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UBC Theses and Dissertations

Essays in capital market equilibrium Harris, Richard Glen 1975

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ESSAYS IN CAPITAL MARKET EQUILIBRIUM by RICHARD GLEN HARRIS B.A. (Hons.) Queen's University, 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE*..OF DOCTOR OF PHILOSOPHY in the Department of Economics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1975 \ In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the 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 that the L i b r a r y s h a l l make i t f r e e l y ava i1 a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree 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 may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d tha 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 not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depa rtment The 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 20 75 Wesbrook P l a c e Vancouver, Canada V6T 1W5 0 - i -Abstract. The thesis consists of three essays dealing with related problems of capital market equilibrium under conditions of uncertainty. The method of analysis is the construction and use of two period general equilibrium models. The f i r s t essay develops a set of necessary and sufficient con-ditions, for the determination of equilibrium prices in the well-known Capital Asset Pricing model. A number of structural properties of the equilibrium are determined and applied in investigation of the comparative statics of the model. In particular, the effect on equilibrium prices of changes in the stochastic parameters and number of investors is established. The second essay deals with problems of financing and investment which occur when production and firms are added to the model. It is demonstrated that a "financial inconsistency" may arise as firms pursuing a particular objective, may attempt to invest more'than the households in the economy are willing to finance. This possibility has implications for the exist-ence of general equilibrium, admissable decision rules for firms, and the firm's problem of choosing a feasible financial policy. In order to deal with these issues a number of properties of the 'stock market economy' are established. The final essay relaxes the assumption, maintained in the f i r s t two essays, that a l l debt in the economy is default free. A number of simple capital market models are developed which determine an equilib-rium interest rate on (risky) debt and an equilibrium probability of de-fault. The comparative statics of these variables are investigated with respect to changes in the riskless rate of interest, the productivity of investment, the expectations of investors, and the transactions costs of default. It is shown that when equity is introduced, then provided - 1 1 -bankruptcy is not costless, the market also determines an equilibrium debt/equity ratio. Acknowledgements I should like to thank John Cragg, my principal supervisor and chairman of my thesis committee to whom I owe a great debt, for the countless conversations and c r i t i c a l counterexamples which made this thesis possible. I should also like to thank the^other members of my thesis committee, Chris Archibald and Erwin Diewert, for perceptive comments on various drafts, usually written in my rather i l l e g i b l e handwriting. In particular I should like to thank Erwin Diewert and Keizo Nagatani whose teaching originally stimulated my interest in capital market problems and whose questions were a continual source of new problems. I am also grateful to the Central Mortgage and Housing Corporation and Canada Council for financial support, to Erwin Diewert and Alan Woodland who employed me as a research assistant, and to Mrs. D. MacKenzie who typed the final draft. Lastly, I should like to thank the faculty and graduate students of the Department of Economics at the University of British Columbia who, during the years •1972 through 1975, made my stay both pleasant and intellectually challenging. - iv -TABLE OF CONTENTS Page ABSTRACT i ACKNOWLEDGEMENTS i i LIST OF TABLES v LIST OF FIGURES Vt Chapter I INTRODUCTION 1 II THE STRUCTURE OF A CAPITAL MARKET EQUILIBRIUM , 8 1. Introduction 8 2. The Model 10 3. A General Equilibrium Equation 12 4. The Structure of Equilibrium Prices ... 16 5. Comparative Statics .. 26 6. Conclusions ., 38 III RISK BEARING, INVESTMENT AND FINANCING IN STOCK MARKETS .. 45 1. Introduction 45 2. The Basic Model 48 3. Financial Consistency 60 4. Some Examples of Firm Behaviour 66 5. The Capital Structure Problem 74 6. Summary and Conclusion ...... 85 IV THE COST AND CONSEQUENCES OF DEFAULT 95 1. Introduction 95 2. Default: Some General Considerations 97 3. Equilibrium in a Loan Market: Demand Determined Loan Price 101 4. Risk Aversion and the Demand for Loans 108 5. Loan Market Equilibrium: Supply Determined Loan Price 111 6. The Introduction of Equity 122 7. Conclusion 134 VITA 145 V -LIST OF TABLES Table Page 1. Chapter IV Summary of Comparative Static Results 136 ? ^ vi -LIST OF FIGURES c Figure Page 1. Chapter II Individual Portfolio Equilibrium . 21 2. Chapter II Individual Preferences with the Property that the Marginal Rate of Substitution Between Risk and Return is Independent of Return 28 1. Chapter IV Expected Return Function 115 2. Chapter IV. Expected Return Functions for Firms 1 and 2 .. 120 - 1 -Chapter I INTRODUCTION The following three essays are concerned/with capital market equil-ibrium under conditions of uncertainty. The emphasis in each case is r. the effect of uncertainty 1 on the individual market participants, and its implications for the economy as a whole. Capital markets, in addition to their risk bearing function, have as a primary task the intertemporal allocation of commodities -- in particular they f a c i l i t a t e the savings investment process. In order to deal with the intertemporal aspect, the models considered in the following essays are two-period models. This is primarily to put the problems in as simple form as possible, but also to preserve comparability between the models constructed herein and those in the fast growing literature on the subject. The models a l l have the common feature of having less than complete markets in the Arrow-Debreu sense. In general, the purchase of some type of security, serves as.a substitute for trading in both futures markets and Arrow's [1964] complete insurance markets. This change in market organization, f i r s t introduced by Diamond [1967], has some rather strong implications for capital theory. The essays are concerned with these implications. Much of the recent interest, both theoretical and empirical, in the effect of uncertainty on capital-market equilibrium is due to the seminal paper of Sharpe [1964] on the general equilibrium pricing of assets under conditions of risk. This paper, which used the mean-variance portfolio model of Tobin-Markowitz, has generated a vast and s t i l l expanding l i t e r -ature on extensions of the model, and of its implications and applications - 2 -It is probably f a i r to say that most current capital market theory is taught in the context of this particular model, now known as the Capital Asset Pricing model. It is with some apologies then that my f i r s t essay is concerned pre-cisely with the Capital Asset Pricing model. My reasons are as follows. All the literature has been concerned with the implications of the market equilibrium assumption for the structure of the asset prices. This type of analysis produced or used, in some form or another, what became known as the Capital Asset Pricing equation: an equation relating equilibrium asset prices to various parameters of the model. The f i r s t essay, 'The Structure of a Capital Market Equilibrium 1, demonstrates, while i t is true that the equilibrium prices must satisfy this equation, i t is not true that this equation determines equilibrium prices, i.e., i t is a necessary but not a sufficient condition. The essay is concerned with developing a complete (i.e., sufficient) determination of equilibrium prices and exploring some of the implications of this derivation, both for the structure of equilibrium prices and portfolios and for the com-parative statics of this general equilibrium model. The Capital Asset Pricing Model is a model of a pure exchange econ-omy in which individuals trade bundles of asset claims in order to attain individual portfolio equilibrium. Diamond [1967] introduced production in this type of model, although he did not employ the mean-variance assumption. The introduction of production and firms raises the issue as to what c r i t e r i a firms should or would use in making their production decisions, as the profits of a firm under uncertainty are a random vari-able and, consequently, cannot be maximized. Diamond argued that stock market value maximization was the appropriate c r i t e r i a and showed that in - 3 -his model the equilibrium was Pareto efficient in a certain restricted sense. This paper prompted a number of responses questioning the gener-al i t y of Diamond's result and attempting to elucidate the conditions under.which i t may or may not be true. In this debate a l l the models have a capital or stock market in which individuals trade the shares of different firms and have the feature that, while production occurs, in-vestment does not. Consequently on the real (production) side of the economy there is no intertemporal allocation problem. The sole function of stock markets in these models is to f a c i l i t a t e risk bearing. They do not serve the traditional role of a capital market as a mechanisrm.which allocates real investment funds. The second essay is concerned with the implications of integrating the risk-bearing function of a stock market with the savings-investment function. This is done by requiring that firms raise the investment funds necessary for production in the same capital market as households trade equity shares. The consequences of this are rather strong. Perhaps the most striking result is that when a firm pursues an objective independent of the preferences for risk held by the household sector an equilibrium may not exist. Therefore, when..a capital market serves both a risk bear-ing function and investment allocation function, the investment policies firms may actually carry out are constrained by what the capital market will allow them to do. In order to examine these and related issues some new properties of the stock market model are developed in the second essay. The securities, which are traded by households in the f i r s t two essays and are issued by firms in the second essay, could be best de-scribed as common stocks. That i s , purchase of the security provides - 4 -a claim on the income stream of the asset against which the security is issued. The risk or uncertainty of holding such a security i s , thus, the uncertainty as to the income produced by the primary asset. Many of the contracts written in capital markets, however, are of the debt type; that i s , the repayment is of fixed amount in value terms and there is no uncer-tainty as to this amount. The sole source of uncertainty on this type of contract is the possibility of default in which case the loan will not be repaid at the terms originally agreed to in the contract. This type of uncertainty, or default risk, is significantly different from the vari-a b i l i t y of the income stream of the equity type of securities, as the probability of default on a debt contract is directly related to the price of the contract. For a given investment project financed by a debt in-strument, the probability of default increases as the price of the debt f a l l s , or equivalently, increases as the gross interest rate on the debt rises. The increase in the interest rate has another effect of course in that when the debt does pay off, i t pays off at a higher rate. This par-ticular feature of contracts on which default is a possibility is dealt with in the third essay. / Some simple equilibrium models of a capital market are constructed in which trade occurs in a single type of debt contract in order to f i n -ance investment in a risky project. Any loans secured by the project have some positive probability of defaulting. The models deal with the deter-mination of the equilibrium loan price and the equilibrium probability of default, and with their relationship to the risk free interest rate, the productivity of investment and the expectations of the capital market participants. In.the latter part of the essay an equity type of security is introduced in addition to the debt type security. The introduction of - 5 - J equity to the model has the consequence that as the capital structure of an investment project becomes more highly levered, the probability of the debt portion of the total capital structure paying off at i t s original price is diminished. Furthermore, when default or bankruptcy occurs there is a real cost involved, in that the returns from the investment project are less than they would have been had default not occurred^ Due to the possibility of costly default, the model has the feature that not only is an equilibrium price of debt determined, but in addition an equilibrium debt/equity ratio. This is in strong contrast of course to the financial irrelevance propositions of the Miller-Modigliani. [1958] sort, which state that when debt is risk free the economy is indifferent between varying debt/equity ratios. The relationship between equilibrium debt/equity ratios and various parameters is investigated. - 6 -Footnote 1. Throughout the thesis the terms uncertainty and risk w i l l be used interchangeably. Both denote the situation in which the consequence of a certain act or 'experiment' may be described by a probability distribution, either objective or subjective. - 7 -References Arrow, K. J. [1964], "The Role of Securities in the Optimal Allocation of Risk-Bearing", Review of Economic /Studies, 31, 91-96. Diamond, P. A. [1967], "The Role of the Stock Market in a General Equil-ibrium Model with Technological Uncertainty", American Economic Review, 57, 759-76. Miller, M. H. and Modigliani, F. [1958], "The Cost of Capital, Corporation Finance, and the Theory of Investment", American Economic Review, 48, Sharpe, W. [1964], "Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk", Journal of Finance, 14, 425-492. 261-297. v - 8 -Chapter II THE STRUCTURE OF A CAPITAL MARKET EQUILIBRIUM 1. Introduction The mean-variance portfolio model of Markowitz and Tobin has been the most substantive contribution to the theory of individual asset demand under uncertainty, in terms of comparative static results and testable implications. 1 Although subject to a number of criticisms at the axio-matic level, i t s t i l l stands as the classic portfolio model. The general equilibrium extension of the Tobin-Markowitz model due to Sharpe [1964], Lintner [1965] and Mossin [1966] has led to important propositions about the nature of risk in general equilibrium and its effect on the pricing of assets, and the model has subsequently been subjected to extensive em-2 pirical testing. It would appear then that the Capital Asset Pricing model, as i t is referred to in the literature, is to remain with us, as i t is the only general equilibrium model of asset pricing under uncertainty which can be econometrically implemented. It has been used for a variety of purposes in areas ranging from corporate-finance theory to the debate 3 on the social discount rate. What is somewhat remarkable, given the extensive use of this model, is that for a long time neither the existence question nor the comparative static properties of the model were investigated. Hart [1974].has recently treated the existence question; i t is the purpose of this paper to investi-gate the comparative static properties of the model. In doing so we shall also establish a number of propositions on the structure of the equilibrium which have not been discussed in the existing literature. - 9 -^In order to carry out the comparative statics, i t proved to be neces-sary to provide a complete description of the determination of the equil-ibrium prices in the mean-variance general equilibrium model. The mode of analysis initiated by Sharpe [1964], Lintner [1965] and Mossin [1966] was to use the first-order conditions of individual portfolio equilibrium and a market clearing assumption to derive a set of linear equations which the equilibrium asset-^prices satisfied. To carry the analysis this far was sufficient to determine a general-equi1ibrium version of the well-known Tobin [1958] separation theorem and to derive some implications about the pricing of ' r i s k 1. These linear equations, or the capital-asset pricing equation, do not, however, determine equilibrium prices, and i t is neces-sary to consider exp l i c i t l y the excess demand functions of the economy in order to give a complete determination of equilibrium prices. It is some-what surprising that, in such a general model of arbitrary dimensions allowing n assets and H investors, we are able to establish some rather strong qualitative propositions, since, as is generally well known, these 4 are usually quite d i f f i c u l t to come by. The rest of the paper proceeds as follows. In section 2 we introduce some notation and the assumptions of the model. Section 3 contains the derivation of a set of equations which the equilibrium prices must satisfy and some of the implications of these equations. In section 4 we estab-li s h the determination of equilibrium prices and portfolios, and some characteristics of the equilibrium. Finally in section 5 we investigate the comparative static properties of the model with section 6 presenting a summary and discussion of the results obtained. - 10 -2 . The Model The model consists of H investors, H f i n i t e and integer valued, who hold portfolios of n + 1 assets, where the Oth asset is riskless. By holding a portfolio at the beginning of the period, an individual realizes a random return on his portfolio at the end of the period. The following assumptions are basic to the Capital Asset Pricing Model. A.l. A l l investors are single-period, expected-utility-of-terminal-wealth maximizers who choose among alternative portfolios on the basis of the mean and variance of the portfolio. A.2. All investors are price takers, and no production takes place, i.e., a classical competitive exchange model. A.3. There are no short sales restrictions on any of the n i l assets. A.4. A l l investors have identical subjective probability distribu-tions on asset returns. A.5. The distribution of the asset endowments across investors is '•A 5 given. Our notation and some further assumptions are as follows: R.l. V denotes the variance-covariance matrix o f the probability distribution of risky asset returns; V is an n x n, real symmetric positive definite matrix. R.2. u denotes the mean vector of returns on the risky assets, y e R", and we assume y » 0^. By definition the return on the risk-less asset is 1. h th R.3. U (i^pjV^) denotes the h individual's u t i l i t y function, defined over mean portfolio return r^, and standard deviation of the portfolio return, v h, for a l l h = .1 H. U is defined - 11 -rp 1_ U U over the interior R with U > 0 and U < 0 everywhere with U + r v r denoting the partial derivative of U with respect to r^ and h h U r denoting the partial derivative of U with respect to v^; U h s t r i c t l y quasi-concave; lim [ U ^ ( r h , v J / u j V h , v h ) ] = 0 v 0 v n n r n n for any r h > 0. R.4. q1=(qQ,q') denotes the price vector on the n + 1 assets; q e R R.5. ^ n =^ sho' sh^ denotes portfolio of the h ^ investor; s^ e R n + 1 by A.3. R.6. Wu=(i, ,iu) denotes the endowment of the h t f l investor; w. » 0 n no n n ~~n+1 u ~ * i ~ n+1 8 R.7. E w. = i , where i ' = (1, ..., 1), i e R is the unit vector. h=l n These assumptions are quite usual and need no explanation here. All vectors are taken to be column vectors, with row vectors indicated by a transpose superscript, i.e., x' is the transpose of x where x is an n x 1 vector. The individual investor maximizes his u t i l i t y with respect to his portfolio vector subject to the restriction that his portfolio be feasible at current prices q. Thus we define: D.l. The h**1 investors budget set at prices q is B n(q) = {s h | s h e Rn A; q' s h < q'w^. An equilibrium is defined as a set of prices and portfolio allocations such that each individual maximizes his u t i l i t y and a l l asset markets clear. Therefore: D.2. An equilibrium is an H + 1 - tuple (q , s l 5 s 2 , s H) such that - 12 -a) s£ e Bh(q*) for a l l h, and s£ maximizes U*1 over a l l s^ e B h(q*), H H # b ) h=l ^ = 1 h=l. S h 0 = U c) e q " * e R n + 1 . We may, of course, normalize q*, but note, since we admit negative returns /v* n+1 9 on portfolios, we may not restrict q to R + . Often we shall normalize q Q = 1 for convenience, and this is clearly admissable, since the asset demand functions have the property of being homogeneous of degree zero in q. We shall further define: D.3. E denotes the set of equilibrium prices q* for the previously described economy. E <= R Using the results of Hart [1974], Appendix 2 we shall assume A.6. E is non-empty. That is we assume that an equilibrium exists. Given the assumption A.4., that a l l probability beliefs are identical, this can be proved. 1 0 E of course sepends on (u,V) and consequently we shall sometimes write E(u,V). The rest of the paper concerns the nature of E and i t s relation to (u,V). 3. A General Equilibrium Equation We shall now proceed on the assumption that we are dealing with equilibrium prices, that is that q e E, and derive a set of linear equations which the equilibrium prices must satisfy. Consider then the individuals, choice problem: max U h(r h,v h) subject to s h e B h(q) (3.1) ^h which can be formulated as a Lagrangean maximization problem as we have - 13 -no non-negativity constraints by A.3. Hence we have the Lagrangean L h = u h < v V - x h ^ ' s h + V o h - *' sh - V o h ) <3-2> giving us the first-order conditions UK + Uh - — L - X. q = 0 , (3,3) r v (s,;vshr2 h ^ - y i o = 0 , (3.4) ^' Sh + V 3 h " q , i h ' V o h = °" ( 3 ' 5 ) In order to make use of the first-order conditions (3.3) - (3.5), we must assume that a solution exists to problem (3.1). However since we allow short sales, the feasible portfolio set B is unbounded below and there is some possibility that a maximum may not exist. Bertsekas [1974] has recently shown that under quite reasonable con-ditions on the underlying expected u t i l i t y function and probability distribution a f i n i t e solution always exists. Hence in assuming A.6 we have implicitly assumed that for a l l q e E, a fi n i t e solution to (3.1) e x i s t s . 1 1 Therefore we may use the first-order conditions (3.3) - (3.5) to derive some implications about q e E. Substituting (3.4) into (3.3) we have U r " which upon re-arranging gives Premultiplying both sides of (3.7) by i ' and summing over h=l, H - 14 -yields, H H q o i 'U £ U h r/U h v(s^Vs h)^ + q Q i "Vi - i 'q ^ fytystflsj'1* =0 (3.8) H where we use the fact that at an equilibrium price £ s. = i (we have h=l n implicitly assumed s^ = s^, an equilibrium portfolio). Following the procedure used in describing the pricing equation of the capital asset pricing model we define: V m = q'i ( = i'q) E(V m) = u'i (3.9) al = i 'Vi, m where the subscript m refers to a market aggregate and E(V m) is the ex-pected value of the return on the aggregate market portfolio. Solving equation (3.8) we get r U j / U ^ s ^ s ^ ^ 2 = q 0(i'Vi) / [i'q - q Q i'u] = q„ al I [vm - q. E(vm)] (3.10) o m m ^o m = -q„ ol / [ q o E ( V j - V j . ^o m ^o m m We now solve equation (3.6) for the individual's risky portfolio and sum over a l l h=l, ..., H; H o ~ i - I h m ^o m m ^ ^o Premultiplying both sides of (3.11) by V gives 2 V 1 = [q E(7n) - V ] ^ " V > " ( 3 ' 1 2 )  L M o v nr mJ - 15 -Le t us d e f i n e m m m S o l v i n g f o r q we get the well-known c a p i t a l a s s e t p r i c i n g e q u a t i o n : q = q Q[u - 6 V i ] . (3.13) U s u a l l y , t h e "market r i s k " o f the jth' a s s e t i s d e f i n e d as V j i = o^m where V. i s the j t h row o f V, and c o n s e q u e n t l y the j t h e q u a t i o n o f (3.13) i s w r i t t e n a s , q d = I+r ^ j - e a j m ^ % = F r ( 3 ' 1 4 ) where r i s c a l l e d t he r i s k ! e s s r a t e o f i n t e r e s t . The term a. i s o f t e n jm c a l l e d the r i s k c h a r a c t e r i s t i c o f the j t h a s s e t , as i t r e p r e s e n t s t h e m a r g i n a l c o n t r i b u t i o n t o ag g r e g a t e market r i s k , by i n c r e a s i n g t h e j t h a s s e t . Hence, i f we l e t am (x) =• ZXo.. x.x., x E R" , (3.15) 3 3a ( i ) t h e n < y i } = = f j i ( 3 - 1 6 ) e v a l u a t e d a t x = i . We now r e - w r i t e e q u a t i o n (3.13) i n a more b a s i c form as, q = q 0 [y - { ( i ' y - l / q 0 T ' q ) / ( i ' V i ) } V i ] , (3.17) which upon r e - a r r a n g i n g g i v e s us ( i ' V i ) q - i ' q V i = i ' V i q Q y - 1 ' y V i q . Ta k i n g the t r a n s p o s e and u s i n g the f a c t t h a t V i s symmetric, ( i ' V i ) q ' - q' i i ' V = q 0 ( i ' V i ) y ' - y' i i ' V q Q , (3.18) - 16 -or q'[I i i 'V i "Vi (3.19) Now define A' = i i 'V i 'Vi (3.20) HenCe, we may write the final equation system as [I - A] q = [I - A] q Q y. (3.21) Let us denote the solution set of (3.21) as Q = {(q,q0) I [I - A] q = . [I - A] q Qy; q £ Rn, q Q e R } . Then we have by construction, the following lemma. Lemma 3.1: E <= Q, i . e., i f q e E, then q e Q. Note that the converse is not true. That is i f q is a solution to (3.21) i t is not necessarily the case that q is an equilibrium price. 4. The Structure of Equilibrium Prices In this section we investigate the nature of the equilibrium set of prices and the associated set of equilibrium portfolios. It will be con-venient in this and subsequent sections to make the normalization q Q = 1, and throughout this section we shall take the statement q e E to imply (q,l) e E and similarly for Q. Crucial to the subsequent theorem is the following assumption made on the variance-covariance matrix. A.7. Vi » 0 . The implication of the above assumption is that in a well defined sense a l l assets are positively risky. Intuitively, i f one examines the capital asset pricing equation (3.14), assuming risk is negatively valued, i.e., n - 17 -e is greater than zero, then A.7. implies a . m > 0 for a l l j = 1, n, and therefore a l l assets contribute positively at the margin to total mar-ket variance. All subsequent lemmas and theorems stated are implicitly understood to have assumptions A.l. - A.6. and R.l - R.7 holding. The following lemma will be quite useful. Lemma 4.1: The matrix A defined by (3.20) has the following properties: (1) A is idempotent. (2) A is of rank 1. And i f A.7. holds* '3> fl>>W (proof): Property (1) follows since AA = A. A is of rank 1, since i i ' is of rank 1, and the fact that the rank of a product of two matrices is not greater than the minimum of the two ranks; recall rank V=n. (3) follows straightforwardly from A.7. Q.E.D. We shall now establish a theorem which puts some bounds on the set E. Theorem 4.1: (a) If q e E, then i) q - y T 4 P^ . (b) If q e E, and A.7 holds then i i ) q << y. (proof): Let x = q-y and y = y-q. The equation system (3.21) may be written as Ax = x (4.1) or Ay = y. (4.2) That i s , x and y are eigenvectors of the matrix A corresponding to an eigenvalue of 1. Since A is idempotent i t can have eigenvalues of only zero or one, and furthermore i t has at least one eigenvalue equal to unity. (Lang [1966] pg. 193 Ex. 3). - 18 -(i j Now suppose that q = y, and we consider an arbitrary individual 1 equilibrium equation (3.6). Re-writing this, assuming q = y, gives us U H Vs. - I — = q " y = . (4.3) U v (sJVs,)* h h Now suppose s h T 4 ^ . A S L ) ^ / U V < 0, the system (4.3) may be written as V <shV sh>' Since s^ , ( s n V s n ) 2 > 0» as V is positive definite, and therefore Vs h = C]^. But V is non-singular. Hence s^ r- giving a contradiction to the supposition that s^ f= 0^. Therefore s^ = .0^  is the equilibrium portfolio i f q = y. Since h was arbitrary, s^ = 0^  for a l l h = 1 H. H But this contradicts the assumption that q e E, as • Z s^ = C^ ¥= i Therefore q + y h=l " " ( ( i i ) From (i) we have that x 7^0^* y i= 0^ - Since A is a positive matrix, we know these exists a vector z » 0^  and a unique positive eigen-value X > 0, such that Az = Az. (Karlin [1959], Theorem 8.2.1.) Further-more since A is of rank 1, and idempotent, X=l is the only non-zero eigen-value, and its associated eigenspace is the image space of A. Using the fact that A is idempotent, q f y, and A is a positive matrix, we have that the Frobenius eigenvalue = 1, and that either x » 0 or y » 0 , or both. ~n J -n Note that A has an eigenvalue 0 of multiplicity n-1. Suppose y » 0^, i.e q » y, and q-y is the eigenvector corresponding to the Frobenius eigen-value 1. Consider an arbitrary individual's equilibrium equation (3.6), assuming q » y. - 19 -Re-arranging this we have r„h Su I = An- ( 4 - 6 ) v Jh i i ' . \ U r ( s h V s h } Since V is non-singular the equation in brackets must be identically zero, and hence U h sh = i ^ V ^ r w - ^ < < Q n - ( 4- 7> v This clearly contradicts the assumption that q e E. Therefore q « y. Q.E.D. Basically, theorem 4.1 says that risk "matters" in a non-trivial sense. All asset prices must be s t r i c t l y less than their expected values. As a by-product of the proof of theorem 4.1 we establish the well-known separ-ation theorem (Tobin [1958], Sharpe [1964], Lintner [1965], Mossin [1966]), which states that in equilibrium a l l individuals hold positive fractions of an identical risky portfolio consisting of the vector of fixed aggregate supplies of risky assets. Theorem 4.2: If q e E, then s n(q) = X^i , X^  > 0 for a l l h=l, ,.. , H. U H (proof): Let X^  = -jj- (s^Vs^)'5, then from (4.5) we have Uv s^(q) = Y hV _ 1(q-y). Note that y n < 0- Since q e E, we have H H E s.(q) = i = V _ 1(q^y) [ E Y J . Therefore V _ 1(q-y) = i [ E y J - 1 h=l n h=l " h and s h(q) = ^ - i . Let x h = ~ > 0, then s u(q) = X u i . Q.E.D. Since short sales were allowed in constructing the individual's portfolio, - 20 -i t is of some interest to consider whether in equilibrium asset prices are such that some individuals choose to go short in some assets. The separ-ation theorem (Theorem 4.2) t e l l s us that in a general equilibrium indiv-iduals will not go short in risky assets. The following theorem gives sufficient conditions to ensure individuals do not go short in the risk-less asset. Theorem 4.3: If q e E, A.7 holds, and either -Uh .. ., (1) - r - > 1 U " V q along the ray - ( i ' V i T 2 r h = Mvh, where M = i ' u / ( i ' V i ) h for a l l h=l, ... , H, or (2) U (r^.v^) is identical for a l l h e {1,...,H} and homothetic in (r,v), then s h o(q) > 0 for a l l h=l,..., H. (proof): (1) Consider the first-order conditions, for the individual's portfolio problem: Max .U h(r h,v h) subject to s h e B h(q) which are from (4.5), and Theorem 4.2.ii. U h ( i ' V i T 2 Consider figure 1. From the hypothesis (1) of the theorem, and the equil-ibrium condition (4.8), the equilibrium point of the individual, e^ in ( rh> v n) space, occurs to the le f t of the line r h = Mvh and tangent to a line r^ = TTV^ + y h where y h = q',s^ + s h Q . From theorem 4.2 we know sL)(q) - A ^ i , and this corresponds to the point ( A ^ i ' V i ) ^ , A ^ i ' u ) ) = - 21 -Individual Portfolio Equilibrium - 22 -in (rn>v^) space, where lie s on the line r^ = Mvn directly below the equilibrium point e n > As r^(q) = A^(i'y) + s n > i t therefore must be the case that s n Q ( p ) > 0. As y^ -> 0, s^ Q -»• 0. Hence i f endowment income is identically zero, s n Q(q) = 0. We have excluded this possibility, however, ! by R.6. (2) Let W(su) = U h(y's h + s h Q , (s^Vs h) %). Since U h is homo-thetic i t is straightforward to verify that W(s^) is (nomothetic, and quasi-concave. The (asset) demand functions may therefore be written as s* = s(q) y n (4.9) where ^ = + % o . > °' ( 4 - 1 0 ) by using the well-known properties of homothetic functions. Since q e E, H H H E s. (q) = I s (q)y. = s ( q ) Z y. = 1. Clearly then s ( q ) > 0, h=l n o h=l 0 n 0 h=l n 0 and hence s^ Q(q) > 0 i f y n > 0. Q.E.D. The conditions of the theorem are quite strong, and they suggest that the likelihood of short sales in the riski ess asset is quite great. An open question which remains is the possibility of finding a weaker set of sufficient, conditions which ensure the absence of short sales in the riskless asset. ..'••„ .'.• Let us denote by Q(V) the following set: Q(V) = {w | Aw = w, w + 0^}. (4.11) This is the set of non-zero eigenvectors of the matrix A. From the pre-vious discussion we have determined that i f / q e E, then q = y - w* for some w* e Q(V). Since Q(V) denotes the solution set to the capital asset pricing equation, we cannot claim in general that the capital asset pricing - 23 -equation "determines" equilibrium prices. An equilibrium corresponds to a particular w* e Q(V). We shall now consider the manner in which w* is determined. Once Q(V) has been determined, we may choose wQ e Q(V), such that II w || = 1, where || || denotes the euclidean norm. Consider then the A i 1 w set of non-negative real numbers II = {irl-n > 0, TT = °y- , A > 0}. (i'Vi)" 2 Dropping the h subscript, the individual's choice problem can be reformu-lated, given w . as o Max U(r,v) subject to r - -rrv <_ y., (4.12) r,v > 0 Recall that y h = q's h + s h Q = y ' i h - AwQ'sh +'i h o» when q = y - AwQ. w's. v Therefore, y h = r h - Aw^sh where rh = y ' s h + sho* L e t A = wV~ ( i , v i ) ^ ' o then TTV^ = ^w^sn. Hence we may write problem (4.12) as Max U(r,v) subject to r - TTV <_ r - TTV. (4.13) r,v >_ 0 This gives us the individual demand functions for (r,v) as r = r (TT,y) = f r ( i r ) , v = v (7r,y) = f v ( T r ) . (4.14) The individual demand functions for 'risk' v, and 'return' r depend upon the 'price of risk' TT and the endowment income y = r - S>v, which in turn depends upon the price of risk. These demand functions satisfy the usual properties implied by Hicks-Allen demand theory. In addition, they have the property that 'return' r is always a normal good, that is 9 r ^ , i y ^ > 0 and 3 r(y'y) > 0. 1 2 Note that i t is not true in general that f ^ ( i r ) .> 0. Of course, f r(iT) and f v ( i r ) are related by the budget constraint, f (TT) -T r f u ( 7 T ) = y and hence the equation system (4.14) has only one independent - 24 -e q u a t i o n . Note t h a t t h e s e demand f u n c t i o n s a r e not d e f i n e d a t T  - 0. I f T  = 0 then by the argument o f theorem 4.2, s ^ q ) = 0, f o r a l l h, and by the p o s i t i v e s l o p e o f i n d i f f e r e n c e c u r v e s fR(0) = y n = l - 1 ' ^ . Now, how-e v e r , fV(0) = 0. But, i n f a c t , t h i s demand i s i n f e a s i b l e i n the a c t u a l commodity s p a c e , as the economy's demand f o r r i s k y a s s e t s i s i d e n t i c a l l y z e r o . T h e r e f o r e i f q = y, then y ^ = s ^ . Suppose t h a t q = y, o r , e q u i v a -l e n t l y , T  = 0. The demand f u n c t i o n s a t t h i s p o i n t a r e g i v e n as the s o l u t i o n t o : Max U ( r , v ) s u b j e c t t o v = _ ( i * V s ) J s (4.15) r , v >_ 0 r < s Q . A t q = y, then a l l i n d i v i d u a l s w i l l be i n e q u i l i b r i u m h o l d i n g t h e i r endow-ments a t t h i s ' c o n s t r a i n e d ' e q u i l i b r i u m , o r e q u i l i b r i u m w i t h r a t i o n i n g which i s not an e q u i l i b r i u m i n the W a l r a s i a n sense o f d e s i r e d s u p p l y b e i n g g r e a t e r than o r equal t o d e s i r e d demand. We d e f i n e the exc e s s demand f u n c t i o n s i n the usual f a s h i o n . The i n d i v i d u a l e x c e s s demand f u n c t i o n f o r mean r e t u r n i s d e f i n e d as ZJ(TT) = fj!(Tr) - y ' i h + IHQ (4.16) and t h e a g g r e g a t e e x c e s s demand f u n c t i o n i s g i v e n by Z :(TT) = E fJ(TT) - i ' y + 1. (4.17) r h r In p r o v i n g the main theorem o f t h i s s e c t i o n we s h a l l have t o make use o f a p a r t i a l c o n v e r s e t o Lemma 3.1. Lemma 4.2: I f w e Q(V), then s h ( q ) = S^i , f o r some s c a l a r 5^ and a l l h e {1, H}, when y - q = w. ( p r o o f ) : C o n s i d e r an a r b i t r a r y i n d i v i d u a l ' s p o r t f o l i o e q u i l i b r i u m e q u a t i o n (4.5), which i s r e p r o d u c e d here f o r c o n v e n i e n c e . - 25 -U h Vs. - £ h — - q - y (4.5) U r W L e t 3^ = — ( s n V s n ) ^» t n e n ( 4 - 5 ) m a-y b e w r i ' t t e n as U r V s h = l/6 hw where w,e Q(V). As B h > 0, t f w h= l / 3 h w , then w he Q(V) from the b a s i c p r o p e r t i e s o f e i g e n v e c t o r s . T h e r e f o r e , V s h = w h = Aw, •Vii-'w h VYT~ • Upon r e - a r r a n g i n g t h i s l a s t e q u a t i o n we have i i 'w. As V i s a p o s i t i v e d e f i n i t e m a t r i x and c o n s e q u e n t l y n o n - s i n g u l a r , the vec-i'w. t o r - i n b r a c k e t s must be the z e r o v e c t o r . L e t t i n g <5h = . ,y., s^ = S ^ i . Q.E.D. The lemma has the f o l l o w i n g i n t e r p r e t a t i o n . I f an a s s e t p r i c e v e c t o r s a t i s f i e s the b a s i c e i g e n v e c t o r e q u a t i o n , then the p o r t f o l i o o f r i s k y a s s e t s h e l d by e v e r y h o u s e h o l d must be p r o p o r t i o n a l t o the ag g r e g a t e endow-ment o f r i s k y a s s e t s . The lemma i s s t r o n g e r i n one sense than theorem 4.2 as t he p r i c e v e c t o r q i s not r e q u i r e d to be an e q u i l i b r i u m p r i c e v e c t o r , but r a t h e r i t must s a t i s f y o n l y the weaker e i g e n v e c t o r p r o p e r t y . We now prove the c e n t r a l theorem which p r o v i d e s a complete d e t e r m i n a -t i o n o f e q u i l i b r i u m p r i c e s . Theorem 4.4: I f Z r(Tr*) = 0 f o r T T * e IT then q e E A*i'w where q = u-A*w . T T * = — , and w* = A*w . o ( i - v i ) " 2 o - 26 -( p r o o f ) : S i n c e Z^(T T*) = 0 , and TT* > 0 , then by Wairas' law 7 _ V ( T T*) = 0 . T h e r e f o r e E f J j U * ) - = ( i ' V i ) % , and fJ(.Tr*) = v u = [ s h ( q ) ' V s h ( q ) f 2 . h As y-q = A W0, and A w Q e Q(V), then by lemma 4.2 s n ( q ) = 5 ^ i , 5^ e R, f o r a l l h = l , ... , H. S u b s t i t u t i n g , we have v ^ = - 6 n ( i ' V i ) 2 and thus market c l e a r i n g i n the ' r i s k ' market i m p l i e s E 6 n = 1 . T h e r e f o r e £ s h ( c i ) = 1 and c o n s e q u e n t l y q e E. Q.E.D. h n What t h i s theorem says i s t h a t , i n a d d i t i o n t o the m a t r i x e q u a t i o n Aw Q = w Q, a d d i t i o n a l i n f o r m a t i o n on the form o f u t i l i t y f u n c t i o n s and the d i s t r i b u t i o n o f endowments a c r o s s i n d i v i d u a l s , o r e q u i v a l e n t l y t he ag g r e -g a t e demand f u n c t i o n s f o r r i s k o r r e t u r n i s needed i n o r d e r t o d e t e r m i n e e q u i l i b r i u m p r i c e s . T h i s a d d i t i o n a l i n f o r m a t i o n i s p a r t i c u l a r l y c r i t i c a l when c o n s i d e r i n g q u e s t i o n s o f co m p a r a t i v e s t a t i c s . 5. Comparative S t a t i c s A fundamental c r i t e r i a o f the u s e f u l n e s s o f a model i n economics has been whether o r not i t y i e l d s unambiguous c o m p a r a t i v e s t a t i c r e s u l t s , and i t i s the purpose o f t h i s s e c t i o n t o see how f a r one can go i n t h i s d i r e c -t i o n w i t h the C a p i t a l A s s e t P r i c i n g Model. In. o r d e r t o do so we s h a l l use a d i f f e r e n t mode o f a n a l y s i s than i s g e n e r a l l y f o l l o w e d . A t t h i s p o i n t i t i s n e c e s s a r y t o make some f u r t h e r assumptions on i n v e s t o r s ' p r e f e r e n c e s i n o r d e r t o p r o c e e d w i t h the" c o m p a r a t i v e s t a t i c s . T h i s i s n e c e s s a r y f o r two r e a s o n s . The f i r s t i s t h a t w i t h o u t a more s t r i n g e n t s e t o f c o n d i t i o n s on the e x c e s s demand f u n c t i o n s Z v ( T T ) , Z ^ T T ) o t h e r than Wairas' law and some d i f f e r e n t i a b i l i t y a s s u m p t i o n s , we a r e not a s s u r e d i n g e n e r a l t h a t the model has a unique e q u i l i b r i u m p r i c e q*. T h i s r e q u i r e m e n t i s g e n e r a l l y needed i n o r d e r t o make our c o m p a r a t i v e s t a t i c - 27 -13 r e s u l t s m e a n i n g f u l . S e c o n d l y , i n o r d e r t o g e t q u a l i t a t i v e r e s u l t s i t i s n e c e s s a r y t o s i g n the d e r i v a t i v e s o f Z v and i n some manner. I The c l a s s o f p r e f e r e n c e s we choose t o work w i t h i s a c o n v e n i e n t gener-a l i z a t i o n o f the c l a s s o f e x p e c t e d u t i l i t y f u n c t i o n s which e x h i b i t c o n s t a n t a b s o l u t e r i s k a v e r s i o n . B a s i c a l l y , the r e q u i r e m e n t i s t h a t the m a r g i n a l r a t e o f s u b s t i t u t i o n between r i s k and r e t u r n i s i ndependent o f r e t u r n . T h e r e f o r e i n ( r , v ) space a l l i n d i f f e r e n c e c u r v e s have the same s l o p e a l o n g v e r t i c a l l i n e s , as i n F i g u r e 2.. We e x p r e s s t h i s r e q u i r e m e n t by the equa-t i o n -U v/Up = k ( v ) . C l e a r l y by the c o n v e x i t y assumption k'(v) > 0, and the assumption o f r i s k a v e r s i o n i m p l i e s k ( v ) > 0. T h e r e f o r e we assume h h A.8. I n v e s t o r s have u t i l i t y f u n c t i o n s , U ( r , v ) , where U ( r ^ v ^ ) ''> s a t i s f i e s the c o n d i t i o n t h a t - u J j / U ^ = k h ( v h ) w i t h k h ( v h ) con-t i n u o u s and l i m k (v.) = 0, f o r a l l h=l, H. v+0 n Thus a l l i n v e s t o r s have p r e f e r e n c e s o v e r r i s k and r e t u r n , which a r e o f the same c l a s s g i v e n by A.8, a l t h o u g h d i f f e r e n t i n v e s t o r s may have d i f f e r e n t 14 p r e f e r e n c e s w i t h i n t h i s c l a s s . C o n s i d e r the problem o f the i n d i v i d u a l i n v e s t o r , g i v e n by > n U h ( r h ' V h ) ^ j e c t t o r h - T r v h < y h . (5.1) r h ' v h - 0 I t i s s t r a i g h t f o r w a r d t o v e r i f y t h a t the demand f u n c t i o n s g e n e r a t e d by t h i s problem take the form h, M - y n (5.2) v h = f v ^ ' U s i n g the budget c o n s t r a i n t , we have t h a t f J U ) " f j d r ) = 0. (5.3) - 28 -F i g u r e 2. I n d i v i d u a l P r e f e r e n c e s w i t h the P r o p e r t y t h a t the M a r g i n a l Rate o f S u b s t i t u t i o n Between R i s k and Return i s Independent o f Return - 2 9 -The e x c e s s demand f u n c t i o n f o r r e t u r n i s g i v e n by Z > ) = 2 f^M + Y - i ' y + 1 ( 5 . 3 ) r h r where Y = £ y . , and the exc e s s demand f o r r i s k i s g i v e n by h n IM = ^ f j ( i r ) - ( i ' V i ) % . ( 5 . 4 ) v h v The f u n c t i o n s fjjfr) and fJ(Tr) have the p r o p e r t y t h a t fjj' > 0 and fJ''' > 0 . 1 5 C o n s i d e r then the e q u a t i o n Z v ( T T * ) = 0 , where TT* i s d e f i n e d as i n theorem 4 . 4 . I f TT i s unique, i n the sense t h a t f o r a l l TT e n, TT ^ TT , Z ( T T ) . T ^ O , then the e q u i l i b r i u m p r i c e i s uniq u e ; q . = y - X wQ. T h e r e f o r e we have Lemma 5 . 1 : G i v e n A . 8 , the s e t E(y,v) = {q*}. The e q u i l i b r i u m p r i c e v e c t o r i s u n ique. ( p r o o f ) : By A . 8 Z^(T T) = ZfJ'•' > 0 f o r a l l TT e n. T h e r e f o r e Z Y i s g l o b - -h a l l y u n i v a l e n t , which i m p l i e s T T * , Z v ( T T * ) = 0 , i s u nique. Q.E.D. Having a s s u r e d o u r s e l v e s t h a t t he e q u i l i b r i u m i s w e l l d e f i n e d , we now c o n s i d e r q u e s t i o n s o f com p a r a t i v e s t a t i c s . The e q u i l i b r i u m p r i c e q* i s det e r m i n e d by two e q u a t i o n s : and where Aw„ = w„ w„ = 1 ( 5 . 5 ) o o o Z R ( T T * ) = 0 o r Z v ( T T * ) = 0 , ( 5 . 6 ) 1 w„ TT . = A ( i ' V i ) We w i l l c o n s i d e r f i r s t changes i n the e q u i l i b r i u m p r i c e w i t h r e s p e c t t o changes i n the mean v e c t o r y . In p a r t i c u l a r we c o n s i d e r a change i n the mean o f the i t h a s s e t , y".. We note f i r s t Lemma 5 . 2 : The p o s i t i v e e i g e n v e c t o r s o f A are i n v a r i a n t w i t h r e s p e c t t o dw changes i n y , and hence = [ 0 ] . - 30 -T h i s i s a t r i v i a l consequence o f the f a c t t h a t A does not depend on y. dw T h e r e f o r e = ^ n ' f o r a 1 1 1 = 1' " " n* A consequence o f t h i s i s the f o l l o w i n g theorem. Theorem 5.1: Given A.8, i f q* e E ( y , V ) , then * = 1 f o r i = l,. ..., n, (5.7) dyn. dq* g-^- = 0 f o r i ^ j i , j = l , n. (5.8) * t h Remark: S i n c e we have n o r m a l i z e d q Q = 1, q.. i s the p r i c e o f the i a s s e t r e l a t i v e t o the r i s k l e s s a s s e t . ( p r o o f ) : By. d e f i n i t i o n q* = y - X*wQ. T h e r e f o r e , u s i n g Lemma 5.2 = 6 i k - o ^ w o ' w h e r e 6 i k = 1 i f i = k » 6 i k = 0 i f 1 * k -* * ( i 'Vi) Now X = TT i w • D i f f e r e n t i a t i n g e q u a t i o n (5.6) and u s i n g (5.4) we have t h a t E f ^ T r * ) = 0. S i n c e f h ' > 0, j £ = 0, and h v a \ v a y k hence ^f— = 0. T h e r e f o r e -p- 1 = 6-, • Q.E.D. d y k d y k i k The r e s u l t o f theorem 5.1 i s q u i t e s t r o n g . I t says t h a t changes i n a s s e t means a r e a s s e t s p e c i f i c , and moreover an a b s o l u t e change i n the mean r e -t u r n o f an a s s e t r e s u l t s i n an equal a b s o l u t e change i n the p r i c e o f t h e a s s e t , o f the same s i g n . R e c a l l t h a t 6 = 1 i ' y i ^ w a s termed the " p r i c e " o f r i s k i n s e c t i o n 3:... A s i m p l e consequence o f the theorem i s t h a t de/dy^ = 0; t h a t i s the " p r i c e " o f r i s k i s u n a f f e c t e d by changes i n the means o f a s s e t r e t u r n s . I f one d e f i n e s t he p r i c e o f r i s k as 1 y ~ > ( i ' V i ) " 2 our TT , then the same r e s u l t h o l d s . We wish now t o c o n s i d e r the consequence o f changes i n the v a r i a n c e -- 31 -c o v a r i a n c e s t r u c t u r e o f the r e t u r n s on a s s e t s . In p a r t i c u l a r we f o c u s on a change i n a.., t h e ( i , j ) t h and ( j , i ) t h elements o f V. H e n c e f o r t h we dx w i l l t a k e dx t o mean - r — w h e r e x may be a s c a l a r , v e c t o r o r m a t r i x w i t h i j t h e c o r r e c t i n t e r p r e t a t i o n c l e a r from the c o n t e x t . In o r d e r t o c a r r y o u t the a n a l y s i s , we break i t down i n t o two s t e p s . We f i r s t examine the changes i n t h e e i g e n v e c t o r w Q o f e q u a t i o n ( 5 . 5 ) , and then the change i n X* o f e q u a t i o n ( 5 . 6 ) . Combining t h e s e two e f f e c t s we g e t the t o t a l e f f e c t on q*. When changing the elements o f the v a r i a n c e - c o v a r i a n c e m a t r i x , i t i s n e c e s s a r y t o r e s t r i c t t h e p o s s i b l e parameter changes. T h e r e f o r e we w i l l assume h e n c e f o r t h A.9: da., i s r e s t r i c t e d i n such a way t h a t V remains a p o s i t i v e "1 J d e f i n i t e m a t r i x . C o n s i d e r then t he e q u a t i o n system ( 5 . 5 ) . D i f f e r e n t i a t i n g w i t h r e -s p e c t t o o.y and n o t i n g t h a t Wq must l i e on the u n i t sphere g i v e s us two e q u a t i o n s : Adw + Dw = dw o o o dw' w o o (5.9) (5.10) D i s a m a t r i x w i t h elements da kl da. . , R e c a l l i n g t h a t A = nxn V i i ' s t r a i g h t f o r w a r d c a l c u l a t i o n s y i e l d C i i ' V i i ' where D = C = i 'Vi 0 . 1 0 ( i ' V i ) ' . 0 1 . i ' C i •0 J nxn (5.11) (5.12) i . e . , the z e r o m a t r i x w i t h e x c e p t i o n o f l ' s i n the ( i , j ) t h and ( j , i ) t h p l a c e s . Note t h a t i ' C i = 1 i f i = j and 2 o t h e r w i s e . - 32 -Thus we may r e - w r i t e (5.9) as C i i ' V i i 1  i , V i " ( i ' V i ) i 'Ci I - V i i ' i 'Vi dw. (5.13) Now e q u a t i o n (5.10) i m p l i e s t h a t dw Q i s o r t h o g o n a l t o w Q, and hence o r t h o g o n a l t o the l i n e a r subspace c o n t a i n i n g non-zero e i g e n v e c t o r s o f A. S i n c e A i s o f rank 1, dw Q i s i n the k e r n e l o f A, i . e . , i t i s o r t h o g o n a l t o the image space o f A. Thus we have Lemma 5.3: I f dw Q i s a s o l u t i o n t o e q u a t i o n s (5.9) and (5.10) then Adw„ = 0 . o -n Us i n g t h i s lemma we-now. d e r i v e some p r o p e r t i e s o f dw Q. P r e m u l t i p l y i n g both s i d e s o f (5.13) by dw^ we g e t dw . dw . oi_ + oj_ , da. . L i j da. i j i 'w rm = d w o d w o > 0 -T h i s i n e q u a l i t y i n t u r n i m p l i e s 17 (5.14) dw 01 dw . da. -91 > 0 i j i f j i 4=1 * n, (5.15) and d w o i da., -> 0; i = l , (5.16) From e q u a t i o n (5.13) we have a l s o dw. "ok i ' V i "ok v). < 0 f o r k ^ i , j (5.17) These r e s u l t s g i v e us the q u a l i t a t i v e i m p l i c a t i o n s f o r changes i n a . , on the non-zero e i g e n v e c t o r s o f A. We c o n s i d e r now e q u a t i o n (5.6) and i n p a r t i c u l a r Z r ( 7 r ) = 0. From (5.3) we may w r i t e t h i s as - 33 -£ f%*) + Y - i 'u - 1 = 0. (5.18) h r D i f f e r e n t i a t i n g w i t h r e s p e c t t o a., we have E f h ' du* + dY = 0. (5.19) h r . i * Now TT* = 1 w ! , and Y = y ' i - i'w* + 1 where q* = y - w*, thus ( i V i T 2 du* , i 1 d w * i'w* /r o n\ d a i j ( i ' V i ) 2 ( i ' V i ) / z and dY = -i'dw*. (5.21) S u b s t i t u t i n g (5.20) and (5.21) i n t o (5.19) and r e - a r r a n g i n g g i v e s [E ft!:(i'Vi) - ( i ' V i ) ' 7 2 ] i'dw* = E fhJ i'w*. (5.22) h r h r Now u s i n g Wairas' law and d i f f e r e n t i a t i n g (5.3) a t TT* i m p l i e s Z-fJV) = E ATT*) + TT* E fhJ (TT*) h r h r h r ( i ' V i ) % + TT* E f n ' (TT*) h r (5.23) S u b s t i t u t i n g t h i s i n t o (5.22) i m p l i e s E f h r ( i ' V i ) V i'dw* = r-r > 0. (5.24) E f£ ' h r E q u a t i o n (5.24) i s the q u a l i t a t i v e r e s t r i c t i o n p l a c e d on the e q u i l i b r i u m system by e q u a t i o n ( 5 . 6 ) . We now wish t o d e v e l o p the t o t a l e f f e c t . S i n c e q* i s an e q u i l i b r i u m , q* = y - A ° w 0 , A ° > 0, and s i n c e q* + dq* i s an e q u i l i b r i u m q* + dq* = y - A 1 ( w Q + dw Q). L e t q* + dq* = y - w*, and by d e f i n i t i o n w* = * 0w Q. Then w*= A 1 ( w Q + d w Q ) , A 1 > 0. Now dw* by d e f i n i t i o n i s w* - w*. 1 o - 34 -S u b s t i t u t i n g and r e - a r r a n g i n g we have then t h a t dw* = A [w + d w j - A w„ 0 0 0 = [ A 1 - A 0 ] w o + A 1 dwr (5.25) o ~"0 P r e - m u l t i p l y i n g (5.25) by i 1 and u s i n g (5.24) g i v e s us [ A 1 - A 0 ] i'w 0 + A 1 i'dw Q > 0. (5.26) C o n s i d e r now the term i'dw Q. U s i n g Lemma 5.3 we may r e - w r i t e (5.13) as C i i ' i 'Vi v i i ' w„ = dw . o o (5.27) ( i ' V i ) P r e - m u l t i p l y i n g both s i d e s o f (5.27) by i ' g i v e s , u s i n g the f a c t t h a t Aw„ = w„ o o i 'Ci i w. 1 w. 0 0 = 0 = i'dw„. o (5.28) Thus s u b s t i t u t i n g i'dw 0 = 0 i n t o (5.26) i m p l i e s 18 A i - A o > °- (5.29) U s i n g ( 5 . 2 5 ) , (5.29) and (5.14) thro u g h (5.16) we have the f o l l o w i n g r e s u l t s dw, dw I. der, 'i dw h - C x i w o i 01 II I I I . 4o.. -> 0 dw*. da, da, i j [ A . - A l[v . + v .] + A , 1 0 J L 01 OJ J ( 1 dw . dw 01 , d a . . da, oj i j (5.30) 1 0 (5.31) dw| 2 doTT = l x i " A i FvT " A o ] w o k Ol J = h^r-jTYj) - A 0 J w, ok k ^ i , j (5.32) Note t h a t the l a s t r e s u l t ( I I I ) i s not s i g n e d due t o the o f f s e t t i n g e f f e c t o f the term .,y. . However i f a g g r e g a t e market v a r i a n c e i s l a r g e , such - 35 -dw* t h a t 2 / i V i i s s u f f i c i e n t l y s m a l l then ^ — > ^ 0 f o r a l l k ^ i , j . U s i n g the above r e s u l t s and n o t i n g t h a t dq* = -dw*, we have the f o l l o w i n g theorem. Theorem 5.2: a) An i n c r e a s e i n the c o v a r i a n c e between a s s e t s i and j has the e f f e c t o f (1) d e c r e a s i n g (weakly) the sum o f the i and j a s s e t p r i c e s (2) d e c r e a s i n g (weakly) a t l e a s t one o f the i t n o r j * * 1 p r i c e s b) I f a g g r e g a t e market v a r i a n c e i s s u f f i c i e n t l y l a r g e such t h a t 2 / i V i i s " c l o s e " t o z e r o i n the sense t h a t A 1(1-2/. i ' V i ) - A _> 0, then an i n c r e a s e i n the c o v a r i a n c e between a s s e t s i and j has the e f f e c t o f d e c r e a s i n g (weakly) the p r i c e o f a l l o t h e r s a s s e t s k^=i o r j . c) The e f f e c t o f an i n c r e a s e i n the c o v a r i a n c e between a s s e t s i and j has the same q u a l i t a t i v e e f f e c t on a l l o t h e r a s s e t s k ^ i o r j . An analogous r e s u l t h o l d s f o r an i n c r e a s e i n the v a r i a n c e o f an a s s e t . We s t a t e o n l y the p a r t analogous t o a) o f Theorem 5.2 but note t h a t s i m i l a r s t a t e m e n t s t o b) and c ) h o l d . J.L. Theorem 5.3: An i n c r e a s e i n the v a r i a n c e o f t h e i a s s e t has the e f f e c t o f d e c r e a s i n g (weakly) the p r i c e o f the i * " n a s s e t . These r e s u l t s seem t o be i n t u i t i v e l y c o n f o r t a b l e . Note t h a t f o r an th ^ o k 1 i n c r e a s e i n v a r i a n c e o f the i a s s e t -j = - -Av^ w , , and t h e r e f o r e t h e do.j i Vi ok r e q u i r e m e n t t h a t •X J(1 - j r ^ p ) - * 0 - 0 i s m u G n w e a k e r t n a n t h e r e q u i r e m e n t t h a t X1(l - . , v . ) - -A >_ 0. Thus an i n c r e a s e i n the v a r i a n c e o f an a s s e t , f o r l a r g e market v a r i a n c e , i s more l i k e l y t o d e c r e a s e the p r i c e o f o t h e r - 36 -a s s e t s than an i n c r e a s e i n the c o v a r i a n c e between two a s s e t s . The " p r i c e " o f r i s k i n our model has been termed 9, o r T T * , depending on one's d e f i n i t i o n . However i t t u r n s out t h a t i n o n l y one case an i n -c r e a s e i n e i t h e r v a r i a n c e o r c o v a r i a n c e has an unambiguous e f f e c t on the " p r i c e " o f r i s k . From (5.1?) and (5.24) du* = - -%- = l ^ w ; > o. (5.33) h r h r C l e a r l y s i n c e 0 = u* ^ = , de = VM- - n" w* ? which i s ( i ' V i ) 2 1 V l 1 V l ( i ' V i ) 2 ambiguous. Thus one might q u e s t i o n whether i n f a c t 9 was the c o r r e c t measure o f t h e p r i c e o f r i s k , s i n c e one f e e l s t h a t w i t h r i s k a v e r s e i n -v e s t o r s , an i n c r e a s e i n r i s k s h o u l d i m p l y t h a t the r i s k margin s h o u l d i n c r e a s e . In any c a s e we have shown Lemma 5.4: The " p r i c e " o f r i s k i n t e r p r e t e d as TT* unambiguously i n c r e a s e s w i t h i n c r e a s e s i n e i t h e r the v a r i a n c e o r c o v a r i a n c e o f a s s e t s . T h i s completes our program o f c o m p a r a t i v e s t a t i c s w i t h r e s p e c t to the v a r i a n c e - c o v a r i a n c e m a t r i x . A t t h i s p o i n t we s h o u l d mention two f u r t h e r c o m p a r a t i v e s t a t i c e x e r c i s e s . One would be t o c o n s i d e r i n c r e a s e s i n r i s k a v e r s i o n on the p a r t o f the i n v e s t o r s , and the o t h e r would be t o c o n s i d e r s u p p l y e f f e c t s ; i . e . , changing t h e a g g r e g a t e endowment o f a s s e t s . A l t h o u g h i n t h e g e n e r a l model t h i s would be q u i t e d i f f i c u l t , i n the c a s e o f a l l con-sumers hav i n g i d e n t i c a l c o n s t a n t a b s o l u t e r i s k a v e r s i o n u t i l i t y f u n c t i o n s , w i t h r i s k a v e r s i o n parameter A, i t i s w e l l known t h a t the e q u i l i b r i u m p r i c e s q, g i v e n q Q = 1 can be w r i t t e n as q = u - A W ( x ' V x ) J s (5.34) where x i s the a g g r e g a t e endowment v e c t o r o f the economy. I t i s s t r a i g h t -f o r w a r d t o v e r i f y t h a t i n c r e a s e s i n r i s k a v e r s i o n , i . e . , i n c r e a s e i n A, - 37 -u n i f o r m l y i n c r e a s e s the r i s k margin on a l l a s s e t s , and hence decreases r a i l a s s e t p r i c e s . S e c o n d l y , an i n c r e a s e i n the s u p p l y o f , say the f i r s t a s s e t , d x : S 0, w i l l i n c r e a s e o r d e c r e a s e the p r i c e o f the j * 1 1 a s s e t , dq. < 0, as ffjj < 0, j = l , n. T h i s r e s u l t i n d i c a t e s t h a t even i n a s t r o n g l y s p e c i f i e d model s u p p l y e f f e c t s a r e dependent on the p a r t i c u l a r parameters o f the model. Another i n t e r e s t i n g c a s e i s t o examine the e f f e c t s o f i n c r e a s i n g t h e number o f i n v e s t o r s H. In p a r t i c u l a r we pose the q u e s t i o n , as H tends t o i n f i n i t y , what happens t o e q u i l i b r i u m p r i c e s , q*. The r e s u l t one o b t a i n s c l e a r l y depends on the assumptions one makes about the b e h a v i o u r o f aggre-gate market v a r i a n c e . L e t t i n d i c a t e an economy w i t h an a g g r e g a t e endow-ment v e c t o r x^ >>.0n> and we assume t h a t t a l s o i n d i c a t e s the number o f households i n each economy. Fur t h e r m o r e , we assume t h a t a l l households o r i n v e s t o r s have i d e n t i c a l p r e f e r e n c e s w i t h i n the c l a s s o f p r e f e r e n c e s d e s c r i b e d i n A.8. Thus as t grows l a r g e we have more i n d i v i d u a l s a l l p o s s e s s i n g the same p r e f e r e n c e s but p o s s i b l y d i f f e r e n t endowments. Thus f o r any t , t f ( i r t ) = ( x | V x t ) 2 . We a l s o know t h a t k ( v t ) •> 0 as v t -> 0, TT^ = k ( 0 , where v. = 1/t [ x ^ V x ^ ] ^ . Suppose t h a t l i m {sup x } <_ M, where t-*» s<t M i s a l a r g e but f i n i t e v e c t o r i n R". Hence we c o n s i d e r an i n c r e a s i n g sequence o f economies, where u l t i m a t e l y r e s o u r c e s a r e bounded. Thus l i m v t = l i m 1/t [x^VXjJ* 5 = 0, t -> oo t 0 0 g i v e n the assumption o f bounded r e s o u r c e s . S i n c e , by A.8, k i s c o n t i n u o u s f u n c t i o n o f v, l i m k ( v t ) = k ( l i m v t ) = 0, and hence l i m T T * = 0. R e c a l l * ^ t * t h a t q. = u — , and thus q. -> y, o r a s s e t p r i c e s tend t o the mean * ( x i V x t ) " * - 38 -v a l u e s o f the a s s e t r e t u r n s as the number o f i d e n t i c a l i n v e s t o r s tends t o i n f i n i t y . Lemma 5.5: In a sequence o f economies, such t h a t a l l households have i d e n t i c a l p r e f e r e n c e s , t h e number o f households tends t o i n f i n i t y , and the sequence o f a g g r e g a t e endowment v e c t o r s i s bounded, the e q u i l i b r i u m p r i c e v e c t o r tends t o the ex p e c t e d v a l u e o f a s s e t r e t u r n s . T h i s argument i s q u i t e s i m i l a r t o one made by Arrow and L i n d [1970] i n a d i f f e r e n t c o n t e x t . I t c o u l d be i n t e r p r e t e d as s a y i n g t h a t f o r l a r g e but f i n i t e economies e x p e c t e d v a l u e s become good a p p r o x i m a t i o n s t o e q u i l -i b r i u m p r i c e s . An i m p o r t a n t q u e s t i o n i s whether such a r e s u l t o b t a i n s i n more g e n e r a l models, because once t he assumption o f f i x e d a s s e t s u p p l i e s i s dropped and a p r o d u c t i o n s i d e i s i n t r o d u c e d t o the model, a fundamental 19 problem i s the d e c i s i o n r u l e f o r a f i r m under u n c e r t a i n t y . A r e s u l t such as Lemma 5.5 i s i m p o r t a n t because i t says t h a t t h e f i r m may maximize e x p e c t e d p r o f i t s i f the economy i s " l a r g e " and t h i s d e c i s i o n c r i t e r i o n w i l l c l o s e l y approximate the n e o c l a s s i c a l c r i t e r i o n o f ma x i m i z i n g market v a l u e . The f i r m may thus behave i n a r i s k n e u t r a l manner w i t h o u t a f f e c t -i n g i t s market v a l u e t o a s i g n i f i c a n t degree. 6. Cone!usions A number o f c o n c l u s i o n s were o b t a i n e d about the n a t u r e o f t h e e q u i l -i b r i u m o f the w e l l known c a p i t a l a s s e t p r i c i n g model which i s based on mean-variance p o r t f o l i o a n a l y s i s . In o r d e r t o c a r r y o u t the a n a l y s i s we assumed the e x i s t e n c e o f an e q u i l i b r i u m and c o n s i d e r e d e x p l i c i t l y the d e t e r m i n a t i o n o f e q u i l i b r i u m a s s e t p r i c e s . T h i s was a c c o m p l i s h e d by d e r i v i n g t h e agg r e g a t e e x c e s s demand f u n c t i o n s f o r r i s k and e x p e c t e d - 39 -r e t u r n w h ich, t o g e t h e r w i t h a m a t r i x e q u a t i o n d e r i v e d from the c a p i t a l -a s s e t p r i c i n g e q u a t i o n , gave a complete d e t e r m i n a t i o n o f e q u i l i b r i u m p r i c e s . Some c o n c l u s i o n s were d e r i v e d about the s t r u c t u r e o f the s e t o f e q u i l i b r i u m p r i c e s . In p a r t i c u l a r , assuming t h a t the p r i c e o f the r i s k -l e s s a s s e t was n o r m a l i z e d t o u n i t y , we have found t h a t the e q u i l i b r i u m p r i c e v e c t o r was not equal t o the v e c t o r o f e x p e c t e d a s s e t r e t u r n s . I f a f u r t h e r assumption i s made t o t h e e f f e c t t h a t a l l a s s e t s c o n t r i b u t e p o s i t i v e l y a t t h e margin t o a g g r e g a t e r i s k , then t he e q u i l i b r i u m p r i c e s e t i s bounded ( s t r i c t l y ) from above by the v e c t o r o f ex p e c t e d a s s e t r e t u r n s . To p r o c e e d w i t h the c o m p a r a t i v e s t a t i c s , we have assumed a l l i n v e s t o r s have p r e f e r e n c e s from a p a r t i c u l a r c l a s s , where the p r e f e r e n c e s a r e such t h a t t he m a r g i n a l r a t e o f s u b s t i t u t i o n between r e t u r n and r i s k i s ind e p e n -dent o f t h e l e v e l o f r e t u r n . T h i s i s a c o n v e n i e n t g e n e r a l i z a t i o n o f the c o n s t a n t a b s o l u t e r i s k a v e r s i o n u t i l i t y f u n c t i o n , The c o m p a r a t i v e s t a t i c r e s u l t s f o r the model were q u i t e s a t i s f a c t o r y i n the sense t h a t they were i n t u i t i v e l y p l a u s i b l e . Changes i n t h e mean r e t u r n . v e c t o r produces e q u a l , a b s o l u t e p e r c e n t a g e changes i n the c o r r e s p o n d i n g a s s e t p r i c e s , and a change i n , f o r example, the j t n mean a f f e c t s o n l y the j t h p r i c e l e a v i n g a l l o t h e r p r i c e s u n a f f e c t e d . A p o s i t i v e i n c r e a s e i n the v a r i a n c e o f a n . a s s e t i n d u c e s a r e d u c t i o n i n t h e p r i c e o f t h e same a s s e t . A l l o t h e r a s s e t p r i c e s a r e a f f e c t e d i n the same q u a l i t a t i v e manner, however the s i g n o f the q u a l i t a t i v e e f f e c t i s ambiguous. An i n c r e a s e i n the c o v a r i a n c e between two a s s e t s d e c r e a s e s t h e sum o f the two a s s e t p r i c e s , and a g a i n i n d u c e s the same q u a l i t a t i v e but lambiguous e f f e c t on a l l o t h e r a s s e t p r i c e s . - 40 -F i n a l l y we c o n s i d e r e d the e f f e c t s o f i n c r e a s i n g t he number o f i n v e s -t o r s i n the model, by examining a sequence o f economies w i t h the number o f i n v e s t o r s i n c r e a s i n g i n such a way t h a t the sequence converges to an economy h a v i n g a f i n i t e a g g r e g a t e endowment o f a s s e t s . I t was shown t h a t as the number o f i n v e s t o r s tends t o i n f i n i t y , e q u i l i b r i u m a s s e t p r i c e s t e n d t o the e x p e c t e d r e t u r n s o f a s s e t s . Thus, i n l a r g e but f i n i t e econo-mies r i s k has a n e g l i g i b l e e f f e c t on e q u i l i b r i u m p r i c e s . The r e s u l t s i n d i c a t e t he power o f the mean-variance assumption, even i n a g e n e r a l e q u i l i b r i u m c o n t e x t . An i m p o r t a n t q u e s t i o n as t o the r o b u s t -ness o f t h e s e r e s u l t s remains t o be answered. Whether s i m i l a r q u a l i t a t i v e r e s u l t s h o l d i n a more g e n e r a l e x p e c t e d u t i l i t y framework w i t h o u t the s t r o n g s t o c h a s t i c assumption o f n o r m a l i t y i s an open q u e s t i o n . To the e x t e n t t h a t the mean-variance model i s , however, a good a p p r o x i m a t i o n t o an a r b i t r a r y c a p i t a l market model, the compa r a t i v e s t a t i c r e s u l t s d e r i v e d can be viewed i n a s i m i l a r approximate f a s h i o n . - 41 -F o o t n o t e s 1. The c o m p a r a t i v e s t a t i c s o f the p o r t f o l i o h o l d i n g s o f i n d i v i d u a l s have been i n v e s t i g a t e d by Bierwag and Grove [ 1 9 6 8 ] , Jones - Lee [ 1 9 7 1 ] , and Royama and Hamada [1967] . 2. For a r e v i e w o f the e m p i r i c a l l i t e r a t u r e and t h e o r e t i c a l c o n t r i b u t i o n s t o t h i s model see J e n s e n [1972]. T h i s paper a l s o c o n t a i n s a b i b l i o g -raphy o f the numerous a p p l i c a t i o n s o f the c a p i t a l a s s e t p r i c i n g model. 3. I b i d . 4. Arrow and Hahn [ 1 9 7 1 ] , C h a p t e r 10 d i s c u s s the c o m p a r a t i v e s t a t i c s o f g e n e r a l e q u i l i b r i u m models, and the r e s u l t s a v a i l a b l e t o d a t e . 5. The model d i f f e r s somewhat from the c o n v e n t i o n a l c a p i t a l a s s e t p r i c i n g model i n t h a t we do n o t assume an i n f i n i t e l y e l a s t i c s u p p l y o f the s a f e a s s e t a t a f i x e d p r i c e . In the model which f o l l o w s the s u p p l y o f the s a f e a s s e t i s f i x e d and i t s p r i c e i s d e t e r m i n e d i n t h e g e n e r a l e q u i l i b r i u m o f the economy: 6. Our v e c t o r n o t a t i o n i s as f o l l o w s . For x, y e R n , x » y i f and o n l y i f x.j > yn- f o r a l l i = 1, n; x > y i f and o n l y i f x.. _> y.. f o r a l l i = 1, ..., n and f o r a t l e a s t one j e l l , 2 , ..., n}, x. > y •; x >^  y i f and o n l y i f x^ _> y^ f o r a l l i = 1, ..., n. 7. T h i s i s a r e g u l a r i t y assumption on the u t i l i t y f u n c t i o n which ensures an i n t e r i o r s o l u t i o n t o the i n d i v i d u a l ' s u t i l i t y m a x i m i z a t i o n problem. 8. Assumption R.6., which r e q u i r e s t h a t each i n d i v i d u a l endowment v e c t o r be s t r i c t l y p o s i t i v e , i s s t r o n g e r than n e c e s s a r y . A c t u a l l y , f o r the purpose o f the a n a l y s i s undertaken h e r e , a l l t h a t i s r e q u i r e d i s t h a t the endowment income o f each i n v e s t o r be s t r i c t l y p o s i t i v e a t e q u i l -i b r i u m p r i c e s . Assumption R.7 which i m p l i e s t h a t a g g r e g a t e a s s e t - 42 -endowments are equal t o u n i t y f o r each a s s e t i s m e r e l y a c o n v e n i e n c e and i t makes no s u b s t a n t i a l d i f f e r e n c e t o the a n a l y s i s . 9. The n e c e s s i t y o f a d m i t t i n g n e g a t i v e p r i c e s when.the r e t u r n s from an a s s e t may be n e g a t i v e , as i n the case o f n o r m a l l y d i s t r i b u t e d r e t u r n s , i s n o ted by H a r t [ 1 9 7 4 ] , Appendix 2. 10. See H a r t [1974], Theorem 2.2 and Appendix 2. 11. There i s an a d d i t i o n a l problem i n t h a t a f i n i t e s o l u t i o n may. e x i s t which i s n o t i n t e r i o r t o ( r , v ) s pace. However, i t can be shown t h a t p r o v i d e d q « y , and g i v e n the r e g u l a r i t y c o n d i t i o n s R.3 on the u t i l i t y f u n c t i o n , the o p t i m a l s o l u t i o n i s always i n t e r i o r t o ( r , v ) s p ace. Theorem 4 . 1 b e l o w a s s u r e s us t h a t , under c e r t a i n c o n d i t i o n s , f o r q e E, q « y. 12. T h i s f a c t i s w e l l known and can be e a s i l y demonstrated i n the mean-v a r i a n c e diagram. See T o b i n [1958]. 13. See Arrow and Hahn [1971] pp. 207. 14. The b a s i c d i f f i c u l t y i n r e l a x i n g t h i s assumption (A.8) and a t t e m p t i n g t o do c o m p a r a t i v e s t a t i c s i s t h a t the e x c e s s demand f u n c t i o n s f o r r i s k and r e t u r n do not decompose as i n (5.3) and (5.4) below. Gener-a l l y we s h o u l d have Z r(Tr,y,V;w 0). Comparative s t a t i c s i n t h i s g e n e r a l case a r e e x c e e d i n g l y c o m p l i c a t e d and do not g i v e unambiguous r e s u l t s . h' 8 f r ( ^ h' 8 f v ^ 15. f ^ denotes — ^ — and f y denotes — ^ . The s t r i c t p o s i t i v i t y o f t h e s e d e r i v a t i v e s f o l l o w s from A.8 and R.3. 16. We s h a l l assume t h r o u g h o u t t h i s s e c t i o n t h a t the e q u i l i b r i u m p r i c e q* i s a c o n t i n u o u s l y d i f f e r e n t i a t e f u n c t i o n o f the parameters ( y , V ) . 17. A c t u a l l y , we need i n a d d i t i o n t h a t i'w^ > 0. As T T * ( i ' V i ) 2 = i'w* > 0 - 43 -and w Q . i s a p o s i t i v e m u l t i p l e o f w*, t h i s c o n d i t i o n h o l d s . 18. In showing A'-A > 0 we have assumed i' w Q > 0. As T T * ( i' V i ) 2 = i'w* > 0 and w Q i s a p o s i t i v e m u l t i p l e o f w*, t h i s c o n d i t i o n i n f a c t h o l d s . 19.V. T h i s problem i s d i s c u s s e d i n . a number o f papers i n a . r e c e n t Symposium on the O p t i m a l i t y o f C o m p e t i t i v e C a p i t a l Markets [1974] . - 44 -Re f e r e n c e s Arrow, K. J . and L i n d , R. C. [ 1 9 7 0 ] , " U n c e r t a i n t y and the E v a l u a t i o n o f P u b l i c Investment D e c i s i o n s " , American Economic Review, 60, 364-78. and Hahn, F. H. [1971], General Competitive Analysis, San F r a n c i s c o : Holden-Day. B e r t s e k a s , D. [19 7 4 ] , "Necessary and S u f f i c i e n t C o n d i t i o n s f o r E x i s t e n c e o f an Optimal P o r t f o l i o " , Journal of Economic Theory, 8, 235-47. Bierwag, G. 0. and Grove, M. A. [1968], " S l u t s k y E q u a t i o n s f o r A s s e t s " , Journal of Political Economy, 76, 114-27. H a r t , 0. D. [19 7 4 ] , "On the E x i s t e n c e o f E q u i l i b r i u m i n a S e c u r i t i e s Model", Journal of Economic Theory, 9, 293-311. J e n s e n , M. C. £ 1 9 7 2 ] , " C a p i t a l markets: t h e o r y and e v i d e n c e " , Bell Journal of Economics and Management Science, 3, 357-98. J o n e s - L e e , M. W. [19 7 1 ] , "Some P o r t f o l i o Adjustment Theorems f o r the Case o f N o n - N e g a t i v i t y C o n s t r a i n t s on S e c u r i t y H o l d i n g s " , Journal of Finance, 26, 763-75. K a r l i n , S. [1959], Mathematical Methods and Theory in Games, Programming and Economics, V o l . I, Reading, Mass.: Addison-Wesley. Lang, S. [1966], Linear Algebra, Reading, Mass.: Addison-Wesley. L i n t n e r , J . [1965], "The V a l u a t i o n o f R i s k A s s e t s and the S e l e c t i o n o f R i s k y Investments i n Stock P o r t f o l i o s and C a p i t a l Budgets", The Review of Economics and Statistics, 47, 13-37. M o s s i n , J . [1966], " E q u i l i b r i u m i n a C a p i t a l A s s e t Market", Econometrica, 34, 768-83. Royama, S. and Hamada, K. [1967], " S u b s t i t u t i o n and Complementarity i n the C h o i c e o f R i s k y A s s e t s " , i n D. H e s t e r and J . T o b i n ( e d s . ) , Risk Aversion and Portfolio Choice, New York: W i l e y . Sharpe, W. F. [19 6 4 ] , " C a p i t a l A s s e t P r i c e s : A Theory o f Market E q u i l i b -rium Under C o n d i t i o n s o f R i s k " , Journal of Finance, 19, 425-42. Symposium on the O p t i m a l i t y o f C o m p e t i t i v e C a p i t a l Markets [1974], The Bell Journal of Economics.and Management Science, 5, 125-85. T o b i n , J . [ 1 9 5 8 ] , " L i q u i d i t y P r e f e r e n c e as Be h a v i o u r towards R i s k " , Review of Economic Studies, 25, 65-85. - 45 -C h a p t e r I I I RISK BEARING, INVESTMENT AND FINANCING  IN STOCK MARKETS 1. I n t r o d u c t i o n R e c e n t l y a number o f c o n t r i b u t i o n s have been made t o the a n a l y s i s o f p r o d u c t i o n d e c i s i o n s under u n c e r t a i n t y . They have a l l been made w i t h i n the c o n t e x t o f a t w o - p e r i o d g e n e r a l e q u i l i b r i u m model under u n c e r t a i n t y i n which the household s e c t o r i s assumed t o make p o r t f o l i o d e c i s i o n s on t h e h o l d i n g o f c l a i m s a g a i n s t d i f f e r e n t f i r m s ' f u t u r e output.''' T h i s t y p e o f economy, which we s h a l l d i s t i n g u i s h by the term 'stock market' economy, i s c h a r a c t e r i z e d by the f e a t u r e t h a t markets a r e assumed t o be i n c o m p l e t e 2 i n the Arrow-Debreu se n s e ; t h a t i s , the number o f c o n t r a c t s t r a d e d i n the economy i s fewer than the number o f s t a t e s o f n a t u r e . T h i s c h a r a c t e r i s t i c has t h e e f f e c t o f l i n k i n g t h e p r o d u c t i o n o r r e a l i n v e s t m e n t d e c i s i o n s o f f i r m s t o the p o r t f o l i o o r f i n a n c i a l d e c i s i o n s o f h o u s e h o l d s . S i n c e the ' e q u i t y s h a r e s ' o f the d i f f e r e n t f i r m s a r e t r a d e d on the s t o c k market by the households a t t e m p t i n g t o a c h i e v e an o p t i m a l p o r t f o l i o b a l a n c e , the market v a l u e o f the f i r m , o r the t o t a l v a l u e o f i t s e q u i t y , becomes a p r i c e d e t e r m i n e d i n the g e n e r a l e q u i l i b r i u m o f the economy. These c o n t r i b u t i o n s t o the s t o c k market economy have c o n c e n t r a t e d s o l e l y upon the r i s k - b e a r i n g f u n c t i o n o f the s t o c k market and on the a l l o -c a t i v e s i g n i f i c a n c e o f a s s e t p r i c e s i n making e f f i c i e n t p r o d u c t i o n d e c i s -i o n s . The c o m p l i c a t i o n s we c o n s i d e r i n t h i s paper a r i s e from a t t e m p t i n g 3 t o i n t e g r a t e the r i s k - b e a r i n g f u n c t i o n s o f t h e c a p i t a l market w i t h i t s more t r a d i t i o n a l r o l e as an i n t e r m e d i a r y i n the s a v i n g s - i n v e s t m e n t p r o c e s s . - 46 -In t h e e x i s t i n g l i t e r a t u r e , two q u e s t i o n s have been asked o f t h i s model. The f i r s t i s s u p p o s i n g t h a t f i r m s make t h e i r p r o d u c t i o n - i n v e s t m e n t d e c i s i o n s such as to maximize the market v a l u e o f the f i r m , would the r e -4 s u i t i n g e q u i l i b r i u m a l l o c a t i o n o f investment be P a r e t o e f f i c i e n t ? The r e a s o n market v a l u e m a x i m i z a t i o n i s p o s t u l a t e d as the f i r m ' s d e c i s i o n r u l e i s t h a t i t c o r r e s p o n d s c l o s e l y t o the n e o - c l a s s i c a l c r i t e r i o n o f m aximizing p r o f i t s under c e r t a i n t y , and has been presumed i n the c o r p o r a t e f i n a n c e l i t e r a t u r e t o be the o n l y goal o f the f i r m g e n e r a l l y c o n s i s t e n t w i t h the owners' i n t e r e s t s . I n t e r e s t i n g l y enough, the answer t o t h i s q u e s t i o n has g e n e r a l l y been i n the n e g a t i v e ; t h a t i s , f i r m s a c t i n g on t h i s c r i t e r i o n 5 make i n e f f i c i e n t p r o d u c t i o n d e c i s i o n s . The n a t u r a l q u e s t i o n which comes out o f t h i s n e g a t i v e c o n c l u s i o n i s whether o r n o t , i n the c o n t e x t o f the s t o c k market economy, t h e r e e x i s t p r o d u c t i o n d e c i s i o n s which would be unanimously approved by a l l s h a r e h o l d e r s o f the f i r m . The c o n d i t i o n s under which such 'unanimity theorems' a r e t r u e t u r n out to be r a t h e r s t r i n g e n t , and one would not e x p e c t t h e s e c o n d i t i o n s t o h o l d i n g e n e r a l . King [1974] has, i n any c a s e , s e v e r e l y c r i t i c i z e d the u s e f u l n e s s o r r e a l -i s t i c n a t u r e o f the c o n c e p t o f u n a n i m i t y , independent o f the c o n d i t i o n s under which i t may o r may not h o l d . The g e n e r a l f e e l i n g one has a f t e r examining t h i s l i t e r a t u r e i s t h a t the c r i t e r i o n which the f i r m s h o u l d use i n making i t s p r o d u c t i o n - i n v e s t m e n t d e c i s i o n s such as t o be c o n s i s t e n t w i t h i t s owners:' i n t e r e s t s i s v e r y much u n s e t t l e d . I t would seem t h a t the d e c i s i o n r u l e s which f i r m s a c t u a l l y use may i n f a c t be q u i t e independent o f any economy-wide e f f i c i e n c y c o n s i d e r -a t i o n s o r may bear no c l o s e r e l a t i o n t o the p r e f e r e n c e s o f t h e i r s t o c k -h o l d e r s . As i s w e l l known, the number o f d e c i s i o n r u l e s proposed as ex-p l a i n i n g f i r m s ' b e h a v i o u r i s q u i t e l a r g e . Under u n c e r t a i n t y the problem - 47 -i s compounded- i n t h a t the f i r m must t a k e an a t t i t u d e towards r i s k and form e x p e c t a t i o n s r e g a r d i n g the p r o b a b i l i t y o f f u t u r e e v e n t s . The d i v e r s i o n o f i n t e r e s t s between management and ownership i s an even g r e a t e r p o s s i b i l i t y under u n c e r t a i n t y . D i f f e r e n t f i r m s may, o f c o u r s e , have d i f f e r e n t a t t i -t i d e s towards r i s k and d i f f e r e n t p r o b a b i l i t y b e l i e f s , a l l l e a d i n g t o t h e l i k e l i h o o d t h a t d i f f e r e n t f i r m s w i l l choose t h e i r p r o d u c t i o n - i n v e s t m e n t d e c i s i o n s based on d i f f e r e n t o b j e c t i v e s . The r e s u l t i n g d i v e r s i t y o f d e c i s i o n r u l e s may be the norm r a t h e r than the e x c e p t i o n i n an economy w i t h s i g n i f i c a n t u n c e r t a i n t i e s . I t appears we a r e l e f t w i t h the i m p r e s s i o n t h a t f i r m s have s u b s t a n t -i a l independence i n t h e i r d e c i s i o n r u l e s whence the c o n n e c t i o n between the p o r t f o l i o d e c i s i o n s o f the household s e c t o r and the ' r e a l ' i n v e s t m e n t d e c i s i o n s o f the f i r m s i n the economy has been s u b s t a n t i a l l y weakened, i f not t o t a l l y e l i m i n a t e d , a t the f i r m d e c i s i o n l e v e l . What we propose t o argue i s t h a t such independence does not i n f a c t e x i s t and t h a t the c a p i t a l market o f the economy f o r c e s some v e r y r e a l c o n s t r a i n t s upon the f i r m . The r e a s o n , o u r a n a l y s i s d i f f e r s from p r e v i o u s ones i s t h a t i n t h e i r con-s i d e r a t i o n o f the s t o c k market economy a l l p r e v i o u s a u t h o r s have i g n o r e d a b a s i c c o n s t r a i n t i n a n a l y z i n g the i n v e s t m e n t d e c i s i o n . 7 The c o n s t r a i n t i s a budget i n e q u a l i t y i n the f i r s t - p e r i o d which r e q u i r e s t h a t the c o s t o f the i n v e s t m e n t p r o j e c t s undertaken by the f i r m cannot exceed the v a l u e o f i n v e s t m e n t funds which must be r a i s e d i n the c a p i t a l market. T h i s e s s e n t i a l i n s t i t u t i o n a l o r market c o n s t r a i n t b i n d s the f i r m ' s b e h a v i o u r , j u s t as the t e c h n o l o g y o f the f i r m i s a c o n s t r a i n t t o be taken i n t o a c c o u n t . Thus the c a p i t a l market o f t h e economy f o r c e s a b a s i c budget c o n s t r a i n t upon t h e f i r m , and the ' r e a l ' d e c i s i o n s o f f i r m s become c l o s e l y r e l a t e d t o the ' f i n a n c i a l ' d e c i s i o n s o f h o u s e h o l d s , r e g a r d l e s s o f whether the f i r m - 48 -d e s i r e s t o a c t i n d e p e n d e n t l y o f i t s s h a r e h o l d e r s o r i n t h e i r i n t e r e s t . We s h a l l show t h a t c o n s i d e r a t i o n o f t h i s b a s i c c o n s t r a i n t has some i n t e r -e s t i n g i m p l i c a t i o n s f o r the e x i s t e n c e o f e q u i l i b r i u m , the c l a s s o f admis-s a b l e d e c i s i o n r u l e s f o r the f i r m , and the problem o f making the f i n a n c i n g d e c i s i o n . The subsequent s e c t i o n s a r e as f o l l o w s . In s e c t i o n 2 we d e v e l o p the b a s i c model and the c o n c e p t o f the f i n a n c i a l - e q u i l i b r i u m c o r r e s p o n d e n c e . In s e c t i o n 3 we i n t r o d u c e the b a s i c c o n s t r a i n t and examine the p r o p e r t y o f f i n a n c i a l c o n s i s t e n c y i n the i n v e s t m e n t - f i n a n c i n g d e c i s i o n o f the f i r m . The next s e c t i o n c o n s i d e r s two examples which demonstrate the non-vacuous n a t u r e o f the c a p i t a l market c o n s t r a i n t on i n v e s t m e n t c h o i c e s . In s e c t i o n 5 we c o n s i d e r the i m p l i c a t i o n s o f the c a p i t a l - m a r k e t c o n s t r a i n t f o r f i n a n -c i a l d e c i s i o n s and the problem o f the c a p i t a l s t r u c t u r e o f the f i r m when 0 t h e r e e x i s t both bonds and e q u i t i e s i n the model. The l a s t s e c t i o n g i v e s an e x p l a n a t i o n o f how the c o n s t r a i n t was c i r c u m v e n t e d i n p r e v i o u s models o f the s t o c k market economy and f i n a l l y a summary o f t h i s e s s a y ' s f i n d i n g s . 2. The B a s i c Model In t h i s s e c t i o n we s h a l l c o n s t r u c t the b a s i c s t o c k market e q u i l i b r i u m model and, i n p a r t i c u l a r , the c o n c e p t o f the f i n a n c i a l - e q u i l i b r i u m c o r r e -ct spondence. The model i s a t w o - p e r i o d model w i t h t e c h n o l o g i c a l u n c e r t a i n t y o c c u r r i n g i n the second p e r i o d . In the f i r s t p e r i o d f i r m s purchase a s i n g l e homogeneous i n v e s t m e n t good from the h o u s e h o l d s . The households i n t u r n both t r a d e t h e i r e x i s t i n g s h a r e s o f the f i r m s ' e q u i t i e s and r e c e i v e new e q u i t y s h a r e s i n exchange f o r s e . l l i n g the f i r m s t h e i r endowment o f the homogeneous i n v e s t m e n t good.' The d i f f e r e n t f i r m s , which we assume t o be f i x e d i n number, a l l produce a s i n g l e homogeneous consumption good i n the - 49 -second p e r i o d , the amount o f the good produced depending upon both the i n v e s t m e n t made i n the f i r s t p e r i o d and the s t a t e o f n a t u r e i n the second p e r i o d . The consumption good i s d i s t r i b u t e d t o s h a r e h o l d e r s i n p r o p o r t i o n t o the p e r c e n t a g e share o f the f i r m t h e y p urchased i n the f i r s t p e r i o d . The e q u i l i b r i u m i n t h i s model r e f e r s t o the e q u i l i b r i u m o f the p o r t f o l i o d e c i s i o n s o f households and i n v e s t m e n t d e c i s i o n s o f f i r m s which o c c u r s i n the f i r s t p e r i o d . In the second p e r i o d no t r a d e t a k e s p l a c e , p r o d u c t i o n and d i s t r i b u t i o n o f the consumption good b e i n g the o n l y a c t i v i t i e s under-t a k e n . I t i s c o n v e n i e n t i n t h i s model t o a n a l y z e the f i n a n c i a l e q u i l i b r i u m , s e p a r a t e l y from the t o t a l e q u i l i b r i u m o f the economy. The b a s i c i d e a i s t h a t t h e market i s o r g a n i z e d i n a s e q u e n t i a l f a s h i o n . Firms announce t h e i r i n v e s t m e n t d e c i s i o n s and t h e i r o f f e r i n g s o f new e q u i t y . Once the i n v e s t m e n t d e c i s i o n o f a f i r m i s known, i t s p a t t e r n o f next p e r i o d ' s o u t -9 put a c r o s s s t a t e s i s known, and, g i v e n i t s announced o f f e r i n g o f new e q u i t y , each i n v e s t o r knows h i s p e r c e n t a g e endowment o f the t o t a l o u t s t a n d -i n g s t o c k o f s h a r e s . Each i n d i v i d u a l , t h e n , knows h i s budget c o n s t r a i n t and can make the n e c e s s a r y a djustments i n h i s p o r t f o l i o . A f i n a n c i a l e q u i l i b r i u m i s c h a r a c t e r i z e d by a v e c t o r o f market v a l u e s , one on each f i r m , and a s e t o f e q u i l i b r i u m p o r t f o l i o h o l d i n g s , one f o r each i n d i v i d u a l , such t h a t t h e s u p p l y o f e q u i t i e s e q u a l s the demand. Note t h a t a f i n a n c i a l e q u i l i b r i u m i s c o n d i t i o n a l upon the announced i n v e s t m e n t and f i n a n c i n g d e c i s i o n s o f f i r m s . I t i s i m p o r t a n t t o r e a l i z e t h a t a f i n a n c i a l e q u i l i b -r i um does not imply t o t a l e q u i l i b r i u m f o r the economy. Fur t h e r m o r e , the n o t i o n t h a t the market i s o r g a n i z e d i n a s e q u e n t i a l f a s h i o n such t h a t f i n a n c i a l markets c l e a r f i r s t i s o n l y a c o n v e n i e n t a n a l y t i c a l d e v i c e and i s not t o be t a k e n as an a c t u a l d e s c r i p t i o n o f the way a s t o c k market - 50 -economy f u n c t i o n s . In a c t u a l f a c t t o t a l e q u i l i b r i u m must be reached s i m u l -t a n e o u s l y on a l l markets.. T h i s p o i n t w i l l be o f p a r t i c u l a r importance i n S e c t i o n 4 below. We i n t r o d u c e now the n o t a t i o n and b a s i c assumptions o f the model. There a r e J+1 f i r m s i n the economy, indexed w i t h s u b s c r i p t s j=0, 1, J . Each f i r m uses the i n p u t o f a s i n g l e homogeneous i n v e s t m e n t good i n the f i r s t p e r i o d t o produce a homogeneous o u t p u t , a f t e r the r e a l i z a t i o n o f t h e s t a t e o f n a t u r e , i n the second p e r i o d . Inputs a r e denoted by x., o u t p u t s by y.:, and the s t a t e o f n a t u r e v a r i a b l e by e , where 6 i s an element o f a measurable space (Q, E ). Here E denotes the B o r e l f i e l d o f e v e n t s i n tt. Note t h a t s i n c e no t r a d e o c c u r s i n the second p e r i o d the o n l y u n c e r t a i n t y which i s p r e s e n t i s t e c h n o l o g i c a l u n c e r t a i n t y ; t h a t i s , the r e a l i z a t i o n o f the v a r i a b l e 6 which cannot be o b s e r v e d u n t i l the second p e r i o d . To each f i r m j , t h e r e i s a t e c h n o l o g y f,-(x-, 9) which g i v e s the o u t p u t y . i n the second p e r i o d , g i v e n t h a t x. was chosen i n the f i r s t p e r i o d , and t h a t J e v e n t 9 o b t a i n s i n the second p e r i o d . We assume t h a t f\. s a t i s f i e s the f o l l o w i n g : A . l . f , : R + X Q + R +. A.2. f . i s t w i c e d i f f e r e n t ! ' a b l e , s t r i c t l y monotone i n c r e a s i n g and J s t r i c t l y concave i n x.. J A.3. l i m fH :x,,9) = + oo, f.(0,9) = 0, f o r a l l 9 e Q. x.->0 J J J 3 A.4. f , i s a measurable f u n c t i o n i n ft.10 Note t h a t assumption A.3. i m p l i e s t h a t i t i s n e c e s s a r y and s u f f i c i e n t t h a t a f i r m make some p o s i t i v e i n v e s t m e n t i n o r d e r t o produce some p o s i t i v e o u t p u t i n a l l s t a t e s o f n a t u r e . The c o n c a v i t y o f f . i m p l i e s t h e r e i s some - 51 -o l d c a p i t a l s t o c k on which the f i r m i s e a r n i n g r e n t s , but the o l d c a p i t a l s t o c k i s taken t o be i m p e r a t i v e w i t h o u t some new i n v e s t m e n t . ' The v e c t o r f u n c t i o n f ( x , e ) , maps R+ + 1 X Q, i n t o R^ + 1 t a k i n g on v a l u e s [ f (x ; 8 ) , f j ( x , 0 ) ] i n R^ + 1; i . e . , the J+1 v e c t o r o f the f i r m s ' o u t -p u t s . The f i r m , o f c o u r s e , i s c h a r a c t e r i z e d by more than j u s t a t e c h n o l o g y and we s h a l l r e t u r n t o t h i s . p o i n t below. The i n v e s t o r s o r households o f the economy a r e assumed t o se-11 t h e i r f i r s t p e r i o d endowments o f the i n v e s t m e n t good p l u s t h e i r . e x i s t i n g owner-s h i p c l a i m s to f i r m s i n o r d e r to purchase a d e s i r e d p o r t f o l i o o f e q u i t i e s , which i s d e t e r m i n e d by m a x i m i z i n g the e x p e c t e d u t i l i t y o f second p e r i o d consumption. I n v e s t o r s are i ndexed by i , where i runs from 1 t o I. We make the f o l l o w i n g assumptions on i n v e s t o r s p r e f e r e n c e s . B . l . Each i n v e s t o r has a s t r i c t l y concave, n o n - d e c r e a s i n g , t w i c e d i f f e r e n t i a t e u t i l i t y f u n c t i o n o f consumption, u . f c . . ) . 1 1 B.2. l i m u ' . ( c ) = + °°. c.-K) 1 1 B.3. u. i s bounded. B.4. Each i n v e s t o r has a s u b j e c t i v e p r o b a b i l i t y measure u. on the s t a t e space ( f i , E ) . Thus each i n v e s t o r ' s e x p e c t a t i o n s a r e c h a r a c t e r i z e d by a p r o b a b i l i t y space (ft,E,u.).. Each i n v e s t o r has an endowment e^ o f the i n v e s t m e n t good i n the f i r s t p e r i o d . We assume B.5. e i > 0 f o r a l l i = l , I. Each i n v e s t o r i , has an i n i t i a l h o l d i n g o f S.. o f s h a r e s o f f i r m j , I _ T J where £ S.. = S. > 0. Each f i r m j , i n a d d i t i o n t o making an i n v e s t m e n t i = l • U J -d e c i s i o n as t o x., must make a f i n a n c i n g d e c i s i o n ' v i n terms o f the q u a n t i t y J o f new e q u i t i e s i t p l a n s t o o f f e r on the market. We s h a l l denote t h i s J - 52 -quantity of new equities by AS,, and we shall require without loss of 12 generality that AS, be non-negative. Thus, the total supply of equities of the jth firm is given by S. = S. + AS. > 0. We can then define I.. = J J J I J S-./S, as the percentage share of jth firm, i n i t i a l l y owned by the ith investor. Similarly we define d. = AS./S, ^  0, as the percentage share J J J of the jth firm, which the jth firm i t s e l f offers in new issues of stock. From the above definitions we have that I Z s\. + d. = 1. (2.1) i=l 1 J J The total market value of the jth firm is denoted by V.; i.e., V. is 3 3 the total value of the outstanding equities S, in the jth firm. We can J now define the budget set' of feasible portfolios of the ith investor. Let V = (V Q, Vj)' e R J + 1 denote the vector of the market values of 13 -the firms in the economy, and s. = (s. , s,,)' denote the vector 1 1 0 1 J of i n i t i a l percentage, ownership shares held by the ith investor. D.l. The budget set of the ith investor is B^V.w.d) = ' {s i | s i e R J + 1; V ' s i < V s . + we.}. Note that since i . depends upon the firms' announcement of AS'.s, the "l • . i 3 budget set depends upon this set of parameters and consequently upon d, the J+1 vector (d Q, dj)'. The price of the first-period investment good is denoted by w. Often we shall normalize w to unity, as we are allowed one price normalization in the model by virtue of homogeneity. Note also that the budget set is unbounded as short sales are allowed, and this allows certain d i f f i c u l t i e s as to the existence of optimal portfolios. We wish now to characterize the investors asset demand functions. - 53 -B e f o r e ' p r o c e e d i n g however we s h a l l make two f u r t h e r assumptions on the n a t u r e o f t h e economy. A. 5. There e x i s t s a ' r i s k l e s s ' f i r m i n the economy, i n the sense t h a t f (x; 9) i s independent o f the r e a l i z a t i o n o f 9 e fl. 15 T h i s f i r m s h a l l be denoted w i t h the s u b s c r i p t 0. B. 6. For each i n v e s t o r i = l , I , u.{9 e fl|f.(x.;0) > 0, f o r any x, > 0} > 0. J B.6. i m p l i e s t h a t i f any f i r m u n d e r t a k e s p o s i t i v e i n v e s t m e n t , each i n d i v -i d u a l a s s i g n s some p o s i t i v e p r o b a b i l i t y t o the s e t o f s t a t e s i n which the f i r m produces some p o s i t i v e o u t p u t . We c o u l d r e p l a c e B.6 by a s t r o n g e r a s s u m p t i o n , t h a t each p r o b a b i l i t y measure u. was 'atomless' which would r e q u i r e i n c o n j u n c t i o n w i t h A.3 t h a t the above i n e q u a l i t y be r e p l a c e d w i t h y,{0 e fl | f.(x . ; 0 ) > 0 f o r any x. > 0} = 1, but t h i s t u r n s out not to be n e c e s s a r y . Note t h a t g i v e n the v e c t o r f u n c t i o n f ( x , e ) , and s., the i n d i v i d u a l ' s p o r t f o l i o h o l d i n g s , h i s consumption i f the s t a t e 0 o b t a i n s i s g i v e n by c..(s.;x;0) = s ^ x ; © ) . Thus the i t h i n v e s t o r has the f o l l o w i n g problem max E y [ u ^ c . ) ] = Oc. ( s i ; x , 0 ) ] d y i  S i 1 s u b j e c t t o s. e B ^ V . w j d ) . (2.2) Here E denotes the e x p e c t a t i o n o p e r a t o r w i t h r e s p e c t t o the p r o b a b i l i t y measure u., and the i n t e g r a l i s taken t o be the usual Lesbegue i n t e g r a l . Now s i n c e B 1 was unbounded, a f i n i t e s o l u t i o n t b (2.2) may not always e x i s t . However, B e r s t e k a s [1974] has shown t h a t , g i v e n some r e a s o n a b l e c o n d i t i o n s on the u t i l i t y f u n c t i o n , a f i n i t e s o l u t i o n always e x i s t s . Thus we assume 54 -C l . Given t h a t (V,w) » Oj +2> a f i n i t e s o l u t i o n t o (2.2) e x i s t s f o r a l l i = l , ..., I . 1 6 L e t g 1 ^ . ;x) = E^ [u.. ( c , ) ] be the 'induced u t i l i t y f u n c t i o n ' o f p o r t -f o l i o h o l d i n g s c o n d i t i o n a l on the v e c t o r o f p r o d u c t i o n d e c i s i o n s x. We then have the f o l l o w i n g lemma. Lemma 2.1: G i v e n A.1-5, B. 1-7 the i n d u c e d u t i l i t y f u n c t i o n g 1(s.';x) has the f o l l o w i n g p r o p e r t i e s ; g ^ s ^ x ) i s 2.1a. concave i n s^ f o r f i x e d x; 2.1b. concave i n x f o r f i x e d s^; 2.1c. m o n o t o n i c a l l y i n c r e a s i n g i n each s,., i f x. > 0; 2.Id. m o n o t o n i c a l l y i n c r e a s i n g i n each x. i f s.. > 0 ; 2.1e. c o n t i n u o u s l y d i f f e r e n t i a b l e i n ( s . j , x ) , on i n t R ^ J + ^ . ( p r o o f ) : C o n c a v i t y f o l l o w s from the f a c t s t h a t i i . ( c ^ ) i s a concave f u n c t i o n , and s . j f ( x , 0 ) i s a l i n e a r f u n c t i o n i n s^ and a concave f u n c t i o n i n x, s i n c e each f . j(x.;0) i s concave. 2.1c. i s proved by n o t i n g t h a t i f x, > 0, then s , . f . ( x , ; 6 ) i s i n c r e a s i n g i n s.. f o r a l l 0 e tt such t h a t J I J J J IJ ^ f . ( x . ; 0 ) > 0. S i n c e y.{9 | f . ( x . ; 9 ) > 0, f o r any x, > 0} > 0 by B.6., i n e v a l u a t i n g u.j [s.|f(x,0)]du.. t h e r e i s a s e t o f p o s i t i v e measure i n tt on which f . ( x . , 0 ) t a k e s on p o s i t i v e v a l u e s . Thus the m o n o t o n i c i t y o f J J u,(c.) e n s u r e s m o n o t o n i c i t y i n s... S i m i l a r l y , m o n o t o n i c i t y i n x. f o l l o w s 3 3 u s i n g the monotone i n c r e a s i n g p r o p e r t y o f f . w i t h r e s p e c t t o x.. D i f f e r -3 3 e n t i a b i l i t y i s ensured by t h e d i f f e r e n t i a b i l i t y o f u ^ c ^ and s ! f ( x , 6 ) i n ( s . , x ) . Q.E.D. The i n v e s t o r ' s problem (2.2) can be e x p r e s s e d as max s_-g 1 ( i ;x) s u b j e c t t o s. e - B ^ V . v t f ) . (2.3) - 55 -\ From t h i s problem we d e r i v e the consumer's demand f u n c t i o n s f o r a s s e t s , which we denote by ' h](V,y.;x,d) j=0, J . (2.4) where y. denotes the income o f the i t h i n d i v i d u a l g i v e n by V s^ + we^. These demand f u n c t i o n s , g i v e n Lemma 2.1, can be shown t o have the usual p r o p e r t i e s . Lemma 2.2: F o r a l l i = l , I, g i v e n A.1-5, B.1-7 and C . l , then t h e demand f u n c t i o n s h 1 ( V , y . ;d,x) s a t i s f y the f o l l o w i n g p r o p e r t i e s : (1) Homogeneity o f degree z e r o i n ( V , y . ) . J . (2) Budget i d e n t i t y , £ V .hl(V,y.;d,x) = y . . j=0 1 (3) Symmetry and n e g a t i v e s e m i - d e f i n i t e n e s s o f the S l u t s k y m a t r i x S 1 = [ b ^ ] i K i dhl w h e r e b k £ = avf + h k 9 y ^ ' k > £ = 0 ' J-K 1 (4) J o i n t l y c o n t i n u o u s i n (d,x) on i n t R 2 ^ + ^ . ( p r o o f ) : The theorem f o l l o w s from the usual theorems i n n e o c l a s s i c a l consumer t h e o r y . 1 7 We can now d e f i n e the agg r e g a t e excess-demand f u n c t i o n f o r a s s e t s h a r e s as I Z(V,w;x,d) = [ z h n(V,y.;d,x) - s.] - d , (2.5) i = l 1 ^ 1 r e c a l l i n g t h a t d i s the J+1 v e c t o r o f p e r c e n t a g e s o f t o t a l e q u i t y o f f e r e d by f i r m s i n new i s s u e s . I t w i l l be c o n v e n i e n t t o n o r m a l i z e w=l, and t h i s i s c l e a r l y a d m i s s a b l e g i v e n the homogeneity p r o p e r t y o f the demand f u n c t i o n s . We can now d e f i n e a f i n a n c i a l e q u i l i b r i u m . D.2. h financial equilibrium, c o n d i t i o n a l upon a s e t o f p r o d u c t i o n - 56 -d e c i s i o n s x e R d + 1 , and a s e t o f f i n a n c i n g d e c i s i o n s d e R d + 1 , i s an 1+1 -t u p l e { s * , ..., s*; V*} such t h a t a) s* i s maximal w i t h r e s p e c t t o E^[u.(c..).] o v e r a l l s e t s^ e B (V , l ; d , x ) , f o r a l l i = l , I. b) V* e RJ++1 and c) z(V*;x,d) £ . 0 j + j . Thus a f i n a n c i a l e q u i l i b r i u m , c o r r e s p o n d s t o a s e t o f market v a l u e s on a l l f i r m s , and p o r t f o l i o d e c i s i o n s by i n v e s t o r s , such t h a t each i n v e s t o r ' s p o r t f o l i o c h o i c e maximizes h i s e x p e c t e d u t i l i t y and i s f e a s i b l e i n h i s budget s e t , and t h a t the market f o r f i n a n c i a l a s s e t s c l e a r s . We r e -emphasize a t t h i s p o i n t t h a t a f i n a n c i a l e q u i l i b r i u m does not i m p l y t o t a l e q u i l i b r i u m i n the economy, as the v e c t o r s (x,d) a r e so f a r r a t h e r a r b i -t r a r y . We s h a l l g e n e r a l l y assume C.2. G iven any v e c t o r (x,d) such t h a t x » 0 J + 1 and d >^  0 j + 1 a f i n a n c i a l e q u i l i b r i u m c o n d i t i o n a l on t h e s e v a l u e s e x i s t s . Note t h a t the assumption t h a t a f i n a n c i a l e q u i l i b r i u m e x i s t s i s much weaker than the assumption t h a t a g l o b a l e q u i l i b r i u m e x i s t s . In the f i n a n c i a l s e c t o r o f t h i s economy a v e r s i o n o f Walras'law i s s a t i s f i e d due t o the r e q u i r e m e n t t h a t households s a t i s f y t h e i r budget c o n s t r a i n t . Walras'law can be w r i t t e n as I J J Z Z V, z!(V;d,x) = z V.d.. (2.6) i = l j=0 J J j=G J J T h i s says s i m p l y t h a t the v a l u e o f t o t a l e q u i t i e s demanded by h o u s e h o l d s , i n e x c e s s o f t h e i r own e q u i t y h o l d i n g s , i s equal t o the v a l u e o f the t o t a l e q u i t i e s s u p p l i e d by the f i r m s . T h i s , o f c o u r s e , h o l d s r e g a r d l e s s o f - 57 -whether the economy i s f i n a n c i a l e q u i l i b r i u m o r n o t , i . e . , some s e c u r i t i e s may be i n e x c e s s s u p p l y and o t h e r s i n e x c e s s demand. I f the economy i s i n f i n a n c i a l e q u i l i b r i u m a much s t r o n g e r r e s u l t o b t a i n s . R e c a l l the i n d i v i -d u a l ' s budget e q u a t i o n , J J S Vii = E V u + er (2-6) j=0 J 1 J j=0 J 1 J 1 Adding t h e s e up a c r o s s i n d i v i d u a l s a t market c l e a r i n g p r i c e s and p o r t -f o l i o s we have t h a t J J I Z V* = Z V* (1-d.) + Z e,, (2.7) j=0 J j=0 J J i = l or upon r e - a r r a n g i n g J I Z d.V* = Z e.. (2.8) j=0 J 3 i = l 1 Now d.V* i s the funds o b t a i n e d by the j t h f i r m i n the s t o c k market a t 3 3 f i n a n c i a l e q u i l i b r i u m market v a l u e s , e^ i s the s u p p l y o f f i r s t p e r i o d i n v e s t m e n t goods by the i t h i n d i v i d u a l . T h e r e f o r e (2.8) i m p l i e s t h a t i n a f i n a n c i a l e q u i l i b r i u m the t o t a l demand f o r funds by the r e a l s e c t o r o r firmstwiMIl equal t h e t o t a l f i x e d s u p p l y o f i n v e s t m e n t goods t o the f i r m s e c t o r . S t a t e d another.way, the consequence o f a f i n a n c i a l e q u i l i b r i u m i s t h a t the t o t a l f l o w o f funds t o the f i r m s e c t o r must equal the t o t a l s u p p l y o f i n v e s t m e n t goods t o t h a t s e c t o r . Hence i f f i r m s chose t o a c t c o l l e c t i v e l y they c o u l d use the funds s u p p l i e d t o them by households t o p urchase t h e a g g r e g a t e endowment o f i n v e s t m e n t goods. . T h i s r e l a t i o n s h i p , w h i l e c e r t a i n l y not i m p l y i n g e q u i l i b r i u m i n the f i r m s e c t o r , i s a r e a l c o n s t r a i n t upon the e q u i l i b r i u m V* i n t h a t , g i v e n J marked v a l u e s , the ( J + l ) t h V* i s a u t o m a t i c a l l y d e t e r m i n e d by ( 2 . 8 ) . Thus, f o r example, i n - 58 -s e a r c h i n g f o r f i n a n c i a l - e q u i l i b r i u m market v a l u e s i n R + , we may r e s t r i c t our s e a r c h t o t h e l i n e a r m a n i f o l d A J ( d , e ) = {V | V e RV ; E d.V. = £ e. = e}. (2.9) j=0 J J i = l 1 Thus i n e f f e c t , the equilibrium assumption a l l o w s us t o make a p a r t i c u l a r n o r m a l i z a t i o n g i v e n by (2.9) above, on e q u i l i b r i u m market v a l u e s . Note t h a t t h i s n o r m a l i z a t i o n i s a d d i t i o n a l t o s e t t i n g w=l, which we d i d by v i r t u e o f the u s u a l homogeneity p r o p e r t y o f the excess demand f u n c t i o n s . A u s e f u l c o n c e p t i s t h a t o f the f i n a n c i a l - e q u i l i b r i u m c o r r e s p o n d e n c e . T h i s i s s i m p l y a mapping from t h e s e t o f i n v e s t m e n t - f i n a n c i n g d e c i s i o n s by f i r m s , ( x , d ) , t o the s e t o f e q u i l i b r i u m market v a l u e s . Hence we d e f i n e D.3. The financial-equilibrium correspondence i s a mapping from (x,d) e R 2 ( J + 1 ) t o the s e t E(x,d) = {V* | V* e A J ( d , e ) ; Z(V*;x,d) < 0 ^ } . (2.10) Assumption C.2 a s s u r e s us t h a t E(x,d) i s non-empty, and f u r t h e r m o r e i t 119' can be shown t h a t E(x,d) i s an upper hemi c o n t i n u o u s c o r r e s p o n d e n c e . ' T h i s i m p l i e s t h a t " s m a l l " changes i n (x,d) produce " s m a l l " changes i n the e q u i l i b r i u m p r i c e s e t . In g e n e r a l E(x,d) w i l l not be s i n g l e - v a l u e d . J u s t as i n the case o f t h e Walras c o r r e s p o n d e n c e , t h e r e i s the g e n e r a l presump-20 t i o n t h a t the number o f e q u i l i b r i a i s , a t l e a s t , f i n i t e . The f a c t t h a t the market v a l u e o f a f i r m i s a p r i c e i n a g e n e r a l e q u i l i b r i u m model has some r a t h e r s t r o n g i m p l i c a t i o n s f o r the t h e o r y o f the f i r m . As noted i n the i n t r o d u c t i o n , a common s u g g e s t i o n has been t h a t the a p p r o p r i a t e d e c i s i o n c r i t e r i o n f o r a f i r m , even under u n c e r t a i n t y , i s the m a x i m i z a t i o n o f the s t o c k market v a l u e o f the f i r m . R e c e n t l y a number o f o b j e c t i o n s has been r a i s e d about t h i s c r i t e r i o n on the b a s i s t h a t i t - 59 -l e a d s t o i n e f f i c i e n t p r o d u c t i o n d e c i s i o n s from the v i e w p o i n t o f s h a r e -h o l d e r s . The o b s e r v a t i o n , however, t h a t the s t o c k market v a l u e o f a f i r m i s one element o f a v e c t o r which l i e s i n the range o f the f i n a n c i a l -e q u i l i b r i u m c o r r e s p o n d e n c e r a i s e s two f u r t h e r p o i n t s about the proposed c r i t e r i o n . Suppose f i r s t t h a t E(x,d) were s i n g l e - v a l u e d so t h a t we c o u l d w r i t e the v e c t o r . f u n c t i o n V = E ( x , d ) . Then the net market v a l u e o f the j t h f i r m would be N. = ( l - d . ) V , = (1-d.) E . ( x , d ) , i . e . , the v a l u e o f the J J J J J f i r m to the o r i g i n a l s h a r e h o l d e r s . Now, viewed as an o l i g o p o l y o r non-c o o p e r a t i v e game problem, the m a x i m i z a t i o n o f N. would be f e a s i b l e p r o -v i d e d the f i r m knows the v e c t o r f u n c t i o n E.(x,d) and the v e c t o r s (x,d) o r J the d e c i s i o n s o f a l l o t h e r f i r m s . T h i s , o f c o u r s e , i s e q u i v a l e n t t o assuming the f i r m can compute the g e n e r a l e q u i l i b r i u m o f the economy. From both an i n f o r m a t i o n and c o m p u t a t i o n a l v i e w p o i n t , m a x i m i z a t i o n o f s t o c k market v a l u e appears t o be a n o n - o p e r a t i o n a l c r i t e r i o n , u n l e s s we endow t h e f i r m w i t h r a t h e r e x c e p t i o n a l a b i l i t i e s t h a t we n o r m a l l y do not r e g a r d economic agents as p o s s e s s i n g . S e c o n d l y , the f a c t t h a t E(x,d) i s g e n e r a l l y not s i n g l e - v a l u e d i m p l i e s t h a t the term m a x i m i z a t i o n here has no meaning. A c o r r e s p o n d e n c e cannot be maximized. For a g i v e n v e c t o r o f p r o d u c t i o n and f i n a n c i n g d e c i s i o n s the e q u i l i b r i u m market v a l u e o f the f i r m may take on a number o f p o s s i b l e v a l u e s depending upon the adjustment p r o c e s s e s which a r e o p e r a t i n g i n the s t o c k market. At b e s t then maximiza-t i o n o f market v a l u e can be viewed as a c r i t e r i o n which might.be used i n a m o n o p o l i s t i c a l l y c o m p e t i t i v e framework where the "market v a l u e f u n c t i o n s " 22 a r e p e r c e i v e d as opposed t o b e i n g a c t u a l market v a l u e f u n c t i o n s . P o s t u -l a t i n g t h i s t y p e o f t h e o r y has i t s own d i f f i c u l t i e s o f c o u r s e , i n t h a t some mechanism must be i n t r o d u c e d which r e l a t e s the p e r c e i v e d demand f u n c t i o n s t o the t r u e demand f u n c t i o n s . I t would seem t h a t the market - 60 -v a l u e c r i t e r i o n poses more problems than s o l u t i o n s , n o t o n l y i n r e l a t i o n t o i t s n o n - o p t i m a l i t y p r o p e r t i e s , but a l s o i n i t s i n t r a c t a b l e n a t u r e . We c o n c l u d e t h i s s e c t i o n w i t h some f i n a l comments on the n a t u r e o f a f i n a n c i a l e q u i l i b r i u m , a c o n c e p t which i s c o n d i t i o n a l upon both the f i n a n c i n g and i n v e s t m e n t d e c i s i o n s made by f i r m s i n the economy. The c o n c e p t o f a f i n a n c i a l e q u i l i b r i u m seems t o have f i r s t been i n t r o d u c e d i n t o economic a n a l y s i s by T o b i n [1969] i n h i s g e n e r a l e q u i l i b r i u m t r e a t -ment o f a monetary economy. T o b i n argued i t was a u s e f u l c o n c e p t i n t h a t i t was n o t u n r e a s o n a b l e t h a t i n the s h o r t - r u n the p r o d u c t i o n d e c i s i o n s o f the economy c o u l d be taken as g i v e n . He seems t o i m p l i c i t l y assume t h a t f i n a n c i a l markets c l e a r much f a s t e r than markets which dea l i n " r e a l " commodities. In the c o n t e x t o f our model t h i s i n t e r p r e t a t i o n may be used but i t i s c e r t a i n l y not n e c e s s a r y . Rather i t may be t r e a t e d as a u s e f u l a n a l y t i c a l d e v i c e t o s e p a r a t e the p r o d u c t i o n / f i r m s i d e o f the economy from the p o r t f o l i o / h o u s e h o l d s i d e . In a c t u a l f a c t , o f c o u r s e , both s i d e s o f t h e economy o p e r a t e s i m u l t a n e o u s l y , and i f a s t o c k market economy has a f u l l e q u i l i b r i u m then a l l markets would have t o c l e a r s i m u l t a n e o u s l y , both r e a l and f i n a n c i a l . Whether o r not a s t o c k market economy has a f u l l e q u i l i b r i u m i s an i s s u e t r e a t e d i n S e c t i o n 4. 3. F i n a n c i a l C o n s i s t e n c y Most a u t h o r s d e a l i n g w i t h models o f the s t o c k market economy have a v o i d e d the f i n a n c i n g d e c i s i o n o f t h e f i r m , which i n the c o n t e x t o f our model i s the c h o i c e o f d,, c o n c e n t r a t i n g s o l e l y upon the r e a l i n v e s t m e n t d e c i s i o n , o r c h o i c e o f x-, and i t s r e l a t i o n t o the r i s k - b e a r i n g f u n c t i o n o f the market. Presumably, however, what makes the s t o c k market economy model an i n t e r e s t i n g one i s the f e a t u r e o f the c a p i t a l market which - 61 -r e q u i r e s t h a t f i r m s r a i s e the n e c e s s a r y i n v e s t m e n t funds i n the c a p i t a l market. The p o s s i b i l i t y then e x i s t s t h a t the f i n a n c i n g d e c i s i o n and i n -vestment d e c i s i o n o f the f i r m a r e i n c o m p a t i b l e , g i v e n a f i n a n c i a l e q u i l -i b r i u m . The amount o f funds a l l o c a t e d t o the f i r m i n a f i n a n c i a l e q u i l -i b r i u m s i m p l y may not be s u f f i c i e n t t o c o v e r the c o s t s o f the proposed i n v e s t m e n t p r o j e c t . In the Arrow-Debreu economy w i t h complete markets, the p o s s i b i l i t y o f i n c o n s i s t e n c y between f i n a n c i n g and p r o d u c t i o n d e c i s i o n s does not a r i s e . G i v e n the r e l e v a n t p r i c e s on a l l date and e v e n t p a i r s , f i r m s choose t h e i r p r o d u c t i o n p l a n s i n o r d e r t o maximize market v a l u e . S i n c e the f i r m i s always assumed t o have the p o s s i b i l i t y o f i n a c t i o n , i t s mar-ket v a l u e must be n o n - n e g a t i v e . F urthermore, the completeness o f markets gua r a n t e e s t h a t the f i r m can meet any o b l i g a t i o n s i t i n c u r s i n the f i r s t p e r i o d by s e l l i n g f o r w a r d c o n t r a c t s on i t s f u t u r e o u t p u t . S i n c e market v a l u e i s n o n - n e g a t i v e , the t o t a l r e c e i p t s from s e l l i n g i t s f u t u r e o u t p u t must exceed i t s t o t a l expenses i n p u r c h a s i n g c u r r e n t i n p u t s . In the s t o c k market economy, however, due t o the absence o f f u t u r e s markets the f i r m must pay f o r i t s c u r r e n t i n p u t s by s e l l i n g c l a i m s t o the f u t u r e o u t -put i n the form o f common s h a r e s . The f i r m has no knowledge as t o the m a r g i n a l change i n i t s market v a l u e by p u r c h a s i n g an a d d i t i o n a l u n i t o f i n v e s t m e n t goods, o r t o the m a r g i n a l change i n i t s market v a l u e by i s s u i n g a n o t h e r s h a r e o f new e q u i t y . C o n s e q u e n t l y , f o r a g i v e n i n v e s t m e n t de-c i s i o n the f i r m has no means o f e v a l u a t i n g how much s t o c k must be i s s u e d i n o r d e r t o c o v e r i n v e s t m e n t c o s t s , o r whether i n f a c t any amount o f s t o c k i s s u e d w i l l c o v e r the c o s t o f i n v e s t m e n t . The c h o i c e o f the i n v e s t m e n t p l a n , which i s i n our model the q u a n t i t y o f i n v e s t m e n t goods x., depends upon the d e c i s i o n c r i t e r i o n o f the f i r m . - 62 -There have been a number proposed i n the l i t e r a t u r e . To mention a few, choose the p r o d u c t i o n p l a n such as t o a) maximize the n e t market v a l u e o f t he f i r m , b) maximize e x p e c t e d p r o f i t s , c ) maximize an ex p e c t e d u t i l i t y f u n c t i o n o f p r o f i t s , d) maximize s t o c k h o l d e r w e l f a r e o r e) m i n i m i z e the 23 p r o b a b i l i t y o f a tak e o v e r . I t may be t h a t some o r a l l o f th e s e d e c i s -i o n r u l e s s u f f e r from the above mentioned problem. Thus we s h a l l d e f i n e a d e c i s i o n r u l e t o have the p r o p e r t y o f financial consistency i f i t l e a d s t o s i m u l t a n e o u s i n v e s t m e n t and f i n a n c i n g d e c i s i o n s which r e s u l t i n s u f f i c -i e n t funds being a v a i l a b l e t o the f i r m i n o r d e r t o c a r r y o u t i t s proposed i n v e s t m e n t p l a n . There i s a c e r t a i n type o f myopia i m p l i c i t i n f i r m s ' b e h a v i o u r i n t h i s w o r l d . I t may not seem u n r e a s o n a b l e t o r e q u i r e t h a t they r e c o g n i z e the e f f e c t t h e i r p r o d u c t i o n d e c i s i o n and f i n a n c i n g d e c i s i o n has on t h e i r market v a l u e , and thus r e q u i r e t h a t they maximize t h e i r o b j e c t i v e f u n c -t i o n , s u b j e c t not o n l y t o t h e i r t e c h n o l o g y , but a l s o t o the c a p i t a l mar-ke t c o n s t r a i n t . There a r e , however, some good reasons f i r m s may not do so. F i r s t , t h e r e a r e the i n f o r m a t i o n and co m p u t a t i o n a l c o s t s i n v o l v e d i n e v a l u a t i n g t he g e n e r a l . e q u i l i b r i u m e f f e c t s . S e c o n d l y , i f f i r m s under-ta k e such a p o l i c y , they have t o c o n s i d e r not o n l y the impact o f t h e i r own d e c i s i o n s , but a l s o the impact o f o t h e r f i r m s ' d e c i s i o n s on the v i a -b i l i t y o f t h e i r p l a n s . T h i s l e a d s i m m e d i a t e l y to o l i g o p o l i s t i c t y p e i m p l i c a t i o n s i n the c o m p e t i t i o n f o r funds i n the c a p i t a l market. An i n t e r e s t i n g q u e s t i o n , which a r i s e s t h e n , i s whether we can t e l l a c o m p e t i -t i v e - l i k e s t o r y i n t h i s economy. We s h a l l say t h a t f i r m s a r e financially myopic i f th e y choose an inv e s t m e n t p l a n w i t h o u t e x p l i c i t l y r e c o g n i z i n g the c a p i t a l market c o n s t r a i n t . A l l the c o n t r i b u t i o n s made thus f a r t o - 63 -the s t o c k - m a r k e t economy have i m p l i c i t l y assumed f i n a n c i a l myopia on the p a r t o f f i r m s . F i n a n c i a l myopia i s c e r t a i n l y one way t o m a i n t a i n a com-p e t i t i v e s t r u c t u r e w i t h i n t h e s t o c k market economy. Firms p e r c e i v e no i n t e r d e p e n d e n c e between t h e i r f i n a n c i n g d e c i s i o n s and o t h e r f i r m s ' a c t i o n s . Of c o u r s e f i n a n c i a l myopia a l s o r e q u i r e s t h a t f i r m s t a k e no c o g n i z a n c e o f the f a c t t h a t t h e i r f i n a n c i n g d e c i s i o n and in v e s t m e n t d e c i s i o n a r e r e l a t e d i n some f a s h i o n . I f we admit t h a t f i r m s do r e c o g n i z e t h i s i n t e r d e p e n d e n c e , then we must a l s o l o g i c a l l y admit t h a t they r e c o g n i z e the g e n e r a l e q u i l -i b r i u m n a t u r e o f the s t o c k market and thus t h e i r i n t e r d e p e n d e n c e w i t h o t h e r f i r m s . T h e r e f o r e , i f we do n o t a l l o w t h a t f i r m s a r e f i n a n c i a l l y myopic then we must drop the c o m p e t i t i v e assumption. For much o f the a n a l y s i s which f o l l o w s , however, we s h a l l m a i n t a i n the assumption o f f i n a n c i a l myopia. In terms o f the n o t a t i o n o f the p r e v i o u s s e c t i o n , the j t h f i r m chooses ( x . , A S . ) , i . e . , a q u a n t i t y o f i n v e s t m e n t goods, x., and a quan-t i t y o f new e q u i t i e s , AS., t o s u p p l y on the c a p i t a l market. The p o s s i -b i l i t y o f bond f i n a n c i n g w i l l be c o n s i d e r e d i n S e c t i o n 5 below. The t o t a l o u t s t a n d i n g s u p p l y o f e q u i t i e s i s g i v e n by S. = S. + AS., and t h e r e -J J J f o r e d- = AS-/S- i s the p r o p o r t i o n o f t o t a l e q u i t y s u p p l i e d which c o n s i s t s o f new i s s u e s by the f i r m . We assume t h r o u g h o u t t h a t s h a r e s a r e i n f i n i t e -l y d i v i s i b l e . I f V* i s the e q u i l i b r i u m market v a l u e f o r the j t h f i r m •k c o n d i t i o n a l on the d e c i s i o n s (x,d) o f a l l f i r m s , and p. i s the e q u i l i b r i u m 3 p r i c e p e r s h a r e , then we have the i d e n t i t y S.p* = V*. J J 3 D.4. The in v e s t m e n t and f i n a n c i n g d e c i s i o n s o f f i r m s , ( x . , d . ) , f o r J 3 . j=0, J are financially consistent i f a) V* e.E(x,d) - 64 -and b) d.vt _>x. f o r j=0, J . J J J The c o n s t r a i n t i n b) says s i m p l y t h a t the t o t a l funds g o i n g t o the f i r m i n a f i n a n c i a l e q u i l i b r i u m must exceed t he c o s t s o f the proposed i n v e s t -ment. C l e a r l y as a l l i n v e s t m e n t i s assumed t o be n o n - n e g a t i v e , x, _> 0, and the n o n - n e g a t i v i t y o f a s s e t p r i c e s r e q u i r e s d, _> 0. S i n c e AS, >_ 0, i . e . , we have e x c l u d e d the p o s s i b i l i t y o f share r e p u r c h a s e , and S. >_ 0, then d. <_ 1. T h e r e f o r e we a l s o have <J 0 < d. < 1 (3.1) J x, > 0 f o r a l l j=0, J . (3.2) A n a t u r a l q u e s t i o n t o ask i s whether a s e t o f f i n a n c i a l l y c o n s i s t e n t d e c i s i o n s e x i s t s . An a f f i r m a t i v e answer to t h i s q u e s t i o n i s n e c e s s a r y t o ensure t h a t t he proposed c r i t e r i o n o f f i n a n c i a l c o n s i s t e n c y i s l o g i c -a l l y c o m p a t i b l e w i t h t he assumptions o f the model. Note t h a t s i n c e we have not s p e c i f i e d f i r m b e h a v i o u r t he e x i s t e n c e o f f i n a n c i a l l y c o n s i s t e n t d e c i s i o n s i s a weaker r e q u i r e m e n t than t he e x i s t e n c e o f a complete g e n e r a l e q u i l i b r i u m . Theorem 3.1: Given assumptions A, B and C, a s e t o f f i n a n c i a l l y c o n s i s t -e n t d e c i s i o n s e x i s t s , ( p r o o f ) : In o r d e r t o prove t h i s theorem i t i s c o n v e n i e n t to use a d i f f e r -e n t n o r m a l i z a t i o n o f p r i c e s , than ,the one used thus f a r , w=l. T h e r e f o r e , by v i r t u e o f t h e homogeneity o f t h e e x c e s s demand f u n c t i o n s , which f o l l o w s from Lemma 2.2 ( 1 ) , we may choose (V,w) e A J + 1 , the u n i t s i m p l e x iii R d + 2 . A J + 1 i s a compact, convex s u b s e t o f R d + 2 . I t i s n e c e s s a r y t o bound the f e a s i b l e s e t o f x, and we may assume w i t h o u t l o s s o f g e n e r a l i t y t h a t J ... I 1 x. < e = E e.. (3.3) j=0 3 ~ i = l 1 - 65 -L e t S =' {(x,d) | such t h a t ( 3 . 1 ) , (3.2) and (3.3) h o l d } . S i s a compact convex s u b s e t o f R ^ J + ^ . Thus T = S X A ^ + 1 i s a compact convex s u b s e t o f R 3 J + ^ . We s h a l l now c o n s t r u c t two c o r r e s p o n d e n c e s . . L e t i|V(Viw) ='{(x,d) | (x,d) E S; d.V. > wx j 5 j=0, J } , be a c o r r e s p o n d e n c e d e f i n e d on AJ+} The r e q u i r e m e n t t h a t d.V. >^wx,, i s , J J J s i m p l y t h e re q u i r e m e n t o f f i n a n c i a l c o n s i s t e n c y . C l e a r l y ^ ( V . w ) i s non-empty. We s h a l l demonstrate t h a t 4'1(V,w) i s convex v a l u e d . L e t x^ = 1 2 2. 2 1 1 2 2 Ax + ( l - A ) x and d^ = Ad + ( l - A ) d , and suppose (x ,d ) and (x ,d ) are 1 1 2 2 i n ^ ( V . w ) : t h a t i s , d,V, ^ w x . and d.V. _> wx. f o r j=0, J . Now J J J J J J [Adj. + ( l - A ) d ^ ] V . > w[Ax*. + (l-X)xJ] and t h e r e f o r e (x,,d,) e i M V . w ) . J J J J J A A 1 Thus ^ ( V j w ) i s convex v a l u e d . ^ ( V j w ) i s an upper hemi-continuous c o r r e -spondence as the graph o f \p1 i s c l o s e d . A c o r r e s p o n d e n c e (x,d) i s con-s t r u c t e d as f o l l o w s . R e c a l l t h a t E(x,d) i s an upper.hemi-continuous c o r r e s p o n d e n c e , which by C.2 i s non-empty, d e f i n e d on S, and takes v a l u e s i n A J + 1 . By a theorem o f M i c h e l [1956] t h e r e e x i s t s a c o n t i n u o u s f u n c t i o n f ( x , d ) , d e f i n e d on S, such t h a t f ( x , d ) e E ( x , d ) . L e t ^ 2 ( x , d ) s f ( x , d ) . C l e a r l y ty2{x,d) i s non-empty, upper hemi-continuous s i n c e i t i s c o n t i n u o u s , and convex v a l u e d s i n c e i t i s a f u n c t i o n . Now d e f i n e ip(x,d,V,w) = i p j ( V , w ) X ii2(x,d). X/J i s non-empty, upper h e m i - c o n t i n u o u s , convex v a l u e d and maps T i n t o i t s e l f . Thus we may a p p l y K a k u t a n i ' s f i x e d p o i n t theorem which says t h e r e e x i s t s a f i x e d p o i n t (x*,d*,V*,w*) e T, such t h a t (x*,d*) £ ^(V*,w*) and (V*,w*) e i p 2 ( x * , d * ) . By c o n s t r u c t i o n (x*,d*) a r e f i n a n c i a l l y c o n s i s t e n t a t p r i c e s (V*,w*). Q.E.D. Thus the d e f i n i t i o n proposed i s non-vacuous i n t h a t the s e t o f f i n a n -c i a l l y c o n s i s t e n t d e c i s i o n s was shown t o be non-empty. However, we have - 66 -s a i d n o t h i n g y e t about the d e c i s i o n r u l e s by which the f i r m a r r i v e s a t i t s c h o i c e o f ( x . , d . ) . I t i s i m p o r t a n t t o note t h a t i n o r d e r t o be v i a b l e the d e c i s i o n r u l e which the f i r m uses must r e s u l t i n f i n a n c i a l l y c o n s i s t e n t d e c i s i o n s . I t seems not u n r e a s o n a b l e t o s u s p e c t t h a t some f i n a n c i a l l y myopic d e c i s i o n r u l e s may r e s u l t i n i n c o n s i s t e n t d e c i s i o n s . , The above e x i s t e n c e theorem, can be i n t e r p r e t e d as s a y i n g t h e r e e x i s t s some d e c i s i o n r u l e s which r e s u l t i n f i n a n c i a l l y c o n s i s t e n t d e c i s i o n s . The next s e c t i o n c o n s i d e r s two i m p o r t a n t examples o f c o n s i s t e n t and i n c o n s i s t e n t b e h a v i o u r on the p a r t o f f i r m s . 4. Some Examples o f Firm B e h a v i o u r We wish t o demonstrate f i r s t t h a t some r e a s o n a b l e f i n a n c i a l l y myopic d e c i s i o n - r u l e s l e a d t o i n c o n s i s t e n t c h o i c e s on the p a r t o f f i r m s and con-s e q u e n t l y the f a i l u r e o f a f u l l g e n e r a l e q u i l i b r i u m t o e x i s t . The second example i l l u s t r a t e s a s i t u a t i o n i n which a s e t o f f i n a n c i a l l y myopic de-c i s i o n r u l e s l e a d t o a s e t o f c o n s i s t e n t i n v e s t m e n t and f i n a n c i n g d e c i s -i o n s . The examples w i l l be c o n s t r u c t e d on the b a s i s o f the e x p l i c i t mean-25 v a r i a n c e a s s e t - p r i c i n g model. Fu r t h e r m o r e , i n o r d e r t o s p e c i f y e x p l i c i t -l y the f i n a n c i n g - i n v e s t m e n t d e c i s i o n i t i s n e c e s s a r y t o choose a p a r t i c u -l a r s e t o f f.:(x.;9) and a c h o i c e r u l e f o r ( x - , d . ) . F i r s t we s h a l l assume t h a t a l l f i r m s r e s o r t t o an extreme s o r t o f new e q u i t y f i n a n c i n g , such t h a t they o f f e r such a l a r g e o f f e r i n g o f new e q u i t i e s , t h a t t o a l l i n t e n t purposes d. = 1. T h i s can a l s o be i n t e r -3 p r e t e d as a case i n which a l l f i r m s a r e j u s t s t a r t i n g up, and f i r m s s e l l t h e i r e q u i t y i n o r d e r t o purchase i n v e s t m e n t goods. In t h i s case the f i n a n c i a l c o n s t r a i n t becomes - 67 -In t h i s case the t o t a l market v a l u e o f the f i r m must exceed i t s i n p u t c o s t s . I t i s somewhat analogous to the M a r s h a l l i a n r u l e t h a t a f i r m under p e r f e c t c o m p e t i t i o n must i n the s h o r t run have t o t a l revenues which exceed t o t a l v a r i a b l e c o s t s . The p r o d u c t i o n s i d e i s s p e c i f i e d by assuming t h a t the s t o c h a s t i c t e c h n o l o g y o f the f i r m s e c t o r i s g i v e n by E C f j f x ^ e ) ] = a ^ J , a. > 0; (4.2) Gov [ f j ( X j ; e ) f ^ s e ) ] - a^xjxj f o r a l l i , j = l , ..., J . (4.3) A l l economic ag e n t s are assumed t o have i d e n t i c a l e x p e c t a t i o n s , both managers and i n v e s t o r s . The 0th f i r m i s assumed tb be r i s k l e s s as i t produces a c e r t a i n second p e r i o d o u t p u t g i v e n by y 0 = a 0 X g . The " r i s k -l e s s " i n t e r e s t r a t e ( g r o s s ) i n the economy i s d e f i n e d by 1+r = y 0 / V Q where Vo i s the market v a l u e o f the r i s k l e s s f i r m . . A l l i n d i v i d u a l s a r e assumed t o have i d e n t i c a l c o n s t a n t a b s o l u t e r i s k - a v e r s i o n u t i l i t y f u n c -t i o n s , w i t h r i s k a v e r s i o n parameter X > 0. Assuming t h e d i s t r i b u t i o n o f f i r m s ' second p e r i o d o u t p u t i s n o r m a l l y d i s t r i b u t e d , the f i n a n c i a l -?fi e q u i l i b r i u m market v a l u e s a r e g i v e n by J Z c o v ( y i , y j . ) a) V. = jL. [ E ( y ) - X ^ 1 p 3 1 + R J (E E c o v ( y . ,y.) i j J E a , . x 2 x 2 1+r ^ i i A / h h\h^ J J (E E a. .x.x?) f o r j = l , J (4.4.1) - 68 -b) 1+r = a 0 x j / VQ•.. (4.4.2) J I c) £ V. = £ e. = e. (4.4.3) j=0 J i = l 1 Given the J + l - v e c t o r o f p r o d u c t i o n d e c i s i o n s x = (x x . ) ' , f i n a n c i a l 3 e q u i l i b r i u m market v a l u e s a r e s p e c i f i e d c o m p l e t e l y by e q u a t i o n s a ) , b) and c) o f 4.4. The l a r g e r i s the parameter X , the more r i s k - a v e r s e i n d i v i d -u a l s a r e s a i d t o be. The f i n a n c i n g c o n s t r a i n t , r e l a t i n g i n v e s t m e n t d e c i s i o n s x, and market v a l u e s V, can thus be w r i t t e n as w - C Y J " x& a u x i 1 ( \ \ a u x i ¥^ i X J ( 4 - 5 ) f o r a l l j=l, ..., J . The f i n a n c i a l c o n s t r a i n t o f the r i s k i ess f i r m i s 2 ^ h ao xo V 0 = f^r- 1 x 0 . (4.6) An a d d i t i o n a l c o n s t r a i n t upon the model i s t h a t the c o v a r i a n c e m a t r i x o f the J r i s k y f i r m s ' o u t p u t p a t t e r n s be p o s i t i v e d e f i n i t e . Hence A(x) = [a.j x\ x j ] J x J (4.7) i s a p o s i t i v e - d e f i n i t e m a t r i x f o r a l l x » O j . A s u f f i c i e n t c o n d i t i o n f o r t h i s t o h o l d i s t h a t A(x) be a p o s i t i v e d i a g o n a l m a t r i x , i . e . a.- = 0 f o r i + jf a.-. > 0 i = l , J . We s h a l l assume t h a t a l l f i r m s a r e e x p e c t e d - p r o f i t m a x i m i z e r s . T h i s assumption i s p r o b a b l y the most common one employed i n the p a r t i a l e q u i l -i b r i u m l i t e r a t u r e on the t h e o r y o f the f i r m under u n c e r t a i n t y and has been - 69 -used e x t e n s i v e l y i n the management s c i e n c e l i t e r a t u r e . Expected p r o f i t s o f the j t h f i r m a r e g i v e n by K J = 1+r Yd " "j • (4'8) M a x i m i z a t i o n o f I I . w i t h r e s p e c t t o x. y i e l d s J J ^2 x* = d + r ) " 2 > 0 • ( 4 - 9 ) S u b s t i t u t i n g (4.9) i n t o the v a l u a t i o n formulae (4.4.1) y i e l d s = ^ • a » t j ' i ' ] i j j V t V ' 1 ' ( 4 - 1 0 ) 2 a. 2 Now i f V. > x* = -f- ( 1 + r ) " , then a j / d + r ) 2 2 - 2 X 1 3 ^ . 3 ^ ^ ( 1 Z a ^ a ^ j ^ d + r ) 2 > a " / ( 1 + r ) 2 4, 1 1 J o r , upon r e - a r r a n g i n g , 2/4 - 2XEa.a.a. ./(E J « J/ 4 - » J « I » J V « J J V i ' j ^ i " - ( 4 - n ) C o n s i d e r now the " d i a g o n a l " c a p i t a l market model, which assumes a ^ = 0 f o r i i= j . Then (4.11) becomes a!/4 - 2 X a j j a 2 / ( E a..a 2) > 0 o r h - 2Xa. ./(E a. .z.)h > 0. (4.12) S i n c e a . . > 0, the second term on the l e f t - h a n d s i d e o f i n e q u a l i t y (4.12) i s p o s i t i v e . C l e a r l y f o r X l a r g e enough t h i s i n e q u a l i t y cannot h o l d , and - 70 -thus f i n a n c i a l c o n s i s t e n c y does not o b t a i n . The more r i s k y a f i r m i s r e l a t i v e to the o t h e r s , t h a t i s the l a r g e r a.., the g r e a t e r the l i k e l i -3 3 hood o f the f i r m c h o o s i n g f i n a n c i a l l y i n c o n s i s t e n t i n v e s t m e n t d e c i s i o n s . We p r e s e n t here a s i m p l e n u m e r i c a l example t o i l l u s t r a t e the problem o f n o n - e x i s t e n c e . The example has two r i s k y f i r m s w i t h t e c h n o l o g i c a l parameters a i = 2 a n = 1 (4.13) d. 2 ~ % ^22 ^ a 1 2 = a 2 1 = 0. I f we l e t R = (1+r) = 1, t h e n the o p t i m a l i n v e s t m e n t by f i r m s one and two i s g i v e n by x* = 1 and x* = 1. U s i n g the v a l u a t i o n f o r m u lae w i t h a r i s k a v e r s i o n c o e f f i c i e n t A = 3 2/6, we g e t the f i n a n c i a l e q u i l i b r i u m market v a l u e s o f V 2 = 4/3 and V 2 = 2/3. As V x = 4/3 > x*, f i r m 1 i s f i n a n c i a l l y c o n s i s t e n t . However V 2 = 2/3 < x* and i s t h e r e f o r e f i n a n c i a l l y i n c o n s i s -t e n t . T h i s example c o r r e s p o n d s t o t h e c a s e i n which the i n t e r e s t r a t e i s a r b i t r a r i l y s e t , f o r example by a government w i l l i n g t o s e l l r i s k l e s s bonds i n any amount a t a f i x e d i n t e r e s t r a t e . T h i s i s the c a s e u s u a l l y c o n s i d e r e d i n the c a p i t a l a s s e t p r i c i n g model. What happens i f the g o v e r n -ment changes the i n t e r e s t r a t e ? The f o l l o w i n g example c o n s i d e r s the case i n which the i n t e r e s t r a t e i s e ndogenously d e t e r m i n e d , and shows f i n a n c i a l i n c o n s i s t e n c y can o c c u r . T h i s example uses the same parameters as used above but i n a d d i t i o n we must s p e c i f y a Q and e, then u s i n g ( 4 . 4 . 1 ) , (4.4.2).and (4.4.3) the com-p l e t e f i n a n c i a l e q u i l i b r i u m can be d e t e r m i n e d . L e t a Q = 2 and e = 2 . S t r a i g h t f o r w a r d c a l c u l a t i o n s y i e l d - 71 -= 2/3 V* = 1 / 3 and c o n s e q u e n t l y (1+r*) = /2 . The r i s k l e s s f i r m and f i r m 1 a r e f i n a n c i a l l y c o n s i s t e n t , w h i l e f i r m 2 i s f i n a n c i a l l y i n c o n s i s t e n t . S i n c e x ' ( r ) < 0 from ( 4 . 9 ) , t h e r e i s a p o s s i b i l i t y e q u i l i b r i u m might be e s t a b l i s h e d by s u f f i c i e n t change i n the i n t e r e s t r a t e . Note however t h a t (4.12) i s independent o f r , and c o n s e q u e n t l y changes i n r w i l l not a f f e c t the i n -c o n s i s t e n c y r e s u l t . In the example c o n s t r u c t e d the demand f o r in v e s t m e n t funds by f i r m s from e q u a t i o n (4.9) depends o n l y upon the r i s k l e s s i n t e r e s t r a t e , and has the p r o p e r t y t h a t changing the i n t e r e s t r a t e changes a l l f i r m s ' demand f o r i n v e s t m e n t p r o p o r t i o n a t e l y . The r i s k f a c t o r o f each f i r m j i s g i v e n by and i t s v a l u e i s equal t o (4.14) m u l t i p l i e s by X / ( l + r ) . The r i s k d i s -c o u n t , f o r a change i n r , t h e r e f o r e changes by the same f a c t o r o f p r o p o r -t i o n a l i t y f o r a l l f i r m s j = l , J . S i m i l a r l y , the v a l u e o f the e x p e c t e d r e t u r n f a c t o r o f each f i r m i s which changes by the same f a c t o r o f p r o p o r t i o n a l i t y f o r a change i n r . T h e r e f o r e , w i t h e x p e c t e d - p r o f i t m a x i m i z i n g f i r m s , changes i n r a f f e c t o n l y t he a b s o l u t e and not the r e l a t i v e market v a l u e s o f the r i s k y f i r m s . I f some r i s k y f i r m i s f i n a n c i a l l y i n c o n s i s t e n t a t some r , t h e n by lower-i n g r you i n c r e a s e i t s demand f o r in v e s t m e n t f u n d s , but s i m u l t a n e o u s l y (4.14) (4.15) - 72 -you i n c r e a s e i t s market v a l u e by the same f a c t o r from e q u a t i o n ( 4 . 1 0 ) . S i m i l a r l y f o r i n c r e a s e s i n r . Thus once f i n a n c i a l l y i n c o n s i s t e n t , always f i n a n c i a l l y i n c o n s i s t e n t . The f a i l u r e o f a s e t o f f i n a n c i a l l y c o n s i s t e n t c h o i c e s on the p a r t o f f i r m s t o e x i s t f o r any s e t o f f i n a n c i a l e q u i l i b r i u m p r i c e s ( r , V ) i m p l i e s t h a t w i t h r e s p e c t to a t l e a s t one f i r m i t s demand f o r i n v e s t m e n t funds w i l l always exceed the s u p p l y o f i n v e s t m e n t funds t o t h a t f i r m by the c a p i t a l market, i n any f i n a n c i a l e q u i l i b r i u m . T h e r e f o r e a l t h o u g h a f i n a n c i a l e q u i l i b r i u m e x i s t s a f u l l g e n e r a l e q u i l i b r i u m does not. A l t e r n a t i v e l y s t a t e d , t h e r e e x i s t s no s e t o f p r i c e s on s e c u r i t i e s and r e a l commodities which c l e a r a l l markets s i m u l t a n e o u s l y . I f a tatonnement p r o c e s s were t o run i n the economy t h e r e i s no p o s s i b i l i t y t h a t i t would e v e r converge as markets would not c l e a r f o r any s e t o f p r i c e s . The s o u r c e o f t h e n o n - e x i s t e n c e o f e q u i l i b r i u m i n the above example i s c l e a r . I t can o b v i o u s l y be a t t r i b u t e d t o t h e f i n a n c i a l myopia o f f i r m s . I f f i r m s were t o maximize e x p e c t e d p r o f i t s s u b j e c t t o ( 4 . 1 ) , t h e n f i r m s c o u l d never demand more i n v e s t m e n t goods than the s t o c k market wished t o f i n a n c e . As n o t e d , i t i s , however, d i f f i c u l t t o d e v e l o p a t h e o r y o f f i r m b e h a v i o u r , which may be d e s c r i b e d as " c o m p e t i t i v e " , when t h i s c o n s t r a i n t i s taken i n t o a c c o u n t . A l t e r n a t i v e l y , the s o u r c e o f n o n - e x i s t e n c e i s the d e c i s i o n r u l e o f the f i r m . In the above example, f i r m s i n a t t e m p t i n g t o maximize e x p e c t e d p r o f i t s wish t o take on more r i s k than the household s e c t o r i s w i l l i n g t o b e ar. The f a c t t h a t f i r m s a r e f i n a n c i a l l y myopic i s e q u i v a l e n t t o s a y i n g t h a t f i r m s do not t a k e i n t o a c c o u n t the economy's w i l l i n g n e s s t o bear r i s k , and t h i s i s the cause o f the b a s i c market , ., 28 f a i l u r e . I f t h e r e were an a d d i t i o n a l f i c t i t i o u s " p r i c e o f r i s k " which f i r m s - 73 -responded t o . i n c a l c u l a t i n g t h e i r i n v e s t m e n t demand f u n c t i o n , then by a p p r o p r i a t e l y m a n i p u l a t i n g t h i s p r i c e the market f o r i n v e s t m e n t funds by a l l f i r m s might be c l e a r e d . " R i s k " , however, i s not a market commodity and t h e r e does not e x i s t a market p r i c e f o r i t . A l t e r n a t i v e l y f i r m s might base t h e i r d e c i s i o n s on t h e i r own market v a l u e s V., but t h i s type o f b e h a v i o u r would v i o l a t e f i n a n c i a l myopia on the p a r t o f f i r m s and the c o m p e t i t i v e assumptions o f the model. The example we c o n s t r u c t now i s s i m i l a r t o the one above, w i t h the e x c e p t i o n t h a t f i r m s use a d i f f e r e n t d e c i s i o n r u l e . T h i s example has the c h a r a c t e r i s t i c t h a t f i n a n c i a l myopia p e r s i s t s , but the r e s u l t i n g i n v e s t m e n t d e c i s i o n s s a t i s f y the c o n s i s t e n c y p r o p e r t y . Suppose t h e d e c i s i o n c r i t e r i a o f a l l f i r m s i s t o maximize 9j<*j> '• iw r y j - - XJ ( 4 - 1 6 ) where 3 >_ 0. The f u n c t i o n g. might be i n t e r p r e t e d as a u t i l i t y f u n c t i o n o f p r o f i t s ( n e t ) w i t h a r i s k a v e r s i o n parameter 3, on the v a r i a n c e o f the f i r m ' s own gr o s s r e t u r n s . In t h i s example we s h a l l m a i n t a i n the ' d i a g o n a l ' model assumption. The m a x i m i z a t i o n o f (4.13) y i e l d s two p o s s i b l e s o l u -t i o n s . L e t C 2 ( 3 a . . + J + r ) ] = x j < r ' 3 ) -J J Then X j = Xj ( r,3) i f g j ( x j [ r , 3 ] ) >-0, (4.17) x* = 0 i f gjCXjIr.-B]) < 0. Now i f t h e r i s k a v e r s i o n parameter o f the f i r m , 3, i s g r e a t e r than A./(££a. . x t ^ x * ^ ) ^ , the community r i s k a v e r s i o n i n d e x , then from ( 4 . 1 7 ) , i t i j 1 J 1 J - 74 -f o l l o w s i m m e d i a t e l y t h a t the r e s u l t i n g p r o d u c t i o n d e c i s i o n s a r e f i n a n c i a l l y c o n s i s t e n t . I f a l l f i r m s are s u f f i c i e n t l y r i s k a v e r s e , then f i n a n c i a l c o n s i s t e n c y i s e n s u r e d . Over t h e l o n g run f i r m s may l e a r n from e x p e r i e n c e t h a t i t i s n e c e s s a r y t o adopt a c e r t a i n r i s k - a v e r s e a t t i t u d e i n o r d e r t o be c o m p e t i t i v e i n the c a p i t a l market f o r f u n d s . F l e x i b i l i t y i n d e c i s i o n r u l e s i s d e s i r a b l e , as the f i r m c o u l d a d j u s t i t s degree o f r i s k a v e r s i o n to be c o n s i s t e n t w i t h t h a t r e f l e c t e d i n the c a p i t a l market. The above examples demonstrate t h a t we a r e i n somewhat o f a dilemma as r e g a r d s the f i r m i n t h i s t y p e o f economy. As the f i r s t example demon-s t r a t e i g n o r i n g t h e c a p i t a l market c o n s t r a i n t l e a d s t o c e r t a i n o b v i o u s d i f f i c u l t i e s . I f the f i r m i s w i l l i n g t o adopt f l e x i b l e a t t i t u d e s towards r i s k , the second example s u g g e s t s t h a t t h e s e d i f f i c u l t i e s might be c i r c u m -v e n t e d . A major problem w i t h the l a t t e r f o r m u l a t i o n i s the problem o f , the f i r m l e a r n i n g t h e " r i g h t " 6. As 6 changes moreover the f i r m ' s behav-i o u r .would change. I t i s not o b v i o u s why the f i r m ' s p r e f e r e n c e s s h o u l d change, r a t h e r one would s u s p e c t t h a t the f i r m would l e a r n from i t s e x per-i e n c e about the c a p i t a l market c o n s t r a i n t . 5. The C a p i t a l S t r u c t u r e Problem U n t i l now we have assumed a s i n g l e s o u r c e o f f i n a n c i n g , by new e q u i -t i e s o n l y . We now examine what happens upon the i n t r o d u c t i o n o f bonds which are i s s u e d by each f i r m . The f i r m f a c e s a problem i n c h o o s i n g a c a p i t a l s t r u c t u r e , i n t h a t i t may f i n a n c e a proposed i n v e s t m e n t p r o j e c t b y . i s s u i n g new e q u i t i e s , bonds, o r c o m b i n a t i o n s o f b o t h . For the moment l e t us r e q u i r e t h a t f i r m s i s s u e bonds o n l y up t o the p o i n t a t which the p r o b a b i l i t y o f d e f a u l t i s z e r o . Thus we may assume t h a t a l l p o t e n t i a l b o n d h o l d e r s r e g a r d the. p r o b a b i l i t y o f d e f a u l t on bond payments' as z e r o . - 75 -I f a f i r m s e l l s a bond i n p e r i o d one f o r one u n i t o f a c c o u n t which i n our model i s a d o l l a r ' s worth o f i n v e s t m e n t goods, then the bond pays (1+r) i n u n i t s o f consumption goods the next p e r i o d , where r i s the r i s k l e s s r a t e o f i n t e r e s t . The j t h f i r m , g i v e n t h a t i t has undertaken the i n v e s t m e n t p l a n x i $ y i e l d s a p a t t e r n o f r e t u r n s g i v e n by f.(x;9) o v e r a l l 0 e fl. The no J b a n k r u p t c y r e q u i r e m e n t i m p l i e s t h a t f,-(x-;0) > (l+r)|B,, f o r a l l 0 e fl (5.1) when the r i s k l e s s i n t e r e s t r a t e i s r and the j t h f i r m i s s u e s B. d o l l a r s worth o f bonds (measured i n terms o f inv e s t m e n t g o o d s ) . Now f o r some i n t e r e s t r a t e s , the s e t o f x. e R + which s a t i s f y (5.1) may be the empty s e t . But i n o r d e r t o a v o i d t h e ba n k r u p t c y i s s u e i n the c a p i t a l - s t r u c t u r e problem, i t i s n e c e s s a r y t h a t t h e r e a r e some i n v e s t m e n t p l a n s which y i e l d s t r i c t l y p o s i t i v e o u t p u t i n a l l s t a t e s o f n a t u r e , and t h e r e f o r e some p o s s i b i l i t y e x i s t s f o r the i s s u i n g o f d e f a u l t f r e e bonds. Assumptions A.3 and B.6 ensure t h i s i s p o s s i b l e . Assume each f i r m has no o u t s t a n d i n g debt o b l i g a t i o n s , and consequent-l y any bonds i s s u e d i n the f i r s t p e r i o d c o n s t i t u t e the t o t a l debt i n the c a p i t a l s t r u c t u r e o f the f i r m . F or each f i r m j , t h e s e bonds w i l l be denoted by B.. R e c a l l t h a t AS. denotes the q u a n t i t y o f new e q u i t i e s i s s u e d by t h e j t h f i r m and p. denotes t he p r i c e o f one sha r e i n the j t h u f i r m . The r e q u i r e m e n t o f f i n a n c i a l c o n s i s t e n c y may be w r i t t e n as AS.p; + B. > x. (5.2) where (5.1) i s assumed t o h o l d under the no ban k r u p t c y assumption. I t might be u s e f u l a t t h i s s t a g e t o r e c a l l the b a s i c M o d i g l i a n i - M i l l e r - 76 -theorem. The t o t a l market v a l u e o f the f i r m i s now V, = E. + B., where 3 3 3 E. i s the v a l u e o f the f i r m ' s e q u i t y , i n our model equal t o S.p..'^u 3 J J Assuming no b a n k r u p t c y and t h a t i n d i v i d u a l s can borrow and l e n d : a t . t h e same r a t e o f i n t e r e s t as f i r m s and i g n o r i n g t r a n s a c t i o n s c o s t s , t h e n , f o r a given p r o d u c t i o n p l a n the t o t a l market v a l u e o f the f i r m , V,, i s un-J changed w i t h r e s p e c t t o changes i n the d e b t / e q u i t y r a t i o B./E.. Note 3 3 t h a t the theorem h o l d s f o r a g i v e n p r o d u c t i o n p l a n . The theorem does not a s s u r e t h a t a g i v e n f i n a n c i a l / i n v e s t m e n t p l a n s a t i s f i e s the p r o p e r t y o f f i n a n c i a l c o n s i s t e n c y . I t i s o f t e n c l a i m e d t h a t the M+M theorem 'proves' the i r r e l e v a n c y o f c o r p o r a t e f i n a n c i a l p o l i c y . Under what con-d i t i o n s might t h i s be t r u e ? C e r t a i n l y i f an economy has a complete s e t o f Arrow-Debreu markets, f i n a n c i a l p o l i c y i s i r r e l e v a n t . In t h i s type o f s i t u a t i o n the m a x i m i z a t i o n o f the market v a l u e o f the f i r m i s a d e s i r -a b l e o b j e c t i v e from the s t o c k h o l d e r s ' v i e w p o i n t , and the M+M theorem t e l l s us the c a p i t a l s t r u c t u r e o f the f i r m makes no d i f f e r e n c e as t o the market value, o f the f i r m . F u r t h e r m o r e , i n an economy w i t h complete mar-ke t s the problem o f f i n a n c i a l c o n s i s t e n c y i s i r r e l e v a n t . In the s t o c k -market economy.the f i r m , however, has no o b v i o u s r e a s o n f o r m a x i m i z i n g i t s market v a l u e , and the f i n a n c i a l c o n s i s t e n c y problem i s v e r y r e a l . The q u e s t i o n then a r i s e s as t o what f i n a n c i a l i r r e l e v a n c e ( o r more a c c u -r a t e l y the i r r e l e v a n c e o f the f i r m ' s c a p i t a l s t r u c t u r e ) means i n the s t o c k - m a r k e t economy. We might make the f o l l o w i n g p r o p o s i t i o n ; if a f i n a n c i a l l y c o n s i s t e n t s e t o f d e c i s i o n s has* been made w i t h r e s p e c t t o x,,AS. and B., then any-; J j 3 i i o t h e r v a l u e s o f (AS., B . ) , such t h a t the no b a n k r u p t c y c o n d i t i o n was met, 3 3 w i l l , c o n t i n u e t o be f i n a n c i a l l y c o n s i s t e n t . A p o s i t i v e answer t o the above p r o p o s i t i o n would go a l o n g way t o j u s t i f y i n g the i r r e l e v a n c e o f - 77 -the f i r m ' s c a p i t a l s t r u c t u r e . For i f i t were t r u e , the problem o f a f i r m c h o o s i n g a c a p i t a l s t r u c t u r e c o u l d be d i v o r c e d from the problem o f a c h i e v -i n g f i n a n c i a l c o n s i s t e n c y . F urthermore, i f a f i r m were f o r t u n a t e enough to have a d e c i s i o n c r i t e r i o n which a l l o w e d i t t o o b t a i n the n e c e s s a r y f i n a n c i n g f o r i t s i n v e s t m e n t p r o j e c t s , then the means by which i t o b t a i n e d t h i s f i n a n c i n g would be a m a t t e r o f i n d i f f e r e n c e ; any c o m b i n a t i o n o f new e q u i t y i s s u e s and debt would s u f f i c e . The main theorem o f t h i s s e c t i o n demonstrates t h a t the p r o p o s i t i o n i s , i n f a c t , t r u e . B e f o r e p r o c e e d i n g w i t h the theorem, l e t us p o i n t out a g a i n the a r g u -ments made i n the l a s t s e c t i o n . As we d i d t h e r e we c o u l d c e r t a i n l y con-s t r u c t examples o f d e c i s i o n r u l e s which a r e f i n a n c i a l l y i n c o n s i s t e n t . The i n t r o d u c t i o n o f bonds i n no way changes t h a t p o s s i b i l i t y . Suppose, f o r example, a f i r m used bonds as i t s s o l e s o u r c e o f f i n a n c i n g . The no-b a n k r u p t c y c o n d i t i o n r e q u i r e s t h a t (5.1) h o l d and f i n a n c i a l c o n s i s t e n c y r e q u i r e s B j ^ X j . (5.3) There a r e c e r t a i n l y d e c i s i o n r u l e s which would v i o l a t e one o f t h e s e con-s t r a i n t s , what i s perhaps more i n t e r e s t i n g i s t h a t t h e r e may e x i s t no x, > 0 which i s c o m p a t i b l e w i t h both (5.3) and ( 5 . 1 ) . Suppose y . ( 6 ) = x . h ( e ) , where h<h(Q) < V z f o r a l l 0 e fl. Then (5.1) r e q u i r e s t h a t J h x. :> (1+r) B,, and (5.3) r e q u i r e s B. _> x. o r (1+r) B, >_ ( l + r ) x . . Com-J J J J J J b i n i n g both y i e l d s h Xj > ( l + r ) X j . (5.4) I f the i n t e r e s t r a t e ( g r o s s ) , 1+r, i s g r e a t e r than J g , the o n l y i n v e s t m e n t c h o i c e which i s f e a s i b l e i s x. = 0 . What i s ' p e c u l i a r ' about t h i s example? F i r s t , o f c o u r s e , the r e q u i r e -- 78 -ment o f no-bankruptcy w i t h p r o b a b i l i t y one i s no doubt u n r e a l i s t i c . A l -most a l l debt has a t t a c h e d t o i t some p r o b a b i l i t y o f d e f a u l t , a l t h o u g h i n p r a c t i c e i t may be so c l o s e t o z e r o t h a t i t may be i g n o r e d . What the example does show i s t h a t . t h e r e q u i r e m e n t o f f i n a n c i a l c o n s i s t e n c y p r o -v i d e s an i n c e n t i v e f o r f i r m s t o d i v e r s i f y t h e i r c a p i t a l s t r u c t u r e . In the above example, the f i r m by i s s u i n g some e q u i t y can c l e a r l y r e a l i z e i n v e s t m e n t c h o i c e s o t h e r than x. = 0, a v o i d i n g t he no-bankruptcy con-s t r a i n t , and s t i l l y i e l d a p o s i t i v e market v a l u e . The example demonstrates t h a t p r o v i d e d a l l bonds i s s u e d must have a z e r o p r o b a b i l i t y o f de f a u l t , ' ' then a r o l e f o r f i n a n c i a l p o l i c y e x i s t s i n c h o o s i n g a c a p i t a l s t r u c t u r e such t h a t t he d e s i r e d i n v e s t m e n t p o l i c y i s a l s o f e a s i b l e w i t h the no bank-31 r u p t c y r e q u i r e m e n t . i Suppose t h a t a f i n a n c i a l l y c o n s i s t e n t s e t o f d e c i s i o n s has been made by t he j t h f i r m g i v e n by the t r i p l e {x.,d,B,} which s a t i s f i e s the b a s i c vJ J J i n e q u a l i t y d.E, .+ B, > x, (5.4) J J J J We wish to prove t he f o l l o w i n g theorem. Theorem 5.1: Gi v e n t he c o n d i t i o n s o f 1) no b a n k r u p t c y , 2) a l l i n d i v i d u a l s can borrow and l e n d a t the r i s k l e s s i n t e r e s t r a t e , 3) c o n d i t i o n s o f f i n a n -c i a l e q u i l i b r i u m h o l d and 4) f o r some f i r m k e {0, 1, J} the i n i t i a l s e t o f d e c i s i o n s ( x ^ , d^, B^) were f i n a n c i a l l y c o n s i s t e n t , then t h e r e e x i s t s a n o t h e r f i n a n c i a l e q u i l i b r i u m w i t h B^/E^ t a k i n g any v a l u e and such t h a t f i n a n c i a l c o n s i s t e n c y c o n t i n u e s t o h o l d f o r the kth f i r m . Remark: The p r o o f o f the theorem r e l i e s on a l e v e r a g e type argument used by M o d i g l i a n i and M i l l e r i n t h e i r o r i g i n a l paper. The o n l y new n o t a t i o n i s D., which i s the debt ( p o s i t i v e o r n e g a t i v e ) h e l d by the i t h i n v e s t o r . The o u t p u t p a t t e r n s produced by f i r m s , g i v e n i n v e s t m e n t d e c i s i o n s x. a r e - 79 -g i v e n by y . ( 9 ) . ' 32 ( p r o o f ) : R e c a l l t h a t t he budget c o n s t r a i n t o f the i t h i n v e s t o r i s g i v e n by J J _ E S..p, + D, = £ S,,p. + e.. (5.5) j=0 1J J 1 j=0 J J 1 Consumption o f the i t h i n v e s t o r i n s t a t e 9 e fl, g i v e n h i s s h a r e - h o l d i n g s {S. , S.,}, d e b t D. and i n t e r e s t r a t e ( g r o s s ) R = 1+r, i s g i v e n by 10 1 u 1 c i ( e ) = Jo ( sij / sj> [ y j ( e ) " R B j ] + R D i = 2 (S../S.) [ y . ( 9 ) - RB.] + R[e. - { I ( S . . - S , , ) p , } ] . j _ Q 1 J J 1 J 1 j _ Q ' J ' J J (5.6) In f i n a n c i a l e q u i l i b r i u m t he demand f o r sh a r e s e q u a l s the s u p p l y o f s h a r e s , I so £ S. . = S. = S. + AS.. Now ad d i n g i n d i v i d u a l budget c o n s t r a i n t s i = l, •j = j ' J j J J .'.., I a t f i n a n c i a l e q u i l i b r i u m p r i c e s p = ( p 0 , p.) and R, we g e t £ £ S..p. + £ D, = £ £ S-.p. + £ e, i j J J i i j J J i £ S.p, + £ D. = £ S.p. + £ e.. (5.7) j i j J J i R e - a r r a n g i n g (5.7) y i e l d s £ AS.p. + £ D, = £ e, , j J J i i The n e t e x c e s s demand f o r bonds i s g i v e n by (5.8) Z B = S [ e . - { £ ( S . . - S )p }J - S B . . (5.9) i j=0 J 1 J J j where the f i r s t - t e r m i n square b r a c k e t s on the r i g h t r e p r e s e n t s £ and - 80 -the second t e r m . r e p r e s e n t s the s u p p l y o f bonds by f i r m s . In e q u i l i b r i u m E S. .p: = E. = V. - B.. Hence u s i n g (5.8) -j "13 3 3 3 3 = E e. + l S.p. - E V. = 0 (5.10) i 1 j J J j J Thus, i f the market f o r e q u i t i e s c l e a r s , t h i s i m p l i e s e q u i l i b r i u m i n the bond market; 33 Now we take the kth f i r m t o be k=0, w i t h o u t l o s s o f g e n e r a l i t y . Suppose t h a t t h i s f i r m i s s u e s no d e b t , but i s s u e s new e q u i t i e s ( i n a d d i t i o n t o those o f f e r e d i n the f i r s t e q u i l i b r i u m ) , A S Q such t h a t A S Q p o = B Q. T h a t i s t h e v a l u e o f new e q u i t i e s a t the e q u i l i b r i u m p r i c e s o f the f i r s t s i t u a -t i o n i s equal t o the v a l u e o f the bonds i t i s s u e d i n the f i r s t s i t u a t i o n . A l l v a r i a b l e s i n the second s i t u a t i o n a r e denoted w i t h c a r e t s . Thus B Q = 0. Now suppose a) R = R b) - p. = Pj j=0, 1, ..., J c) S . - S . J>1 d) Bj = B. j > 1. (5.11) That i s we h o l d the p r i c e s o f a l l s e c u r i t i e s c o n s t a n t and h o l d the s u p p l i e s o f a l l s e c u r i t i e s by f i r m s o t h e r t h a n j=0, a t the same l e v e l . Now V = S p + B o o H o o o o"o 0^0 0 - 81 -u s i n g the above assumptions. Hence we change the d e b t / e q u i t y r a t i o o f the oth f i r m , but by assumption we do not change the t o t a l market v a l u e o f the f i r m . More i m p o r t a n t l y we do not change the p r i c e per s h a r e o f the oth f i r m ' s e q u i t y . T h i s i m p l i e s t h a t s i n c e The c o n s t r u c t i o n i s such t h a t the new f i n a n c i a l d e c i s i o n s o f the f i r m remain f i n a n c i a l l y c o n s i s t e n t . . The p r o o f i s completed by showing t h a t t h e c o n s t r u c t e d v a r i a b l e s a r e i n f a c t e q u i l i b r i u m ones. The consumption p o s s i b i l i t i e s o f the i t h i n v e s t o r i n the second s i t u a t i o n are g i v e n by Assume now t h a t the i n d i v i d u a l u n d e r t a k e s some home-made l e v e r a g e . S p e c i f i c a l l y , assume t h a t f o r e v e r y d o l l a r o f e q u i t y the i n v e s t o r owned i n the o t h f i r m i n the i n i t i a l s i t u a t i o n , the i n d i v i d u a l borrows B Q / p o S i n a d d i t i o n t o Dn.. T h e r e f o r e and w i t h the proceeds o f t h i s l o a n he i n c r e a s e s h i s h o l d i n g s o f e q u i t y i n the f i r s t f i r m . Thus AS„p„ + AS D > x„ . o r o o r o — o (5.13) (5.15) S. p = S. p + (.-i o K o i o K o v = S. p + S. ( c 2 - ) . i o H o i o v S „ ; 0 (5.16) - 82 -Now u s i n g assumptions (5.11) and the above l e v e r a g e we r e - c a l c u l a t e the i n d i v i d u a l ' s consumption o p p o r t u n i t y s e t g i v e n by (5.14). S. J S.. J c,(e) = ^ - y n ( e ) + z ^ [y-.(e) - R B.] + R[e. - z (s.. -s..) P. - { S i o P 0 + S i o ^ > - V o " = . ^ y 0 ( e ) - R p B + z ^ i i [y . ( e ) - R B,] 5 o 0 V 0 j=l 5j 1 J + R [ e i I ( S . . - S )p ] . (5.17) j=l B S p + B From (5.16) we have t h a t S i Q p o = S i Q ( p 0 + f-) = S i o ( - 9 - | But S o P 0 + B o = ' V i = J i = V o f r o m ( 5' 1 2 )- T h e r e f o r e S . o / S 0 = S i o / S 0 -S u b s t i t u t i n g t h i s i n (5.17) we have t h a t J S.. J £.(0) = Z ^ [y (0) - R B.] + R[e. - Z (S..-S.. p , ] , (5.18) and t h e r e f o r e from (5.6) c\(0) = c.j ( 0 ) . Thus the i n d i v i d u a l ' s o p p o r t u n i t y s e t has not changed as a r e s u l t o f the f i r m c h a n g i n g i t s c a p i t a l s t r u c t u r e . S i n c e the i n i t i a l s i t u a t i o n was o p t i m a l f o r a l l i n v e s t o r s as they were a l l i n p o r t f o l i o e q u i l i b r i u m , the f i n a l s i t u a t i o n must be o p t i m a l . To e s t a b -l i s h t h a t t h e second s i t u a t i o n i s an e q u i l i b r i u m , i t o n l y remains t o show t h a t markets a r e c l e a r i n g . The demand f o r the e q u i t y s h a r e s o f the oth f i r m by the i t h i n d i v i d -B ual i s g i v e n by S i Q = S i o + s . n . Summing o v e r a l l i n d i v i d u a l s i = l , , I we g e t t h a t p S„ io' *o o - 83 -i^o • <z Sio> { P 1 ^ ) * £ Sio <VEo'' <5'1 9» Thus the demand f o r s h a r e s has i n c r e a s e d by a f a c t o r o f ( B Q / E 0 ) . However AS /S„ = B / p „ S „ s i n c e AS D = B . and thus t he s u p p l y o f s h a r e s f o r t h e o o o o o o o o f i r s t f i r m i n c r e a s e s by the same f a c t o r and hence a l l s h a r e markets c l e a r . S i m i l a r l y i n the bond market, the d e c r e a s e i n the demand f o r bonds by the o t h f i r m i s B , but the i n c r e a s e i n the demand f o r bonds by i n d i v i d u a l s i s B E'S. [•?-] = B from (5.15) and thus the bond market c l e a r s . Q.E.D. i 1 0 ^o Thus i t i s p o s s i b l e t h a t the f i r m , h a v i n g i n i t i a l l y made a f i n a n c i a l l y c o n s i s t e n t s e t o f d e c i s i o n s , can change i t s d e b t / e q u i t y r a t i o . a n d s t i l l r emain f i n a n c i a l l y c o n s i s t e n t . The crux o f the p r o o f here i s t h a t not o n l y t o t a l market v a l u e remains c o n s t a n t as i n the M o d i g l i a n i - M i l l e r theorem, but the a c t u a l s h a r e p r i c e remains c o n s t a n t , i n s p i t e o f changes i n the c a p i t a l s t r u c t u r e o f the f i r m . An i n t e r e s t i n g q u e s t i o n then a r i s e s as t o the r o b u s t n e s s o f the above 1 i r r e l e v a n c e theorem. Is i t p o s s i b l e t h a t , when the f i r m s u b s t i t u t e s new e q u i t y f o r debt i n i t s f i n a n c i n g mix, the economy w i l l s e t t l e a t an e q u i l i b r i u m w i t h the same t o t a l market v a l u e , but a d i f f e r e n t s h a r e p r i c e ? The answer i s no, p r o v i d e d the s u b s t i t u t i o n o f e q u i t y f o r debt takes p l a c e a t a p a r t i c u l a r r a t i o . C o n s i d e r the s e t o f p o s s i b l e e q u i l i b r i a w i t h c o n s t a n t market v a l u e . Suppose t he f i r m t a k e s - AB^ d o l l a r s o f bonds o f f the market. I f i t i s s u e s AS^ = AB k/P| < new e q u i t i e s , where p^ i s the o r i g i n a l e q u i l i b r i u m p r i c e o f e q u i t y , then the new e q u i l i b r i u m p r i c e o f e q u i t y s h a r e s must be p^ = p^. The ~ over v a r i -a b l e s denotes the e q u i l i b r i u m v a l u e s a f t e r the s u b s t i t u t i o n o f e q u i t y f o r debt by the f i r m . The p r o o f i s s t r a i g h t f o r w a r d . U s i n g t he i d e n t i t y = S k p k + B k ' a n d t n e ^ a c t t n a t w e s t r i c t o u r s e l v e s t o e q u i l i b r i u m w i t h - 84 -c o n s t a n t market v a l u e , we have ( S k + A S k ) ( p k + A p k ) = -AB k•+ V k - B k o r S k P k + AVk + S k A P k + A S k A P k = - A B k + \ " B k . S u b s t i t u t i n g A S k = - B k / p k , V|< = V k = S k p k + B k ' w e n a v e S k A p k - B k A p k / p k = 0 or ( S k + A S k ) A P | < = 0 which i m p l i e s A p k = 0, s i n c e S k + A S k > 0. T h e r e f o r e p k = p k - The p o s s i -b i l i t y o f an e q u i l i b r i u m w i t h A p k = 0 i s ensured by the main theorem o f t h i s s e c t i o n . Thus h o l d i n g t he in v e s t m e n t d e c i s i o n s o f f i r m s and t h e i r t o t a l market v a l u e s c o n s t a n t , we can s u b s t i t u t e new e q u i t y f o r debt i n the r a t i o A S k / A B k = , and m a i n t a i n share p r i c e c o n s t a n t and ensure f i n a n c i a l c o n s i s t e n c y i s m a i n t a i n e d , once e s t a b l i s h e d . I f the f i r m does not f o l l o w t h i s p a r t i c u l a r f i n a n c i a l s t r a t e g y t h e n , even comparing e q u i l -i b r i a w i t h c o n s t a n t market v a l u e s , t h e r e i s no a s s u r a n c e t h e s h a r e p r i c e o f the f i r m w i l l remain c o n s t a n t . C o n s i d e r an extreme case i n which the j t h f i r m reduces i t s demand f o r d e b t , w i t h no o f f s e t t i n g i n c r e a s e i n new e q u i t y . The M+M theorem t e l l s us t h a t t h e r e e x i s t s a new e q u i l i b r i u m w i t h the same market v a l u e on the j t h f i r m as i n the o r i g i n a l e q u i l i b r i u m ; c o n s e q u e n t l y we may have dV. = dE. + dB. = 0. The funds a v a i l a b l e f o r vJ J J i n v e s t m e n t a re F. = d.E. + B.. In t h i s case d. i s c o n s t a n t so dF. = J J J J J J d,- (-dB.) + dB. = ( l - d - ) d B . < 0. Thus f i n a n c i a l c o n s i s t e n c y i s v i o l a t e d , J J J J J the- t o t a l i n c r e a s e i n the market v a l u e o f the e q u i t y g o i n g t o e x i s t i n g s h a r e h o l d e r s . There i s a p o s s i b i l i t y o f i n t e r m e d i a t e c a s e s i n which t he - 85 -f i r m i s s u e s some new e q u i t y , but some o f the i n c r e a s e i n e q u i t y v a l u e goes t o the f i r m , and some t o e x i s t i n g s h a r e h o l d e r s . In t h i s c a s e a s e t o f d e c i s i o n s which were p r e v i o u s l y f i n a n c i a l l y c o n s i s t e n t c o u l d now be i n c o n s i s t e n t . Thus the f o r c e o f the i r r e l e v a n c e theorem i s d i m i n i s h e d somewhat as the f i r m has no way o f e s t a b l i s h i n g a p r i o r what i t s s h a r e p r i c e w i l l be i n e q u i l i b r i u m . Of c o u r s e we have r e s t r i c t e d o u r s e l v e s so f a r to comparing e q u i l i b r i a w i t h c o n s t a n t market v a l u e s on a l l f i r m s . As the economy may w e l l have m u l t i p l e f i n a n c i a l e q u i l i b r i a , t h e r e i s no a s s u r -ance t h a t when a f i r m changes i t s f i n a n c i n g mix, the same e q u i l i b r i u m w i l l be r e - e s t a b l i s h e d . T h e r e f o r e t h e i d e a o f comparing e q u i l i b r i a w i t h con-s t a n t market v a l u e s i s not c o m p l e t e l y j u s t i f i e d w i t h o u t s p e c i f y i n g t he market adjustment p r o c e s s e s . F o r " s m a l l " parameter changes we might assume, however, t h a t f i n a n c i a l e q u i l i b r i a a r e l o c a l l y unique and s t a b l e , and c o n s e q u e n t l y our c o m p a r a t i v e s t a t i c e x e r c i s e s would be v a l i d . The a c t u a l p r o c e s s by which the f i r m s e l e c t s ( d . , B.) has been l e f t J J u n s p e c i f i e d i n t h i s s e c t i o n . T h i s i s the most u n s a t i s f a c t o r y a s p e c t o f t h e s t o c k market economy, and f u r t h e r work i n t h i s a r e a i s c l e a r l y d e s i r -a b l e . 6. Summary and C o n c l u s i o n We compare now the t y p e o f market arrangement o f t h i s model, w i t h p r e v i o u s l m o d e l s o f the s t o c k market. ( i ) The o r i g i n a l s t o c k market model was t h a t o f Diamond [ 1 9 6 7 ] . Diamond's model had i n c o m p l e t e m a r k e t s , but the t e c h n o l o g i c a l / s t o c h a s t i c s p e c i f i c a t i o n was such t h a t t h e f i r m made e f f i c i e n t d e c i s i o n s by maximiz-i n g market v a l u e . In t h i s model, p a r t i c i p a t i o n i n the s t o c k market amounted e s s e n t i a l l y t o t a k i n g p a r t i n a gambling game. I n d i v i d u a l s . t r a d e - 86 -e q u i t y s h a r e s and p r i c e s a r e e s t a b l i s h e d but no r e a l p r o d u c t i v e a c t i v i t y t a k e s p l a c e u n t i l t h e s t a t e o f n a t u r e i s r e a l i z e d . A f t e r the s t a t e o f n a t u r e o b t a i n s i n the second p e r i o d , f i r m s engage i n p r o d u c t i o n u s i n g i n -puts which were c o n t r a c t e d i n the f i r s t p e r i o d . Each f i r m i s assumed t o a v o i d becoming i n s o l v e n t i n t h a t i t never c o n t r a c t s i n the f i r s t p e r i o d t o h i r e more i n p u t s than i t c o u l d pay f o r i n the second p e r i o d w i t h o u t -put i n each s t a t e o f n a t u r e . Thus the f i r m makes no r e a l i n v e s t m e n t d e c i s i o n , and c o n s e q u e n t l y the s t o c k market puts no r e a l c o n s t r a i n t on the f i r m . ( i i ) The s t o c k market models o f S t i g l i t z [1972] and Radner [1972] d i f f e r from the Diamond-type market i n t h a t r e a l i n v e s t m e n t i s undertaken by t h e f i r m i n the f i r s t p e r i o d i n o r d e r t o produce o u t p u t ( u n c e r t a i n ) i n the subsequent p e r i o d . The method o f f i n a n c i n g i n t h e s e models i s , how-e v e r , r a t h e r odd. The i n i t i a l s h a r e h o l d e r s are assumed t o b ear the f u l l c o s t s o f the i n v e s t m e n t d e c i s i o n o f the f i r m , each s h a r e h o l d e r b e a r i n g a p e r c e n t a g e o f the c o s t i n p r o p o r t i o n t o h i s i n i t i a l s h a r e h o l d i n g s . In t h i s manner the budget c o n s t r a i n t on the f i r m i n the f i r s t p e r i o d i s s h i f t e d t o the s h a r e h o l d e r s , i t being assumed i m p l i c i t l y t h a t the f i r m n e ver under-t a k e s so much i n v e s t m e n t as t o d r i v e a s h a r e h o l d e r ' s n e t w e a l t h t o z e r o . In t h e s e models t h e n , no f i n a n c i n g by the f i r m t a k e s p l a c e , and consequent-l y the f i r m f a c e s no c o n s t r a i n t s i n the c a p i t a l market. C o n s i d e r an econ-omy which o p e r a t e s i n the f o l l o w i n g manner. Firms pay f o r i n v e s t m e n t by i s s u i n g r i s k l e s s bonds o n l y , and no f i r m i s s u e s so much debt as t o v i o l a t e the b a n k r u p t c y - c o n d i t i o n . T h i s t y p e o f f i n a n c i n g arrangement i s e q u i v a l e n t to t h a t i n the S t i g l i t z and Radner economies, i n the sense t h a t i n e q u i l -i b r i u m the p o r t f o l i o h o l d i n g s o f i n d i v i d u a l s w i l l be i d e n t i c a l i n b o t h s i t u a t i o n s . - 87 -( i i i ) A t h i r d t y p e o f model i s L e i and,1 s [ 1 9 7 4 ] . H i s s t o c h a s t i c s p e c i f i c a t i o n i s such t h a t a c e r t a i n c o s t ' i s s u b t r a c t e d from a random o u t p u t . A l t h o u g h he does not s t a t e the type o f market o r g a n i z a t i o n he has i n mind t h e r e are two p o s s i b l e i n t e r p r e t a t i o n s . One i s t h a t the mar-k e t works i n the same way as i n Diamond's model, a l l c o n t r a c t s b e i n g p a i d f o r a f t e r the u n c e r t a i n outcome. An a l t e r n a t i v e i n t e r p r e t a t i o n i s t h a t a l l f i n a n c i n g i s bond f i n a n c i n g w i t h the i m p l i c i t assumption t h a t no debt i s i s s u e d beyond the p o i n t o f no b a n k r u p t c y . Thus the e x i s t i n g models o f the s t o c k market economy have c o m p l e t e l y a v o i d e d t r e a t i n g the c a p i t a l market as an i n s t i t u t i o n which s e r v e s to f a c i l i t a t e the s a v i n g / i n v e s t m e n t p r o c e s s . Rather the t r e a t m e n t o f ,the s t o c k market i n t h e s e models f o c u s e s o n l y upon the r i s k - b e a r i n g f u n c t i o n o f e q u i t y m a r k e t s , and as such o n l y h a l f o f the s t o r y has been t o l d . We have attempted i n . t h i s paper t o s u g g e s t a number o f i n t e r e s t i n g problems which a r i s e when the f u n c t i o n s o f the s t o c k market a r e expanded t o i n c l u d e both r i s k b e a r i n g and the f i n a n c i n g o f r e a l i n v e s t m e n t . The examples p r e s e n t e d s u g g e s t t h a t t h e s e two f u n c t i o n s a r e c l o s e l y r e l a t e d . The h o u s e h o l d s e c t o r ' s d e s i r e t o bear r i s k c l e a r l y a f f e c t s the type o f i n v e s t m e n t p r o j e c t s i t i s w i l l i n g t o f u n d , and the q u a n t i t i e s o f f u n d i n g a v a i l a b l e t o d i f f e r e n t i n v e s t m e n t p r o j e c t s . The f i r m s i n the economy can o n l y a v o i d the i n f l u e n c e o f the h o u s e h o l d s e c t o r t o the e x t e n t t h a t they can f i n a n c e t h e i r p r o d u c t i o n a c t i v i t i e s by i s s u i n g r i s k l e s s d e b t . The p o s s i b i l i t i e s f o r t h i s may be e x t r e m e l y l i m i t e d . The major problem i n c o n s t r u c t i n g a complete model o f the s t o c k mar-ke t economy i n which i n v e s t m e n t i s f i n a n c e d i n the s t o c k market i s i n p r o v i d i n g a t h e o r y o f f i r m b e h a v i o u r which a v o i d s f i n a n c i a l i n c o n s i s t e n c y . That i s , f i n a n c i a l myopia i s a p r o p e r t y which must c l e a r l y be e l i m i n a t e d , - 88 -g i v e n the p o s s i b i l i t i e s f o r n o n - e x i s t e n c e o f e q u i l i b r i u m . I t would seem t h a t t h e r e a r e two p o s s i b i l i t i e s h e r e . The f i r s t i s a type o f m o n o p o l i s t i c c o m p e t i t i o n t h e o r y i n which the f i r m maximizes i t s o b j e c t i v e f u n c t i o n sub-j e c t t o a p e r c e i v e d f i n a n c i a l c o n s t r a i n t . We have d i s c u s s e d t h i s t y p e o f a t h e o r y a l r e a d y . The second p o s s i b i l i t y i s more i n t h e c o m p e t i t i v e frame-work, i n which i t i s assumed t h a t ' t h e f i r m f a c e s a budget c o n s t r a i n t which r e q u i r e s t h a t the f i r m c annot i n v e s t more than the market c u r r e n t l y a l l o -c a t e s t o i t i n i n v e s t m e n t f u n d s . The f i r m then a c t s as i f t h i s amount were f i x e d . E q u i l i b r i u m i n t h i s s o r t o f c a p i t a l - r a t i o n i n g model would be c h a r a c t e r i z e d by an e q u i l i b r i u m r a t i o n i n g o f i n v e s t m e n t f u n d s , t o g e t h e r w i t h the usual e q u i l i b r i u m c o n d i t i o n s and f i n a n c i a l c o n s i s t e n c y would be a u t o m a t i c a l l y e n s u r e d . There may o f c o u r s e be o t h e r p o s s i b l e d e s c r i p t i o n s o f f i r m b e h a v i o u r . - 89 -F o o t n o t e s 1. The o r i g i n a l c o n t r i b u t i o n was t h a t o f Diamond [1967]. Subsequent c o n t r i b u t i o n s have been made by Ekern and W i l s o n [1974], Jensen and Long [ 1 9 7 2 ] , L e l a n d [ 1 9 7 3 ] [ 1 9 7 4 ] , Radner [1972][1974] and S t i g l i t z [ 1 972]. 2. The s t a n d a r d r e f e r e n c e t o the Arrow-Debreu economy i s Debreu [1959]. C h a p t e r 7 i n p a r t i c u l a r i n t r o d u c e s u n c e r t a i n t y and d e a l s w i t h the case o f complete c o n t i n g e n t markets. 3. In t h i s paper the terms ' c a p i t a l market' and 'stock market' a r e used synomously, as t h i s market s e r v e s both a r i s k - b e a r i n g and i n v e s t m e n t -a l l o c a t i v e f u n c t i o n , a f e a t u r e which d i s t i n g u i s h e s ' r e a l w o r l d ' s t o c k markets. 4. T h i s i s s u e was r a i s e d i n i t i a l l y by Diamond [1967] and g i v e n an answer i n t h e a f f i r m a t i v e . S t i g l i t z [1972] and Jensen and Long [1972] both came t o the c o n t r a r y c o n c l u s i o n by r e l a x i n g some o f Diamond's s t r i n -gent assumptions. 5. Op. eit. 6. I n v e s t i g a t i o n s a l o n g t h i s l i n e were taken by L e l a n d [1974], Ekern and W i l s o n [1974] and Radner [1974]. L e l a n d [1973] p r o v i d e s a summary o f t h i s l i t e r a t u r e . 7. See s e c t i o n 3.6 below f o r an e x p l a n a t i o n o f how the c o n s t r a i n t was a v o i d e d . 8. F o r the d i s t i n c t i o n between t e c h n o l o g i c a l and p r i c e u n c e r t a i n t y see Diamond [ 1 9 6 7 ] , page 760. 9. N o t i c e the s t r o n g i n f o r m a t i o n a l r e q u i r e m e n t s o f t h i s model. I t must be assumed e i t h e r t h a t a l l i n v e s t o r s know both the t e c h n o l o g i e s and - 90 -i n v e s t m e n t d e c i s i o n s o f a l l f i r m s , o r t h a t f i r m s announce t h e i r p a t -t e r n s o f o u t p u t a c r o s s s t a t e s o f n a t u r e , and t h a t t h e y a r e t r u t h f u l i n t h e i r announcements. 10. R + X fl denotes t he c r o s s p r o d u c t o f R + and fl. l i m + denotes t he l e f t -x+y hand l i m i t as x approaches y from the r i g h t . A.4 i s a t e c h n i c a l assumption which e n s u r e s t h a t i n an i n d i v i d u a l ' s m a x i m i z a t i o n prob-lem, (2.2) below, h i s u t i l i t y f u n c t i o n i s measurable w i t h r e s p e c t t o h i s s u b j e c t i v e p r o b a b i l i t y measure. < 11. Note we a r e assuming the u t i l i t y f u n c t i o n i s s t a t e - i n d e p e n d e n t . By d e a l i n g w i t h a f i n i t e - s t a t e model the e x p e c t e d u t i l i t y h y p o t h e s i s c o u l d be r e l a x e d , as i n Debreu [1959], c h a p t e r 7. 12. T h i s s i m p l y e x c l u d e s the p o s s i b i l i t y o f s h a r e r e p u r c h a s e . T h i s i s not a s e r i o u s r e s t r i c t i o n g i v e n t he t w o - p e r i o d n a t u r e o f the model and the absence o f a l t e r n a t i v e means o f f i n a n c e . 13. x' denotes t he t r a n s p o s e o f the v e c t o r x. 14. See B e r s t e k a s [1974]. 15. The r i s k l e s s f i r m assumption i s r e a l l y q u i t e i n e s s e n t i a l . We use i t o n l y when d e a l i n g w i t h the examples o f s e c t i o n 4. 16. Our v e c t o r n o t a t i o n i s as f o l l o w s . For x,y e R n, x » y i f and o n l y i f x^ > y i f o r a l l i = l , n; x > y i f and o n l y i f x. >_ y^ f o r a l l i = l , ..., n and f o r a t l e a s t one j e {1,2, ..., n}, x. > y . ; x _> y 3 i f and o n l y i f x^ >_y. f o r a l l i = l , n. 0^ denotes t he z e r o v e c t o r i n R n. 17. F o r example see Ka t z n e r [ 1 9 7 0 ] , c h a p t e r 3. P r o p e r t y (4) f o l l o w s from the maximum theorem. See Debreu [1959] p. 19. 18. The e x i s t e n c e o f a f i n a n c i a l e q u i l i b r i u m c o u l d be proved i n a s t r a i g h t f o r w a r d f a s h i o n , from a theorem o f Hart [1974]. Note t h a t - 91 -from A.3 a l l s e c u r i t y r e t u r n s a r e s t r i c t l y p o s i t i v e , and thus e q u i l -i b r i u m market v a l u e s can be taken t o be n o n - n e g a t i v e . 19. A c o r r e s p o n d e n c e ¥ from a m e t r i c space T i n t o a m e t r i c space S i s s a i d t o be upper hemi-continuous a t x e S, i f ¥(x) i* $ and i f f o r e v e r y neighbourhood V o f V(x) t h e r e e x i s t s a neighbourhood V o f x such t h a t V(z) C U f o r e v e r y z e V. The c o r r e s p o n d e n c e ¥ i s s a i d to be upper hemi-continuous i f i t i s upper hemi-continuous a t e v e r y x £ S. The upper h e m i - c o n t i n u i t y o f E(x,d) on i n t - R 2 ^ + ^ f o l l o w s i m m e d i a t e l y from the c o n t i n u i t y o f Z(V,x,d) on i n t R ^ J + ^ . 20. See Debreu [1970]. 21. For example S t i g l i t z [1972] and Jensen and Long [ 1 9 7 2 ] . 22. Radner [1972] appears t o be one o f the f i r s t t o have noted t h i s problem. 23. For a) see M i l l e r and M o d i g l i a n i 1958 ; b) and c ) see the l i t e r a -t u r e on the f i r m under u n c e r t a i n t y , a r e p r e s e n t a t i v e sample b e i n g Baron [ 1 9 7 0 ] , L e l a n d [1972] and Sandmo [1 9 7 1 ] ; d) see the l i t e r a t u r e on u n a n i m i t y ; note 6 above; e) see King [1974]. 24. A c t u a l l y , t he d e f i n i t i o n D.3. r e q u i r e d t h a t E(x,d) t a k e v a l u e s i n R^ + 1. With the a l t e r n a t i v e n o r m a l i z a t i o n on p r i c e s used h e r e , how-e v e r , E(x,d) i s un d e r s t o o d t o take v a l u e s i n A d + 1 . 25. The b a s i c developments o f t h i s model were made by Sharpe [1974], L i n t n e r [1965] and Mossin [1966]. 26. T h i s i s the b a s i c v a l u a t i o n e q u a t i o n o f the c a p i t a l a s s e t p r i c i n g model. See note 25 above. 27. I f the t e c h n o l o g y o f the r i s k l e s s f i r m was c o n s t a n t r e t u r n s t o s c a l e , f o r example y 0 = a Q x 0 , then the o n l y i n t e r e s t r a t e ( r i s k l e s s ) compat-i b l e w i t h f u l l e q u i l i b r i u m would be 1+r = a 0 . As the t e c h n o l o g y used - 92 -i n t h i s example i s s t r i c t l y c oncave, t h e r e i s no reason t h i s s h o u l d be t he c a s e . 28. When the assumption o f i d e n t i c a l e x p e c t a t i o n s f o r f i r m s and i n v e s t o r s i s r e l a x e d a n o t h e r p o s s i b i l i t y o f market f a i l u r e a r i s e s . I f both f i r m managers and i n v e s t o r s have i d e n t i c a l degrees o f r i s k a v e r s i o n , but d i f f e r i n t h e i r p r o b a b i l i t y b e l i e f s , f i r m s w i l l attempt t o i n v e s t i n p r o j e c t s which y i e l d r e t u r n s i n s t a t e s o f n a t u r e which have a hi g h p r o b a b i l i t y o f o c c u r r e n c e r e l a t i v e t o the f i r m s ' b e l i e f s . I n v e s t o r s w i l l , however, v a l u e t h o s e f i r m s lower than o t h e r w i s e , i f th e y do not a s s i g n t he same s t a t e s the same h i g h degree o f p r o b a b i l i t y . T h i s t y p e o f p o s s i b i l i t y c o u l d e a s i l y be a so u r c e o f f i n a n c i a l i n c o n s i s t -ency. 29. The o r i g i n a l M o d i g l i a n i - M i l l e r theorem was p a r t i a l e q u i l i b r i u m propo-s i t i o n . See M o d i g l i a n i and M i l l e r [1958]. S t i g l i t z [1969] gave a g e n e r a l e q u i l i b r i u m v e r s i o n o f the theorem. 30. In p r e v i o u s s e c t i o n s V. = E. as the f i r m i s s u e d e q u i t y o n l y . J J 31. The no b a n k r u p t c y assumption i s s e v e r e l y r e s t r i c t i v e . The consequences o f d r o p p i n g t h i s assumption are q u i t e s t r o n g . See c h a p t e r f o u r i n t h i s t h e s i s . 32. T h i s theorem i s s i m i l a r i n i t s method o f p r o o f t o the g e n e r a l e q u i l -i b r i u m v e r s i o n o f the M+M theorem proved by S t i g l i t z [1969]. 33. Note t h a t we do n o t mean t o imply t h a t A.5. h o l d s . T h i s can be an a r b i t r a r y f i r m w i t h u n c e r t a i n t y as t o i t s d i s t r i b u t i o n o f o u t p u t a c r o s s s t a t e s . - 93 -Ref e r e n c e s Baron, D. [1970], " P r i c e U n c e r t a i n t y , U t i l i t y , and I n d u s t r y E q u i l i b r i u m i n Pure C o m p e t i t i o n " , International Economic Review, 11, 463-80. B e r t s e k a s , D. [1974], "Necessary and S u f f i c i e n t C o n d i t i o n s f o r E x i s t e n c e o f an Optimal P o r t f o l i o " , Journal of Economic Theory, 8, 235-47. Debreu, G. [1959], Theory of Value, New York: W i l e y . [1970], "Economies w i t h a F i n i t e S e t o f E q u i l i b r i a " , Econo-metrica, 28, 387-392. Diamond, P.A. [1967], "The Rol e o f the Stock Market i n a General E q u i l i b -rium Model w i t h T e c h n o l o g i c a l U n c e r t a i n t y " , American Economic Review, 57, 759-776. . Eker n , S. and W i l s o n , R. [ 1 9 7 4 ] , "On the Theory o f the Fi r m i n an Economy w i t h Incomplete M a r k e t s " , Bell Journal of Economics and Management Science, 5, 171-180. H a r t , O.D. [19 7 4 ] , "On the E x i s t e n c e o f E q u i l i b r i u m i n a S e c u r i t i e s Model", Journal of Economic Theory, 9, 293-311. J e n s e n , M.C. and Long, J.B. [1972], " C o r p o r a t e Investment under U n c e r t a i n t y and P a r e t o O p t i m a l i t y i n the C a p i t a l M a r k e t s " , Bell Journal of Economics and Management Science, 3, 151-174. K a t z n e r , D.W. [1 9 7 0 ] , Static Demand Theory, New York: M a c m i l l a n . K i n g , M.A. [19 7 4 ] , " C o r p o r a t e P o l i c y , U n c e r t a i n t y , and the Stock Market", paper p r e s e n t e d t o the c o n f e r e n c e on " E q u i l i b r i u m and D i s e q u i l i b r i u m i n Economic Theory" a t V i e n n a , J u l y 1974. L e l a n d , H.E. [1 9 7 2 ] , "Theory o f the Firm F a c i n g Random Demand", American Economic Review, 62, 238-291. [ 1 9 7 3 ] , " C a p i t a l A s s e t M a r k e t s , P r o d u c t i o n , and O p t i m a l i t y : A S y n t h e s i s " , T e c h n i c a l R e p o r t No. 115, I n s t i t u t e f o r Mathematical S t u d i e s i n the S o c i a l S c i e n c e s , S t a n f o r d U n i v e r s i t y . [ 1 9 7 4 ] , " P r o d u c t i o n Theory and the Stock Market", Bell Journal of Economics and Management Science, 5, 125-143. L i n t n e r , J . [1965], "The V a l u a t i o n o f R i s k A s s e t s and the S e l e c t i o n o f R i s k y Investments i n Stock P o r t f o l i o s and C a p i t a l Budgets", The Review of Economics and Statistics, 47, 13-37. M i c h e l , E. [19 5 6 ] , "Continuous S e l e c t i o n s I " , Annals of'Mathematics,- 63, 361-382. - 94 -M i l l e r , M. and M o d i g l i a n i , F. [1958], "The Cost o f C a p i t a l , C o r p o r a t i o n F i n a n c e , and the Theory o f Investment", American Economic Review, 48, 261-297. M o s s i n , J . [1966], " E q u i l i b r i u m i n a C a p i t a l A s s e t Market", Econometrica, 34, 768-783. Radner, R. [1 9 7 2 ] , " E x i s t e n c e o f E q u i l i b r i u m P l a n s , P r i c e s and P r i c e E x p e c t a t i o n s ,in a Sequence o f Ma r k e t s " , Econometrica, 40, 289-303. [1974], "A Note on the Unani m i t y o f S t o c k h o l d e r s ' P r e f e r e n c e s Among A l t e r n a t i v e P r o d u c t i o n P l a n s : A R e f o r m u l a t i o n o f the Ekern-W i l s o n Model", Bell Journal of Economics and Management Science, 5, 181-184. Sandmo, A. [1971], "On the Theory o f the C o m p e t i t i v e F i r m under P r i c e U n c e r t a i n t y " , American Economic Review, 61, 65-73. Sharpe, W. [1964], " C a p i t a l A s s e t P r i c e s : A Theory o f Market E q u i l i b r i u m under C o n d i t i o n s o f R i s k " , Journal of Finance, 14, 425-492. S t i g l i t z , J . E . [1 9 6 9 ] , "A Re-Examination o f the M o d i g l i a n i - M i l l e r Theorem", American Economic Review, 59, 78-93. [ 1 9 7 2 ] , "On the O p t i m a l i t y o f the Stock Market A l l o c a t i o n o f Investment", Quarterly Journal of Economics,- 86, 25-60. T o b i n , J . [ 1 9 6 9 ] , "A General E q u i l i b r i u m Approach t o Monetary T h e o r y " , Journal of Money, Credit and Banking,- I, 15-30. - 95 -C h a p t e r IV THE COSTS AND CONSEQUENCES OF DEFAULT 1. I n t r o d u c t i o n The a n a l y s i s o f d e f a u l t on c o n t r a c t s i n c a p i t a l markets i s a s u b j e c t o f i n c r e a s i n g i n t e r e s t t o e c o n o m i s t s . * The reasons f o r t h i s i n t e r e s t a r e many. B e f o r e enumerating them, a b r i e f d i s c u s s i o n o f c o n t r a c t t y p e s may, however, be i n o r d e r . C a p i t a l - m a r k e t r e l a t i o n s between economic agents t y p i c a l l y d e a l w i t h f a c i l i t a t i n g t he s a v i n g s - i n v e s t m e n t p r o c e s s ; as a consequence, an e s s e n -t i a l f e a t u r e o f a c a p i t a l market c o n t r a c t i s the s p e c i f i c a t i o n o f i n t e r -temporal terms o f exchange. Because o f the absence o f complete markets i n the Arrow-Debreu sense and s i n c e markets o p e r a t e s e q u e n t i a l l y a t each d a t e , two i m p o r t a n t f e a t u r e s o f a c a p i t a l market c o n t r a c t a r e u n c e r t a i n t y and i m p e r f e c t i n f o r m a t i o n . The u n c e r t a i n t y and i m p e r f e c t i n f o r m a t i o n may r e l a t e t o the date o f re-payment, the i d e n t i t y o f t h e o t h e r p a r t y , the p r o b a b i l i t y o f re-payment, the amount o f payment, and so f o r t h . Most c a p i t a l - m a r k e t c o n t r a c t s can be c l a s s i f i e d i n t o two t y p e s . The t y p e o f c o n t r a c t most f a m i l i a r i n economic t h e o r y i s the i n s u r a n c e con-t r a c t . T h i s s t a t e s t h a t , c o n d i t i o n a l on a c e r t a i n e v e n t o c c u r r i n g which i s i d e n t i f i a b l e t o both p a r t i e s , a c e r t a i n amount o f money o r goods s h a l l be p a i d o r d e l i v e r e d . A second ty p e o f c o n t r a c t i s the u n c o n d i t i o n a l c o n t r a c t which merely p r o v i d e s a c l a i m a g a i n s t some income stream and u s u a l l y s i g n i f i e s ownership o f some i d e n t i f i a b l e a s s e t . The most common type o f c o n t r a c t o f t h i s s o r t i s the common s t o c k o r e q u i t y o f a c o r p o r -a t i o n . T h e r e i s n e i t h e r an e v e n t o r d a t e c o n d i t i o n i n g c l a u s e t o t h i s type - 96 -o f c o n t r a c t which promises payment o f any s e t amount. Some c o n t r a c t s a r e o f c o u r s e a m i x t u r e o f the two. An i m p o r t a n t f e a t u r e o f c o n d i t i o n a l c o n t r a c t s i s the p o s s i b i l i t y o f d e f a u l t ; t h a t i s , the terms o f t h e c o n t r a c t w i l l not be met by one p a r t y o r the o t h e r . Much o f the u n c e r t a i n t y i n v o l v e d i n t h e s e type o f c o n t r a c t s r e l a t e s t o the p o s s i b i l i t y and the c i r c u m s t a n c e s under which d e f a u l t may o c c u r . As a r e s u l t o f t h e d e f a u l t f e a t u r e many so c a l l e d c a p i t a l - m a r k e t i m p e r f e c t i o n s can be r e a d i l y e x p l a i n e d . F o r example, the c o l l a t e r a l i n a b o r r o w e r - l e n d e r r e l a t i o n becomes i m p o r t a n t as a means o f p r o v i d i n g i n s u r -ance a g a i n s t d e f a u l t , and as a means o f p r o v i d i n g i n f o r m a t i o n t o the l e n -3 d e r about the p r o b a b i l i t y o f the borrower r e - p a y i n g . The purpose o f t h i s c h a p t e r i s t o c o n s i d e r the n a t u r e o f the e q u i l i b -rium i n a c a p i t a l market when d e f a u l t i s a p o s s i b i l i t y . In p a r t i c u l a r , we s h a l l be c o n c e r n e d w i t h the d e t e r m i n a t i o n o f the e q u i l i b r i u m p r i c e o f l o a n s on which d e f a u l t may o c c u r and the d e t e r m i n a t i o n o f the p r o b a b i l i t y o f d e f a u l t , which i s a market-determined v a r i a b l e i n the models c o n s i d e r e d . F u r t h e r m o r e , we s h a l l c o n s i d e r how t h e s e v a r i a b l e s r e l a t e t o c e r t a i n p a r -ameters, such as the r i s k l e s s i n t e r e s t r a t e , the p r o d u c t i v i t y o f i n v e s t -ment, the c o l l a t e r a l v a l u e o f the l o a n , and the e x p e c t a t i o n s o f borrowers and l e n d e r s i n the market. The a n a l y s i s i s p a r t i a l e q u i l i b r i u m to the e x t e n t t h a t t h e c a p i t a l market i s t r e a t e d i n i s o l a t i o n , but i t c o u l d be 4 g i v e n a s i m p l e g e n e r a l e q u i l i b r i u m i n t e r p r e t a t i o n . The r e s t o f t h e e s s a y proceeds as f o l l o w s . In s e c t i o n 2 t h e r e i s a g e n e r a l d i s c u s s i o n o f d e f a u l t and i t s economic s i g n i f i c a n c e . S e c t i o n 3 d e v e l o p s a s i m p l e model o f a c a p i t a l market w i t h one t y p e o f s e c u r i t y a v a i l a b l e f o r i n v e s t m e n t p u r p o s e s . The f o c u s here i s on the demand s i d e o f the l o a n market. S e c t i o n 4 c o n s i d e r s the r o l e o f r i s k a v e r s i o n and - 97 -the n a t u r e o f the demand f u n c t i o n f o r i n v e s t m e n t l o a n s when d e f a u l t i s a p o s s i b i l i t y . The next s e c t i o n c o n s i d e r s the same market, o n l y t h i s time i t has the f e a t u r e t h a t the e q u i l i b r i u m p r i c e and p r o b a b i l i t y o f d e f a u l t a r e . d e t e r m i n e d by s u p p l y f a c t o r s . S e c t i o n 6 then c o n s i d e r s the c o m p l i -c a t i o n s when an a d d i t i o n a l s e c u r i t y , e q u i t y , i s i n t r o d u c e d . The l a s t s e c t i o n p r e s e n t s some c o n c l u s i o n s . 2. D e f a u l t : Some General C o n s i d e r a t i o n s , The s i g n i f i c a n c e o f d e f a u l t f o r m a t t e r s which have t r a d i t i o n a l l y con-c e r n e d e c o n o m i s t s , namely r e s o u r c e a l l o c a t i o n , w e l f a r e , and d i s t r i b u t i o n i s not g e n e r a l l y a p p r e c i a t e d . Some might argue t h a t the o r d e r s o f magni-tude i n v o l v e d a r e a t the second o r t h i r d d ecimal p l a c e ; c o n s e q u e n t l y , i t i s not a m a t t e r we s h o u l d be much concerned about. There a r e , however, reaso n s t o b e l i e v e such a view i s not c o r r e c t . The reasons a r e as f o l l o w s . 1. E m p i r i c a l l y d e f a u l t i s c l o s e l y c o r r e l a t e d t o the b u s i n e s s c y c l e and economic growth. C o r p o r a t e b a n k r u p t c i e s a r e a common phenomena. F o r example, i n the U n i t e d S t a t e s a p p r o x i m a t e l y 44 per 10,000 b u s i n e s s e s f a i l e d i n 1970, w i t h an a v e r a g e l i a b i l i t y p e r f a i l u r e o f $175,000. De-f a u l t on consumer c r e d i t and mortgages i s a phenomenon we a r e a l l aware o f , and s p e c i a l i z e d i n s t i t u t i o n s have a r i s e n t o d e a l w i t h i t . There appears t o be a d i f f i c u l t i d e n t i f i c a t i o n problem i n d e t e r m i n i n g the o r d e r o f magnitude i n v o l v e d i n d e f a u l t . Some t a k e o v e r s and mergers, f o r example, may i n f a c t be p a r t i a l s u b s t i t u t e s f o r d e f a u l t . F i n a n c i a l i n s t i t u t i o n s o f t e n extend c r e d i t t o t h e i r borrowers i n s i t u a t i o n s t h e y would n o r m a l l y f e e l i t was not prudent t o do s o ; the r e a s o n b e i n g , t h a t were c r e d i t not e x t e n d e d , the borrower would d e f a u l t on h i s l o a n and the f i n a n c i a l company i n v o l v e d would have t o bear the s u p e r v i s i o n and a d m i n i s t r a t i o n c o s t s - 98 -a s s o c i a t e d w i t h the d e f a u l t . T h e r e f o r e many c o n t r a c t s which do d e f a u l t may not be i d e n t i f i e d as so d o i n g because i n s t i t u t i o n s have d e v e l o p e d which e f f e c t i v e l y d i s g u i s e d e f a u l t by w r i t i n g new c o n t r a c t s or m o d i f y i n g t h e terms o f the o l d c o n t r a c t s . 2. The l i t e r a t u r e on c o r p o r a t e f i n a n c e has c o n s i d e r e d the e f f e c t o f bank-r u p t c y on t h e f i r m ' s o p t i m a l f i n a n c i a l mix. As soon as t h e p o s s i b i l i t y o f bonds becoming r i s k y a s s e t s i s a d m i t t e d , the M o d i g l i a n i - M i l l e r theorem i s not g e n e r a l l y t r u e and by v a r y i n g the d e b t / e q u i t y r a t i o t h e f i r m can a f f e c t i t s market v a l u e . ^ Thus d e f a u l t on c o r p o r a t e bonds s h o u l d be an i m p o r t a n t f a c t o r i n e x p l a i n i n g t h e i r v a l u a t i o n and i n the r e l a t i v e s u p p l i e s o f debt and e q u i t y , S e c o n d l y S t i g l i t z [1972] has demonstrated t h a t w i t h b a n k r u p t c y the r e a l p r o d u c t i o n d e c i s i o n s o f the f i r m a r e not independent o f i t s f i n -a n c i n g d e c i s i o n s . Thus i n e x p l a i n i n g the i n v e s t m e n t b e h a v i o u r o f the c o r p o r a t e s e c t o r , one s h o u l d pay a t t e n t i o n t o the c a p i t a l s t r u c t u r e o f c o r p o r a t i o n s and the d e f a u l t r i s k on t h e i r d e b t . 3. From t h e p o i n t o f view o f g e n e r a l e q u i l i b r i u m t h e o r y , d e f a u l t on con-t r a c t s has i m p o r t a n t i m p l i c a t i o n s i n economies which have markets o p e r a t -i n g s e q u e n t i a l l y o v e r t i m e . 7 A t any d a t e t h e r e a r e p r e - e x i s t i n g c o n t r a c t s which were w r i t t e n a t an e a r l i e r d a t e and whose terms must be met i n the c u r r e n t p e r i o d . Depending upon the v a l u e s o f c u r r e n t ( s p o t ) e q u i l i b r i u m p r i c e s some o f t h e s e p r e - e x i s t i n g c o n t r a c t s may d e f a u l t . An economy con-s i s t s . ; n o t o n l y o f a s e t o f markets, but a l s o a system o f r u l e s which de-t e r m i n e s the d i s t r i b u t i o n o f the l o s s on d e f a u l t i n g c o n t r a c t s amongst agents i n the economy. In a d d i t i o n the s e t o f r u l e s must be c o n s i s t e n t ; once a g e n t s t a k e t h e s e r u l e s i n t o a c c o u n t i n making c o n t r a c t s the r e s u l t -i n g demand and s u p p l y c o r r e s p o n d e n c e s must have the a p p r o p r i a t e p r o p e r t i e s - 99 -i n o r d e r t h a t e x i s t e n c e o f an e q u i l i b r i u m can be p r o v e d . 4 . A g r e a t d e a l o f the r e s o u r c e c o s t s o f t r a n s a c t i o n s i n v o l v e d ' i n . r u n n i n g c a p i t a l markets a r e due t o the d e f a u l t f e a t u r e o f c o n t r a c t s . P r e -sumably, i f d e f a u l t were not a p o s s i b i l i t y , then anyone e n t e r i n g a c o n t r a c t a t the e x i s t i n g p r i c e would be 100% c e r t a i n o f meeting the terms o f t h e c o n t r a c t . Under what c o n d i t i o n s might t h i s be t r u e ? C l e a r l y i t would r e q u i r e an i n d i v i d u a l h a v i n g v e r y d e t a i l e d i n f o r m a t i o n as t o h i s f u t u r e p o s i t i o n and the r e l a t i v e p r i c e s a t f u t u r e d a t e s . S e c o n d l y , i t would r e -q u i r e a code o f e t h i c s adhered t o by a l l agents p a r t i c i p a t i n g i n the market under which no one would ohoose t o d e f a u l t on a c o n t r a c t . The f a c t t h a t both f e a t u r e s a r e h a r d l y c h a r a c t e r i s t i c o f any ' r e a l w o r l d ' c a p i t a l markets i n d i c a t e s t h a t d e f a u l t i s a s i g n i f i c a n t p o s s i b i l i t y . Because o f the p o s s i b i l i t y o f d e f a u l t a f i r m o r i n d i v i d u a l making a o l o a n d e s i r e s c e r t a i n i n f o r m a t i o n which he would not o t h e r w i s e r e q u i r e . F i r s t he needs i n f o r m a t i o n about the c i r c u m s t a n c e s under which d e f a u l t w i l l o c c u r . These r e l a t e t o both e x t e r n a l c i r c u m s t a n c e s , i . e . , t h o s e be-yond the c o n t r o l o f the borrower, and t h o s e c i r c u m s t a n c e s which the behav-i o u r o f the borrower might m o d i f y . For example, i f the borrower i s a f i r m then some i n f o r m a t i o n about i t s c h o i c e o f t e c h n i q u e i s r e l e v a n t t o d e t e r -m i n i n g the p o s s i b i l i t y o f d e f a u l t . A problem o f the 'moral h a z a r d ' v a r i e t y may f a c e the l e n d e r i n t h i s c a s e s i n c e a borrower may choose t o d e f a u l t i n c i r c u m s t a n c e s he c o u l d a v o i d , were he more d i r e c t l y r e s p o n s i b l e f o r t h e - o u t -come o f h i s a c t i o n s . The l o a n company, c o n s e q u e n t l y , may i n c u r c o s t s m o n i t o r i n g t h e borrower:'s."actions i n an attempt t o reduce the p r o b a b i l i t y o f d e f a u l t . Second, i n f o r m a t i o n i s r e q u i r e d about the e x t e n t t o which the l o a n company can r e c o v e r i t s l o s s e s i f a borrower d e f a u l t s . T h i s i n v o l v e s - 100 -some d e t a i l e d knowledge o f the c o l l a t e r a l s e c u r i n g the l o a n , i t s f u t u r e g v a l u e , and the ease w i t h which i t can be marketed. 5. Investment i n r e a l c a p i t a l , i n c l u d i n g consumer d u r a b l e s , r e a l e s t a t e , r e s o u r c e e x t r a c t i o n , and p l a n t and equipment, i s s i g n i f i c a n t l y a f f e c t e d by the d e f a u l t r i s k a s s o c i a t e d w i t h the f i n a n c i a l c o n t r a c t s u n d e r l y i n g the r e a l c a p i t a l f o r m a t i o n . The p r o b a b i l i t y o f d e f a u l t i s , o f c o u r s e , a s u b j e c t i v e m a t t e r , and thus d i f f e r e n c e s i n e x p e c t a t i o n s between borrowers and l e n d e r s , and d i f f e r e n c e s i n r i s k a v e r s i o n u n d o u b t e d l y a f f e c t the o u t -come o f the i n v e s t m e n t p r o c e s s . For example, c o n s i d e r the s i t u a t i o n K e y n e s 1 0 c o n s i d e r e d o f a ' c o l l a p s e ' i n i n v e s t o r s e x p e c t a t i o n s ; t h i s might be though o f as a deepening pessimism, e i t h e r on the p a r t o f s a v e r s o r on the p a r t o f f i r m s o r i n d i v i d u a l s u n d e r t a k i n g the r e a l i n v e s t m e n t . S u r e l y the q u a n t i t y and a l l o c a t i o n o f i n v e s t m e n t s h o u l d depend upon who t h i n k s what. The s i g n i f i c a n c e o f d e f a u l t i n a f f e c t i n g r e a l c a p i t a l a c c u m u l a t i o n i s enhanced by t h e o b s e r v a t i o n t h a t t h e r e are v e r y few second-hand c a p i t a l good markets o r r e n t a l markets f o r c a p i t a l goods as t r a d i t i o n a l n e o c l a s s i -c a l i n v e s t m e n t t h e o r y assumes. I f a purchase o f a c a p i t a l good i s i r r e -v e r s i b l e , o r : c a p i t a l i s o f a p u t t y - c l a y n a t u r e , and t h i s purchase must be f i n a n c e d w i t h a l o a n , t h e n , s h o u l d c o n d i t i o n s t u r n o u t t o be u n f a v o u r a b l e t o the i n v e s t m e n t , t h e r e would be l i t t l e the l o a n company c o u l d do t o c o v e r i t s l o s s e s ; the c a p i t a l good i s n e i t h e r s a l e a b l e , nor would i t be p o s s i b l e t o r e n t i t o u t t o f i r m s e x p e r i e n c i n g more f a v o u r a b l e c o n d i t i o n s . Summarizing t h i s s e c t i o n , i t i s f a i r t o say d e f a u l t on c o n t r a c t s i s not an i n s i g n i f i c a n t m a t t e r and a complete e x a m i n a t i o n o f the causes and consequences o f d e f a u l t i s n e c e s s a r y t o a b e t t e r u n d e r s t a n d i n g o f the manner i n which c a p i t a l markets f u n c t i o n . - 101 -3. E q u i l i b r i u m i n a Loan Market: Demand Determined Loan P r i c e In t h i s s e c t i o n a s i m p l e model o f a c a p i t a l market w i t h a s i n g l e t y p e o f c o n t r a c t i s d e v e l o p e d i n a p a r t i a l e q u i l i b r i u m c o n t e x t , a l t h o u g h i t c o u l d be extended t o a s i m p l e g e n e r a l e q u i l i b r i u m c o n t e x t . The-model o f t h i s s e c t i o n has the c h a r a c t e r i s t i c t h a t t he e q u i l i b r i u m l o a n p r i c e and p r o b a b i l i t y o f d e f a u l t a r e d e t e r m i n e d by the demand s i d e o f the c a p i t a l market, t h a t i s , the borrowers o f in v e s t m e n t f u n d s . Borrowers have a c c e s s t o r e a l i n v e s t m e n t o p p o r t u n i t i e s , but th e y need t o a c q u i r e t he n e c e s s a r y funds i n o r d e r t o undertake t he in v e s t m e n t . In most o f the a n a l y s i s we s h a l l m a i n t a i n t he assumption o f r i s k n e u t r a l i t y i n o r d e r t o f o c u s on the d e f a u l t i s s u e and the r o l e o f e x p e c t a t i o n s . 1 1 The type o f l o a n a v a i l a b l e i s a s i m p l e o n e - p e r i o d debt c o n t r a c t which i s - r e p a i d i n the second p e r i o d , p r o v i d e d t he borrower does not d e f a u l t . Borrowers have a c c e s s t o a c o n s t a n t - r e t u r n s - t o - s c a l e t e c h n o l o g y o r inv e s t m e n t p r o j e c t . For a r e a l i n v e s t m e n t o r i n p u t i n t h e f i r s t p e r i o d , denoted by x, the i n v e s t m e n t p r o j e c t g i v e s a r e t u r n i n the second p e r i o d o f y ( x , G ) , where 9 denotes a s t a t e o f n a t u r e v a r i a b l e , an element o f the s t a t e space fl. More s p e c i f i c a l l y i t i s assumed A . l . y ( x , e ) = r ( 6 ) x , x >_ 0 r ( e ) > 0 f o r a l l 6efl. A.2. r ( 6 ) = r s , r > 0 se[0,M]. The p o s i t i v e s c a l a r s p a r a m e t e r i z e s the s t a t e space and ta k e s v a l u e s i n the i n t e r v a l [0,M], I t i s assumed t h a t the economy c o n s i s t s o f many borrowers o r f i r m s w i t h a c c e s s t o the i d e n t i c a l t e c h n o l o g y g i v e n by A . l and A.2. E n t r y and - 102 -e x i t i . o f f i r m s i s c o s t l e s s and i s presumed t o take p l a c e i f the e x p e c t e d r a t e o f p r o f i t on the i n v e s t m e n t p r o j e c t s d i f f e r s from an exogenous r a t e o f r e t u r n , p, which i s t a k e n t o be g r e a t e r than z e r o . We s h a l l show t h a t a p equal t o z e r o p r e c l u d e s the e x i s t e n c e o f an e q u i l i b r i u m i n the c a p i t a l market when d e f a u l t on l o a n c o n t r a c t s i s a d m i s s a b l e . A p o s i t i v e oppor-t u n i t y c o s t can be r a t i o n a l i z e d on two grounds. I f one views t h e s e bor-rowers as r i s k n e u t r a l f i r m s , then p r e p r e s e n t s the p r o f i t r a t e t o the i n d u s t r y on f i x e d f a c t o r s . F o r example, i f a u n i t o f i n v e s t m e n t r e q u i r e s a u n i t o f l a n d , then p r e p r e s e n t s the c o m p e t i t i v e r e n t a l t o l a n d . On the o t h e r hand i f borrowers a r e i n d i v i d u a l s , the p can be viewed s i m p l e as t h e i r o p p o r t u n i t y c o s t t o i n v e s t i n g , perhaps g i v e n by the r a t e o f r e t u r n on a r i s k l e s s s e c u r i t y . We s h a l l assume t h a t the c o m p e t i t i v e p r o c e s s , through the demand f o r i n v e s t m e n t f u n d s , f o r c e s the l o a n p r i c e , denoted by q, t o such a l e v e l t h a t the e x p e c t e d p r o f i t r a t e e q u a l s p. Borrowers may choose t o d e f a u l t on t h e i r l o a n s i f t h e y w i s h , the o n l y p e n a l t y b e i n g t h a t they l o s e a l l c l a i m s to r e t u r n s from t h e i n v e s t m e n t p r o j e c t , i t b e i n g assumed t h a t the l e n d e r s i n the market have a c q u i r e d t h e s e c l a i m s . T h i s t r a n s f e r o f ownership w i l l be t r e a t e d more e x p l i c i t l y i n s e c t i o n 5. With r e g a r d t o e x p e c t a t i o n s we assume: B . l . A l l borrowers have i d e n t i c a l e x p e c t a t i o n s as t o the s t a t e o f the w o r l d . These a r e r e p r e s e n t e d by the d i s t r i b u t i o n f u n c t i o n F ( s ) , w i t h s u p p o r t i n the i n t e r v a l [ 0,M], The d i s t r i b u t i o n may a l s o be r e p r e s e n t e d by a c o n t i n u o u s p r o b a b i l i t y d e n s i t y f u n c t i o n , p ( s ) d s = d F ( s ) . In much o f the subsequent a n a l y s i s we s h a l l be c o n c e r n e d w i t h changes i n e x p e c t a t i o n s , i n which case the d i s t r i b u t i o n i s p a r a m e t e r i z e d w i t h a s c a l a r - 103 -t , F ( s , t ) a l s o b e i n g presumed d i f f e r e n t i a t e i n t . The g r o s s r e t u r n s t o an i n v e s t m e n t p r o j e c t , g i v e n an outcome s and i n p u t x, a r e by A . l and A.2 r s x . The borrowing c o s t s o r the amount prom-i s e d t o repay on a l o a n o f x a r e g i v e n by qx, where 1/q i s the p r i c e o f a l o a n or q = 1+r i s t h e g r o s s i n t e r e s t r a t e on t h e s e l o a n s . The p r i c e o f i n v e s t m e n t goods w i l l always be s e t equal t o u n i t y . As borrowers a r e assumed t o maximize e x p e c t e d e n d - o f - p e r i o d p r o f i t s , t h e y w i l l choose t o d e f a u l t on t h e i r l o a n s i n any s t a t e i n which net r e c e i p t s a r e n e g a t i v e o r r s x < qx. Thus the n e t r e t u r n s t o a borrower from u n d e r t a k i n g an i n v e s t -ment p r o j e c t a r e g i v e n by !r s x - qx i f s >_ q/r 0 i f s < q/r,. T h e r e f o r e , e x p e c t e d p r o f i t s t o a borrower a r e g i v e n by M En = / q / r [ r s x - q x ] d F ( s ) M M = r x / q / r s d F ( s ) - qx / q / r d F ( s ) , (3.1) 1 p where the c a p i t a l E denotes the e x p e c t a t i o n o p e r a t o r . With the f r e e e n t r y / e x i t a ssumption, i n o r d e r f o r the demand f o r l o a n s to be a p o s i t i v e f i n i t e amount, which i s a n e c e s s a r y c o n d i t i o n f o r l o a n market e q u i l i b r i u m , the e x p e c t e d r a t e o f p r o f i t must be equal t o the exogenous r a t e o f r e t u r n p. M M r / q / r s d F ( s ) - q / q / r d F ( s ) = p. (3.2) The p r i c e q i s now presumed t o be the e q u i l i b r i u m p r i c e . L e t q / r = z and p/r = v. From b a s i c p r o b a b i l i t y t h e o r y we can r e - w r i t e (3.2) as - 104 -r E ( s | s > z) = q + p/R(z) (3.3) where E ( s | s _> z) i s the c o n d i t i o n a l mean o f s, c o n d i t i o n e d on the f a c t M t h a t the l o a n i s not d e f a u l t e d , and R(z) = / d F ( s ) , i s the p r o b a b i l i t y o f not d e f a u l t i n g . (3.2) can a l s o be w r i t t e n as M M f ( z ) = fz s d F ( s ) - z /_, d F ( s ) = v. (3.4) Thus, the market has an e q u i l i b r i u m i f t h e r e e x i s t s a z*e[0,M] such t h a t f ( z * ) = v. Note t h a t once z* i s d e t e r m i n e d , both an e q u i l i b r i u m l o a n p r i c e q = z * r , and an e q u i l i b r i u m p r o b a b i l i t y o f d e f a u l t F ( z * ) a r e d e t e r -mined. H e n c e f o r t h z* w i l l be r e f e r r e d t o as the d e f a u l t p o i n t . N o t i c e a l s o t h a t changes i n the l o a n r a t e q have two e f f e c t s . F i r s t , q a f f e c t s the d i r e c t p r o d u c t i v i t y o f the i n v e s t m e n t by a f f e c t i n g net r e c e i p t s ; s econd, i t a f f e c t s t he p r o b a b i l i t y w i t h which d e f a u l t o c c u r s . T h i s second f e a t u r e makes the d e f a u l t problem, and the s u p p l y and demand o f s e c u r i t i e s on which d e f a u l t can o c c u r , d i f f e r e n t from the t r e a t m e n t o f s e c u r i t i e s on which d e f a u l t c annot o c c u r . Now f ( z ) by assumption B . l i s c o n t i n u o u s and d i f f e r e n t i a t e on the i n t e r v a l [0,M]. F u r t h e r m o r e , f ( 0 ) = E ( s ) = s and f(M) = 0 . Thus, by the mean-value theorem f ( z ) t a k e s on a l l v a l u e s i n [ 0 , s ] . T a k i n g t he d e r i v a -t i v e o f f ( z ) we have M f ' ( z ) = - z d F ( z ) - d F ( s ) + z d F ( z ) M = -/ d F ( s ) < 0, f o r a l l zefO.M]. (3.5) T h i s i m p l i e s t h a t f ( z ) i s u n i q u e l y i n v e r t i b l e on [0,M] and the o n l y s o l u -t i o n o f f ( z ) = 0 , i s z = M. T h e r e f o r e i f p = 0, the o n l y l o a n r a t e com-p a t i b l e w i t h e q u i l i b r i u m i s q = Mr, w i t h the p r o b a b i l i t y o f d e f a u l t b e i n g - 105 -equal t o one. Thus a l l l o a n s would d e f a u l t w i t h c e r t a i n t y , and t h i s would not appear t o be a r e a s o n a b l e s o r t o f c a p i t a l - m a r k e t e q u i l i b r i u m . Now i f v e ( 0 , s ) , then f ( z ) = v p o s s e s s e s a unique s o l u t i o n z*e(0,M) and q = z * r . Hence we have demonstrated P r o p o s i t i o n 3.1: I f p e ( 0 , r l ) , then the l o a n market has a unique e q u i l i b -rium l o a n r a t e and a unique e q u i l i b r i u m p r o b a b i l i t y o f d e f a u l t , F ( z * ) , such t h a t 0 < F ( z * ) < 1. Thus d e f a u l t o c c u r s w i t h p o s i t i v e p r o b a b i l i t y but not w i t h c e r t a i n t y . T h i s a n a l y s i s g i v e s us a model o f a l o a n market which has i t s p r i c e d e t e r m i n e d s o l e l y by demand c o n d i t i o n s . N o t i c e t h a t because o f the con-s t a n t r e t u r n s assumption the i n d i v i d u a l l e v e l s o f x a r e i n d e t e r m i n a t e . I n t r o d u c t i o n o f a s u p p l y s i d e would determine the a g g r e g a t e amount o f i n v e s t m e n t . We wish now t o c o n s i d e r the e f f e c t s o f changes i n the parameters o f the market on the e q u i l i b r i u m v a r i a b l e s q* and F ( z * ) . G i v e n the assumption o f r i s k n e u t r a l i t y the parameters a r e r , the p r o d u c t i v i t y o f i n v e s t m e n t , p, the exogenous r a t e o f p r o f i t o r o p p o r t u n i t y c o s t o f i n v e s t i n g , and F ( s , t ) , the e x p e c t a t i o n s o f the b o r r o w e r s . In o r d e r t o do the c o m p a r a t i v e s t a t i c s we d i f f e r e n t i a t e the e q u i l i b -rium e q u a t i o n f ( z * ) = v and the i d e n t i t i e s q* = z * r and p = v r . f ' ( z * ) d z * = dv (3.6) dq = z*dr + r d z * (3.7) dp = c d r + r d v . (3.8) C o n s i d e r now changes i n r , i . e . , d r > 0, dp = 0. R e c a l l i n g t h a t f ' ( z * ) < 0 we have d r = — < 0 and thus d z * > 0. From (3.7) then dq* > 0. - 106 -P r o p o s i t i o n 3.2: An i n c r e a s e i n the p r o d u c t i v i t y o f i n v e s t m e n t causes an i n c r e a s e i n the e q u i l i b r i u m l o a n r a t e and an i n c r e a s e i n the e q u i l i b r i u m p r o b a b i l i t y o f d e f a u l t . T h i s p r o p o s i t i o n i s r a t h e r i n t e r e s t i n g i n t h a t the more p r o d u c t i v e the i n v e s t m e n t , the h i g h e r the p r o b a b i l i t y o f d e f a u l t . I t i s sometimes argued on the b a s i s o f c a s u a l e m p i r i c i s m t h a t p r o j e c t s w i t h h i g h e x p e c t e d r e t u r n s tend to be " r i s k i e r " . I f the r i s k o f a p r o j e c t t o the l e n d e r i s i n some sense p o s i t i v e l y r e l a t e d t o the s i z e o f the p r o b a b i l i t y o f d e f a u l t , then the above p r o p o s i t i o n shows t h a t the c o r r e l a t i o n between " r i s k " and p r o -d i c t i v i t y may be a r e s u l t o f the n a t u r e o f e q u i l i b r i u m i n the c a p i t a l market; t h a t i s , it.may be an endogenously g e n e r a t e d phenomenon r a t h e r than a t e c h n o l o g i c a l f a c t . C o n s i d e r i n g changes i n p (dp > 0, d r = 0 ) we have u s i n g (3.6) through (3.7) t h a t d z * < 0 and dq* < 0. Hence P r o p o s i t i o n 3.3: An i n c r e a s e i n the r e q u i r e d p r o f i t r a t e o f the i n d u s t r y ( o r the o p p o r t u n i t y c o s t o f i n v e s t i n g ) d e c r e a s e s the e q u i l i b r i u m l o a n r a t e and d e c r e a s e s the e q u i l i b r i u m p r o b a b i l i t y o f d e f a u l t . I f we view p as " t h e " r i s k l e s s i n t e r e s t r a t e ( g r o s s ) o f the economy then p r o p o s i t i o n 3.3 says t h a t h i g h e r i n t e r e s t r a t e s on r i s k l e s s v e n t u r e s i s a s s o c i a t e d w i t h lower i n t e r e s t r a t e s on r i s k y i n v e s t m e n t s , an i n v e r s e r e -l a t i o n s h i p . N o t i c e t h a t the h i g h e r the r i s k l e s s i n t e r e s t r a t e the l e s s r i s k y a r e the p r o j e c t s as measured by the p r o b a b i l i t y o f d e f a u l t . In t h i s c a s e , the r i s e i n the e x p e c t e d r e t u r n t o i n v e s t i n g by d e c r e a s i n g the prob-a b i l i t y o f d e f a u l t i s g r e a t e r than the f a l l i n the e x p e c t e d r e t u r n due t o the l o w e r i n g o f the l o a n r a t e . The d e f a u l t e f f e c t o f l o w e r i n g t h e l o a n - 107 -r a t e swamps the revenue e f f e c t . We have been a s s o c i a t i n g the term " r i s k " w i t h the p r o b a b i l i t y o f de-f a u l t . We w ish now t o c o n s i d e r changes i n the e x p e c t a t i o n s o f borrowers such t h a t from t h e economy's v i e w p o i n t the i n v e s t m e n t p r o j e c t appears r i s k i e r . To do t h i s we s h a l l use the R o t h s c h i l d - S t i g l i t z n o t i o n o f a m e a n - p r e s e r v i n g i n c r e a s e i n r i s k . L e t F ( s , t ) be a f a m i l y o f d i s t r i b u t i o n f u n c t i o n s w i t h s u p p o r t i n [0,M] and i n d e x e d by t . Then as Diamond and S t i g l i t z [1974] have shown, i n c r e a s e s i n t ccorrespond t o i n c r e a s e s i n r i s k i n the sense o f R o t h s c h i l d and S t i g l i t z [1970] i f M / Q d F ( s i t ) = 1 , x fQ F t ( s , t ) d s > 0, 0 < x < M, and M / Q F t ( s , t ) d s = 0. Now, u s i n g i n t e g r a t i o n by p a r t s we r e - w r i t e the e q u i l i b r i u m e q u a t i o n (3.4) as M • M - z * F ( z * , t ) - / * F ( s , t ) d s - z * [ l - F ( z * , t ) ] = v. (3.9) C a n c e l l i n g terms and r e - w r i t i n g M M - / * F ( s , t ) d s - z* = v. (3.10) Now d i f f e r e n t i a t i n g (3.10) w i t h r e s p e c t t o z* and t we have M F ( z * , t ) d z * - dz* - ifz* F t ( s , t ) d s } d t = 0 or . * M [ F ( z * , t ) - = fzit F t ( s , t ) d s - 108 -z* =• -/0 F t ( s , t ) d s < 0. (3.11) where the l a s t i n e q u a l i t y f o l l o w s from the d e f i n i t i o n o f an i n c r e a s e i n r i s k . Now as F ( z * , t ) < 1, j ^ — — °* T n u s w e n a v e s n o w n P r o p o s i t i o n 3.4: A change i n the e x p e c t a t i o n s o f borrowers i n the sense o f an i n c r e a s e i n r i s k does not d e c r e a s e the e q u i l i b r i u m l o a n r a t e . Thus, from the p o i n t o f view o f a c a p i t a l market whose l o a n r a t e i s d e t e r -mined on the demand s i d e , a change i n e x p e c t a t i o n s o f b o r r o w e r s , i n t h a t t h e y p e r c e i v e t h e i n v e s t m e n t p r o j e c t s t o be r i s k i e r , w i l l not d e c r e a s e t h e i n t e r e s t r a t e on r i s k y l o a n s . T h i s i s , p e r h a p s , somewhat p a r a d o x i c a l i n t h a t one might s u s p e c t a c o l l a p s e i n the e x p e c t a t i o n s o f borrowers might cause a d e c r e a s e i n t h e demand f o r i n v e s t m e n t funds and hence a drop i n t he i n t e r e s t r a t e on r i s k y l o a n s . The reason t h i s i s not t r u e , i s t h a t the change i n e x p e c t a t i o n s i n the sense o f an i n c r e a s e i n r i s k a c t u a l l y increases e x p e c t e d p r o f i t s . Because o f the asymmetry o f the r e t u r n s o f the d i s t r i b u t i o n but h o l d i n g the mean c o n s t a n t , we i n c r e a s e t he p r o b a b i l i t y o f s t a t e s which pay o f f w e l l i n the upper ends o f the d i s t r i b u t i o n . But the i n c r e a s e d p r o b a b i l i t y o f s t a t e s w i t h a low p a y - o f f does n o t a f f e c t ex-p e c t e d p r o f i t s as the borrower can choose t o d e f a u l t i n those s t a t e s . Thus, as a consequence o f d e f a u l t we have a s i t u a t i o n where from t h e econ-omy's v i e w p o i n t the i n v e s t m e n t p r o j e c t i s r i s k i e r , y e t from the i n d i v i d u a l borrower's v i e w p o i n t t he p r o j e c t i s more d e s i r a b l e i n terms o f e x p e c t e d r e t u r n . 4. R i s k A v e r s i o n and the Demand f o r Loans T h i s s e c t i o n i s a b r i e f d i g r e s s i o n on t h e p r o p e r t i e s o f t h e demand f u n c t i o n f o r l o a n s when borrowers a r e r i s k a v e r s e . In the l a s t s e c t i o n - 109 -u s i n g t h e assumptions o f r i s k n e u t r a l i t y , s t o c h a s t i c c o n s t a n t r e t u r n s t o s c a l e , and no b a r r i e r s t o e n t r y o r e x i t we i n e f f e c t g e n e r a t e d a demand c u r v e f o r l o a n s which was h o r i z o n t a l a t the e q u i l i b r i u m l o a n r a t e . The e q u i l i b r i u m q u a n t i t y o f i n v e s t m e n t was l e f t i n d e t e r m i n a t e . Here we assume e s s e n t i a l l y the same s i t u a t i o n w i t h the f o l l o w i n g changes. A l l borrowers a r e a l i k e i n the sense t h a t they have i d e n t i c a l u t i l i t y f u n c t i o n s and e x p e c t a t i o n s . The number o f borrowers i s f i x e d and they maximize the e x p e c t e d u t i l i t y o f net r e c e i p t s from the i n v e s t m e n t p r o j e c t . There i s i n t h i s model no exogenous r a t e o f p r o f i t , and i n s t e a d o f g e n e r a t i n g an e q u i l i b r i u m l o a n p r i c e we g e n e r a t e a demand cu r v e f o r l o a n s . The f o l l o w i n g assumptions a r e made on the u t i l i t y f u n c t i o n s o f bor-rowers, U ( j l ) . C l U(n) i s a bounded, s t r i c t l y c oncave, t w i c e d i f f e r e n t i a t e non-decreas-i n g f u n c t i o n d e f i n e d on R + w i t h f i n i t e f i r s t and second d e r i v a t i v e s e v e r y -where. Note t h a t this i s not a p o r t f o l i o c h o i c e problem. The b o r r o w e r s , or f i r m s as you may choose t o c a l l them, have o n l y one d e c i s i o n to make - how much to i n v e s t ; T h e i r n e t r e c e i p t s , n, a r e never n e g a t i v e as t h e y may always choose to d e f a u l t on the l o a n . R e c a l l t h a t net p r o f i t s a r e g i v e n by !r s x - qx i f s > q/r 0 i f s < q / r . Thus the e x p e c t e d u t i l i t y o f p r o f i t s i s g i v e n by M EU(n) = / q / r U [ r s x - q x ] d F ( s ) , (4.1) which can be r e - w r i t t e n as - 110 -EU(n) = fz U [ r x ( s - z ) ] d F ( s ) . (4.2) M a x i m i z i n g w i t h r e s p e c t t o x, we have the f i r s t - o r d e r c o n d i t i o n , assuming 13 an i n t e r i o r s o l u t i o n ( x * > 0), = / V [ r x * ( s - z ) ] r ( s - z ) d F ( s ) = 0, (4.3) where x* i s the o p t i m a l q u a n t i t y o f i n v e s t m e n t o r s i z e o f l o a n . Of some i n t e r e s t i s whether o r not the demand c u r v e f o r l o a n s , x * ( q , r ) , i s downward s l o p i n g w i t h r e s p e c t t o the l o a n p r i c e q and i n c r e a s i n g w i t h r e s p e c t t o the p r o d u c t i v i t y o f i n v e s t m e n t . From (4.3) we see t h a t x* i s a f u n c t i o n o f z and r . F i r s t we d i f f e r e n t i a t e (4.3) w i t h r e s p e c t t o x* and z. M fz [U"r ( s - z ) dx* - U " r ( s - z ) r x * d z - U ' r d z ] d F ( s ) - U ' [ r x * ( z - z ) ] r ( z - z ) d F ( z ) = 0. (4.4) The l a s t term i n (4.4) w i l l v a n i s h p r o v i d e d U'(0) i s f i n i t e which i s assumed by C l . Thus (4.4) can be r e - w r i t t e n as M M {/ U " r 2 ( s - z ) 2 d F ( s ) W x * = {/ [U"r ( s - z ) x * + U ' r ] d F ( s ) } d z . (4.5) As IT < 0 by c o n c a v i t y the term i n b r a c k e t s on the l e f t - h a n d s i d e o f (4.5) i s n e g a t i v e . C o n s i d e r the term U " r 2 ( s - z ) x * + U'r. (4.6) The term i n b r a c k e t s on the r i g h t - h a n d s i d e o f (4.5) i s p o s i t i v e o r nega-t i v e as (4.6) i s g r e a t e r o r l e s s than z e r o . Suppose U"n + U' > 0. Then n i l " TTU"(TT) we have - TJT- < 1. But R(n) = - jji i s n o t h i n g but the A r r o w - P r a t t - I l l -measure o f r e l a t i v e r i s k a v e r s i o n . We have thus shown P r o p o s i t i o n 4.1: The demand f o r l o a n s i s an i n c r e a s i n g o r d e c r e a s i n g f u n c t i o n o f the l o a n r a t e as the c o e f f i c i e n t o f r e l a t i v e r i s k a v e r s i o n i s g r e a t e r o r l e s s than u n i t y everywhere. T h e r e f o r e , we have the s u r p r i s i n g p r o p o s i t i o n t h a t the demand f u n c t i o n f o r l o a n s may i n f a c t be upward s l o p i n g i f the u t i l i t y f u n c t i o n has a c o e f f i c i e n t o f r e l a t i v e r i s k a v e r s i o n g r e a t e r than u n i t y . There i s no g e n e r a l way one can p r e c l u d e such a s i t u a t i o n . A c o r o l l a r y o f p r o p o s i t i o n 4.1 i s t h a t i f the u t i l i t y f u n c t i o n e x h i b i t s c o n s t a n t r e l a t i v e r i s k a v e r -s i o n , w i t h a r e l a t i v e r i s k a v e r s i o n c o e f f i c i e n t equal t o u n i t y then t he demand f o r l o a n s i s p r i c e i n e l a s t i c . In g e n e r a l i t i s not p o s s i b l e to s i g n t he e f f e c t o f x* w i t h r e s p e c t t o changes i n r . In subsequent s e c t i o n s when d e a l i n g w i t h a s u p p l y d e t e r -mined l o a n p r i c e , the e f f e c t o f parameter changes on the e q u i l i b r i u m q u a n t i t y o f i n v e s t m e n t w i l l depend upon whether the demand c u r v e f o r l o a n s has p o s i t i v e o r n e g a t i v e s l o p e . 5. Loan Market E q u i l i b r i u m : S upply Determined Loan P r i c e In t h i s s e c t i o n we c o n s i d e r a c a p i t a l market, a g a i n w i t h a s i n g l e type o f l o a n c o n t r a c t a v a i l a b l e , o n l y i n t h i s s i t u a t i o n the e q u i l i b r i u m l o a n p r i c e and e q u i l i b r i u m p r o b a b i l i t y o f d e f a u l t i s d e t e r m i n e d by the s u p p l y s i d e o f the market. The s u p p l i e r s o f i n v e s t m e n t funds a r e assumed t o be r i s k n e u t r a l and to be s u f f i c i e n t l y l a r g e i n number t h a t the market i s c o n s i d e r e d t o be com-p e t i t i v e . These l e n d e r s may put t h e i r funds i n e i t h e r a r i s k l e s s s e c u r i t y paying a g r o s s r a t e o f i n t e r e s t p , o r i n l o a n s t o i n d i v i d u a l s o r f i r m s - 112 -h a v i n g a c c e s s t o t e c h n o l o g i e s such as t h o s e d e s c r i b e d by A . l and A.2 i n s e c t i o n 3. These l o a n s a r e r i s k y i n t h a t the borrowers may choose to de-f a u l t , and, as we assume the l e n d e r s know the t e c h n o l o g i c a l c o e f f i c i e n t r , t h e y r e a l i z e t h a t i f s, the ' s t a t e - o f - t h e - w o r l d ' parameter s h o u l d t a k e a v a l u e l e s s than q / r , the l o a n w i l l not pay the promised amount. Should d e f a u l t o c c u r , however, t h e r i g h t s t o i n v e s t m e n t p r o j e c t and the c a p i t a l a s s o c i a t e d w i t h i t a r e t r a n s f e r r e d t o the l e n d e r . We assume though t h a t the i n v e s t m e n t p r o j e c t t o the l e n d e r i s worth l e s s than t o the o r i g i n a l borrower. T h i s c o u l d be f o r two r e a s o n s . I f a l e n d e r has t o take o v e r an i n v e s t m e n t p r o j e c t and a c t u a l l y s u p e r v i s e i t s o p e r a t i o n he may i n c u r s u b s t a n t i a l c o s t s d o i n g so, o r he may l a c k the e x p e r t i s e n e c e s s a r y f o r a f u l l y e f f i c i e n t o p e r a t i o n . A l t e r n a t i v e l y , i f t h e l e n d e r chooses not t o complete the i n v e s t m e n t p r o j e c t but r a t h e r t o s e l l o f f the c a p i t a l e q u i p -ment a s s o c i a t e d w i t h i t , he may i n c u r m a r k e t i n g c o s t s i n the p r o c e s s and i n any c a s e w i l l not r e c o v e r the f u l l amount due on the l o a n . We s h a l l term the v a l u e o f the i n v e s t m e n t p r o j e c t t o the l e n d e r , s h o u l d he have t o take i t over, the collateral value. The c o l l a t e r a l r e t u r n f u n c -t i o n i s c ( x , s ) = c x s , where c > 0. The assumption t h a t the c o l l a t e r a l v a l u e o f the p r o j e c t i s always l e s s than i t s i n v e s t m e n t v a l u e i m p l i e s t h a t c i s l e s s than r . A l l l e n d e r s a r e assumed t o have the same c o l l a t e r a l f u n c t i o n . The e x p e c t a t i o n s o f l e n d e r s a r e r e p r e s e n t e d by a d i s t r i b u t i o n f u n c t i o n F ( s ) w i t h s u p p o r t i n the i n t e r v a l [ 0 , M ] . 1 4 A l l l e n d e r s have the same ex-p e c t a t i o n s . A g a i n , changes i n e x p e c t a t i o n s w i l l be r e p r e s e n t e d by changes i n t , where F ( s , t ) i s a f a m i l y o f d i s t r i b u t i o n s . The d e f i n i t i o n o f an i n c r e a s e i n r i s k used i n t h i s s e c t i o n i s the same as i n the p r e v i o u s s e c -t i o n w i t h the a d d i t i o n a l r e q u i r e m e n t t h a t F . ( z * , t ) .>_ 0. That i s , we assume - 113 -t h a t t he i n c r e a s e i n r i s k i s always such t h a t , from t he v i e w p o i n t o f l e n d e r s , a t the e q u i l i b r i u m d e f a u l t p o i n t , z*, the p r o b a b i l i t y o f d e f a u l t does not d e c r e a s e . Sometimes we s h a l l r e f e r t o t h i s type o f i n c r e a s e i n r i s k as an i n c r e a s e i n d e f a u l t r i s k . C o m p e t i t i o n f o r c e s a l l s u p p l i e r s o f i n v e s t m e n t funds t o charge the same p r i c e q on r i s k y l o a n s , and c a p i t a l - m a r k e t e q u i l i b r i u m r e q u i r e s t h a t the l o w e s t q c o m p a t i b l e w i t h t he e x p e c t e d r e t u r n on r i s k y l o a n s b e i n g equal t o p be c h a r g e d . Should t h i s not be t r u e , f o r example i f the ex-p e c t e d r e t u r n t o thes e l o a n s exceeded p, then t h e r e would be an exc e s s s u p p l y o f funds f o r r i s k y i n v e s t m e n t p r o j e c t s d r i v i n g down the l o a n r a t e so as t o r e s t o r e c a p i t a l market e q u i l i b r i u m . The r e t u r n f u n c t i o n t o l e n d e r s i s denoted by $(q,s,x) where !qx i f s >_ q/r cxs i f s < q / r . T h e r e f o r e t he e x p e c t e d r e t u r n p er d o l l a r l o a n e d out i s g i v e n by q/r M c / 0 s d F ( s ) + q / q / r d F ( s ) . (5.1) A g a i n we l e t z = q / r , and c a p i t a l market e q u i l i b r i u m r e q u i r e s t h a t (5.1) be equal t o p. Thus we have z M g ( z ) = c / 0 s d F ( s ) + r z / z d F ( s ) = p (5.2) as the b a s i c e q u i l i b r i u m e q u a t i o n . We a r e l o o k i n g f o r s o l u t i o n s z*e[0,M], to the e q u a t i o n g ( z * ) = p. Note t h a t g ( z ) i s c o n t i n u o u s ' a n d d i f f e r e n t i a t e , g(0) = 0 and g(M) = c s , and thus by the mean-value theorem g ( z ) t a k e s on a l l v a l u e s i n the i n t e r v a l £ 0 , c s ] . T h e r e f o r e , i f pe(0,cs) a s o l u t i o n - 114 -z*e(0,M) exists for equation (5.2). What of uniqueness? Evaluating the derivative of g(z) we have M g'(z) = czdF(z) + rf dF(s) - rzdF(z) M = (c-r)zdF(z) + rf dF(s). (5.3) As r > c, since investment value exceeds collateral value, (5.3) is of indeterminate sign. Changing the default point has two effects of oppo-M site sign. The term r/ zdF(s) is the direct p r o f i t a b i l i t y effect and is positive. This term accounts for the added return by raising the loan price. The term (c-r)zdF(z) is the default effect, and is always negative. As the default point is raised collateral value is substituted for loan value and as c is less than r the terms of this substitution effect are negative. As g'(z) cannot be signed i t appears we cannot claim that a unique equilibrium price exists on technical arguments alone. In Figure 1 g(z) is graphed. Note that i t s slope may have both posi-tive and negative values. For an exogenous rate of return p the equation g(z) = p has three solutions z1> z 2, and z 3. Now, we shall argue zl is the only possible equilibrium solution. Why? Suppose for example a l l lenders are charging q 2 = r z 2 . Then any lender could charge q, where q x < q < q 2, attract a large number of loan customers and earn an expec-ted rate of.return above p. Thus z x is the only possible equilibrium solution, and a l l lenders must be charging this price. Notice at z1 that g'Uj) > 0. Is i t possible that at some equilibrium z*, g'(z*) < 0? The M answer is no. Notice f i r s t that from (5.3) g'(0) = r/ QdF(s) .= r > 0. Thus the function g(z) always starts at 0 with a rising segment. Now sup-pose there existed an equilibrium z* such that g'(z*) < 0. But then by - 115 -F i g u r e 1. Expected Return F u n c t i o n - 116 -the mean v a l u e theorem t h e r e would e x i s t a z < z*, such t h a t g ( z ) = p, and g ' ( z ) > 0. Thus z* c o u l d n o t have been an e q u i l i b r i u m , p r o v i n g o u r 15 p r o p o s i t i o n . T h e r e f o r e we have shown P r o p o s i t i o n 5.1: There e x i s t s a unique e q u i l i b r i u m d e f a u l t p o i n t z*, and f u r t h e r m o r e i n t h e neighbourhood o f t h i s e q u i l i b r i u m t he e x p e c t e d r e t u r n on r i s k y l o a n s , g ( z ) i s i n c r e a s i n g w i t h z. Now d i f f e r e n t i a t i n g (5.2) a t the e q u i l i b r i u m p o i n t z* we have g ' ( z * ) d z * = dp. (5.4) S i n c e g ' ( z * ) > 0, the f o l l o w i n g p r o p o s i t i o n h o l d s . P r o p o s i t i o n 5.2: An i n c r e a s e i n the exogenous r a t e o f r e t u r n causes an i n c r e a s e i n the e q u i l i b r i u m l o a n r a t e and an i n c r e a s e i n the e q u i l i b r i u m p r o b a b i l i t y o f d e f a u l t . N o t i c e t h a t i f we have a downward s l o p i n g demand cu r v e f o r r i s k y l o a n s , say v i a the a n a l y s i s o f s e c t i o n 4, then an i n c r e a s e i n the r i s k l e s s i n t e r -e s t r a t e would cause a d e c l i n e i n the e q u i l i b r i u m q u a n t i t y o f i n v e s t m e n t . C o n t r a r y t o the demand-determined model, an i n c r e a s e i n the i n t e r e s t r a t e ( r i s k l e s s ) c auses an i n c r e a s e i n the p r o b a b i l i t y o f d e f a u l t . The s u p p l y d e t e r m i n e d model thus p o s i t s a p o s i t i v e c o r r e l a t i o n between h i g h i n t e r e s t r a t e s and h i g h l e v e l s o f d e f a u l t . We t u r n now t o c o n s i d e r a t i o n o f a change i n the p r o d u c t i v i t y c o e f f i c -i e n t r . D i f f e r e n t i a t i n g (5.2) w i t h r e s p e c t t o r and z* we g e t the e q u a t i o n M g ' ( z * ) d z * + {z* / z * d F ( s ) } d r = 0 . (5.5) As g 1 Cz*)''•> 0, 'w"e have -r- < 0. D i f f e r e n t i a t i n g the i d e n t i t y q = z * r we - 117 -have ^ = ^  + z*. From (5.5) M ^ - z * / ^ d F ( s ) d r = g'(z*) and from (5.3) M g'(z*) = ( c - r ) z * d F ( z * ) + - r / ^ d F ( s ) . T h e r e f o r e M ,„* - r z * / * d F ( s ) 53*. = z * + 9 (z* ) f d r g ' ( z * ) g ' ( z * ) M - r z * / z ^ d F ( s ) + ( c - r ) z * 2 d F ( z * ) + r z * d F ( z * ) ( c - r ) z * d F ( z * ) n g'(z*) < ° ' Thus we have shown P r o p o s i t i o n 5.3: An i n c r e a s e i n the p r o d u c t i v i t y o f i n v e s t m e n t causes a) a d e c r e a s e i n the e q u i l i b r i u m d e f a u l t r a t e and b) a d e c r e a s e i n the e q u i l -i b r i u m l o a n r a t e . N o t i c e a g a i n t h a t the r e s u l t s o f the s u p p l y model a r e c o m p l e t e l y o p p o s i t e t o t h o s e o f the demand model ( p r o p o s i t i o n 3.2). In t h i s c ase more produc-t i v e i n v e s t m e n t s a r e a s s o c i a t e d w i t h lower d e f a u l t p r o b a b i l i t i e s . I t would seem t h a t i n a s u p p l y - d e t e r m i n e d case h i g h e r p r o d u c t i v i t y p r o j e c t s have a lower c o s t o f c a p i t a l ; t h a t i s , a lower c o s t o f funds t o b o r r o w e r s . Changes i n the c o l l a t e r a l v a l u e o f the p r o j e c t a r e t r e a t e d s i m i l a r l y . D i f f e r e n t i a t i n g (5.2) w i t h r e s p e c t t o c and z* we g e t - 118 -z* g.'(z*)dz* +' {/ s d F ( s ) } d c = 0 which i m p l i e s < 0. Hence P r o p o s i t i o n 5.4: An i n c r e a s e i n the c o l l a t e r a l v a l u e o f p r o j e c t s causes a decrease i n the e q u i l i b r i u m p r o b a b i l i t y o f d e f a u l t and a d e c r e a s e i n the e q u i l i b r i u m l o a n r a t e . T h i s r e s u l t makes a g r e a t d e a l o f i n t u i t i v e s e n s e . I t seems not unreason-a b l e t h a t the b e t t e r the c o l l a t e r a l v a l u e o f the p r o j e c t t h e lower the l o a n r a t e s h o u l d be. In f a c t , t h i s i s a r a t h e r common phenomena ob s e r v e d i n c a p i t a l m a r k e t s . F i n a l l y c o n s i d e r changes i n the e x p e c t a t i o n s o f l e n d e r s . The type o f change we wish t o c o n s i d e r i s an i n c r e a s e i n r i s k i n the R o t h s c h i l d -S t i g l i t z [1970] s e n s e , such t h a t p r o b a b i l i t y o f d e f a u l t does n o t d e c r e a s e a t t h e e q u i l i b r i u m d e f a u l t p o i n t . We c h a r a c t e r i z e t h i s by the c o n d i t i o n s x ( i ) /0. F t ( s , t ) d s _> 0 f o r 0 < x < M and ( i i ) F t ( z * , t ) > 0 . U s i n g i n t e g r a t i o n by p a r t s (5.2) may be r e - w r i t t e h as z* g ( z * , t ) = c [ z * F ( z * , t ) - / Q F ( s , t ) d s ] + r z * [ l - F ( z * , t ) ] = p. (5.6) Now g z ( z * , t ) > 0 from p r o p o s i t i o n 5.1. D i f f e r e n t i a t i n g (5.6) w i t h r e s p e c t t o t we have z* g t ( z * , t ) = ( c - r ) z * F t ( z * , t ) - c / Q F t ( s , t ) d s < 0, where the i n e q u a l i t y f o l l o w s from the d e f i n i t i o n o f an i n c r e a s e i n d e f a u l t - 119 -r i s k . Thus d i f f e r e n t i a t i o n o f (5.6) w i t h r e s p e c t t o z* and t i m p l i e s -rt- > 0. Now F ( z * , t ) i s the p r o b a b i l i t y o f d e f a u l t as viewed by l e n d e r s , P r o p o s i t i o n 5.5: A change i n the e x p e c t a t i o n s o f l e n d e r s , r e p r e s e n t e d by an i n c r e a s e i n d e f a u l t r i s k , does n o t d e c r e a s e the e q u i l i b r i u m l o a n r a t e and does not d e c r e a s e the e q u i l i b r i u m p r o b a b i l i t y o f d e f a u l t . Thus a c o l l a p s e i n the e x p e c t a t i o n s o f l e n d e r s may cause an i n c r e a s e i n the e q u i l i b r i u m l o a n r a t e . I f the demand c u r v e . f o r l o a n s were downward s l o p i n g t h i s would cause a r e d u c t i o n i n the e q u i l i b r i u m q u a n t i t y o f r i s k y i n v e s t m e n t . R e c a l l , however, t h e r e i s no a priori reason f o r the demand cu r v e f o r r i s k y l o a n s t o have a n e g a t i v e s l o p e . T h i s completes our program o f c o m p a r a t i v e s t a t i c s . We wish t o ask one o t h e r q u e s t i o n o f t h i s model. Suppose t h e r e are two t y p e s o f f i r m s o r t e c h n o l o g i e s i n e x i s t e n c e , one unambiguously more p r o d u c t i v e than the o t h e r , for.example w i t h p r o d u c t i v i t y c o e f f i c i e n t s r1 and r 2 , where r x > r 2 . Does t h e r e e x i s t a c o m p e t i t i v e c a p i t a l market e q u i l i b r i u m w i t h d e f a u l t , i n which the market a l l o c a t e s some in v e s t m e n t funds t o both f i r m s , o r t e c h -n o l o g i e s ? L e t g 1 ( z ) and g 2 ( z ) be the e x p e c t e d r e t u r n f u n c t i o n s f o r f i r m s 1 and 2 r e s p e c t i v e l y . I t i s s t r a i g h t f o r w a r d t o v e r i f y and hence L = F s ( z * , t ) ^ | * - + F t ( z * , t ) _> 0. T h e r e f o r e we have shown g a ( o ) v g 2 ( o ) > g 2(M) = . g 2 ( M ) , and g 2 ( z ) > g 2 ( z ) f o r 0 < z < M. These f u n c t i o n s a r e graphed i n F i g u r e 2. - 120 -F i g u r e 2. Expected Return F u n c t i o n s f o r Firms 1 and 2 - 12.1 -I f o n l y type two f i r m s e x i s t e d q 2 = z 2 r 2 would be'the e q u i l i b r i u m l o a n p r i c e . S i n c e type one f i r m s e x i s . t , however, and l e n d e r s can d i s t i n g u i s h type one from type.two f i r m s , an e q u i l i b r i u m w i t h two l o a n r a t e s q x = z : r x and q 2 can e x i s t , w i t h both types o f f i r m s y i e l d i n g e x p e c t e d r a t e s o f r e -t u r n on t h e i r l o a n s equal t o p. T h i s p o s s i b i l i t y i n d i c a t e s t h a t c ompeti-t i v e e q u i l i b r i u m w i t h d e f a u l t may i n a v e r y s t r o n g sense be i n e f f i c i e n t . Any i n v e s t m e n t t a k e n out o f t y p e two f i r m s and put i n type one f i r m s would r a i s e the t o t a l e x p e c t e d r e t u r n t o a f i x e d q u a n t i t y o f i n v e s t m e n t goods. From an e f f i c i e n c y v i e w p o i n t the b e s t s o l u t i o n i s t o a l l o c a t e a l l i n v e s t -ment t o t y p e one f i r m s . The i n t r o d u c t i o n o f d e f a u l t on l o a n c o n t r a c t s i s one s t e p towards i n e f f i c i e n c y , i n t h a t i f the l o a n i s d e f a u l t e d , then the c o l l a t e r a l r e t u r n i s l e s s than the i n v e s t m e n t r e t u r n . The problem i s compounded f u r t h e r , however, as f i r m s o f d i f f e r e n t b a s i c t e c h n o l o g i c a l p r o d u c t i v i t y , may y i e l d the same e x p e c t e d r e t u r n on t h e i r l o a n s . T h i s o c c u r s as l e n d e r s may change f i r m s o f d i f f e r e n t t y p e s d i f f e r e n t l o a n r a t e s . In a c o m p e t i t i v e e q u i l i b r i u m any attempt t o charge a h i g h e r p r o d u c t i v i t y f i r m a lower l o a n r a t e would r e s u l t , as a consequence o f p r o p o s i t i o n 5.1, i n the l e n d e r e a r n i n g l e s s than the o p p o r t u n i t y c o s t , p, on h i s l o a n . The s o l u t i o n mechanism o f the c a p i t a l market s u s t a i n s a c o m p e t i t i v e i n e f -f i c i e n c y ^ Note t h a t a government w i t h the same i n f o r m a t i o n as the market p a r t i c i p a n t s c o u l d , by s i m p l y e x c l u d i n g the l o a n r e q u e s t s o f t y p e two f i r m s and g u a r a n t e e i n g the'rloans o f t y p e one f i r m s , r a i s e a g g r e g a t e ex-p e c t e d r e t u r n . T h i s b r i e f d i g r e s s i o n on the e f f i c i e n c y a s p e c t s o f d e f a u l t i n d i c a t e s t h a t t h e r e may be some s e r i o u s q u e s t i o n s as to the o p t i m a l i t y o f the c a p i -t a l market a l l o c a t i o n o f i n v e s t m e n t , q u i t e i n d e p e n d e n t as t o any i s s u e s o f r i s k b e a r i n g . C l e a r l y a more complete e q u i l i b r i u m t r e a t m e n t i s - 122 -c a l l e d f o r . 6. The I n t r o d u c t i o n o f E q u i t y Our a n a l y s i s so f a r has c o n c e n t r a t e d on a c a p i t a l market w i t h a s i n g l e type o f c o n t r a c t which f a c i l i t a t e s the s a v i n g s - i n v e s t m e n t p r o c e s s . In t h i s s e c t i o n we c o n s i d e r some p o s s i b l e c o m p l i c a t i o n s o f i n t r o d u c i n g an a l t e r n a t i v e means o f i n v e s t i n g i n r i s k y i n v e s t m e n t p r o j e c t s — t h e purchase o f e q u i t y o r l i m i t e d l i a b i l i t y s t o c k s i n r i s k y f i r m s . The model i n t h i s s e c t i o n w i l l be a s i m p l e g e n e r a l model o f the s t a n -d a r d t w o - p e r i o d v a r i e t y , i n v e s t m e n t t a k i n g p l a c e i n the f i r s t p e r i o d and o u t p u t (random) o c c u r i n g i n the second p e r i o d . 1 7 The economy c o n s i s t s o f two t y p e s o f f i r m s w i t h a l l f i r m s o f each ty p e h a v i n g i d e n t i c a l t e c h n o l o -g i e s . The r i s k l e s s f i r m s have a c o n s t a n t - r e t u r n s - t o - s c a l e t e c h n o l o g y which f i x e s the r i s k l e s s i n t e r e s t r a t e ( g r o s s ) p p r o v i d e d some p o s i t i v e i n v e s t m e n t o c c u r s i n the r i s k l e s s t e c h n o l o g y . R i s k y f i r m s have t e c h n o l o -g i e s i d e n t i c a l t o t h o s e i n v e s t m e n t p r o j e c t s o u t l i n e d i n s e c t i o n 3. We s h a l l r e q u i r e , however, t h a t i n s t a t e s i n which the f i r m goes b a n k r u p t , i . e . , d e f a u l t s on i t s payments t o b o n d h o l d e r s , a change i n t e c h n i q u e o c c u r s . That i s , i n s t a t e s o f b a n k r u p t c y the f i r m does no t o p e r a t e w i t h the same t e c h n o l o g y as i n t h o s e s t a t e s i n which the f i r m i s s o l v e n t . The t e c h n o l o g y i n bankrupt s t a t e s w i l l be i d e n t i c a l t o the c o l l a t e r a l r e t u r n f u n c t i o n o f s e c t i o n 5, and has the p r o p e r t y t h a t i t pays o f f a t a r a t e below t h a t o f the t e c h n o l o g y i n s o l v e n t s t a t e s . The reason f o r making the above assumption i s t h a t i t i n t r o d u c e s a deadweight c o s t t o the economy o f b a n k r u p t c y . In p r e v i o u s a n a l y s e s o f b a n k r u p t c y , the b a n k r u p t c y always o c c u r s a t z e r o r e s o u r c e c o s t t o the 18 economy. Yet, i t would seem t h a t the t r a n s a c t i o n s c o s t s i n v o l v e d i n - 123 -a d m i n i s t e r i n g b a n k r u p t c i e s , and perhaps more i m p o r t a n t the l o s s o f o u t p u t due t o the change o f management d u r i n g a b a n k r u p t c y , would not be i n s i g n i f -i c a n t . In many ca s e s a f i r m which goes bankrupt w i l l a c t u a l l y s h u t down, and i f the i n v e s t m e n t made was o f the p u t t y - c l a y v a r i e t y the c o s t t o the economy o f b a n k r u p t c y i s the e n t i r e o u t p u t l o s t , as the c a p i t a l i n v e s t e d i n the f i r m has no consumption o r p r o d u c t i o n v a l u e . Of c o u r s e s i n c e bank-r u p t c i e s i n c u r r e a l c o s t s on the economy, t h i s i m p l i e s t h a t from the s o c i a l - e f f i c i e n c y v i e w p o i n t the o r g a n i z a t i o n o f p r o d u c t i o n s h o u l d be such as t o e l i m i n a t e b a n k r u p t c i e s . How t h i s might be done i s a d i f f i c u l t q u e s t i o n and we s h a l l not be concerned w i t h i t h e r e . B a n k r u p t c i e s a r e an i n s t i t u t i o n a l f e a t u r e o f market economies and our p r i m a r y c o n c e r n i s the p o s i t i v e i m p l i c a t i o n s o f t h i s f a c t . As a r e s u l t o f b a n k r u p t c y i n d u c i n g r e a l c o s t s , we s h a l l show i n t h i s s e c t i o n t h a t i t i s p o s s i b l e f o r the economy t o have e q u i l i b r i a w i t h a d e t e r m i n a t e d e b t - e q u i t y r a t i o , w h i l e m a i n t a i n i n g the assumptions o f i d e n t i -c a l e x p e c t a t i o n s and r i s k n e u t r a l i t y on t h e p a r t o f i n v e s t o r s . T h i s i s i n c o n t r a s t to the well-known M o d i g l i a n i - M i l l e r p r o p o s i t i o n , which s t a t e s t h a t p r o v i d e d the debt i s s u e d has no d e f a u l t r i s k the economy as a whole i s i n d i f f e r e n t between a l t e r n a t i v e d e b t / e q u i t y r a t i o s o f the f i r m s i n the 19 economy. We s h a l l show t h a t , even w i t h the s t r o n g assumptions made, the d e b t / e q u i t y r a t i o o f the economy may be a market d e t e r m i n e d v a r i a b l e . In a d d i t i o n t o the r e s o u r c e c o s t s o f b a n k r u p t c y , the r e s u l t h i n g e s on the f a c t t h a t any bond i s s u e d by a r i s k y f i r m has a p o s i t i v e p r o b a b i l i t y o f not p a y i n g o f f a t the nominal i n t e r e s t r a t e , as i n some s t a t e s a r b i t r a r i l y s m a l l amounts o f o u t p u t a r e o b t a i n e d . Furthermore, changes i n the d e b t / e q u i t y r a t i o o f a r i s k y f i r m a f f e c t the d e f a u l t p o i n t and t h i s becomes an a d d i t i o n a l mechanism by which e x p e c t e d r a t e s o f r e t u r n are e q u a l i z e d . - 124 -R e c a l l t h a t the g r o s s r e t u r n on an i n v e s t m e n t p r o j e c t o r t o a r i s k y f i r m ' s t e c h n o l o g y i s g i v e n by r s x , where x i s t o t a l i n v e s t m e n t , p r o v i d e d the f i r m does not d e f a u l t . I f the f i r m does d e f a u l t on i t s bond payments, then i n t h o s e s t a t e s s i n which i t i s bankrupt o u t p u t i s g i v e n by c s x . We have assumed t h a t b a n k r u p t c y has r e a l r e s o u r c e c o s t s , o r e q u i v a l e n t l y i n d u c e s deadweight l o s s e s which i s e x p r e s s e d by the assumption t h a t c i s l e s s than r . L e t B s t a n d f o r d o l l a r s worth o f bonds (measured i n u n i t s o f i n v e s t m e n t goods) and E the d o l l a r s amount o f e q u i t y f o r a p a r t i c u l a r f i r m . Thus x = B + E, o r the amount i n v e s t e d by a s i n g l e f i r m i s equal t o the v a l u e o f i t s d e b t p l u s e q u i t y . T h i s assumes a l l f i r m s s t a r t up i n the f i r s t p e r i o d and a r e d i s s o l v e d i n the second p e r i o d . Bondholders are t o r e c e i v e qB i n the second p e r i o d where q i s t h e nominal i n t e r e s t r a t e ( g r o s s ) on bonds. T h e r e f o r e n(q, B, s, x ) , the r e t u r n t o t o t a l e q u i t y E, i s g i v e n by !r s x - qB i f s n — rx (6.1) o i f , < £ . R e c a l l t h a t q / r = z. L e t B/x = w, where w i s the p r o p o r t i o n o f i n v e s t m e n t which i s debt f i n a n c e d , and (1 - w) the p r o p o r t i o n which i s e q u i t y f i n a n c e d ; an i n c r e a s e i n w w i l l c o r r e s p o n d t o an i n c r e a s e i n t h e d e b t / e q u i t y r a t i o . U s i n g t h i s n o t a t i o n the r e t u r n on a d o l l a r ' s worth o f i n v e s t m e n t i n e q u i t y , R E ( z , w), i s g i v e n by R E ( z , w) r s gw_ i f s > zw 1 - w 1 - w (6.2) 0 i f s < zw. - 125 -Note a g a i n t h a t the c o n s t a n t - r e t u r n s assumption makes the r e t u r n t o e q u i t y independent o f - t h e s c a l e o f i n v e s t m e n t . The d e f a u l t p o i n t i s g i v e n by zw i n t h i s model. T h e r e f o r e , u s i n g (6.2), the e x p e c t e d r a t e o f r e t u r n on i n v e s t i n g i n the e q u i t y o f r i s k y f i r m s i s g i v e n by M M E ¥ z ' w> - r h '» 5dF<s> - '„ dF<s> <6-3> where F ( s ) i s d i s t r i b u t i o n f u n c t i o n r e p r e s e n t i n g the e x p e c t a t i o n s o f i n v e s t o r s . The r e t u r n to a d o l l a r ' s i n v e s t m e n t i n the bonds o f a r i s k y f i r m i s I q i f s ,>zw R B ( z , w) = (6.4) [ — i f s < zw. \ w Hence the e x p e c t e d r e t u r n t o i n v e s t m e n t i n r i s k y bonds i s g i v e n by zw M E R B ( z , w) = - ^ / 0 s d F ( s ) + q / z w d F ( s ) . (6.5) The r a t e o f r e t u r n t o i n v e s t i n g i n e i t h e r the bonds o r e q u i t y o f the r i s k l e s s f i r m i s g i v e n by p, where px i s the p r o d u c t i o n f u n c t i o n f o r r i s k -l e s s f i r m s . C a p i t a l market e q u i l i b r i u m , g i v e n the assumption o f r i s k n e u t r a l i t y , r e q u i r e s t h a t the e x p e c t e d r a t e o f r e t u r n on a l l s e c u r i t i e s i n which p o s i t i v e i n v e s t m e n t t a k e s p l a c e be e q u a l . Supposing t h a t i n v e s t m e n t o c c u r s i n a l l s e c u r i t i e s the market e q u i l i b r i u m c o n d i t i o n s can be w r i t t e n as E R £ ( z , w) = p E R B ( z , w) = p. (6.6) - 126 -D i v i d i n g both o f t h e s e e q u a t i o n s through by r , and l e t t i n g p / r = v, we get Mz> w> s rhi 'zw sdF<s> - r ^ V ^w d F ( s > = v < 6- 7> 1 zw M Mz> w) = fw-'o s d F ( s ) + z / z w d F ( s ) = v.. (6.8) We wish t o i n v e s t i g a t e the p o s s i b i l i t y o f an e q u i l i b r i u m i n which i n v e s t m e n t o c c u r s i n r i s k y bonds, e q u i t y and the r i s k l e s s f i r m and thus the e q u a t i o n s (6.7) and (6.8) have a s o l u t i o n ( z * , w*). I f t h i s i s the c a s e , then c a p i t a l market e q u i l i b r i u m d e t e r m i n e s both a nominal i n t e r e s t r a t e on debt and an e q u i l i b r i u m d e b t / e q u i t y r a t i o . I t i s q u i t e c o n c e i v a b l e o f c o u r s e t h a t e q u i l i b r i u m may e n t a i l i n v e s t m e n t i n a l l o f one type o f s e c u r i t y . Note t h a t i f an i n v e s t o r h o l d s a p o r t f o l i o w i t h the f r a c t i o n w h e l d i n r i s k y bonds and the r e m a i n d e r - ( 1 - w) i n e q u i t i e s o f the r i s k y f i r m h i s e x p e c t e d r e t u r n on the p o r t f o l i o i s g i v e n by zw M E R p ( z , vi) = c f Q s d F ( s ) + r s d F ( s ) . (6.9) A g a i n , p r o v i d e d i n v e s t m e n t o c c u r s i n a l l s e c u r i t i e s , c a p i t a l - m a r k e t e q u i l i b r i u m r e q u i r e s E R f z w ) = p. L e t zw = y , then the f u n c t i o n h(y) = y M C/Q s d F ( s ) + r / s d F ( s ) , can e a s i l y be shown t o t a k e v a l u e s i n the i n t e r -20 v a l [ c s , r s ] and t o be u n i q u e l y i n v e r t i b l e on [ 0 , M]. T h e r e f o r e , a n e c e s s a r y c o n d i t i o n f o r such an e q u i l i b r i u m t o e x i s t i s t h a t p e ( c i , r s ) , 21 and we.assume t h i s c o n d i t i o n h o l d s h e n c e f o r t h . C l e a r l y one p o s s i b l e e q u i l i b r i u m i s f o r no i n v e s t m e n t t o o c c u r i n e i t h e r the r i s k l e s s f i r m o r i n the bonds o f r i s k y f i r m s , i n which case the e x p e c t e d r e t u r n on a l l - 127 -p o r t f o l i o s i s g i v e n by r s . In terms o f e f f i c i e n c y t h i s e q u i l i b r i u m i s the most d e s i r a b l e , p r o v i d e d p < r l . The c a s e w* = 1 i s e x c l u d e d i n t h i s s e c t i o n as i t c o r r e s p o n d s t o a c o m p l e t e l y debt f i n a n c e d e q u i l i b r i u m , which we t r e a t e d i n s e c t i o n 5. Such an e q u i l i b r i u m i s p o s s i b l e as demonstrated i n t h a t s e c t i o n . In o r d e r t o prove t h a t an i n t e r m e d i a t e case i s p o s s i b l e , we must show t h e r e e x i s t s w*, 0 < w* < 1 and z* > 0, 0 < z*w* < M, such t h a t (6.7) and (6.8) h o l d . In f a c t t h i s i s the c a s e . P r o p o s i t i o n 6.1: I f p e ( c s , r s ) t h e r e e x i s t s a c a p i t a l market e q u i l i b r l rium w i t h 0 < w* .< 1, i . e . , a s t r i c t l y p o s i t i v e debt e q u i t y r a t i o . ( P r o o f ) : The p r o o f i s based on the f a c t t h a t w r H 2 ( z , w) •+ (1 - w)r H ^ z , w) = E R p ( z w ) . (6.10) T h e r e f o r e o f the t h r e e e q u a t i o n s ( 6 . 7 ) , (6.8) and (6.9) o n l y two are i n -dependent, the t h i r d b e i n g d e t e r m i n e d by (6.10). As p e ( c s , r i ) , we know t h e r e e x i s t s a unique y * , 0 < y * < M, such t h a t y * M E R p ( y * ) = c / 0 s d F ( s ) + r /y* s d F ( s ) = p. (6.11) As y * = zw, we may w r i t e z as an i m p l i c i t f u n c t i o n o f w, z ( w ) , w i t h the p r o p e r t y t h a t z(w)w = y * . T a k i n g the s o l u t i o n y * from (6.11) and sub-s t i t u t i n g i n t o H ( z , w) we have H j z t w ) , w] = I / y * s d F ( s ) - y * /y* d F ( s ) ] . (6.12) We s h a l l show now t h e r e e x i s t s a w*, 0 < w* < 1 such t h a t H 1 [ z ( w * ) , w*] = v. 128 -C o n s i d e r the e x p r e s s i o n i n b r a c k e t s on the r i g h t - h a n d s i d e o f ( 6 . 1 2 ) , M M fyit s d F ( s ) - y * fyie d F ( s ) . (6.13) D i v i d i n g t h r o u g h by [1 - F ( y * ) J > 0 we g e t E ( s | s > y*) - y * > y * - y * = 0. Hence the e x p r e s s i o n i n b r a c k e t s i s s t r i c t l y p o s i t i v e i n (6.12). Now from (6.11) r y * M ± f Q s d F ( s i + fyic s d F ( s ) = p / r = v. (6.14) S u b t r a c t i n g (6.13) from ( 6 . 1 4 ) , we have c y * M ±JQ s d F ( s ) + y * fyie d F ( s ) > 0, (6.15) which i m p l i e s t h a t (6.13) i s s t r i c t l y l e s s than v. L e t t i n g (6.13) equal a we have shown 0 < a < v. (6.12) can be w r i t t e n as H 1 [ z ( w ) , w] = . (6.16) Now 1 im + H ^ z t w ) , w] = a w -»• 0 and l i m _ H 1 [ z ( w ) , w] = + <» w •-»• 3i~ where the p l u s and minus s u p e r s c r i p t s denote r i g h t and l e f t hand l i m i t s - 129 -r e s p e c t i v e l y . S i n c e H 1 [ z ( w ) , w] i s a c o n t i n u o u s f u n c t i o n o f w and a < v, t h e r e e x i s t s aw*, 0 < w* < 1 such t h a t H 1 [ z ( w * ) , w*] = v. U s i n g (6.10) and s u b s t i t u t i n g f o r w* and z* •= z(w*) we have t h a t H 2 ( z * , w*) = v, and the p r o p o s i t i o n i s p r o v e d . Q.E.D. Thus we have e s t a b l i s h e d t h a t p r o v i d e d the r i s k l e s s f i r m s ' t e c h n o l o -g i e s have a p r o d u c t i v i t y o f an a p p r o p r i a t e v a l u e , t h e r e e x i s t s a c a p i t a l market e q u i l i b r i u m w i t h a d e t e r m i n a t e d e b t / e q u i t y r a t i o . N o t i c e t h a t the d e b t / e q u i t y r a t i o o f t h i s economy i s an e q u i l i b r i u m one, and any attempt by f i r m s t o change t h e i r d e b t / e q u i t y r a t i o s w i l l r e s u l t i n a r b i t r a g e t a k i n g p l a c e i n the c a p i t a l market such as t o f o r c e the d e b t / e q u i t y r a t i o back t o i t s e q u i l i b r i u m v a l u e . In e q u i l i b r i u m the market v a l u e o f any f i r m i n v e s t i n g an amount x i s px. In o r d e r t o do the comparative, s t a t i c s o f the g e n e r a l e q u i l i b r i u m , i t i s n e c e s s a r y t o work w i t h two o f the t h r e e e q u i l i b r i u m e q u a t i o n s . We s h a l l use (6.8) m u l t i p l i e d by r and (6.11) t o g i v e us z*w* M G ^ z * , w*) = s d F ( s ) + r z * d F ( s ) = p, (6,17) z*w* M G 2 ( z * , w*) = c / Q s d F ( s ) + r s d F ( s ) = p. (6.18) E v a l u a t i n g the p a r t i a l d e r i v a t i v e s o f t h e s e e q u a t i o n s we have M G u(z,w) = (c - r)zw dF(zw) + r f m d F ( s ) , z w G 1 ? ( z , w ) = s d F ( s ) + (c - r ) z dF(zw) < 0, i Z w u G 2 I ( z , w ) = (c - r)zw dF(zw) < 0, and G,„(z,w) = (c - r ) z 2 w dF(zw) < 0, - 130 -where G.. denotes the p a r t i a l d e r i v a t i v e o f G., i = 1, 2 w i t h r e s p e c t t o "I J 1 j = z, w. The l a s t t h r e e i n e q u a l i t i e s f o l l o w from t h e f a c t t h a t c < r . G i r i s o f i n d e t e r m i n a t e s i g n as the f i r s t term i s n e g a t i v e and the second p o s i t i v e . R e c a l l , however, t h a t G a ( z , w) i s the e x p e c t e d r e t u r n t o i n v e s t -i n g i n the bonds o f r i s k y f i r m s a t the nominal i n t e r e s t r a t e o f q = r z . Now, j u s t as was argued i n s e c t i o n 5 w i t h r e g a r d t o p r o p o s i t i o n 5.1, com-p e t i t i o n amongst l e n d e r s w i l l f o r c e q = r z * , t he i n t e r e s t r a t e on bonds, to the lo w e s t one c o m p a t i b l e w i t h (6.17) f o r any v a l u e o f w, which t o the i n d i v i d u a l i n v e s t o r i s take n as g i v e n . Note t h a t G^O, w) = 0, G^M, w) > 0 and G ^ O , w) = 1 > 0. Thus by the same r e a s o n i n g as used t o prove p r o p o s i t i o n 5.1, i t must be the c a s e t h a t a t an e q u i l i b r i u m ( z * , w*), G j ^ z * , w*) > 0. T h e r e f o r e the J a c o b i a n m a t r i x G l l G12 G 2 1 G 2 2 has the s i g n p a t t e r n < + a t e q u i l i b r i u m v a l u e s o f ( z * , w*). L e t t i n g |G| denote the d e t e r m i n a n t o f t h i s m a t r i x we see t h a t |G| < 0.. P r o c e e d i n g w i t h the c o m p a r a t i v e s t a t i c s we d i f f e r e n t i a t e (6.17) and (6.18) w i t h r e s p e c t t o z, w and p a t the e q u i l i b r i u m p o i n t ( z * , w*) t o g e t - f i l l " d z * . " " d p " G 2 1 G 2 2 dw*; dp - 131 -Using Cramer's r u l e we have d z * ^22 " ^12 dp dw* _ G n " G 2 i ^ n - r?n < U. Now dp zw G 2 2 - G 1 2 = (c - r ) z 2 w dF(zw) + fQ s d F ( s ) - (c - r ) z 2 dF(zw) w zw r J w* = (c - r ) z z dF(zw) [w - 1] + \ / n s d F ( s ) > 0 a t ( z * , w*) as 0 < w* < 1. T h e r e f o r e < 0 and as q* = z * r , < 0. Hence P r o p o s i t i o n 6.2: An i n c r e a s e i n the r i s k l e s s i n t e r e s t r a t e p ( t h e p r o -d u c t i v i t y o f t h e r i s k l e s s f i r m ) c auses a) a d e c r e a s e i n the nominal r a t e o f i n t e r e s t on r i s k y bonds and b) a d e c r e a s e i n the e q u i l i b r i u m d e b t / e q u i t y r a t i o . T h i s p r o p o s i t i o n : , s t a t e s t h a t as a r e s u l t o f market e q u i l i b r i u m the nominal i n t e r e s t r a t e on r i s k y bonds i s n e g a t i v e l y c o r r e l a t e d w i t h the r i s k l e s s i n t e r e s t r a t e . F urthermore, r i s k y f i r m s w i l l tend t o have r e l a t i v e l y h i g h l y l e v e r e d c a p i t a l s t r u c t u r e s when i n t e r e s t r a t e s a r e low, even though the nominal i n t e r e s t r a t e s on r i s k y bonds w i l l tend t o be h i g h . We c o n s i d e r now i n c r e a s e s i n the p r o d u c t i v i t y o f r i s k y f i r m s . L e t t i n g G and G 2 r s t a n d f o r the p a r t i a l d e r i v a t i v e s o f (6.17) and (6.18) w i t h r e s p e c t t o r we g e t - 132 -and ; 2 r = / z w s d F ( s ) > o . A g a i n u s i n g Cramer's r u l e we have t h a t dw* d r G n . - G i r G 2 i " G 2 r > 0. T h e r e f o r e P r o p o s i t i o n 6.3: An i n c r e a s e i n the p r o d u c t i v i t y o f r i s k y f i r m s causes an i n c r e a s e i n the e q u i l i b r i u m d e b t / e q u i t y r a t i o . Thus economies w i t h more p r o d u c t i v e r i s k y t e c h n o l o g i e s w i l l have more h i g h l y l e v e r e d f i n a n c i a l s t r u c t u r e s . The e f f e c t o f changes i n p r o d u c t i v i t y on the p r i c e o f bonds i s ambiguous i n t h i s model. We t u r n now t o changes i n the c o s t o f ban k r u p t c y . R e c a l l t h a t an i n c r e a s e i n c c o r r e s p o n d s t o l o w e r i n g t h e c o s t o f ba n k r u p t c y . Now 1 zw ' i c W/ Q s d F ( s ) > 0 and So 2C dz^ dc zw / 0 s d F ( s ) > 0. -G • G,„ ic 12 "G2C G22 - 133 -zw / 0 s d F ( s ) zw / 0 s d F ( s ) Tel [- ±-fn sdF(s)] > 0. S i m i l a r l y dw* > 0. We have shown dc P r o p o s i t i o n 6.3: A d e c r e a s e i n the c o s t o f b a n k r u p t c y causes a) an i n -c r e a s e i n the nominal i n t e r e s t r a t e on r i s k y bonds and b) an i n c r e a s e i n the e q u i l i b r i u m d e b t / e q u i t y r a t i o . T h i s p r o p o s i t i o n makes a good deal o f i n t u i t i v e s e n s e . As the c o s t s o f d e f a u l t t o b o n d h o l d e r s d i m i n i s h e s , both t h e r a t e o f i n t e r e s t p a i d on bonds goes up, and the economy's r e l a t i v e s e c u r i t y mix s h i f t s so as t o r a i s e t h e p r o p o r t i o n o f r i s k y i n v e s t m e n t f i n a n c e d by i s s u i n g d e b t . Note t h a t i n the l i m i t i n g c ase o f z e r o c o s t s t o b a n k r u p t c y , c = r , the M o d i g l i a n i - M i l l e r theorem i s t r u e and t h e r e i s no e q u i l i b r i u m d e t e r m i n e d d e b t / e q u i t y r a t i o ' f o r the economy, i n the sense t h a t any d e b t / e q u i t y r a t i o i s c o m p a t i b l e w i t h an i n v e s t o r e a r n i n g an e x p e c t e d r a t e o f r e t u r n o f r s on h i s p o r t f o l i o . We propose now t o examine the changes i n t h e e q u i l i b r i u m p r o b a b i l i t y o f d e f a u l t , w i t h r e s p e c t t o the parameters o f the model. In o r d e r t o do so i t s u f f i c e s t o examine the e q u i l i b r i u m e q u a t i o n (6.14) l e t t i n g z*w* = y * , the e q u i l i b r i u m d e f a u l t p o i n t . y * M h(y*) = c / 0 s d F ( s ) + r / * s d F ( s ) = p. (6.20) Note t h a t 3h(y*) 3y < 0, and i t i s s t r a i g h t f o r w a r d t o show - 134 -P r o p o s i t i o n 6.4: The e q u i l i b r i u m p r o b a b i l i t y o f d e f a u l t o r b a n k r u p t c y o f r i s k y f i r m s i n c r e a s e s w i t h a) d e c r e a s e s i n the r i s k l e s s i n t e r e s t r a t e , b) i n c r e a s e s i n the p r o d u c t i v i t y o f . r i s k y i n v e s t m e n t and c) d e c r e a s e s i n the c o s t o f b a n k r u p t c y . T h e r e f o r e low i n t e r e s t r a t e s on r i s k l e s s s e c u r i t i e s a r e a s s o c i a t e d w i t h h i g h b a n k r u p t c y r a t e s on r i s k y f i r m s , the r e a s o n b e i n g t h a t , from propo-s i t i o n 6.2, w i t h low i n t e r e s t r a t e s f i r m s tend t o have h i g h l y l e v e r e d c a p i t a l s t r u c t u r e s and t h e r e f o r e b a n k r u p t c y becomes more l i k e l y . Note a l s o t h a t economies w i t h m o r e ' h i g h l y p r o d u c t i v e t e c h n o l o g i e s tend to have h i g h e r b a n k r u p t c y r a t e s . T h i s completes the a n a l y s i s o f the two s e c u r i t y market: There i s one problem we have l e f t u n t r e a t e d . T h i s i s the q u e s t i o n o f the p r o d u c t i o n d e c i s i o n on the p a r t o f the f i r m . I f the f i r m has a p r o d u c t i o n d e c i s i o n t o make, f o r example the s c a l e o f i n v e s t m e n t o r c h o i c e o f t e c h n i q u e , which i t d o e s n ' t i n the s i m p l e economy c o n s i d e r e d h e r e , t h e n t h i s d e c i s i o n may 22 be c l o s e l y r e l a t e d t o the d e b t / e q u i t y r a t i o and the p r i c e o f d e b t . T h i s i s an i m p o r t a n t q u e s t i o n t o t a c k l e , as r e a l d e c i s i o n s become c l o s e l y r e -l a t e d t o f i n a n c i a l v a r i a b l e s , and the presumed independence o f t h e s e v a r i a b l e s i n n e o c l a s s i c a l t h e o r y i s no l o n g e r v a l i d . 7. C o n c l u s i o n We have c o n s i d e r e d t h r e e models o f a c a p i t a l market which has a t y p e o f d e b t c o n t r a c t through which i n v e s t m e n t i s made on a r i s k y i n v e s t m e n t p r o j e c t . The s i g n i f i c a n t f e a t u r e o f t h i s c o n t r a c t i s t h a t , g i v e n some' p o s i t i v e amount o f i n v e s t m e n t was u n d e r t a k e n , t h e r e i s always some p o s i -t i v e p r o b a b i l i t y t h a t the borrower w i l l d e f a u l t on the l o a n . Thus, l e n d -i n g v i a t h i s type o f debt c o n t r a c t becomes a r i s k y v e n t u r e as w e l l . In - 135 -each case we c o n s i d e r e d the d e t e r m i n a t i o n o f the e q u i l i b r i u m i n t e r e s t r a t e on the l o a n and the e q u i l i b r i u m p r o b a b i l i t y o f d e f a u l t . A number o f com-p a r a t i v e s t a t i c r e s u l t s were d e v e l o p e d and t h e s e a r e summarized i n T a b l e 1 below. Some o b s e r v a t i o n s on t h e s e r e s u l t s f o l l o w . 1. Note t h a t w i t h r e s p e c t t o changes i n the i n t e r e s t r a t e and t h e p r o -d u c t i v i t y o f i n v e s t m e n t the r e s u l t s o f the s u p p l y and demand models a r e c o m p l e t e l y o p p o s i t e . T h i s might s u g g e s t t h a t i n some i n t e r m e d i a t e c a s e s , f o r example where both demand and s u p p l y c u r v e s were not p e r f e c t l y e l a s t i c the r e s u l t s would p r o b a b l y be ambiguous. U n l e s s one can make a good case f o r one s i d e o f the market d o m i n a t i n g i t would seem d i f f i c u l t t o argue on t h e o r e t i c a l grounds any q u a l i t a t i v e c o m p a r a t i v e s t a t i c r e l a t i o n s h i p be-tween r i s k l e s s i n t e r e s t r a t e s and the p r i c e o f l o a n s i n the money-market.- -2. N o t i c e t h a t i n the s u p p l y and demand models the e f f e c t o f an i n c r e a s e i n r i s k i s t o r a i s e the e q u i l i b r i u m i n t e r e s t r a t e on l o a n s . T h e r e f o r e , even though we have assumed r i s k n e u t r a l i t y on the p a r t o f f i r m s and i n -v e s t o r s , i n c r e a s e s i n r i s k on i n v e s t m e n t p r o j e c t s c o r r e s p o n d t o h i g h e r l o a n r a t e s . Hence the c o s t o f c a p i t a l t o i n v e s t m e n t p r o j e c t s unambiguous-l y c o n t a i n s a r i s k d i s c o u n t when the p o s s i b i l i t y o f d e f a u l t i s a d m i t t e d . In o t h e r words, when a l l r e a l i n v e s t m e n t must be f i n a n c e d by i s s u i n g debt or t a k i n g out a l o a n , then comparing two f i r m s , one r i s k i e r than the o t h e r i n t h e R o t h s c h i l d - S t i g l i t z s e n s e , the r i s k i e r f i r m w i l l have t o pay a h i g h e r i n t e r e s t r a t e on i t s l o a n s o r the debt i t i s s u e s than the l e s s 23 r i s k y f i r m . R e c a l l t h a t the i n c r e a s e s i n r i s k c o n s i d e r e d h o l d the ex-p e c t e d v a l u e o f the i n v e s t m e n t p r o j e c t c o n s t a n t , and thus d i f f e r e n c e s i n the l o a n r a t e cannot be a t t r i b u t e d t o d i f f e r e n c e s i n e x p e c t e d v a l u e s . - 136 -T a b l e 1 SUMMARY OF COMPARATIVE STATIC RESULTS A. Demand Determined Model Parameter I n c r e a s e s E q u i l i b r i u m V a r i a b l e s Response q P r r P r i s k B. Supply Determined Model E q u i l i b r i u m V a r i a b l e Response Parameter I n c r e a s e s q r - -p + c - -r i s k + + C. D e b t - E q u i t y Model/Supply Determined E q u i l i b r i u m V a r i a b l e Response q w p rz—i r~ r p c ? + + - - -+ + . + NOTE: The p i n each column denotes the e q u i l i b r i u m p r o b a b i l i t y o f d e f a u l t . A " ?" s i g n i f i e s an ambiguous c o m p a r a t i v e s t a t i c r e s u l t . - 137 -3. The i n t r o d u c t i o n o f e q u i t y i n t o the model g i v e s the market an a d d i -t i o n a l e q u i l i b r a t i n g mechanism, i n t h a t movement between bonds, e q u i t y and the r i s k l e s s s e c u r i t y a f f e c t not o n l y the r e l a t i v e q u a n t i t i e s o f t h e s e s e c u r i t i e s but a l s o the e x p e c t e d r e t u r n t o h o l d i n g them by a l t e r i n g the d e f a u l t p o i n t on d e b t . 4. In the s u p p l y model and t w o - s e c u r i t y model t h e c o s t s a s s o c i a t e d w i t h d e f a u l t on l o a n s o r the b a n k r u p t c y o f l i m i t e d l i a b i l i t y f i r m s have r a t h e r d i f f e r e n t i m p l i c a t i o n s . In the o n e - s e c u r i t y l o a n model l o w e r i n g the c o s t s o f d e f a u l t t o l e n d e r s lowers both the nominal i n t e r e s t r a t e on l o a n s and the e q u i l i b r i u m p r o b a b i l i t y o f d e f a u l t . C o n v e r s e l y i n the two s e c u r i t y model, l o w e r i n g the r e s o u r c e c o s t o f b a n k r u p t c y r a i s e s nominal i n t e r e s t r a t e s on bonds and r a i s e s the p r o b a b i l i t y o f b a n k r u p t c y . T h i s s u g g e s t s t h a t the i n t r o d u c t i o n o f a l i m i t e d l i a b i l i t y s e c u r i t y as a means o f f i n -a n c i n g i n v e s t m e n t may be a mixed b l e s s i n g from a w e l f a r e p o i n t o f view. Should l o w e r i n g t h e c o s t s o f b a n k r u p t c y become a p o l i c y i s s u e , f o r example by a r e f o r m o f the b a n k r u p t c y laws, then t h i s w i l l have the u n f o r t u n a t e e f f e c t o f r a i s i n g the p r o b a b i l i t y o f d e f a u l t or b a n k r u p t c y . The e x p e c t e d g a i n from such an a c t i o n may be e i t h e r p o s i t i v e o r n e g a t i v e . 5. The i n t r o d u c t i o n o f t r a n s a c t i o n s c o s t s t o b a n k r u p t c y has the i m p l i -c a t i o n t h a t a c o m p e t i t i v e c a p i t a l market may.have e q u i l i b r i a w i t h a d e t e r -minate d e b t / e q u i t y r a t i o . The i n a b i l i t y t o d e v e l o p models which p r o v i d e an e x p l a n a t i o n o f d e b t / e q u i t y r a t i o d e t e r m i n a t i o n has l o n g been a problem i n economic t h e o r y and c o r p o r a t i o n f i n a n c e . S t i g l i t z [1972] has r e c e n t l y 24 g i v e n an e x p l a n a t i o n i n terms o f d i v e r g e n t e x p e c t a t i o n s . The e x p l a n a t i o n o f f e r e d i n s e c t i o n 6 i s based on a c o s t t o b a n k r u p t c y argument, i n a d d i -t i o n t o the n e c e s s a r y assumption t h a t a l l c o r p o r a t e debt i n r i s k y f i r m s - 138 -has some p o s i t i v e p r o b a b i l i t y o f d e f a u l t . I t was e s t a b l i s h e d t h a t the h i g h e r the t r a n s a c t i o n s c o s t s o f b a n k r u p t c y a r e , the lower i s the e q u i l -i b r i u m d e b t / e q u i t y r a t i o . F u r t h e r e f f o r t i n d e v e l o p i n g t e s t a b l e hypoth-e s e s which p r o v i d e an e x p l a n a t i o n as t o how t h i s i m p o r t a n t economic v a r i a b l e i s d e t e r m i n e d seems c a l l e d f o r . - 139 -F o o t n o t e s 1. See notes 5-10 below. The i n c r e a s i n g a t t e n t i o n p a i d t o r i s k by econo-m i s t s s t u d y i n g c a p i t a l markets i s no doubt r e s p o n s i b l e f o r t h i s con-c e r n , as t h e r i s k o f d e f a u l t i s o f t e n the most s i g n i f i c a n t r i s k f a c i n g the l e n d e r . The l i t e r a t u r e which t r e a t s t h i s t o p i c most e x t e n s i v e l y i s the c r e d i t r a t i o n i n g l i t e r a t u r e , a l t h o u g h i t i s u s u a l l y a t the i n d i v i d u a l f i r m o r i n v e s t o r l e v e l . See f o r example Hodgman [1960], J a f f e e and M o d i g l i a n i [ 1 9 6 9 ] , J a f f e e [1971] and the r e f e r e n c e s con-t a i n e d t h e r e i n . 2. For example p r e f e r r e d s t o c k , c o n v e r t i b l e d e b e n t u r e s , w a r r a n t s o r s t o c k o p t i o n s . 3. T h i s p a r t i c u l a r a s p e c t o f the l o a n market has been t r e a t e d by B a r r o [ 1 9 7 4 ] . 4. The g e n e r a l e q u i l i b r i u m e x t e n s i o n c o u l d be g i v e n by i n t r o d u c i n g a r i s k -l e s s p r o d u c t i o n t e c h n o l o g y w i t h c o n s t a n t r e t u r n s t o s c a l e i n c a p i t a l i n v e s t m e n t . T h i s would t i e down the r i s k l e s s i n t e r e s t r a t e . T h i s type o f g e n e r a l e q u i l i b r i u m t r e a t m e n t i s used i n s e c t i o n 6 below. 5. Altman [1971] documents the e m p i r i c a l r e l a t i o n s h i p between c o r p o r a t e b a n k r u p t c y and g e n e r a l economic v a r i a b l e s . These f i g u r e s a r e from h i s T a b l e 1-3, page 15. 6. The l i t e r a t u r e here i s v e r y l a r g e . For example see M i l l e r and M o d i g l i a n i [ 1 9 58], Smith [ 1 9 7 2 ] , and S t i g l i t z [ 1 9 6 9 ] , [1974]. 7. General e q u i l i b r i u m models which o p e r a t e s e q u e n t i a l l y o v e r time have been d e v e l o p e d by Radner [ 1 9 7 2 ] , Hahn [ 1 9 7 1 ] , D i e w e r t [ 1 9 7 4 ] , Grandmont [ 1 9 7 4 ] , and Green [197 3 ] . Green [1972] and Arrow and Hahn [ 1 9 7 1 ] , pp. 119-122 have d e a l t p a r t i c u l a r l y w i t h the i s s u e o f p r e - e x i s t i n g c o n t r a c t s . - 140 -8. The i n t r o d u c t i o n o f i n f o r m a t i o n i n t o economic a n a l y s i s i s s t i l l i n i t s r e l a t i v e i n f a n c y . Some p r o g r e s s has been made i n d e v e l o p i n g the co n c e p t o f ' s c r e e n i n g ' . See Spence [1974] and Arrow [1973] f o r example. 9. B a r r o [1974] d e a l s w i t h t h i s . S e c t i o n 5 below t r e a t s the r o l e o f c o l l a t e r a l and i t s r e l a t i o n t o d e f a u l t . 10. • See Keynes [ 1 9 3 6 ] , C h a p t e r 12. 11. The case o f r i s k a v e r s e borrowers i s t r e a t e d i n S e c t i o n 4 below. 12. A l l i n t e g r a l s can be taken t o be Riemann i n t e g r a l s as the d i s t r i b u t i o n F ( s ) has a c o n t i n u o u s d e n s i t y f u n c t i o n and the r e g i o n o f i n t e g r a t i o n i s a compact i n t e r v a l . 13. In f a c t an i n t e r i o r s o l u t i o n can always be shown t o e x i s t , p r o v i d e d M M = U'(0) [/, r s d F ( s ) - / r z d F ( s ) ] > 0. dEU dx n Z X=0 From C . l . U'(0) > 0. T h e r e f o r e d E U dx i s g r e a t e r than z e r o i f x=0 M M / s d F ( s ) - z / dF.(s) > 0. M D i v i d i n g both s i d e s by / d F ( s ) we have E ( s | s > z) - z > 0 p r o v i d e d z > 0. T h e r e f o r e the demand f u n c t i o n f o r l o a n s i s d e f i n e d i m p l i c i t l y by (4.3) p r o v i d e d q > 0. 14. We m a i n t a i n the assumption t h a t F ( s ) has a c o n t i n u o u s d e n s i t y f u n c t i o n f o r reasons mentioned above. See note 12. 15. T h i s p r o p o s i t i o n e x c l u d e s t he d e g e n e r a t e case g ' ( z * ) = 0 a t an e q u i l -i b r i u m d e f a u l t p o i n t g ( z * ) = p. T h i s case i s de g e n e r a t e i n the sense t h a t the s e t o f p o i n t s i n [ 0 , M] on which i t may o c c u r i s o f measure z e r o by Sard's theorem. - 141 -16. As we have m a i n t a i n e d the assumption o f r i s k n e u t r a l i t y i n t h i s s e c -t i o n , e x p e c t e d v a l u e i s the o n l y r e l e v a n t moment o f the d i s t r i b u t i o n which needs t o be c o n s i d e r e d . The o p t i m a l i t y o f the c a p i t a l market a l l o c a t i o n o f i n v e s t m e n t when i n v e s t o r s a r e r i s k a v e r s e has been c a l l e d i n t o q u e s t i o n by a number o f p e o p l e . See the r e c e n t Symposium on t he O p t i m a l i t y o f C o m p e t i t i v e C a p i t a l Markets [1974]. They are con c e r n e d s o l e l y w i t h t h e ' o p t i m a l a l l o c a t i o n o f r i s k b e a r i n g , however, and d e a l w i t h c a p i t a l markets i n which common s t o c k s a re the o n l y type o f s e c u r i t i e s a v a i l a b l e ; c o n s e q u e n t l y d e f a u l t i s not a p o s s i b i l i t y . 17. As f o r example o u t l i n e d i n C h a p t e r I I I , s e c t i o n 2 i n t h i s t h e s i s . 18. See f o r example Smith [1972] and S t i g l i t z [ 1 9 7 2 ] . S t i g l e r [1967] has an i n t e r e s t i n g d i s c u s s i o n o f t r a n s a c t i o n c o s t s i n c a p i t a l markets and t t h e i r r e l a t i o n t o the much used term ' c a p i t a l , market i m p e r f e c t i o n s ' . . 19. The o r i g i n a l M+M theorem was a p a r t i a l e q u i l i b r i u m p r o p o s i t i o n . See M i l l e r and M o d i g l i a n i [ 1 9 5 8 ] . S t i g l i t z [ 1 9 6 9 ] , [1974] gave a g e n e r a l e q u i l i b r i u m v e r s i o n o f the theorem. 20. T h i s f o l l o w s as h(y) i s a c o n t i n u o u s l y d i f f e r e n t i a t e f u n c t i o n on (0, M) and h'(y) = (c - r ) y d F ( y ) < 0, f o r y e (0, M). 21. I f p > r s , then e q u i l i b r i u m r e q u i r e s t h a t a l l i n v e s t m e n t o c c u r i n the r i s k l e s s f i r m s . I f p = r s , then p r o v i d e d no debt i s i s s u e d t he a l l o -c a t i o n o f i n v e s t m e n t between r i s k y and r i s k l e s s f i r m s i s a m a t t e r o f i n d i f f e r e n c e t o r i s k n e u t r a l i n v e s t o r s . If.p < c s , then no i n v e s t m e n t w i l l be made i n the r i s k l e s s i n d u s t r y . In t h i s case i t might be p o s s i b l e t o have an e q u i l i b r i u m w i t h a p o s i t i v e d e b t / e q u i t y r a t i o , but i n t h i s case e q u a t i n g the e x p e c t e d r e t u r n s o f bonds and e q u i t y would d e t e r m i n e o n l y one o f the two v a r i a b l e s , q and w. T h e r e f o r e the nominal i n t e r e s t r a t e on bonds would be a f u n c t i o n o f t h e d e b t / e q u i t y - 142 -r a t i o , and t h e r e would e x i s t a continuum o f e q u i l i b r i a w i t h v a r y i n g d e b t / e q u i t y r a t i o s . 22. S t i g l i t z [1972] and Smith [1969] have made some p r o g r e s s on t h i s i s s u e . 23. S t r i c t l y s p e a k i n g t h i s c o n c l u s i o n does n o t f o l l o w a u t o m a t i c a l l y from the c o m p a r a t i v e s t a t i c p r o p o s i t i o n s 3.4 and 5.5. I t can be shown, however, t h a t i f t h e r e a r e two f i r m s which d i f f e r i n t h e i r r i s k c h a r a c t e r i s t i c s i n the R o t h s c h i l d - S t i g l i t z s e n s e , then c a p i t a l market e q u i l i b r i u m i s c h a r a c t e r i z e d by two l o a n r a t e s which bear t he same q u a l i t a t i v e r e l a t i o n s h i p t o one a n o t h e r as g i v e n i n p r o p o s i t i o n s 3.4 and 5.5. T h i s argument can be extended t o any number o f f i r m s . 24. As S t i g l i t z n o t e s , one o f the problems w i t h t he d i v e r g e n t e x p e c t a t i o n s a s s u m p t i o n , i s .that i t can e x p l a i n a l m o s t a n y t h i n g , and e m p i r i c a l l y i s v e r y d i f f i c u l t t o t e s t f o r . L i n t n e r [1969] a l s o d e a l s w i t h d i v e r s e e x p e c t a t i o n s i n a d i f f e r e n t c o n t e x t . - 143 -Re f e r e n c e s Altman, E. I. [1971], Corporate Bankruptcy in America, L e x i n g t o n , M a s s a c h u s e t t s : D. C. Heath and Company. Arrow, K.J. [1 9 7 3 ] , "Higher E d u c a t i o n as a F i l t e r " , Journal of Public Economics, 2, 193-216. and Hahn, F.H. [ 1 9 7 1 ] , General Competitive Analysis, San F r a n c i s c o : Hoi den-Day. Barrow R.J. [ 1 9 7 4 ] , "The Loan Market, C o l l a t e r a l and Rates o f I n t e r e s t " , Report 7401, C e n t r e f o r Mathematical S t u d i e s i n B u s i n e s s and Econo-m i c s , U n i v e r s i t y o f C h i c a g o . Diamond, P.A. and S t i g l i t z , J . E . [1 9 7 4 ] , " I n c r e a s e s i n R i s k and R i s k A v e r s i o n " , Journal of Economic Theory, 8, 337-360. D i e w e r t , W.E. [1 9 7 4 ] , " W a i r a s 1 Theory o f C a p i t a l F o r m a t i o n and the E x i s t -ence o f a Temporary E q u i l i b r i u m " , paper p r e s e n t e d t o the c o n f e r e n c e ' on " E q u i l i b r i u m and D i s e q u i l i b r i u m i n Economic Theory" a t V i e n n a , J u l y 1974. Grandmont, J.M. [1974], "On the S h o r t Run E q u i l i b r i u m i n a Monetary Economy", i n J . Dreze ( e d . ) , Allocation under Uncertainty, Equilibrium and Gptimality, f o r t h c o m i n g M a c m i l l a n . Green, J.R. [1 9 7 3 ] , "Temporary General E q u i l i b r i u m i n a S e q u e n t i a l T r a d i n g Model w i t h Spot and F u t u r e s T r a n s a c t i o n s " , Econometrica, 41, 1103-1124. [1972], " P r e - e x i s t i n g C o n t r a c t s and Temporary General E q u i l i b -r i u m " , D i s c u s s i o n Paper Number 246, Harvard I n s t i t u t e o f Economic R e s e a r c h , Harvard U n i v e r s i t y . Hahn, F.H. [1 9 7 1 ] , " E q u i l i b r i u m w i t h T r a n s a c t i o n C o s t s " , Econometrica, 39, 417-439. Hodgman, D.R. [19 6 0 ] , " C r e d i t R i s k and C r e d i t R a t i o n i n g " , Quarterly Journal of Economics, 74, 258-278. J a f f e e , D.M.. [1971] , Credit Rationing and the Commercial Loan Market, New York: W i l e y . and M o d i g l i a n i , F. [ 1 9 6 9 ] , "A Theory and T e s t o f C r e d i t R a t i o n i n g " , American Economic Review, 59, 850-872. Keynes, J.M. [1 9 3 6 J , The General Theory of Employment Interest and Money, New York: H a r c o u r t Brace. L i n t n e r , J . [ 1 9 6 9 ] , "The A g g r e g a t i o n o f I n v e s t o r ' s D i v e r s e Judgments and P r e f e r e n c e s i n P u r e l y C o m p e t i t i v e S e c u r i t i e s M a r k e t s " , Journal of Financial and Quantitative Analysis, 4, 347-400. - 144 -M i l l e r , M.H. and M o d i g l i a n i , F. [19 5 8 ] , "The C o s t o f C a p i t a l , C o r p o r a t i o n F i n a n c e , and the Theory o f Investment", American Economic Review, 48, 261-297.• Radner, R. [1972], " E x i s t e n c e o f E q u i l i b r i u m P l a n s , P r i c e s and P r i c e E x p e c t a t i o n s i n a Sequence o f Mar k e t s " , Econometrica, 40, 289-303. R o t h s c h i l d , M. and J . E . S t i g l i t z [ 1 9 7 0], " I n c r e a s i n g R i s k . I. A D e f i n i -t i o n " , Journal of Economic Theory, 2, 225-243. Smith, V.L. [1 9 7 2 ] , " D e f a u l t R i s k , S c a l e , and the Homemade Leverage Theorem", American Economic Review, 62, 66-76. Spence, A.M. [19 7 3 ] , "Job Market S i g n a l l i n g " , Quarterly Journal of Economics^ 87, 355-374. S t i g l e r , C.J. [1 9 6 7 ] , " I m p e r f e c t i o n s i n the C a p i t a l Market", Journal of Political Economy, 75, 287-292. S t i g l i t z , J . E . [ 1 9 6 9 ] , "A Re-Examination o f t h e M o d i g l i a n i - M i H e r Theorem", American Economic Review, 59, 78-93. [1972], "Some A s p e c t s o f the Pure Theory o f C o r p o r a t e F i n a n c e ; B a n k r u p t c i e s and T a k e o v e r s " , Bell Journal of Economics and Management Science, 3, 458-82. [ 1 9 7 4 ] , "On the I r r e l e v a n c e o f C o r p o r a t e F i n a n c i a l P o l i c y " , American Economic Review, 64, 851-866. Symposium on the O p t i m a l i t y o f C o m p e t i t i v e C a p i t a l M a r k e t s , Bell Journal of Economics and Management Science, 5, 125-186. 

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