs mainly when the capital stock decreases. This reflect the fact that experiencing a financing constraint becomes more likely as the value of 9 declines. The impact of a negative shock on the capital stock is smoothed and dampened by overinvestment; it does not fall as fast and as much as in the full enforcement case. Figure 3.2 traces the evolution of the financial flows following the same shocks. The entrepreneur is not financially constrained when a positive shock occurs. He receives the same amount of funds as in the full enforcement case and makes lower net repayments thereafter. Note however that self-financing increases after the fourth period. This increase in self-financing coincides with the augmentation of overinvestment seen in figure 3.1. After a negative shock, the entrepreneur finances a portion of overinvestment through a reduction of his consumption level. Negative shocks are costly to the entrepreneur; self-finance is increased in a significant way because the entrepreneur does not receive as much funds from the lender as in the full enforcement case. It is seen from the top-right panel that the entrepreneur initially makes higher net payments than in the full enforcement case; thereafter, net borrowing (the negative value of transfers) is persistently inferior to the desired level. The entrepreneur has clearly a limited financial capacity. Overall, the asymmetric behavior introduced by financing constraints seems more apparent in financial variables. These results show that downturns are costly to the entrepreneur; he is compelled to self-finance a portion of the firm's activities in this situation. Overinvesting when external finance is relatively abundant provides him with additional internal funds in the future and allows him to reduce future increases in self-finance. 3 . 9 Conclusion. This essay has used dynamic contract theory to study the joint behavior of investment decisions and financial flows when financing constraints arise from the fact that contracts signed between lenders and borrowers cannot be fully enforced. While this approach has typically been followed in the study of international financial relationships, it has been argued that partial enforcement may also pose a problem in a domestic context. The main 75 result of this essay is that partial enforcement may lead firms to overinvest. Simulations using a reasonable parameterization of the model show that overinvestment appears as a significant feature of investment decisions even if financing constraints are effectively binding only on a few occasions. However, in spite of its significance, overinvestment does not introduce much endogenous persistence nor propagation. In fact, overinvestment is shown to dampen rather than exacerbate business fluctuations. Nevertheless, the pattern followed by investment expenditures in this model is consistent with the empirical finding that investment spending is excessively sensitive to variations in internal funds' abundance. This exercise stresses the importance of being cautious with interpretations of this empirical finding based on models with a limited dynamic environment or models where shocks are in fact zero-probability events. Even though the model studied in this essay is able to replicate Fazzari, Hubbard, and Peterson's (1988) empirical finding, it does not warrant the presence of a financial accelerator. This is at odds with the macroeconomic implications usually inferred from this empirical finding. However, this result is obtained in an infinite horizon framework where agents' anticipations are consistent with the stochas-tic environment. Overinvestment in this model constitutes firms' rational response in a dynamic environment where they fully anticipate the possibility of experiencing a financ-ing constraint. Overinvestment is used to generate additional funds in the future. It is thus akin to precautionary saving in the sense that it allows firms to alleviate an eventual shortage of funds if external finance ever becomes limited in the future. Overinvestment usually occurs at the end of upturns, when experiencing a financing constraint becomes more likely. As the desired capital stock level is also declining at the end of upturns, overinvestment slows down the pace at which the actual capital stock declines. Moreover, this essay raises interesting issues: What is the mechanism responsible for the apparent excess sensitivity of investment spending to variations in internal funds' abun-dance? In particular, does this empirical finding reflect the fact that firms underinvest or overinvest? The results presented in this essay seem sufficient to challenge the general belief that financing constraints invariably limit firms' capacity to invest. Determining whether financing constraints lead to underinvestment or overinvestment is a difficult question to answer in view of the fact the investment level which uniquely reflects investment opportu-nities is unobservable in practice. Nevertheless, an answer to this question would provide important insights into the determinants of firms' investment spending. Further study of these issues is certainly warranted. It is however beyond the scope of this essay. 76 Appendix 3.1 The first part of this appendix describes the technical details surrounding the solution of the dynamic programme (6). Solving programme (6) is nonstandard. The strategy followed is explained in some details. The value function /(•) is characterized by lemma 1, lemma 2, and lemma 3 below. Proofs of the lemmata found in the text follow in the second part of this appendix. Let K, = [fc, k] be the set of feasible capital stock levels.45 In addition, let U* be the maximal expected utility level that can be attained by the entrepreneur in the full-commitment case (i.e. when both enforcement constraints never bind with equality) when the investor is exactly given its reservation level W(k, 9) for any fc € £ and 9 G ©. Then, U = [V(k, 9\), U*] contains the set of feasible expected utility levels in the problem with partial commitment since V(fc, 9{) is the lowest utility level which can be granted to the entrepreneur in that case. Finally, let S = U x K x 0. This set contains all possible values taken by the vector of state variables (U, fc, 9) in the problem with partial commitment. Suppose that F is a function in the space of bounded continuous functions on S (denoted by C(S) hereafter) such that F(U, fc, 9) > W(k, 9) for some (U, k,9) £ S and define the operator T: C(S) -> C(S) by n T(F)(U, fc, 9S) = sup{<7(fc, ks-(l - 6)k, 83) + (1 - 6)k - ks - cs + p^T, F(u*>k" *«>«•} n S.t. u(cs) + P Ui**i ^ U ' (A-1) i=l Ui>V(k3,9i) fori = l , 2 , . . . , n , (A.2) FiUuk^yWik^i) fori = l , 2 , . . . , n , (A3) k3 > fc, fc > k3, c3 > 0, (A4) g{k,93) + (l-6)k-k3-cs>-X (A5) where the maximization is performed by choice of {k3, c3, {C/"j}"=1}.46 Standard contraction mapping arguments cannot be used to analyze the properties of the operator T. The presence of inequality constraints involving the function F itself creates some difficulties. In general, a sequence of functions {Tm(F)} generated by successive iterations on the 4 5 In general, k < CO follows from standard assumptions on the production technology and xl > — X. However, the existence of fc > 0 may have to be assumed explicitly. 4 6 Note that the constraints in (AA) will not bind in general for sufficiently low values of fc. 77 operator T will not converge to a fixed point of T unless it is initialized with a proper initial function F. The reason is that the domain of TM(F) may "shrink" as m increases. This may occur because nothing guarantees that the function TM(F) = T(TM~L(F)) is greater or equal to W for all the values in S where r m _ 1 ( F ) is: even if the constraint set in T is nonempty, the supremum may attain a value less than W. However, only the portion of the domain for which T™^) > W is relevant in T(TM(F)), and so on for T n , + 2 ( .F ) , T m + 3 ( .F ) , etc... (an arbitrary negative value is assigned to T m ( F ) when it is less than W) This fact permits to locate a fixed point of T provided that the sequence of functions { T ^ F ) } is initialized with a function F having a "larger" domain than the one possessed by the limiting function of this sequence. As this last function is not known a priori, the first-best Pareto frontier (the function that would obtained if enforcement constraints were absent from T) is a prime candidate to initialize this sequence. For a given vector of state variables (Ut,kT~L,OT) € S in history tf, define the Pareto frontier / ( U 1 , fct_1, 6t) characterizing the problem as the function obtained by solving oo f(Ut,kt-\§t) = sup{ET j=t s.t. xj = g{kp-x,ej) + (1 - 8)kj-1 - kj - c> > t, oo j=t oo ET ]T ^'-tu(cJ') > v{kT-\eT) WlT, T>t, j=T O O ET ] T Pj~fxj > W(kT-\ BT) V / T , T > t, j=T xT > -X, kT >k, k> kT, cT > 0, V r > t, where the supremum is found by choosing appropriate sequences of capital stock levels and nature-contingent consumption plans. Lemma 1 shows that the function / defined above is a fixed point of T. It shows that the sequence of functions {TM(F)} obtained by m successive applications of T converges pointwise to / when it is initialized with the first-best Pareto frontier /*. In general, the first-best Pareto frontier is readily available since that in the absence of enforcement constraints, standard contraction mapping arguments apply to the operator T. Lemma 1 provides the basis for an algorithm permitting to obtain a numerical approximation to / . Lemma 2 proves the concavity of / under appropriate functional form restrictions on W and V. These restrictions are quite strong but are 78 only sufficient conditions. The cases considered in this paper satisfy these requirements. Lemma 3 establishes the differentiability of the value function / . The concavity and the differentiability of / validates the use of first-order and Kuhn-Tucker conditions to identify optimal policies. The proofs of lemmata 1 and 2 closely follow the proofs of lemmata 1 and 2 found in Thomas and Worrall (1994). For each 0 G 0 and any function FGC(S) define a set S(9;F) = {{U,k) | (U,k,9)eS, U> V(k,B) and F{U,k,9) >W(k,9)}. These sets give the pairs of capital stock levels and entrepreneur's expected utility levels for which mutually beneficial trades exist between the agents, conditional on the Pareto frontier being given by the function F. Given the value functions W and V describing the set of alternatives to a long-term contract and a Pareto frontier F, a self-enforcing contract exists only if the sets S(9; F) are nonempty for every possible 9GQ. Thus, a self-enforcing contract characterized by / exists only when S(9; f) ^ 0 for all 9 € 0 . Unfortunately, for most value functions V and W, it may not be possible to verify if this is true a priori. The iterative scheme proposed by lemma 1 however allows one to determine whether a self-enforcing contract exists for a set of alternatives described by W and V based on the sets S{9; F) generated at each iteration on the operator T. Lemma 1. Let /* be the first-best Pareto frontier characterizing the problem when enforcement constraints are absent. Then, the sequence of functions {Tm(/*)} converges pointwise to / as ra —> oo if and only if S(9;Tm(f*)) ^ 0 for all 9 G 0 , for every func-tion T m ( /*) of this sequence. Proof. 1) Assume that {Tm(f*)} is a sequence of functions for which S(9; T m(/*)) ^ 0 for all 9 £ 0 , for each ra. This guarantees that the supremum in each Tm(f*) is seeked over a nonempty constraint set and validates the following arguments: The sequence {Tm(f*)} is non-increasing: Suppose that T m *(/*) > T m ( /*) . Then, any feasible policy {k3,c3,{Ui}?=l} in T(Tm(f*)) is also feasible in T(T m - 1 ( /*)) since the fact that T m ( /*) > W implies that Tm~l(f*) > W (i.e. constraint (A3) is not re-laxed as ra is increased). Therefore, we have that T(T r"- 1(/*)) > T(Tm(f*)); That is, T^if*) > Tm+1(f*). This fact, together with the one that by definition of/*, /* > Tl(f*) verifies the initial induction assumption. The sequence Tm(f*) converges pointwise to some limiting function: Note that con-straint (A3) implies that the sequence {Tm(f*)} is bounded below by W (only values 79 of (U,k,6) G S for which T(F) > W are relevant). Thus, Tm(f*) belongs to a closed and bounded space. Since it is also a non-increasing sequence, it must therefore converge pointwise to some limiting function /* > f°° > W as m —» oo. The limiting function f°° is a fixed point of T(F): As T m _ 1 (/*) > Tm(f*) > • • • > f°°, we have T m _ 1 ( / * ) > f°°. Thus, any feasible policy in T(/°°) is also feasible in T(T m - 1 ( /*))-This implies that T(T m " 1 ( /*)) > T(/°°) and by taking the limit on both sides, it follows that /°° > T(f°°). Moreover, for any fixed vector (U, k, 6) G S, build the following limiting contract: Apply T successively and choose a sequence of policies {k™, c™, {U™}i=l} such that each one of its elements is a solution corresponding to each problem T(Tm(f*)). Define the quantity U7(fc) as the one solving Tm(f*)(\%{k),k,9i) = W{k,9i). Clearly, since T" 1 -^/*) > Tm(f*), ^(k) > \%(k), and we have GT^" 1) > Uf1 > V(km,9i) for all i. Therefore, as k™ and c^1 are bounded above by constraint (A.5) and below by constraint (A.4), the sequence of policies {k™, c™, {U™}?^} is bounded. Hence, it has a convergent subsequence which converges to some limit {kf*, cf, {U?°}™=1}. For each element of this subsequence, Tm(/*)(C/? n, km, 9{) > W(km,9i) for every i and in the limit, / ^ ( C / f 0 , ^ 0 0 , ^ ) > W(k°°,9i) for every i. Thus, a contract which gives the investor a total gain of f°°(U,k,9) exists and satisfies all the constraints in the problem T(f°°)(U,k,9). Therefore, T(f°°) > f°° and consequently f°° = T(f°°). For every fixed point / of T, there exists a self-enforcing contract giving a discounted expected utility level U to the entrepreneur and a discounted expected gain f(U,k,B) to the investor provided that the pair of endogenous state variables (U, k) belongs to the set S(9;f): Consider the contract built from successive applications of T(/) , start-ing from (U, k) E S(9;f) for some 9 G 0. This contract is clearly self-enforcing the first time T(f) is applied. The forward-looking self-enforcing constraints (A.2) and (A.3) guar-antee that the values taken by the endogenous state variables at the beginning of the second period will also belong to S(9; /) for every possible realization of 9 at that time. Therefore, the contract is also self-enforcing in the second period and T(f) can be applied once more. Again, constraints (A.2) and (A.3) will ensure that it is also the case for the third period, and so on. Hence, a self-enforcing contract exists from the first period onward. Finally, /°° = / : The supremum function / clearly satisfies T(f) = f, that is, it is a fixed point of T (it is bounded and the constraint set in T(f)(U,k,9) is non-empty provided that (U,k) G S(9;f) in the first place). Then, by the previous argument, there exists a self-enforcing contract corresponding to / , starting from any self-enforcing utility levels. Since / is the supremum function however, this contract dominates the one formed 80 using f°°. Thus, f > f°° obtains. In addition, T m ( /*) > Tm{f): Simply make the in-duction assumption that J" 7 1 -^/*) > T m _ 1 ( / ) , compare T m ( /*) and T m ( / ) , and observe that /* > / . Therefore, taking limits on both sides of the inequality T m ( /*) > T m ( / ) , we have that f°°>f and it follows that / = /°°. 2) Assume that / is the pointwise limit of the sequence {Tm(f*)} and suppose there exists an indice M such that S(9,TM(f*)) = 0 for some 9 and S(9,Tm(f*)) ^ 0 for all 9, for m < M. Let Z be any arbitrary large positive number greater than — infjt^ge W(k, 9) and define the operator Tm(f*) as follows: mrMU k 9 \ - l T m(r)(U,k,9), UT™(r)(U,k,9) > W(k,9); 1 U ){U,k,U)-<_^ otherwise. Clearly, f(U, k9)={ 0)' i f f(U>e"> ^ w(k>9^ \—Z, otherwise. is the pointwise limit of the sequence {Tm(f*)} generated from the sequence {Tm(f*)}. Obviously, iterating on the latter or the former sequence is equivalent as constraint (A.3) ensures that only the portion of the Pareto frontier no less than W is relevant. However, as S(9,TM(f*)) = 0 for some 9, fm{f*) = -Z everywhere for all m > M , thereby con-tradicting the fact that / is the pointwise limit of the sequence {Tm(/*)}. Therefore, if / exists, it must be the case that S{8;Tm(f*)) ^ 0 for all Be 0, for every function T m ( /*) in this sequence. || Thus, lemma 1 demonstrates that the Pareto frontier may be obtained by iterating on the recursion Fm+1 = T(Fm), provided that F° = f*. This procedure ought to be stopped however, whenever one of these sets S(B; Fm) is empty for some Be 0. In this case, we may conclude that a self-enforcing contract does not exist for the set of alternatives described by W and V. Lemma 2. Let W{k, B) = a0(B) + ax(B)k and V(k,B) = b0{B) + bx(B)k. Then, the value function f(U, k, B) is concave on S(B; / ) for any given Be 0. Proof. Assume that the function T" 1 - ^ /* ) is concave. The functional form re-strictions on the value functions W and V guarantee that the maximization problem in T™^*)^, k, 8) has a convex constraint set. Choose any two pairs of endogenous state variables (U',kr) and (U",k") in S(93;f) for a given Bs e 0, and let the corre-sponding contracts obtained as a solution to Tm(f*)(U',k',9S) and Tm(f*)(U",k",9S) be respectively given by {k'3,d3,{U^i} and d's,{U'^x}. For any A e (0,1), form 81 the following contract {k$, {U?}?=1}: Set k9 = Xk'a + (1 - A)A;", U? = ALT/ + (1 - A)C/J', and = Ac^ + (1 — A)c". Since the constraint set is convex, this contract is feasible in !*"(/*) (ALT' + (1 - X)U",Xk' + (1 - X)k",9a) and offers both agents no less than the average utility provided by {k'3,c's,{U^=l} and {k",da,{t/l''}"=1}. Therefore, Tm(f*) is concave. As /* is itself concave, any element of the sequence {Tm(f*)} is concave and so is its pointwise limit / . || Lemma 3. The value function / is continuously differentiable on the interior of S(9\ /) for any given 0G0. Proof. Consider any pair (U, k) € int S(9; f) and let the optimal policies associated with (U,k,6) be given by {A;*,c*, {U?}?=1}. Let B(6) C S(9; f) be a small open neighbor-hood around (U,k) and define a function z on (U°,k°) EB(9) by n n z(U°, k°, 9a) = g(k°, 9a) + (1 - 8)k° - K - u-\U° - 0 £ U*irai) + 0 ] T f(U*, k*a, 9^ai. This function is concave and continuously differentiable once since u(-) and g(-,9) are both differentiable and concave functions. Moreover, there exists a neighborhood B(9) small enough such that U° — 0 52"=i U*7vai > 0, as c* > 0 is guaranteed by the assumption that lim^oUcCc) = oo. Thus, for (U°,k°)£B(9) we have that z(U°,k°,9) < f(U°,k°,9), with equality at (U, k). Therefore, the function z satisfies the conditions stated in lemma 1 of Benveniste and Scheinkman (1979). It follows that the value function / is differentiable on the interior of S(9; f) for all 9e 0. || Proof of proposition 1. (see Thomas and Worrall (1988), proposition 2) As the self-enforcing constraints require that U\al £ \^Ji(k\) ,\Ji(k\)), by definition of (^(A;*) and c;(A;*), it follows that c^ 1 G [&(*:*), since fu(-) is continuous. From the equilibrium condi-tions (7), (9), and (15) we have Ucicl)'1 = (1 + C V e t e f ) - 1 - 4 + 1 US) for all i = 1,2, ...,n. Then, when i) c* > c;(A;*) > C - + 1 , we have uc(cs)-1 > u^c** 1) - 1 . Therefore, equation (A.6) can only be satisfied with ipj^1 > 0. It follows from (12) that£/? + 1 =Ui(A;*). Hence c\ = Cj(fc$). Whenii) (^JfeJ) > c* > c{(A;*), equation (A.6) is sat-isfied for = 0 and = 0. For if V>- 3 + 1 > 0, u^c*)" 1 < (1 + $ , + 1 K(^(fc*) )" 1 obtains; contradicting (A.6). Similarly, c ^ 1 > 0 implies that uc(4)-1 > W C (CJ(A:*)) - 1 - c ^ 1 ; still contradicting (A.6). Therefore, necessarily c^"1 = c .^ Finally, when iii) c^ 1 > Cj(A;*) > c* requires > 0 in view of (A.6) since ^ (c -* 1 ) - 1 > u c(c*) - 1. Thus, c*+* = c,-(ifcj). || 82 Proof of proposition 2. Write y* — c* explicitly using the expression for c* in proposition 1. Using m 4 = y* — c4, the definitions of mt(A:) and ^(k), and the fact that y* — c* - 1 = m1'1 + (y* — y t _ 1 ) yield the desired expression. || Proof of proposition 3. Consider equation (8) in history tfa under the assumptions made for this specification of the model. Upon rearranging (8), we obtain "> " C^fiy^'tl + ± ^ M E , ^ ) ^ (A7) Let k* = (a0BEaz/(l — 0(1 — S)))1^1^. This is the full-commitment optimal capital stock level. Moreover, let d(Ul,9s) = (1 + ^ ^ s + 1 ( 0 i / ^ 0 ) 7 T 3 i ) 1 / ( 1 - a ) - 1. 1=1 Using d(Ul,93) in (A7), substracting (1 — 8)kl~l on both sides, and dividing by tf~l yield the desired expression after rearranging the right-hand side. Moreover, since d( •) is necessarily greater or equal to zero, it > i * follows directly. || Proof of proposition 4- Note that from the envelope condition (16), the derivative of the net gain with respect to k is fk(U, k, 9i) — (1 — 8) = a9iBkQ~l > 0 for this specification of the model. That is, it increases in k. Differentiating f(0i(k),k,9i) = (1 — 8)[k — rj\ with respect to k yields fu{Ui(k),k,Oi)XJl(k) — —a9iBka~l. Since fu(^i(k),k,9i) is negative, \Ji(k) is increasing in k. Evaluate fu(Uj(k),k,#i)tj/(k) at ko and ki, ko > ki. Then, fu(\Ji(k0),k0,0i)U/(fco) > fu(^i{h),h,^)U/(^i)- Since \Ji(k) is increasing, fu(-) is independent of k, and / (•) is concave, Ui(fco) < mh)- That is, XJi(k) is concave. Proof of proposition 5. From (8), it is seen that the expected marginal benefit of installing one more unit of capital is 8 X)"=i fkiUJ^1•>K'®i)'Kai- The expected marginal cost is thus equal to 1-0£?=1 ^[fk^iiK), k\, 9{)-(l-8)]7vai. W h e n ^ s + 1 > 0, equations (7), (9), and (15) imply that ^ f M ^ h K A i ) = fc'j1,**) - /„(Ui(fcJ), k\, 9{). Since by definition of Ui(^) , fk{^i{K), kla, 9{) - (1 - 8) = - / U ( U { ( ^ ) , fc*, ^ )U/(fci), the expected marginal cost becomes 1 - / 9 E ? = i [ - / u ( ^ , ^ - \ 9 a ) + /«(Ui(fc*},k\,0$^. Then, using the definition of the set I yields the desired expression. || Proof of proposition 6. The first statement of proposition 6 follows directly from dynamic programme (6): Decision rules in history tf are only functions of the current value of the state variables. Note that U1 depends on zt because its actual value is chosen 83 from a state-contingent menu of utilities after the current realization of z becomes known to both agents. Let the log-linear approximations of the functions U( •) and /C( •) be given by h = 70 + l\Ut + 72^4-1 + 73#t + (A.8) Ut = uo + uiUt-i +.W2*t-2 + w3(9 t_i + uA9t + vt. (A.9) Note that in the above equations, all variables are in logarithm. This is not explicitly written only for clarity. Substituting Ut in the equation for the capital stock and collecting terms, we obtain where the value of each coefficient above is equal to <f>0 = 70 (1 — wi) + 71 w0, <f>x = 7 2 + ^ 1 ) <f>2 = 7iw 2 - 72W1, = 73 + 7 i ^ 4 , <f>4 = 7 i ^ 3 - 7 3 ^ 1 , and et = 71 vt + u t - w i U t - i - Then, since the log of output in history ht+1 is yt+i = B + 9t+\ + akt, it has the following representation, yt+l(l - faL - fcL2) = y + 9t+1(l + faL + ifoL2) + et, where L is the lag operator and where y = B(l — §\ — (f)2) + ctfo, ifri — afo — <f>l, and ip2 = Oi(j>i — <j>2. Using the law of motion of 9, we obtain the approximate autoregressive-moving average representation yt(l - faL - <1>2L2)(1 - pL) ~ y(l - p) + (1 + i\>xL + ip2L2)et (A.IO) for output. Hence, with partial commitment, output has an ARMA(3,2) representation whereas it has only a ARMA(2,1) representation with full commitment since in this case, we have UT — UQ = UQ (because the entrepreneur's utility is constant) which implies that (j>2 = 0 and ip2 — 0. Hence, the additional output dynamics originate from the dynamic behavior of the entrepreneur's expected utility level. || Appendix 3.2 This appendix describes how the set 0 = {9\, 92,..., 9n} and the associated transition matrix H are chosen in order to approximate the autoregressive process 9 t + 1 -9 = p(9t-9) + ait+1, for \p\ < 1 and e t +i ~ N(0,1). The goal is to approximate this continuous process with a discrete n-points Markov process. The procedure used follows Deaton (1991). 84 First, the area under the standard Normal p.d.f. is partitioned into n regions of equal size. Let {e0, e2,..., en} be the set of points forming this partition. These points are found by solving the expression $(ej) = i/n for Cj for each i = 0 , 1 , . . . , n (note that $(•) is the standard Normal c.d.f.). We then compute the n conditional means within each interval [e,-_i,ei], denoted ii, by evaluating numerically i i r _xV2, Let 7 2 = cr 2/(l — p2); It is easily seen that 0 t + 1 ~ N(#, 7 2). The elements in 0 are chosen to be di = 9 + 7<fj for i = 1,2,.. . , n. The elements in the transition matrix II are chosen to be the transition probabilities from the interval [#;_!,0j] to the interval [9j_x,9j] implied by the continuous process 9t+i. That is, we define 7Tjj by the following expression: try = Pr(7e; > §t+i -9> 7 C j _ i | 7et- > 9t - 9 > 7€i_i) Note that the conditional probability PT(7CJ > 6t+i Itj-i \0t-9 = x)= Pr ( 7 e j - > px + ait+1 > 7 e i _ 1 | 9t - 9 = x) v cr cr ' ^ a ' ^ a ' Therefore, TT^ can be written as V i i = „ % % x 1 x f1" [*(ZLZ£1) _ $ ( ^ - ^ ) ] e - ^ d x . 1 These [7Ty] are found by evaluating the above expression numerically. 85 Table 3.1 Value of parameters Parameter Value 0 0.9615 a 0.36 V 0.3 B 0.389* 6 0.10 P 0.862 a 0.063 "0.766 0.208 0.025 0.001 o.ooo-0.207 0.495 0.253 0.044 0.001 0.025 0.253 0.444 0.253 0.025 0.001 0.044 0.253 0.495 0.207 .0.000 0.001 0.025 0.208 0.766. 0 = {0.815,0.930,1.000,1.070,1.185} * Chosen to obtain an average capital stock of unity 86 Table 3.2 Investment and internal funds' abundance = CXQ + ax • C t - l + t*2! r- 0:3 ; h et 'kt-t-2 coefficient reversible investment irreversible investment an -0.1639* -0.2071* (0.023) (0.009) a i 0.9340* 0.9409* (0.005) (0.002) -0.0163* -0.0128* (0.001) (0.001) « 3 0.1360* 0.1685* (0.019) (0.007) i? 2 0.997 0.999 Note: * Statistically different from zero at the 5% level. Standard errors are in parentheses. 87 Table 3.3 Overinvestment statistics statistic reversible irreversible % of periods with binding financing constraint % of periods with overinvestment % of periods with underinvestment 11.9% 28.0% 0.0% 11.8% 57.5% 1.8% consecutive periods with overinvestment reversible irreversible average 2.2 3.5 median 2.0 2.0 maximum 10.0 20.0 % deviation of it from i\ reversible irreversible average 11.4% 7.2% standard deviation 18.1% 13.2% maximum 80.2% 77.6% median 2.7% 1.4% note: Al l statistics are computed conditional on i* > 0. 88 Table 3.4 Summary Statistics a. Reversible investment Partial Enforcement Full Enforcement Variable Mean Standard Mean Standard Deviation Deviation output (yt) 0.3951* 0.0699 0.3946 0.0703 Investment (it) 0.1020* 0.0804 0.1018 0.0843 Capital (kt) 1.0177* 0.1657 1.0128 0.1691 Consumption (Q) 0.2613* 0.0190 0.2606 0.0000 Transfers (xt) 0.0317 0.0803 0.0321 0.0928 b. Irreversible investment Partial Enforcement Full Enforcement Variable Mean Standard Mean Standard Deviation Deviation output (yt) 0.3950* 0.0693 0.3944 0.0696 Investment (it) 0.1020* 0.0732 0.1018 0.0752 Capital (kt) 1.0172* 0.1617 1.0124 0.1637 Consumption (ct) 0.26133* 0.0190 0.2606 0.0000 Transfers (xt) 0.0316 0.0728 0.0320 0.0839 Note: * Statistically different from full-enforcement case at the 5% level. 89 Table 3.5 Moving-average representation of output y* ~ [#* (L)/$* (L)]et (full enforcement) yt ~ [ty(L)/$(L)}et (partial enforcement) 100 * [*(1)$*(1)/$(1)^*(1) - 1] (% difference in long-run impact) a. Reversible investment T + 0.4823L" 1 - 0.0000L 1 - 0.8496L" 1 - 0.8515L 100 * hp(l)$*( l )/$( l)#*( l ) - 1] = 0.08% b. Irreversible Investment 1 + 0.2651L' 1 - 0.1405L 1 - 0.8507L 1 - 0.8525L 100 * [#(l)$*( l)/$(l)<r( l) - 1] = 0.29% Note: y~t is the deviation of output from its mean for the partial enforcement case. y* is the deviation of output from its mean for the full enforcement case. y*t^ y t 1 - 0.8621Z, yt - 1 - 0.8621L 1 + 0.4286L 1 - 0.0247L y t 1 - 0.8621L Vt ^ 1 - 0.8621L 1 + 0.2437L 1 - 0.1472L 90 Figure 3.1 Impulse responses — capital stock posit ive shock negat ive shock partial enforcement - full enforcement U O CL O U partial enforcement 10 12 14 16 18 20 full enforcement 10 12 14 16 18 20 posit ive shock negat ive shock 91 Figure 3.2 Impulse responses — posit ive shock financial flows negat ive shock I i i — i i i — i — i — r -c D partial enforcement full enforcement 10 12 O) C D I i 1 r — i 1 r — i 1 1 r-portial enforcement full enforcement 12 20 posit ive shock negat ive shock 9 2 Chapter 4 A Procedure for Investigating the Presence of Self-Enforcing Contraints Empirically 4.1 Introduction. This essay proposes a framework for investigating empirically whether contractual rela-tionships are influenced by risk-sharing considerations in a manner consistent with the presence of enforcement problems. Enforcement issues arise when agents are unable to precommit themselves to fulfill their contractual engagements. This situation may occur for instance if enforcing the terms of a contract through a third party is very costly. In this case, agents may be tempted to renege on a contract, after it has been signed, if breaching the contract allows them to take advantage of more profitable opportunities. A feasible contract in this context must be self-enforcing; it must offer to each agent, in every period and after any realization of the state of nature, at least as much value as the one offered by any alternative opportunity to the contract. This way, none of the agents will ever have an incentive to breach the contract; abiding by its terms always remains in their self-interest. Self-enforcing contracts have been studied in various contexts by Thomas and Worrall (1988), Kletzer and Wright (1995), Gauthier, Poitevin, and Gonzalez (1997), Kocherlakota (1996), and Sigouin (1997b). Requiring a contract to be self-renforcing leads to restrictions on the set of possible trades which may take place between agents. Over time, the quantity subject to the terms of a self-enforcing contract exhibits a behavior characterized by regime switches: Perfect risk-sharing trades occur between agents when enforcement issues are not of immediate concern. On the contrary, if enforcement issues threaten the contract's viability, trades between agents are governed by the imperative that none of them reneges on the contract. This is a key implication of the literature on self-enforcing contracts. This essay describes how this implication may be challenged empirically using an example drawn from Thomas and Worrall (1988). Thomas and Worrall (1988) study an employment contract between a risk-averse worker and a risk-neutral employer in the presence of low mobility costs and high enforcement costs. The fact that mobility costs are low and enforcement costs are high implies that both the worker and the employer may abandon an on-going contractual relationship to trade with another agent at the current spot market wage. The wage prescribed by the contract must thus ensure that neither of them will ever have an incentive 93 to do so, that is, the wage level must be self-enforcing. This requirement has a specific effect on wage dynamics. Enforcement constraints restrict the path followed over time by contract wages: Each period, self-enforcing constraints confine the set of possible contract wages to an interval around the current spot market wage. The upper-end of this interval represents the highest optimal wage level which may be paid to the worker while keeping the employer from reneging on the contract (by hiring another worker at the current spot market wage). Similarly, the lower-end of this interval is the lowest optimal wage level which prevents the worker from reneging (by offering his services to another employer at the current spot market wage). Any wage level outside this interval would prompt either the worker or the employer to breach the contract. The equilibrium contract wage in any period is determined through a simple updating rule based on the interval of possible contract wages and the value of the full-enforcement wage. The full-enforcement wage in any period is the wage level that the worker would receive if the contract terms were fully enforceable. The contract pays the full-enforcement wage to the worker if this wage level belongs to the interval of feasible contract wages associated with the current spot market's conditions. Paying the full-enforcement wage is not feasible, however, if it lies outside this interval; this would prompt one of the agents to renege. In this case, the contract wage is set as close as possible to the full-enforcement wage level. That is, if on the one hand, the full-enforcement wage level is above the interval of feasible contract wages, the contract wage attains the interval's upper-end. On the other hand, if the full-enforcement lies below this interval, the contract wage is set to its lower-end. Given the value of the current spot market wage, the contract wage process is thus in one of three regimes. Its behavior is akin to the one of a two-sided switching regression model where switching between regimes is triggered by the relative position of the full-enforcement wage with respect to the interval of feasible contract wages. Estimating the two-sided switching regression model implied by a self-enforcing con-tract amounts to estimating the value taken by the lower and upper bound which define the interval of feasible contract wages. These bounds are parameterized as functions of the spot market wage since one interval of potential self-enforcing wages exists for each possible level of the spot market wage. The estimation of the parameters characterizing these bounds is carried out using maximum likelihood. It is shown that specific restric-tions on the value taken by these parameters allow for wage dynamics consistent with a variety of theories. Standard hypothesis testing procedures for nested models may thus 94 be used to discriminate between different theories of wage dynamics within the framework proposed in this essay. For example, one may use this framework to test if the standard spot market model may be rejected in favor of a model with perfect risk-sharing, in favor of the two-sided incomplete enforcement model of Thomas and Worrall (1988), or in favor of Harris and Holmstrom's (1982) model of downward rigid wages. Beaudry and DiNardo (1991, 1995) find evidence suggesting.that wages behave in a manner consistent with the prediction of a contract model with enforcement constraints. Their empirical evidence is however only indirect in the sense that they do not explicitly test the specific restrictions imposed by enforcement constraints on the wage process. Rather, they investigate some collateral implications linked to the presence of enforcement problems: In Beaudry and DiNardo (1995), it is shown that holding constant productivity, variations in wages only affect hours worked through an income effect in an enforcement-constrained contract model. This prediction is at odds with what is observed in traditional models. The authors provide evidence consistent with this prediction by showing that hours worked are negatively correlated with variations in wages, once changes in productivity have been accounted for.1 In Beaudry and DiNardo (1991), the authors find evidence suggesting that wages exhibit the kind of history dependence consistent with the existence of enforcement con-straints. They show that labor market conditions, parameterized by the unemployment rate, have a significant impact on current wages through a fixed and time-varying time-of-entry effect. This is at odds with the spot market model where current wages are only influenced by current market conditions. Similarly, Kocherlakota (1996) proposes a set of empirical predictions linked to the absence of commitment in the context of a risk-sharing arrangement between individuals receiving random endowments each period. The empirical strategy used by Kocherlakota (1996) also exploits the specific manner by which history dependence is introduced by enforcement constraints. It is shown that enforcement constraints imply that all relevant past information is summarized by a vector composed of the marginal utility ratio of each individual's consumption with respect to the one of a given individual. Testing for the presence of enforcement constraints amounts to testing the sufficiency of this summary statistic. This essay adopts a more direct approach than the one of Beaudry and DiNardo (1991, In order to identify unobservable changes in productivity, the authors use an instrumental variables strategy based on the type of history dependence generated by a contract model with enforcement constraints. 95 1995) or Kocherlakota (1996); estimating the interval of feasible contract wages, given the current spot market wage, constitutes a direct appraisal of the restrictions imposed on the wage process by enforcement constraints. The chief advantage of this approach is that it is not only applicable to a wage contract or an insurance scheme. A wage contract is ex-amined in this essay only for the ease of exposition; the framework proposed in this essay may potentially be used to investigate the relevance of commitment issues for dynamic contractual relationships in a wide variety of contexts. There are several theoretical analy-ses illustrating in various settings the importance of considering commitment issues when studying contractual relationships2. Yet, there are few attempts to measure empirically the economic significance of these issues. This essay constitutes a step in this direction: First, it provides the tools that one may use to conduct an empirical analysis allowing to quantify the importance and the significance of enforcement problems. Second, it gives an assessment of the degree of success that one may hope for in detecting the presence of enforcement constraints empirically, if a contract model with two-sided commitment problems actually generates the data under scrutiny. The essay is organized as follows: Section 4.2 describes a self-enforcing labor contract similar to the one found in Thomas and Worrall (1988). However, the model set forth in this essay allows for variations in the workers' marginal productivity as a result of the acquisition of new skills. This model is used to illustrate the specific restrictions imposed on the dynamic behavior of a variable subject to the terms of a self-enforcing contract. Section 4.3 describes how the relevance of these restrictions for the evolution of wages may be tested empirically using panel data. Section 4.4 uses data obtained by simulating the model presented in section 4.2 in order to evaluate the accuracy of this test for a small sample size and a small number of cross-sectional units. Various exercises are performed in order to determine under which circumstances one is more likely to correctly detect the presence of enforcement constraints in practice. The fifth section offers concluding remarks. 4.2 A self-enforcing wage contract. Consider an economy populated with two types of infinitely-lived agents; risk-averse work-ers and risk-neutral entrepreneurs. Entrepreneurs are all identical. Workers are indexed by See for example, Albuquerque and Hopenhayn (1997), Gauthier, Poitevin and Gonzalez (1997), Hart and Moore (1994), Kehoe and Levine (1993), Kletzer and Wright (1995), Kocherlakota (1996), Marcet and Marimon (1992), Sigouin (1997b), Thomas and Worrall (1988, 1990, 1994) 96 the subscript j\ they difFer from one other in personal characteristics Xj which affect their marginal productivity. There is only one final good in this economy and it is produced us-ing labor uniquely. Each of the entrepreneurs has access to a production technology which requires one worker. A production technology employing worker j in period t transforms one unit of labor into Ajt units of the final good. It is assumed that Ajt = OjtSjt. That is, the marginal productivity of each worker is composed of two components; a temporary component, 9jt >0, which reflects fluctuations in labor market conditions and a permanent component, Sjt, which reflects the acquisition of skills by each individual worker. The val-ues of 9jt and Sjt are observed by any pair of agents. The productivity history of worker j from the initial period up to period t is denoted by hjt = {(#1, S i ) , (92, s2),..., (9t, st)} Although entrepreneurs and workers observe both components of Ajt, they treat them pa-rameterically. The spot market wage for worker j in period t is thus equal to his marginal productivity Ajt. The temporary fluctuations in labor productivity are the result of aggregate or sec-toral shocks arising mainly from variations in the demand for labor; even though they are denoted as worker-specific fluctuations, changes in 9jt may be correlated across workers. Each period, the value of 9jt for worker j is drawn from a discrete and finite set G with probability p(9jt). Permanent variations in each worker's marginal productivity arise from the acquisition of skills through learning-by-doing or through the accumulation of human capital. For simplicity, it is assumed that permanent changes in productivity occur exoge-nously. Every period, each worker come across ideas or techniques that were previously unknown to him; the knowledge of these new techniques permanently influences his pro-ductivity. It is assumed that the experience variable Sjt evolves according to the following relation Sjt = Sj t - i 7 e x p ( e j t ) (1) where 7 = 7/E[exp(e j t)]- The random variable exp(eJt) captures the impact of learning about new techniques on each worker's marginal productivity. Fluctuations in e J t are identically and independently distributed with mean zero and variance cr2. The average quality of new techniques, common to all workers, is given by 7. This parameter reflects the average growth in productivity due to the general increase in knowledge. Entrepreneurs have access to a perfect capital market but workers do not. In this context, entrepreneurs (firms) have an incentive to offer long-term wage contracts which insure workers against the random fluctuations in their productivity. In addition, as in Thomas and Worrall (1988), it is assumed that workers and entrepreneurs may always work 97 or hire at the spot market wage.3 Hiring or working at the spot market wage therefore constitutes, for both type of agents, the alternative opportunity to any risk-sharing wage contract. It is costless for workers to move from one firm to another. Hence, once his productivity is known, a worker may quit any wage contract to seek employment on the spot market. Similarly, entrepreneurs do not incur any cost when they hire or fire workers. An entrepreneur may thus fire a worker in order to hire on the spot market another one with a similar productivity level4. Entrepreneurs and workers cannot commit themselves not to adopt this opportunistic behavior. They may abandon any on-going employment contract to trade with other agents at the spot market wage. Nothing forbids them to do so; it is assumed that bilateral employment contracts cannot be enforced through the actions of a third party. Therefore, the only feasible employment contracts in this context must be self-enforcing, that is, they must be such that neither the entrepreneurs nor the workers ever have incentives to renege on the contracts. A self-enforcing employment contract between worker j and any entrepreneur is developed in what follows under the assumption is that if an agent reneges on the contract, he must trade on the spot market forever after. This is the most severe punishment an agent may possibly suffer in this economy. An employment contract beginning in period t is completely described by a sequence of functions {w(hjT)}%t, where w(hjT) is the wage paid to worker j following a productivity history hjT. A sequence of wages {w(hjT)}fLt is evaluated by worker j in history hjt according to the following expected utility function oo EtlY^wihj^r/a], (2) i=0 where a is his subjective discount factor, a < 1 is the coefficient of relative risk aversion, and E t is the expectation operator taken over all possible future histories conditional on history hjt. An entrepreneur's expected benefit from paying this sequence of wages to worker j is given by 0 0 E t [ Y ?[dit+isJt+i ~ w(hjt+i)} ], (3) i=0 Thomas and Worrall (1988) show that there is always trading at the spot market wage, in spite of the possibility that risk-sharing contracts may occur, provided that some entrepreneurs and some workers discount the future heavily. 4 This implicitly assumes that the pool of workers is large enough so that an entrepreneur may always find another worker with the same productivity level. 98 where /3 is his subjective discount factor. A self-enforcing wage contract is designed by making sure that, for any history and every continuation of the contract, the value of (2) and (3) never falls below the value of their counterpart in the case where agents trade on the spot market uniquely. The optimal self-enforcing contract between these two agents is found by choosing among the set of feasible (self-enforcing) allocations, the ones that maximize the en-trepreneur's expected benefit (3) for any feasible level of worker j's expected utility5. That is, the optimal contract in history hjt is the solution to the maximization problem oo f(UjUAjt) = max Et^FiAjt+i-wihjt+i)}] {w(hit+i)}oLo ~ £ oo s.t. Et [ £ V w(hjt+iy/a ] > Ujt and for all A: > 0, (4) oo oo E t + f c [ Y <*'-" ™(hjt+iy/a ] > Et+k [ Y <xi-k-A%+i/a ] i=k i=k oo Et+k[Y,Pi~k[Ajt+i-w(hjt+i)]] >0 where Ujt is worker j ' s reservation utility. The first constraint stipulates that the contract must provide to worker j an utility level of at least Ujt. The second and third constraints are the self-enforcing constraints. They ensure that the worker and the entrepreneur never behave opportunistically. The second constraint states that, in any continuation of the contract, for all possible future histories, it must always be in the worker's interest to pursue the contractual relationship; he may never obtain more utility by reneging on the contract and trading on the spot market from then on. The third constraint imposes a similar requirement on the entrepreneur's expected benefit. The value of f(Ujt,Ajt) corresponds to a point on the Pareto frontier. Varying the value of Ujt allows one to trace the entire Pareto frontier. Note however that not every value of Ujt is feasible in (4); the second and third constraints taken when k = 0 require this reservation utility level to be itself self-enforcing. It is shown in Thomas and Worrall (1988) that treating the worker's utility level allows one to express the Pareto frontier recursively as the solution to a dynamic programming problem. However, notice that given the law of 5 The alternative definition where the worker's utility is maximized is equivalent. 99 motion of Sjt, worker fs output grows at a stochastic rate over time. The maximization problem (4) thereby has a solution which is not time invariant and does not allow for a-recursive definition of the Pareto frontier. A recursive solution to this maximization problem is achieved through a change of variables. Divide each expression in (4) by the proper growth component, Sjt or sj t accordingly, and note that sjt+i = sJt7* exp^^t+i efc)- Define uijt = w(hjt)/sjt and Ujt = Ujt/s?t. Us-ing these definitions, one may rewrite equation (4) as oo f(Ujt,9jt) = max E t [ Y,^0)%t+i-wjt+i] ] 00 s.t. EtlYtWcxyw°t+i/a] >Ujt i=0 and for all k > 0, (4') 00 00 E t + * [ £ > ^ r * w ° t J < r ] > Et+k[J2(^rk 9Jt+i/a] i=k t=fc 00 Et+k[Yd{l0)i~k[Oit+i-wjt+i]] >0 i=k where f(Ujt,9jt) corresponds to f(Ujt,Ajt)/sjt. This maximization problem is stationary. It may thus be written as a dynamic programming problem. The Pareto frontier, independent of time and of j, may be expressed as f(U,9jt)=_ max {9jt-wjt + i0Et[f(U(9jt+l),9jt+1)]} s.t. w^a + ^Et[U(9jt+1)] >U U(9jt+l)>V(9jt+l) V % t + 1 G 0 f(U(9jt+1),9jt+1)>0 V% t + 1e9 (5) where V(9jt+i) = E t+i [X^lt+i Q; l - t_1^Jt/<7]- When U = Ujt, the dynamic programming problem (5) is equivalent to (4). It is assumed that competition among entrepreneurs is such that an entrepreneur makes zero expected profits viewed from the beginning of the contract. Therefore, the expected utility level obtained by worker j in the initial period of the contract, Ujt, is the one solving f(Ujt, 9jt) = 0.6 In this context, a contract is Thomas and Worrall (1988) consider a similar contract. They prove that the constraint set is convex and nonempty. The Pareto frontier is decreasing, strictly concave, and continuously differentiable in the interior of its domain (see Lemma 1). 100 signed in history hjt, between worker j and an entrepreneur, if this expected utility level exceeds V(9jt). The set {U(9jt+i)} in (5) is a menu of future expected utility levels promised to worker j in period t contingent on his future marginal productivity. The idea is the following: The entrepreneur enters each period t having previously promised to the worker a certain utility level for each possible productivity. At the beginning of period t, both agents learn the current productivity. This determines the amount of expected utility received by the worker in period t. The first constraint in (5) ensures that the worker obtains from the contract at least the amount of expected utility level he was promised previously. Taking this utility level as given, the entrepreneur maximizes expected profits by choosing a wage level and a contingent menu of future expected utility levels. The second and third constraints, the self-enforcing constraints, guarantee that these utility levels are such that neither the entrepreneur nor the worker will have an incentive to breach the contract in period t + 1. When the state of nature is revealed in period t + 1, an utility level is chosen from this menu. The entrepreneur is then faced with a similar problem at that time, and so forth for all subsequent periods. Proposition 1 below describes the behavior displayed over time by the optimal wage paid under the self-enforcing contract. Proposition 1. There exists a pair of real-valued functions w(#Jt) < 1 and w(0jt) > 1 for each 9jteQ such that the optimal contract wage in period t lies within the inter-val [0jtw(Qjt),6jtw(8jt)]. Moreover, the contract wage varies over time according to the updating rule where fx = (c*//3) ( 1 / ( 1 _ £ r ) )/7-Corollary 1. The optimal contract wage in the absence of enforcement constraints 9jtw(9jt), if flwjt-i > 0jtvf(0jt), iiwjt-u if 9jtw(9jt) > flwjt-i > 9jtw(9jt), Qjtw(0jt), if 9jtw(9jt) > fiwjt-i, (6) is given by wjt = fJ,Wjt-i (7) for any 9jteQ. 101 Corollary 2 . The spot market wage 6jt lies within the interval [9jtYf{0jt),0jtw(Qjt)] in every period. Proof. See appendix 4.1. || The smallest and greatest wage it is optimal to pay to worker j in period t are respec-tively given by 6jtw(9jt) and %tw(0Jt). When worker j is paid the wage level 0 J tw(0j t), the entrepreneur's expected profits from pursuing the relationship equal the expected profits he would obtain by trading on the spot market from then on; a higher wage level would prompt him to renege. Similarly, 0jtw(0jt) is the smallest wage that can be offered to the worker by a self-enforcing contract in period t. At this wage level, he is just indifferent between reneging and pursuing the relationship; a lower wage level would induce him to breach the contract. The self-enforcing constraints act so as to bound the path that may possibly be followed over time by the contract wage. Proposition 1 actually corresponds to one of the key results in the literature on self-enforcing; namely, that the quantity subject to the terms of a self-enforcing contract (the wage level in the present case) has a behavior over time which is characterized by regime switches. The contract wage follows a simple updating rule; the wage in period t, Wjt, is set to its fist-best level flwjt-i if it is feasible to do so, that is, if requiring the entrepreneur to pay p, Wjt-i to worker j in period t does not induce either one of them to renege. Otherwise, since the worker is risk-averse, Wjt is moved by the smallest amount possible which keeps either agents from breaching the contract. This updating rule does not explicitly depend on all worker j ' s past productivity levels (i.e. on hjt-i). The productivity history of worker j is summarized by the wage level he was paid in the previous period; conditional on Wjt_i, the optimal wage in period t is only influenced by the current productivity level of worker j. There is no additional information contained in worker fs productivity history that is not already included in his previous wage level. 4.3 Empirical implementation. Proposition 1 shows that the self-enforcing constraints gives rise to restrictions on the process followed by wages over time. In order to determine empirically if individual wages are influenced by factors linked to the difficulty to commit not to renege on a contract, one needs to estimate the updating rule (6). Notice however that the wage level on the left-hand side of equation (6) is the wage paid to worker j per unit of skills Sjt in period t. In practice, the skill level of worker j is not observable. To overcome this difficulty, both 102 sides of equation (6) are multiplied by Sjt. Using the definitions of Wjt and of Ajt, one may rewrite equation (6) in terms of the observed wage level as follows: { Ajtw(9jt), if pwjt-iexr)(ejt) > Ajtvf(Qjt), /xty J t_1exp(e J t), if Ajtvf(Ojt) > pwjt-i exp(eJt) > Ajtw(9jt), (8) Ajtw(Qjt), if Ajtw(9jt) > fj,wjt^iexp(ejt) where Ji = 7/2. Some observations are in order. First, observe that in the absence of enforcement problems, the log wage would fol-low a random walk with drift. Enforcement concerns bound the evolution of this random walk. Every time a self-enforcing bound is attained, the course of the random walk is reini-tialized. It resumes its normal course thereafter, and follows it, until a self-enforcing is reached again. Second, notice that the upper and lower bounds on Wjt in (8) are composed of two components; a stationary and a nonstationary component. The nonstationary component, Ajt, reflects worker j's overall productivity whereas the stationary compo-nents w(0) and w(0) reflect current labor market conditions uniquely. For any 9 £ 0, the interval [9jtw(9jt), 9jtw(9jt)] remains unchanged over the course of the contract; only its lo-cation in the wage-market conditions space changes over time. This interval moves up and down with Ajt- Worker j ' s productivity only has a level effect on the range of self-enforcing wage levels in any period. In fact, since w(9jt) < 1 and w(#Jt) > 1 (see proposition 1), the interval of self-enforcing wages is always around the spot market wage Ajt. Third, note that Ajt = 9jtSjt-i'yexp(ejt). Thus, the rule determining in which regime the wage process is does not depend on exp(eJt). In fact, equation (8) could be rewritten in terms of Wjt/ exp(ejt) rather than in terms of Wjt. This new expression would not involve exp(e.,t) on its right-hand side. Therefore, the contract wage varies one-for-one with exp(eyt). The higher exp(eJt) is, the higher the contract wage is, and conversely. Permanent fluctuations in worker fs productivity level are not insured by the contract; only temporary fluctuations are. Fourth, corollaries 1 and 2 imply that two other models are nested within equation (8): If w(0) = 1 and w(0) = 1 regardless of the value of 9, then Wjt = Ajt and (8) corresponds to a model where entrepreneurs and workers trade labor through the spot market uniquely. If there exists an arbitrary large number W such that W > fiWjt-i exp(e;i) > 0 for which w(#) < 0 and w(9) > W regardless of the value of 9, the upper and lower bounds in (8) are never attained, and (8) corresponds to a contracting model with perfect risk-sharing, Equation (8) therefore allows for a test permitting to reject these two models in favor of the model with self-enforcing constraints. 103 Fifth, the functions w(0) and w(#) are unknown. These functions have to be approx-imated. Note however that one has to keep in mind that the bounds in equation (8) only make sense when w(0) > w(0) for all 9. Sixth, the productivity of any worker j is not observable in general. Even if it was, it would be difficult to disentangle the labor market conditions 9jt from Ajt. Therefore, both the productivity level of worker j and the labor market conditions facing him need to be approximated using some observable indicators. It is assumed that individual indicators of Sjt and 9jt are available. Moreover, it is as-sumed that the logarithm of w(#) and w(9) are well approximated by linear functions of the logarithm of 9. These assumptions allow one to derive the empirical counterpart of equation (8): Using equation (1), the logarithm of the upper and lower bounds in equation (8) may be written as ln Ajtw(9jt) = ln w(0 ; t) + lnfyt + In Sjt-i + In 7 + ejt (9) In Ajtw(9jt) = \nw(9jt) + In 9jt + lns j t - i + In 7 + ejt. The first two terms on the right-hand side of the expressions in (9) reflect market condi-tions or the stance of the demand for labor whereas lns J t _i accounts for the amount of skills acquired by worker j prior to entering period t. The last two terms reflect the new acquisition of skills by worker j in period t. In equation (9), lnw(#) and lnw(0) may be replaced by linear functions of ln9jt while \n9jt itself and syt-i may respectively be proxied by some variables Zjt and Xjt. Let Zjt be a vector of variables measuring the strenght of the demand for the labor of worker j (or other workers with similar characteristics). For example, Zjt may be the employment rate in the industry or the region where worker j works; the higher the employment rate is, the stronger the demand for labor is susceptible to be. Let Xjt be a vector of variables providing an indication of the amount of skills acquired so far by worker j in period t. These are variables related to worker j's work experience. By assuming that lnw(#) and lnw(#) are linear functions of ln#, one may rewrite (9) as \nAjt\v(9jt) = ZjtQ+Xjt-iTj—Vjt+ejt In Ajt\v(9jt) = ZjtLJ+Xjt-xn-Vjt + ejt. where Vjt is an error incurred by approximating lnf?Jt with Zjt and l n s J t - i with Xjt-\,7 and Two additional approximation errors are incurred by respectively evaluating each functions ln w( •) and ln w( • ) with linear functions. For simplicity, these errors are assumed identical for both approximations. Allowing the upper and lower bounds to be affected by different approximation errors would lead to the possibility that they form an empty interval, even if Zjttj exceeds z;tw. 104 where n, u and u> are vectors of parameters.8 It is assumed that Vjt is a Gaussian white noise process with variance a2. Equation (10) allows one to express the logarithm of equation (8) as a two-sided endogenous switching regression model (see Maddala and Forrest 1975 or Maddala 1986 for a complete review of switching regression models). That is, the logarithm of Wjt may be written as { ZjtQ+Xjt-irj-Vjt+ejt, if vjt > vju \nn+\n.Wjt-i+ejt, if vjt > vjt > yjt, (11) ZjtUJ+Xjt-ifj-Vjt+ejt, if yjt > Vjt where Vjt = ZjtUJ+Xjt-ir)—In// — lnwjt-\ and Vjt = ZjtuJ+Xjt-ir) — ln/x — lntoyt-i- The fact that the random component Vjt influences switches between the three different regimes makes equation (11) an endogenous switching regression model. When a\ tends to zero, switches between regimes become deterministic and equation (11) reduced to Goldfeld and Quandt's (1972) D-method where the step functions indicating regimes are approximated with Gaussian cumulative distribution functions. Suppose that T data points are available on the variables Wjt, Xjt, and Zjt pertaining to J different workers. Under the assumption that each eyt is a Gaussian white noise process, appendix 4.2 shows that fjt, the density of \nvjjt conditional on the information available in period t, is given by fjt = D{vjt-yjt)[(l/ae)<j)(mjt/ae)[${vjt/<Tv) - ${yjt/<Tv)} + {\lam)(f>{[mjt-Vjt}l(Tm)[\ - $(crm[TJJt + (cr2 /a 2 n )(mj t - t ; i t )] /(cr 1 , (7 e ))] (12) where mjt = Inwjt —in(i—InWjt-\, and a ^ n = alJra1. The functions •) and $(•) respec-tively denote the probability density function and the cumulative distribution function of a standardized Gaussian process. The function D ( •) is an indicator function. Its value is unity when Vjt exceeds u- t, and zero otherwise. It takes into account the fact that equation (11) is only meaningful when the interval [yjt,Vjt] is nonempty, that is, when ZjtCiJ > Zjtu. Q The constant term In 7 cannot be determined independently from the constant terms used in approxi-mating In w( • ) and In w( •). Therefore, if Zjt allows for a constant term, then Xjt cannot allow for one; In7 is "absorbed" in Zjtto and Zjtu). It follows that productivity levels may not be identified; only productivity changes may be measured. 105 Given the parameter vector Cl = (ln/x, cre, crv, UJ, UJ, 77), the log likelihood of the J sam-ples of T observations may be calculated as9 J T (^«) = £X>/#- (13) j=l t=2 The switching regression model (11) may be estimated by finding the parameter vector Cl which maximizes (13). Note however that D( •) is not a continuous function. This function may be approximated in practice using a Gaussian cumulative distribution function, that is, by replacing D(k) in (12) by §(k/<Jk) for an arbitrary small value of 0 * . This strategy has been used in a similar context by Goldfeld and Quandt (1972). Several models are nested within equation (11). Each model implies specific restric-tions on the parameter vector Cl. Likelihood ratio tests may thus be used to discrim-inate these models against the alternative hypothesis that a two-sided partial commit-ment model generates the wage level observed for each worker. Equation (11) reflects a situation where workers are always paid their marginal productivity if Cl is restricted to Cl3 = (0,crm,0,u,L),r)). The fact that av is restricted to zero is an identifying assunn> tion; under the spot market specification, only the variance of e J t — Vjt is identifiable. Even if contractual considerations are important, a one-sided partial commitment model may reflect the reality more accurately. If commitment problems only impose a lower bound on the wage level, then Cl is restricted to Cli = (Infx, ae, av, u>, W, 77) where W is an arbitrary large positive number; if they only impose an upper bound, Cl is restricted to Clu = (n,cre,ov, —W,Q,n). In the case where the first-best risk-sharing allocation is al-ways attained, Clf — (ln/x,cre,0, — W, W,0) imposes the appropriate restrictions on Cl. The likelihood ratio statistics allowing one to discriminate between models may be computed from L(Cl) and Z(0;) for i = s, l,u, or / , obtained by maximizing (13). 4.4 Small sample properties. This section examines the circumstances under which one is more likely to correctly detect the presence of enforcement constraints in an applied context. In practice, it is unlikely that variations in labor demand and in each worker's skill level may be measured accurately. The goal of this section is to evaluate the accuracy of the empirical method based on equation (11) in the case where variables are measured imprecisely. It proceeds as follows: Equations (1) and (8) are simulated several times for a panel of J sample realizations 9 The log likelihood function is conditioned on the first observation for all J workers. 106 of Ojt and e ; t of lenght T. The two-sided switching regression model (11) is then estimated for each of these simulations by maximizing (13). The results are used to uncover the properties of this maximum likelihood estimator. Various exercises are performed. Note that throughout these exercises, a realistic sample size and a small number of cross-sectional units are utilized. The properties of a benchmark case where variables are measured precisely are studied first. The theoretical model of section 4.2 is simulated at an annual frequency. Simulating this model requires that the value of the exogenous variables 9jt and Sjt be supplied for each worker. The functional forms of the self-enforcing bounds w( •) and w( •) in (8) also have to be specified. The temporary variations in labor productivity are assumed to be generated by the following stationary autoregressive process of order one 0jt = (1 - <p) + <p 0jt-i + tjt for j = 1,2,..., J, (14) where \<p\ < 1, E[eJt] = 0, E[e2t] = of, and E[eyteit] = pa2 for i ^ j. The idiosyncratic fluctuations in skill level e J t are generated with a Gaussian white noise process. The initial value of Sjt is set to unity for all workers. It is assumed that the econometrician only has access to imprecise measures of 0jt and Sjt, that is, only Zjt and Xjt are available. These variables are generated by Zjt = ln 9jt + 8e i/jt, (15) Xjt = Xt + (1 - A) ln Sjt + 63 Sjt, (16) where Ujt and e J t are Gaussian white noise processes with mean zero and unit variance. The parameters A, 8g, and 8S control the degree of imprecision of these measures. It is assumed that the self-enforcing bounds in equation (8) are log-linear functions on average, that is, w(0jt) = exp(wo)0- t1_1 exp(-Ujt), (17) w(9jt) = exp(wo)0j1 e x p ( - U j t ) , for a given set of parameters w 0 , w 0, wi, and wj. The variable Ujt is a Gaussian ran-dom disturbance with variance a2 and mean Divergences from a log-linear function are controlled through a2. When a2 = 0, the self-enforcing bounds are log-linear func-tions. For small cr2 > 0, their value is within plus or minus 2 au per cent of the value of a log-linear function, 95 per cent of the time. Deviations from a log-linear functional form 107 are necessary in order to avoid that av tends to zero when the benchmark case where all variables are measured with precision is estimated. This would occur in this case be-cause equation (11) was derived under the assumption that the self-enforcing bounds are log-linear functions. There would therefore be no sources of discrepancy between the simu-lated theoretical model and the empirical model; equation (11) would collapse to Goldfeld and Quandt's (1972) deterministic switching model where the binary variables indicat-ing regime switches are approximated with Gaussian cumulative distribution functions. Setting <J\ > 0 avoids this situation. In all, the values of eleven parameters have to be set in order to simulate the model de-scribed by equations (1), (8), (14) and (17). These are (7, cre, jl, w 0 , wi, wo, wi, au, <j>, p, cre). In addition, the values of A,8g, and 83 must be supplied before the estimation is carried out. The benchmark case where A = 0, 8g = 0, and 8a = 0 is simulated using the parameters' values found in the first column of table 4.1. The values of 7 and <re are such that the annual growth rate of each worker's skill level is 1.6 per cent on average and that it lies between 0 and 4.2 per cent, 95 per cent of the time. The values of wo, wi, w 0, wi, and jl are chosen such that, on average, a binding self-enforcing constraint is observed in 30 per cent of the periods of a typical sample (see table 4.2). The value of au is such that the self-enforcing bounds' functional form diverges from a log-linear function by less than 2 per cent, 95 per cent of the time. Several variations from the benchmark case are studied. The parameters' values used in each one of these variations are also presented in table 4.1. The second and third columns of table 4.1 correspond to the cases where the effects of varying the number of time periods and the number of cross-sectional units are studied. The fourth column corresponds to the case where the self-enforcing bounds are parameterized in such a way that the frequency of a binding commitment constraint is reduced by half, to an average of 15 per cent of the periods (see table 4.2). The fifth and sixth columns correspond to cases where only 9jt is measured imprecisely. The value of 8$ in case 5 and in case 6 is such that the average correlation between \n6jt and Zjt in each panel of data is respectively 0.6 and 0.3. The situation where only Sjt is measured imprecisely is studied using the parameters' values of case 7 and 8. The value of 8S and A in the seventh column leads to an average correlation of 0.5 between the first difference of lns J t and the first difference of Xjt. The variable In Sjt is measured with a time trend uniquely in case 8. In this case, measurement errors account for 17 per cent of the variance of observed skill levels. The magnitude of the measurement 108 errors in this case is similar to the one reported in Altonji and Siow (1987) for data of the Panel Survey of Income Dynamics (PSID). 1 0 The ninth column corresponds to the case where both 0jt and Sjt are measured with errors. This more realistic case combines case 5 and case 8. In practice, one may have to work with data of this poor quality. Finally, the tenth case examines if the results obtained in case 9 improve substantially when the number of cross-sectional units is increased fourfold. The empirical model (11) allows the estimation of 8 parameters.11 These parameters are Cl = (ln/i, a e , wo+ln7 — | cr 2 , tJ i^o+ln-y — |cr 2,<Di,77). Their respective value is also found in table 4.1 for each case studied hereafter. Each case considered in table 4.1 is simulated five hundreds times using the parameters found in the first thirteen rows. The data is then generated according to equation (15) and (16) using the value of the next three parameters. Finally, the last eight parameters are estimated. The estimation is carried out by finding the vector Cl which maximizes (13) using a standard Newton's algorithm. Convergence to a maximum is deemed attained when the absolute value of all first derivatives of (13) is less than 10~4, the Hessian matrix is negative definite, and the change in (13) from one iteration to the other is less than I O - 1 0 . Statistics pertaining to the convergence of the estimation procedure are reported in the first panel of table 4.3. Failures to sucessfully estimate the switching-regression model were usually due to ill-conditioned Hessian matrices. The value of Cl used to initialize each maximization exercise is determined under the assumption that each worker's wage level evolves according to a two-sided switching regression model where the distance between the bounds triggering regime switches is constant and independent of 9jt: The parameters wi and wi are initially set to unity while the values of wo+ln7—|cr2 and wo+ln-y— \a\ are respectively set to — W and W, where W£ {0.01,0.02,0.03,0.05} is chosen to maximize the initial value of (13). The initial estimate of n is obtained from a pooled regression of \nwt on a constant term, zt, and xt_\. The parameters av and ae are both set to half the standard error of this regression's residuals. Finally, the initial value of In// is set to zero. The estimation results are presented in table 4.4. Other statistics are examined in tables 4.5, 4.6 and 4.7.12 Two types of results are considered. The first set of results examines the accuracy of the parameters' estimates and the model's goodness of fit. The second set of results evaluates 1 0 Altonji and Siow (1987) compute this figure based on the statistics reported in Duncan and Hill (1985). The measurement error pertains to reported income and not skill levels. 1 1 More parameters may be estimated if x and z are vectors rather than scalar. 1 *} All reported statistics were computed using the results obtained from successful estimations uniquely. 109 the capacity of the empirical model to detect the presence of self-enforcing constraints. Table 4.4 reports for each case the mean of the parameters estimates, their standard deviation, their asymptotic standard errors computed from the information matrix, and 95% confidence intervals. For the first four cases, the estimate of av evaluates the value of <Tu. In spite of the fact that a small number of time periods and a small number of cross-sectional units are used, the parameters' estimates are fairly accurate in the benchmark case. The estimates of wi and wi display the highest degree of imprecision; the hypothesis that the distance between the lower and the upper self-enforcing bounds is constant for all values of 9jt would probably not be rejected at standard significance levels.13 Cases 2 and 3 study the impact of doubling the number of data points by doubling the value of J and T respectively. In both cases, the precision of the parameters' estimates is increased; the point estimates are closer to their true value on average and the confidence intervals tighten substantially. The gains appear more significant when the number of time periods is increased (case 3). This may seem to suggest that expanding the length of each time series rather than their number may be more rewarding in practice. However, a relatively high degree of correlation is assumed between variations in 9jt for each individual worker (p = 0.8). As 9jt varies quite similarly from one worker to the other, self-enforcing constraints tend to bind around the same time for all workers. Hence, increasing the number of cross-sectional units in this context is not necessarily informative. Increasing the number of cross-sectional units may produce greater efficiency gains when variations in labor market's conditions faced by each individual are not as correlated.14 It is worth noting that in all three first cases considered, the asymptotic standard errors reveal themselves as good estimates of the precision with which Cl evaluates the value of 17. The asymptotic standard errors' values correspond roughly to the ones of the standard deviations of the parameters' estimates. The fourth case nevertheless stresses the importance of using with caution this measure of fl's precision. This may be seen in panel D of table 4.4. Panel D of table 4.4 reports estimation results when the frequency at which a binding self-enforcing constraint occurs is on average reduced by half with respect to the benchmark case. A binding self-enforcing constraint may be observed on average almost every three periods or so in the benchmark case. This happens every seven periods in the fourth case. Unreported results show that it is the case 1 4 Unreported results show that reducing the value of p to 0.4 leads to gains in accuracy of the same order of magnitude as doubling the value of T. 110 In both cases, the lower self-enforcing bound is attained more frequently (see table 4.2). Reducing the frequency at which a binding constraint is observed has a dramatic impact on the precision of the parameters' estimates. The precision with which the values of w0, wo, wi, wi, and 77 are estimated is especially affected (the estimates of In//, ae and av are still fairly accurate); their confidence intervals widen substantially and their standard deviations are roughly increased by a factor of three. This is not surprising since less data points are used to evaluate the self-enforcing bounds. Note however that going from case 1 to case 4 has a more significant impact than going from case 2 or 3 to case 1. This suggests that if self-enforcing constraints bind rather infrequently in practice, one may need to use a panel of data where JxT is fairly large in order to achieve a reasonable degree of accuracy. In addition, notice that the usefulness of the asymptotic standard errors as a measure of accuracy is greatly reduced; their values for the estimates of wo, wo, wi, and wi appear quite erratic. Panels E and F consider the cases where Zjt imprecisely measures 9jt. In these cases (and all remaining ones), the estimate of av does not only evaluate au\ it is also a measure of the imprecision with which the self-enforcing bounds are approximated. Reducing the correlation between Zjt and 6jt more than doubles the standard error of vt. The results for cases 5 and 6 are qualitatively similar. Again, the estimates of wo, wo, wi, and wi are affected the most; they are biased and they span a wide range. Estimating the switching regression model with an imprecise measure of Sjt seems to have a lesser impact on the parameters' estimates than measuring 6jt with imprecision. The results for this case are reported in panels G and H. While the point estimates are also quite imprecise in these cases, most of the confidence intervals still include the parameters' true value (except for r) in panel H). Using a time trend to account for the value of lns J t leads to poor results. This is to be expected since a time trend does not allow for cross-sectional variation in Xjt while there is a fair amount of variability in the actual acquisition of skills by the workers. Using a time trend to approximate Sjt is only appropriate if the value of cre is relatively small compared to the value of 7. Indeed, simply observe that lns J t may be written as lns_,o+*lu7-)-<7e 2~2\=o &k where e is a Gaussian white noise with unit variance. While the results in panel G suggest that the parameters' estimates are influenced to a lesser extent when Sjt is measured with errors rather than when 9jt is, it is rather unlikely that such a "good" measure of skills or experience be available for each individual worker. In practice, one may have to rely on a time trend to approximate In Sjt. This may seriously jeopardize the accuracy of the estimation if the actual value of ae is large compared to the one of 7. I l l The nineth panel of table 4.4 provides an indication of what one may expect to obtain if both In Sjt and \nQjt are approximated with imprecise measures. As might be expected from the results of cases 5 and 8, the parameters' estimates are quite imprecise. The last panel shows that even if the number of cross-sectional units is increased fourfold, there is no significant improvement of the accuracy with which the parameters are estimated. Table 4.5 and 4.6 examine the goodness of fit of the switching regression model (11) for all nine cases considered so far. The goodness of fit of (11) is evaluated by its capacity to accurately identify regime switches. The identification of regime switches are based on the values of $(Vjt/crv) and 1 — <&(vjt/crv). These are respectively the probability that worker j ' s self-enforcing constraint is binding and the probability that the firm's self-enforcing constraint is biding. Table 4.5 reports the root mean squared error (RMSE) entailed in approximating these probabilities. That is, the first and second row of table 4.5 respectively report the square root of j=l t=2 j=l t=2 The quantities Vjt, Ujt, and av respectively denote the estimated values of t>Jt, Vjt, av. Note that these RMSE constitute an overall appraisal of the goodness of fit of the switching regression model (11) since all the estimated parameters' values, except for cre, are involved in their computation. Actual RMSE are reported in panel A and the ratios of each RMSE with respect to the one of the benchmark case are reported in panel B. It may be seen from table 4.5 that the probability estimates are quite imprecise for the cases with measurement errors (cases 5-10). Equation (11) appears to fit the data quite poorly in these cases. Table 4.6 examines the accuracy with which model (11) identifies regimes properly. The rule used to identify regimes assumes that the current regime is the one which is assigned the highest probability by the switching-regression model. Panel A reports the proportion of the actual regimes accurately identified using this rule. Note that even though the estimation would produce the true parameters' values, by construction, regimes could never be predicted with perfect accuracy. This situation occurs because assigning a relatively high value to au implies that uncertainty as to which regime actually occurs always remains in (11). At best, regimes with the wage level corresponding to the upper self-enforcing bound's value are identified correctly only half of the time. This proportion 112 drops dramatically for the cases with measurement errors. This regime is properly iden-tified only 2% of the time in case 9. Increasing the number of cross-sectional units four times in this case does not produce any substantial improvement. The regimes where the wage level attains the lower self-enforcing bound are identified with slightly more precision. This reflects the fact that this regime occurs more frequently (see table 4.2); the lower self-enforcing bound may thereby be approximated with more accuracy. The fact that the regimes where the wage attains its full-enforcement level are predicted quite accurately (for cases 8 and 9 for instance) only means that the rule identifies this particular regime most of the times. This fact is confirmed by the results presented in panel B. This panel reports in which proportion each regime was improperly identified. The second row of panel B shows that even though the regimes where In Wjt is actually equal to Inn+\nwjt-i+ejt are in general identified properly more then nine times out of ten, a substantial proportion of these regimes' identifications are wrong, especially when measurement errors are introduced. This suggests that it is increasingly difficult to distinguish if the data was generated by a model with partial enforcement or a model with full enforcement as Sjt and 9jt become more imprecisely measured. Note however that in all cases, the identification of any of the three regimes is accurate at least 60% of the time. The most striking results of this essay are found in table 4.7. Table 4.7 reports the power of likelihood ratio tests in discriminating the null hypotheses Ho : Wjt is determined by the spot market model, Ho : Wjt is determined by the perfect risk-sharing model, from the alternative hypothesis Hi : Wjt is determined by the partial enforcement model, within the framework of equation (11). The likelihood ratio statistic has a \ 2 distribution with four degrees of freedom for the first null hypothesis and six degrees of freedom for the second one. The tests are performed at the 95% significance level. Panel A shows that the null hypothesis that the wage is determined by the spot market model is always rejected in favor of the partial enforcement model. Except for case 9, the model with full enforcement is accurately rejected at least 99% of the time. Case 9 is the worst with the test unable to reject the full-enforcement model only 5% of the time. Note however from case 10 that the power of the test is increased substantially when the 113 number of cross-sectional units is increased. Panel B presents the same proportions as in panel A , assuming that the estimations that did not converge (see table 4.3 panel A) lead to the non-rejection of either one of the null hypotheses. These proportion constitute lower bounds on the power of the test for all cases. Even under this extreme assumption,15 the test performs relatively well. Overall, the test seems to have a high power in spite of the fact that the self-enforcing bounds in (11) are poorly estimated when variables are measured with errors. Table 4.3, panel B, shows that the value of the log likelihood function at the maximum is on average substantially higher for the partial-enforcement model than for the two other models in all nine cases. This fact seems to explain why the test has a high power. In view of the test's relatively high power, one may wonder if the test does in fact reject the null hypothesis most of the time, even when the null hypothesis is actually true. This question remains unanswered. Attempts to estimate the endogenous switching regression model (11) using data actually generated by a spot market model or by a contract model under full-commitment usually failed; the maximization of (13) in these cases converged quite rarely and, when it did, the Hessian matrix was usually not negative definite. This renders Monte Carlo exercises aimed at answering this question quite difficult to conduct. Nevertheless, some information was obtained from these failed attempts: It seems unlikely that the test would overwhelmingly reject true null hypotheses when there is no measurement errors. However, when measurement errors are substantial, a tendency to identify a contract model under fullrcommitment seems to dominate. This happens even if the true underlying model is the spot market model. This is a result akin to one found in time series analysis; unrestricted autoregressive processes usually have better forecasting abilities than structural models. In the case of equation (11), if the upper and lower bounds are never attained, the log wage follows a random walk with drift term ln //. This corresponds to the full-commitment model. Equa-tion (11) corresponds to the spot market model if the upper and lower self-enforcing bounds are equal to each other. In the presence of large measurement errors, the self-enforcing bounds can only be measured with great imprecision. In this case, it becomes futile to restrict the behavior of the log wage; a random walk may perform just as well, or even better because wages are not measured with errors. It is thus likely that a test procedure based on equation (11) will not be able to adequately distinguish between the spot market model and the full-commitment model when measurement errors are substantial. 1 5 In general, the maximizations failed to converge because of hill-conditioned Hessian matrices. 114 4.5 Conclusion. This essay has proposed a framework for investigating empirically if enforcement issues are important in the context of dynamic contractual relationships. A key implication of the enforcement-constrained theory of contracts is that the variable subject to the terms of a self-enforcing contract evolves over time within a bounded path. It has been shown that this implication may be implemented empirically as a two-sided endogenous switching re-gression model. Using the example of Thomas and Worrall's (1988) wage contract, it has been shown that the conventional contract models abstracting from enforcement issues, involve restrictions on the variations in wages that may be nested within this switching re-gression model. This allows one to use standard test procedures to investigate the presence of enforcement constraints. Several Monte-Carlo exercises have been performed in order to evaluate the accuracy of the switching-regression model in estimating the self-enforcing bounds. In particular, the performance of the estimation strategy is examined for the case where the econometrician only has access to variables measured with errors. In general, the presence of measurement errors—a likely occurrence in the case of contractual relationships—seriously jeopadizes the accuracy of the estimation results: Parameters estimates are imprecise and proper identifications of actual regimes happen rather infrequently. Nevertheless, in spite of the poor goodness of fit of the switching regression model when variables are measured with errors, tests for the presence of enforcement constraints still perform relatively well. The tests have a high power in rejecting the false null hypotheses that conventional contract models without enforcement constraints generate the data under scrutiny. This is an encouraging finding. If enforcement issues effectively characterize contractual relationships, one may actually be able to detect the presence of enforcement constraints using actual data. A natural sequel to this essay is thus to conduct an actual empirical investigation of the presence of self-enforcing constraints. 115 Appendix 4.1 Let X(6jt), 7(3p{Qjt+i)(j){9jt+i), and i/3p(9jt+i)ip(0jt+i) be the multipliers respectively asso-ciated with the first, second and third constraint in equation (5); the first-order conditions are16 -1 + \{9jt)w°-1 = 0 (Al) [1 + ^(9jt+l))fu(U(9jt+l), Ojt+i) + </>(9jt+i) + Y~\a/fi)X{9jt) = 0 for all 9jM € 6, (A2) X(9jt)[wya+raE[U(9jt+l)] - U(9jt)} = 0, (A3) w^a+'foc E [ U(9jt+1) ] > U(9jt) \(9jt) > 0, and for all 9jt+x GG, <j>(9jt+l)[U{9jt+l) - V(9jt+l)} = 0, U(9jt+l) > V(9jt+l), Wjt+i) > 0, (A4) 1>(9ft+x)f{U(9jM),eit+x) = 0, f(U(9jt+1),9jt+1) > 0, i>(9jt+l) > 0. (A5) Moreover, the envelope condition is fu(U(9jt),9jt) = -\(9jt). (A6) Proof of proposition 1. The contract w{hjt+i) — Ajt+i for i > 0 is feasible in (4) and gives the entrepreneur zero expected profits in every periods. Therefore, f(V(9jt)sf, Ajt) > 0, and f(V(9jt),9jt) > 0 follows. Let U(9jt) be worker j ' s expected utility level for which f(U(9jt),9jt) = 0. Then, since /(•) is continuous and decreasing, f(V(9jt),9jt) > f(U{9jt),9jt) implies that any self-enforcing expected utility level U(9jt) is contained within the interval [V(9jt),U(9jt)]. Using equation (Al) we may write (A6) as —fu(U(9jt),9jt) = w]^a. Let 9jtvr(9jt) and 9jtvr(9jt) be a pair of functions such that the equations —fu(U(9jt), 9jt) = [0jtw(0jt)]1-<r and -fu{V(9jt),9jt) = [ f y w ^ ) ] 1 ^ are satisfied. Since U(9jt G [V(9jt),U(9jt)} and /(•) is strictly concave, the optimal contract wage Wjt is contained within [9jtY[(9jt),9jtVf(9jt)]. This allows (A4) and (A5) to be restated as (f>{0jt+i)[u)jt+i ~ %t+iw(0it+i)] = 0, Wjt+i > 9jt+lw(9jt+1), <t>(9jt+i) > 0, (A4') *l>(9jt+i)[0jt+i™{9jt+i) -u>jt+i] = 0, 9jt+1w(9jt+i) > Wjt+i, il>(9jt+1) > 0. (A5;) 1 6 The value function is strictly concave and differentiable. See Thomas and Worrall (1988). 116 Moreover, note that 9jt E[9jtw(9jt), 9jtw(9jt)] since w(#Jt) > 1 and w(0 J t) < 1. Indeed, on the one hand, when U(9jt) = U(9jt), 9jt - 9jtw(9jt) + yPE[f(U(9jtu),9jt+1)] = 0. Since f(U(9jt+i),9jt+\) > 0 for all 9jt+i is guaranteed by the third constraint in equa-tion (5), 9jt - 9jtw(9jt) < 0, and w(0Jt) > 1- On the other hand, when U(9jt) = V(9jt), [Qjiw(Qjt)Y/<r + y<TaE[U(9jt+l)} = V{9jt). But by definition, the value of V(9jt) is also equal to 9jja + 7°a E [ V(9jt+i) ]. Substracting these two expressions for V(9jt) and using the second constraint in (5), one obtains that w(0jt) < 1. The contract wage evolves over time according to equation (6): By substituting (Al) and (A6) (at time t and t + 1) into (A2), one obtains [1 + ip{9jt+l)]w)-^ - <f>(9jt+l) = \JMWjt]1-*, for all 9jt+1 G0, (A7) where p, — (a/P)^^x~a^/y. Notice that since a < 1, w1^ is an increasing function of iv. Except for the presence of /2, equation (A7) is essentially the same as equation (6) in Thomas and Worrall (1988). The remainder of the proof of proposition 1 is thus identical to their proof of proposition 2 when fivjjt is used rather than Wjt: If pWjt > 9jt+i-w(9jt+i), \jlwjty~'7 > since % t + iw(0 ; t + i ) > Wjt+i. Therefore, equa-tion (A7) may only be satisfied for tp(9jt+i) > 0. Hence, given (A51), Wjt+i = 9jt+iw(9jt+i). Similarly, if pwjt < %t+iw(%t+1), [pWjt]1-* < w1^ as wjt+i > 9jt+1y/(9jt+1). Equa-tion (A7) may only be satisfied with equality for (f)(Bjt+i) > 0. Therefore, {AA') im-plies that Wjt+i = 9jtjr\vf(9jt+i). Finally, if 9jt+ivj(9jM) > jiwjt > 0jt+iw(0.,t+i), equa-tion (A7) is only satisfied with equality for ip(9jt+\) = 0 and (j>(9jt+i) = 0: If on the one hand ip(9jt+i) > 0, equation (A51) requires that Wjt+i = 9jt+iw(9jt+i) and the in-equality [1 + ip(9jt+i)]w^l > [p.Wjt]1"'7 follows, contradicting (A7). If on the other hand, <j)(9jt+i) > 0, Wjt+i = 9jt+iw(9jt+i) is prescribed by equation (AA') and it fol-lows that the inequality Wjt+i — <j>(9jt+i) < [jiwjt]1'* holds, again a contradiction of (A7). Thus, wjt+1 = jiwjt. || Appendix 4.2 This appendix derives the density of ln Wj when its evolution over time is described by equation (11). Throughout, it is assumed that ej and Vj are both mutually independent Gaussian white noise processes. This assumption is not essential but greatly simplifies the following derivations. The following result will be useful: Lemma 1. Let e and v be two mutually independent Gaussian white noise process with means zero and variances a2 and a2 respectively. Then, the density of y = e — v 117 c o n d i t i o n a l o n v < a, denoted b y fe-v{y | a > v), is p<j>{py) ${[p<Te<Tv]-l[a + p V 2 y ] ) / $ ( a ) where p = 1 / \ J o \ + c r 2 a n d where $ ( •) a n d <j){ •) are respect ive ly the c u m u l a t i v e a n d the p r o b a b i l i t y d i s t r i b u t i o n funct ions o f a G a u s s i a n process w i t h m e a n zero a n d un i t var iance . Proof. S ince e a n d v are m u t u a l l y independent G a u s s i a n processes, one m a y wr i te fe-v{y\a>v) = $ ( a ) _ 1 f (j>((y + x)/o-e)(j)(x/o-v)dx, J — oo = $ ( a ) - : f [2TT(Tvae)~l e x p ( - 0 . 5 [ ( y + x)2/a2e + x2/a2])dx, J — oo = $ ( a ) - 1 ( 2 7 r ) - a 5 p e x p ( - 0 . 5 p 2 y 2 ) x f ( 2 7 r ) - a 5 ( p < 7 e ( 7 u ) - 1 e x p ( - 0 . 5 [ p 2 < 7 2 c r 2 ] - 1 [ a ; - r - c 7 2 p 2 y ] 2 ) d x , J —oo = ^{a)-lp<f>{py)^{[paeav}-\a^ p2a2vy\). \\ G i v e n e q u a t i o n (11), the densi ty o f the t — th observa t ion of InWj, c o n d i t i o n a l o n al l i n f o r m a t i o n avai lable i n p e r i o d r, is / ( I n wjt) = / ( I n Wjt | Vjt > Vjt > y i t ) P r ( U j t > vjt > v j t ) + / ( I n Wjt | Vjt > Vjt)Pr(vjt > vjt) + f(\nwjt | Vjt <yjt)Pr(vjt <yjt). However , note that / ( I n Wjt | Vjt > Vjt > yjt) = fe. (In wjt - lap-In Wjt-\ \ vjt > vjt > yjt) f{\nwjt | vjt>Vjt) = fej-vjilnwjt-ZjtCJ-Xjt-ir) \ vjt>Vjt), f(lnwjt | Vjt<yjt) = fe._v.(\nWjt-Zjtu-Xjt-in \ vjt<yjt). T h e r e f o r e , u s i n g l e m m a 1, one obta ins tha t f(lnwjt) = (l/ae)(j)(mjt/ae)[$(vjt/av) - $(yjt/crv)] + pHp[™jt-Vjt])[l - ®([vjt+o-tp2(mjt-Vjt)]/{p(Tvo-e))} + P<!>(p[mjt ~ Vjt})®{ \Vjt +o-lp2 {mjt-Vjt))f{pavo-e)), where m J t = In Wjt — In p—In Wjt~\. 118 Cfl O .a u M a v a .2 01 V U -U "8 cd a .2 M 0) «2 •8 01 9 a I ed 13 a JS C M 0 » 3 case 10 T T T T T T T T T T T o T CM T T T T T T T T T T T case 9 T T T T T T T T T T T T T 0.02500 —> T T T T T T T T case 8 T T T T T T T T T T T T T 1.00000 0.00000 T T T T T T T T case 7 T T T T T T T T T T T T T 0.00000 0.01000 T T T T T T T T case 6 T T T T T T T T T T T T T 0.06000 T T T T T T T T case 5 © © T T T 1 T § T T T T T T T © q © ©' i 0.02500 —¥ -0.02916 0.01984 case 4 —> -0.06000 0.02000 —• 15 T T T —• —» -0.04416 0.03584 case 3 T T T T T T T T T T T g g T T r T T T T T T T T case 2 T T T T T T T T T T T o T i—t T T T T T T T T T T T case 1 1.01600 0.00800 0.99806 -0.04500 0.00400 -0.12000 0.01000 0.85000 0.80000 0.01500 50 15 0.00000 0.00000 0.00000 0.01390 0.00800 0.01000 -0.02916 0.01984 0.88000 1.00000 parameter *- b '3. H £ i | £ ba ^ b ^ EH cs a cs a b b " I C S f-<|CS 1 1 1 b" b= t 5 1 '3 * + + o o 3' "3 CQ 3 8 V 1 CD JS CD JS <L> $ CD s ed a CD JS a3 JS cd O a o CD JS © © T f o o o o o « © O CO o o q q q rH rH O 0) CS cd 1 s o r d I u M <s a CB V U V •* co T3 _o *C a cd o u M 4) CN •2 r Q o CT) '1 W lO rH T f oi cd CN rH CM °r °r H~ © © in CN co in 00 CO T f 8 cu' JS W a ° m cu M 1 l i CO CO CN CO CN T f m CJ> T T rH O! H O r H r H CO fc. S CD rC S a 2 J2 p -o ca u cd +> ca a .2 +> cd J ca w co CD -a o 6 -6 cd CD J3 O rQ o o o m Vi a _o 03 T3 O O JS "S J< 00 _o CD c o o a o CM O O CN O O CJ) © © q rH rH O o o © © © q rH rH © © © CN O © O) q q q rH rH © © O © o o m © q q rH rH © © O T f © © CJ) © © q rH rH © O © T f o o o q © q rH rH © © © © © © O) © © q rH rH © CD r * a 3 c CD E CD O fc. a case 10 5631.7 8565.1 8678.4 case 9 1415.6 2131.8 2177.8 case 8 1441.2 2136.2 2207.5 case 7 1862.9 2138.3 2267.3 case 6 1852.2 2137.5 2195.9 case 5 1891.1 2138.2 2219.5 case 4 1981.0 2264.3 2348.9 case 3 4062.8 4374.8 4761.4 case 2 3994.2 4253.1 4598.0 case 1 2003.2 2139.0 2303.3 model spot market full enforcement partial enforcement Table 4.4 Estimation results A. Case 1 - benchmark parameter estimate std dev. asy. s.e. 95% confidence int. In fj, 0;01389 0.00037 0.00042 [ 0.01299, 0.01470 ] &e -0.00799 0.00025 0.00028 [ 0.00743, 0.00855 ] 0v 0.00969 0.00106 0.00118 [ 0.00709, 0.01184 j -0.02965 0.00431 0.00472 [-0.04077,-0.01913 ] W l 1.16190 0.08781 0.09677 [ 0.95822, 1.39718 ] (Do+ln7- | cr2 0.02674 0.00525 0.00570 [ 0.00926, 0.04006 ] 0.95321 0.11359 0.12564 [ 0.55139, 1.21406 ] V 0.99949 0.01643 0.01841 [ 0.95931, 1.04188 ] B. Case 2 - J = 100 parameter estimate std dev. asy. s.e. 95% confidence int. In// 0.01388 0.00027 0.00030 [ 0.01322, 0.01451 ] 0.00798 0.00017 0.00020 [ 0.00762, 0.00838 ] 0.00986 0.00072 0.00081 [ 0.00812, 0.01150 ] -0.02934 0.00307 0.00335 [-0.03665,-0:02168 ] yi 1.15275 0.05931 0.06665 [ 1.01762, 1.30791 ] w 0 + l n 7 - 5 < r 2 0.02183 0.00388 0.00372 [ 0.01258, 0.03759 ] W l 0.90882 0.07566 0^.08089 [ 0.69449, 1.15552 ] 0.99967 0.01214 0.01292 [ 0.97019, 1.02790 ] C Case 3 - T = 30 parameter estimate std dev. asy. s.e. 95% confidence int. In// 0.01390 0.00026 0.00029 [ 0.01330, 0.01449 ) 0.00799 0.00018 0.00019 [ 0.00751, 0.00839 ] 0.00989 0.00072 0.00078 [ 0.00832, 0.01146 ] wo+In 7 - i a 2 -0.02913 0.00226 0.00251 [-0.03414,-0.02428 ] W l 1.15146 0.04590 0.05190 [ 1.03620, 1.26167 ] i D 0 - l - l n 7 - | < j 2 0.02069 0.00265 0.00285 [ 0.01446, 0.02762 ] W l 0.89302 0.05945 0.06202 [ 0.73297, 1.04205 ] 0.99963 0.00540 0.00605 [ 0.98661, 1.01174 ] D. Case 4 - proportion of binding constraints is half of benchmark parameter estimate std dev. asy. s.e. 95% confidence int. In// 0i01390 0.00031 0.00035 [ 0.01315, 0.01456 ] 0:00799 0.00021 0.00024 [ 0.00749, 0.00848 ] 0.00934 0.00183 0.00183 [ 0.00442, 0.01356 ] W o + l n 7 - | f j 2 -0.05032 0.01524 0.90054 [-0.10850,-0.01852 ] W l 1.23376 0.31705 35.7908 [ 0.27714, 2.96405 ] w o + l n 7 - i c r 2 0.04304 0.01427 0.90880 [ 0.01548, 0.10294 ] W l 0.99225 0.30281 37.2233 [ 0.25046, 2.35681 ] 0.99809 0.03212 0.03201 [ 0.91361, 1.08708 ] 121 Table 4. Estimation results (...continued) E . Case 5 - coTT(zjt,\n0jt) = 0.6 parameter estimate std dev. asy. s.e. 95% confidence int. ln/i 0.01401 0.00055 0.00046 [ 0.01282, 0.01528 ] o-e 0.00835 0.00035 0.00033 [ 0.00759, 0.00929 ] 0.02061 0.00252 0.00220 [ 0.01486, 0.02662 ] (±) 0-r-ln7-|«r2 -0.04220 0.02578 0.00805 [-0.10852, 0.00990 ] y i 0.47556 0.11250 0.08470 [ 0.21681, 0.78596 ] uj 0 -r-ln7- |«r2 0.06028 0.03264 0.10097 [-0.00927, 0.25201 ] 0.60010 0.26687 1.10314 [ 0.09327, 2.28874 ] V 1.00408 0.10300 0.03070 [ 0.80383, 1.21970 ] F. Case 6 - corr(^jt,ln^() =  0.3 parameter estimate std dev. asy. s.e. 95% confidence int. ln/i 0.01411 0.00067 0.00046 [ 0.01268, 0.01582 ] 0.00864 0.00045 0.00034 [ 0.00774, 0.00993 ] 0.02488 0.00380 0.00293 [ 0.01597, 0.03362 ] y 0+ln7- |<T2 -0.05270 0.04122 0.01069 [-0.16647, 0.02858 ] W l 0.15102 0.06472 0.05210 [ 0.00272, 0.47543 ] 0.07596 0.04786 6.43343 [-0.02274, 0.27340 ] W l 0.23447 0.14082 53.7968 [-0.13198, 0.96879 ] 1.01163 0.16131 0.03781 [ 0.68525, 1.33843 ] G. Case 7 -coxx{xjt-xjt-x In —In S j t --i) = 0.5 parameter estimate std dev. asy. s.e. 95% confidence int. In// 0.01396 0.00044 0.00042 [ 0.01292, 0.01488 ] 0.00815 0.00027 0.00029 [ 0.00755, 0.00872 ] <r« 0.01532 0.00147 0.00161 [ 0.01213, 0.01919 ] y 0 +ln7 - | ( r2 -0.03671 0.00958 0.00724 [-0.06180,-0.01660 ] W l 1.37397 0.15089 0.14800 [ 1.02712, 1.85490 ] (D0-r-ln7-i(r2 0:04792 0.01306 0.01005 [ 0.02266, 0.12138 ] W l 1.12386 0.22241 0.21590 [ 0.53296, 1.94320 ] »7 0.97158 0.03543 0.02753 [ 0.88662, 1.04846 ] H. Case 8 -Xjt = t parameter estimate std dev. asy. s.e. 95% confidence int. In// 0.01434 0.00086 0.00041 [ 0.01257, 0.01685 ] 0.00901 0.00049 0.00031 [ 0.00808, 0.01106 ] 0.03292 0.00774 0.00422 [ 0.00544, 0.04789 ] -0.03698 0.05462 0.02221 [-0.21259, 0.04611 ] W l 2.41154 0.66088 0.46451 [ 0.42303, 4.60416 ] w0+ln7-|o-2 0.19985 0.05916 0.90950 [ 0.10435, 0.40807 ] W l 1.85394 0.81073 21.3527 [-0.45531, 4.91143 ] »7 0.01645 0.00391 0.00157 [ 0.00765, 0.02614 ] 122 Table 4. Estimation results (...continued) I. Case 9 - corr(zyt, In Ojt) = 0.6 and Xjt = t parameter estimate std dev. asy. s.e. 95% confidence int. lap 0.01455 0.00106 0.00042 [ 0.01234, 0.01708 ] 0.00949 0.00063 0.00033 [ 0.00828, 0.01129 j <r» 0.04092 0.01290 0.00616 [ 0.00000, 0.06423 ] y 0 + l n 7 - | o - 2 -0.05758 0.07360 0.71887 [-0.39353, 0.06990 ] 0.89848 0.35989 19.4967 [-0.14071, 2.07314 ] (1*0+1117-2^ 0.23118 0.07985 7.01128 [ 0.10131, 0.48166 ] 0.70307 0.54274 142.688 [-0.82125, 2.23288 ] V 0.01683 0.00535 0.00196 [ 0.00324, 0.03271 ] J. Case 10 - corr(zjt,ln0jt) = 0.6, = t, and J = 200 parameter estimate std dev. asy. s.e. 95% confidence int. In// 0.01460 0.00100 0.00021 [ 0.01244, 0.01726 ] 0.00946 0.00047 0.00017 [ 0.00864, 0.01070 ] crv 0.04747 0.05580 0.00348 [ 0.02827, 0.06297 ] w 0 + l n 7 - | c r 2 -0.05840 0.05563 0.01330 [-0.26758, 0.04885 ] 0.91745 0.22435 0.13772 [ 0.25468, 1.57928 ] iD 0 +ln7 - la -2 0.22940 0.06292 1.39260 [ 0.11483, 0.48884 ] U>1 0.80721 0.34764 19.4024 [-0.12877, 2.08993 ] V 0.01728 0.00462 0.00096 [ 0.00731, 0.03003 ] 123 CO u O H M V 0) 3 cr CO a CO a O O m ,0) E5 bo a 3 '3 CO C bO cc3 r C •s o CD r f l i f SB H O l CU V T f cu CN CU C O C O C O i n C O m o o CN cq © d co co C - r H O o o CN CO d d oo oo 00 t -01 t-oo i n r H CN d d 0 o 01 00 C O © CN m rH rH © © os C-C S 00 00 C O rH C S © © lO C M O l C O C N 00 O l t~ o t - CO © © d d C O © OS C O C N C N © © d d m os rH t> co C N o o d © m co C N C N T f O l T f C O © © © © I? P i A V P H PH os m co C O os r> cs m t - T f 00 00 C O 00 C S 00 00 t> © rH t - 00 C O C O © © © © © o IP Pi A V p P P H P H co M V M V J3 0) M "B 3 •p u ca M a c "3 u cO H 3 u u < CO _5 cu cC cu cu s 'Ei cu a o o a o T f o m rH o i m q q © o d d H N O l C N 00 C O © os q d o © rH C O rH CO CO o © q rH d o © C N co o i w Tf C S O l © © © © OS cs T f C O rH © q rH d o © 00 rH t> os m cs © q cs © © © OS rH m © C O co T f q i n o d d T f IO rH Tf rH C S i n OS t~ o d d N N s O rH 00 m o i co o d d OS C O OS OS r H f -Tf q co o d d Pi IP A l A p p Al IP a cd cu >-cd ts r=3 in a cd O eg -w C cu 1 •a cu o a o P3 N M O l C S © T f C S C O C O o d d m m co C O OS rH cs cs co C O T f O t> t> © cs C S C S n N rH © rH C O cs cs m i n cs co oo © C S C S T f o d d m co © © C O C O cq cs cq © © d © i n co M t o N co © cs C S 00 © C O C O © C N rH C S rH O l CO 00 C O rH C S rH C S 00 t -00 C O C S rH © © Pi IP A l A p p* A l IP cs v 8 o 00 CJ eg I - 4 - ? e S 8 c a C O .3 to in fl O XJ CQ < CM •a o a a © CO O CO O OS o m o os o o © OS o OS o o o o o © CO OS OS OS OS © 00 © OS © OS rH © © 00 © os © OS © © © © © © © © o © © © © © o © © © a V •s 6 s p o o CQ © 00 00 to OS OS o to OS 00 to to o to OS OS © © CM CM OS OS OS OS CM 00 F3 CM OS OS © o CM © OS OS OS OS O 00 l O T f OS OS © © OS OS OS OS T f T f OS OS OS OS © © OS OS OS OS © © a s .2 S "H 0. 3 CM References Albuquerque, Rui and Hugo A. 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