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International tax competition : the effects of transportation costs Ritchie, James Robert. 1995

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I N T E R N A T I O N A L T A X C O M P E T I T I O N : T H E E F F E C T S O F T R A N S P O R T A T I O N C O S T S By J A M E S R O B E R T RITCHIE A. Sc. (Mathematics and Engineering), Queen's University, 1989 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF T H E REQUIREMENTS FOR T H E D E G R E E OF M A S T E R OF SCIENCE in T H E FACULTY OF GRADUATE STUDIES DEPARTMENT OF MATHEMATICS INSTITUTE OF APPLIED MATHEMATICS We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH COLUMBIA August 1995 © J A M E S R O B E R T RITCHIE, 1995 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract We examine a four player stochastic game in which two of the players are representative households and two of the agents are governments. The game is set in a two country general equil ibr ium model where it is costly to transport goods between nations. A t the beginning of each period, each country receives a random endowment of a single non-storable consumption good. Consumers determine optimal consumption levels and goods shipments based on tax rates and the amount of endowments received that period. Governments determine optimal tax rates and expenditure levels in a strategic form stage game. W i t h complete financial markets, the terms of trade are determined by the relative amounts of goods remaining in each country after taxation and thus governments can manipulate the pattern of trade through tax policies. The choice between cooperating or acting independently in the stage game creates a Prisoners' Di lemma. In the short run governments refuse to cooperate and resort to "beggar thy neighbour" tax policies; each government attempts to better its own domestic situation by over-taxing in order to prey on the resources of its opponent. In an application of the Folk Theorem we find that the long run behaviour is cooperative and over-taxation is avoided, provided that governments and households are sufficiently patient. The addition of transportation costs adds another dimension to the short run be-haviour of governments. In the same model without transportation costs, governments implement a single optimal policy strategy that is independent of the direction of net flow of goods and a unique pure strategy Nash Equi l ib r ium always exists for the stage game. When transportation costs are introduced the optimal behaviour of households partitions the endowment space into three separate shipping regions: a foreign export, n domestic export and no shipping region. In this setting governments can set policy rules contingent upon the current direction of trade, and are thus able discriminate against unfavourable patterns of trade. The ability to manipulate particular directional flows of goods combined with governments acting in their own self-interests results in the non-existence of a pure strategy Nash equil ibrium in the stage game for certain distributions of endowments. This outcome can be interpreted as a trade disagreement between the two countries; a scenario where governments disagree as to which shipping state currently ex-ists. Repeti t ion however can lead to cooperation. If the governments and households are sufficiently patient, then, for the repeated game, there is a unique cooperative subgame perfect equil ibrium that prevents trade disputes from occurring. 111 Table of Contents Abstract ii List of Tables vi List of Figures vii Acknowledgement viii 1 Introduction 1 2 The Model 4 3 The Consumers' Optimization Problem 7 4 Government Policy Game 13 5 Policy Game with No Transportation Costs 17 5.1 Nash Equi l ibr ia for Stage Game with No Transportation Costs 17 5.2 Cooperative Equi l ibr ia for the Stage Game with No Transportation Costs 23 5.3 Equ i l ib r ium for Repeated Game with No Transportation Costs 28 6 Tax Policy Game with Transportation Costs 33 6.1 The Consumers' Optimizat ion Problem: Solution to the C P P 33 6.2 Nash Correspondences for Stage Game with Transportation Costs . . . . 37 6.3 Nash Equi l ibr ia for the Stage Game wi th Transportation Costs 51 6.4 Cooperative Equi l ibr ia for the Stage Game wi th Transportation Costs . . 63 iv Appendices 68 A Proofs for Chapter 5 69 B Proofs for Chapter 6 76 Bibliography 93 v List o f Tables 5.1 Nash Equi l ib r ia in the Policy Stage Game without Shipping Costs 22 5.2 Nash Tax Rates in the Policy Stage Game without Shipping Costs. . . . 22 5.3 Cooperative Equi l ib r ia in the Pol icy Stage Game without Shipping Costs. 25 5.4 Cooperative Tax Rates in the Policy Stage Game without Shipping Costs. 25 6.1 Opt ima l Shipping and Consumption Plans for u(c) = log(c) 34 6.2 Nash Equi l ib r ia in the Pol icy Stage Game wi th Shipping Costs when (1 + s)2 > - f r 53 6.3 Nash Tax Rates in the Policy Stage Game Shipping Costs when (1 + s)2 > - f r 54 6.4 Nash Equi l ib r ia in the Policy Stage Game with Shipping Costs when (1 + s)2 < 55 6.5 Nash Tax Rates in the Policy Stage Game wi th Shipping Costs when ( l + * ) 2 < i ) 5 6 6.6 Cooperative Equi l ibr ia in the Policy Stage Game wi th Shipping Costs. . 65 6.7 Cooperative Tax Rates in the Policy Stage Game with Shipping Costs. . 66 v i List of Figures 5.1 Nash Equi l ib r ium for Policy Stage Game with No Shipping Costs 20 6.1 Part i t ioning of Strategy Space into Shipping Regions 36 6.2 Nash Strategy for the Home Government in the No Shipping Region. . . 40 6.3 Opt ima l Regional Strategies for the Home Government wi th ?/* > 77*. . . 44 6.4 Opt ima l Regional Strategies for the Home Government wi th 77* < 77*. . . 45 6.5 Home Government Nash Reaction Correspondence when (1 + s)2 > 48 6.6 Home Government Nash Reaction Correspondence when (1 + s)2 < 49 6.7 A No Shipping Nash Equi l ibr ium in the Policy Stage Game wi th Shipping Costs 57 6.8 Non-existence of a Pure Strategy Nash Equi l ib r ium in the Pol icy Game wi th Shipping Costs. 58 6.9 Part i t ioning of the Endowment Space in the Pol icy game wi th Shipping Costs when (1 + s)2 > K(-J) 61 v i i Acknowledgement I would like to thank both of my supervisors, Dr . Col in W . Clark of the Department of Mathematics and Dr . Raman Uppal of the Faculty of Commerce and Business A d m i n -istration for their help and guidance in this project. A special note of acknowledgement goes out to Anders Svensson, the author of the kuvio.tex macro package which was used to create the diagrams in this paper. Last of all I would like to thank my fellow stu-dents in the Department of Mathematics for their constant support and encouragement throughout the duration of my studies at the University of Br i t i sh Columbia . v m Chapter 1 Introduction As the world economy becomes more and more unified, the study of international fiscal competition and coordination becomes increasingly important. Economic structures such as integrated financial markets and free trade zones provide conduits for the transmission of national policies. In this state of heightened economic interdependence, the fiscal policy of one country wi l l have a marked effect on the economy of its neighbour. The nature of these effects however depend largely on the method of policy transmission. In this paper, we examine the role that securities markets and transportation costs play in the design of national fiscal policies in integrated economies. The importance of this study has not gone unoticed; there is a large literature that already exists in this field. The standard model consists of two or more regional gov-ernments who each provide a local public good by levying a single tax on some type of mobile capital [20], [21]. In these models, the strategic interaction between governments is provided v ia capital movements; the tax rate of one government alters the distribution of capital allocation between countries, thereby affecting the tax base of its neighbours. Governments, fearing the possibility of capital flight, are lenient when determining tax rates and thus the standard prediction of a model of this type is that in a competitive equil ibrium, there is a lack of public goods and that government coordinated policies are Pareto improving 1 . Our model differs substantially from the standard class of fiscal competition models, ^ehoe [9] constructs a tax competition model where policy coordination is undesirable 1 Chapter 1. Introduction 2 Our model is based on a stochastic endowment process, while the standard fiscal com-petit ion model is based on a production economy. As a result, the mechanism of policy transmission in our model is not the same. Where the standard model emphasizes the role of real capital investment on the government policy, our model emphasizes the role of security markets. This difference is at the root of the contradiction in the predictions of our model and the standard policy competition model. Where the standard model produces an equil ibrium where there is a shortage of public goods, our economy has an excess of public goods. Other policy models that incorporate a stochastic element in-clude: Frenkel and Raz in [5] who examine a two country stochastic taxation model, in order to assess the transmission of fiscal shocks via stock markets (their economy however is a production economy) and Barari and Lapan [1] who study a tariff competition model (but only as an open economy, not in game theoretic terms). Secondly, our model has an infinite time horizon. The standard fiscal competition model is static or two periods at most. The infinte time horizon permits richer strategic behaviour on the part of governments that cannot be realized in models of shorter du-ration. Coates [3] examines tax competition in a repeated game setting. Coates' model is based on a production economy and thus the lengthened time horizon only serves to exaggerate the lack of public goods when governments do not cooperate, however his model does permit collusion between governments. Final ly , our model incorporates a shipping cost. Most fiscal competition models have perfectly mobile capital, an assumption that forces governments to set tax rates according to an arbitrage equation [14]. A more realistic model should include sunk costs for investments. M i n t z and Tulkens [13] as well as de Crombrugghe and Tulkens [4]) examine a model with costly transportation for labour. The geometry of their results are strikingly similar to that our results, but as they are dealing with an investment economy their predictions are opposite to ours. Chapter 1. Introduction 3 In the analysis that follows we find that due to the unique nature of our model, our results contradict the results of the standard fiscal competition model. In our economy, endowments are over taxed and public goods are too plentiful when governments are competing fiscally. The addition of transportation costs does not change this fact. It does however permit governments to set policies that discriminate against directional flows of goods which in turn leads to trade disagreements. In an application of the Folk Theorem we show that coordinated fiscal policies are more efficient and are enforcible in our stochastic game v ia a G r i m Trigger strategy. Chapter 2 The Model We consider a two country general equil ibrium model - an extension of of the endow-ment economy found in Secru, Uppa l and van Hulle [15] which was, in turn an extension of Lucas ' tree economy [11]. In this economy, each country has a random endowment process of a consumable good, as well as two agents: a representative household and a government. The endowment processes are taken to be identical, but independent of one another. The endowed good is non-storable and can be shipped between countries, but shipping is costly and a portion of the shipment is lost in transit. This dissipative cost is understood to be a proxy for any impediments to trade, such as time lags and informa-tional costs as well as the physical costs incurred in making shipments. Representative households hold ownership claims to the endowment process in their respective countries and use these claims to create securities for trading with their peers in order to obtain claims to foreign consumption goods. In this model, households derive ut i l i ty from both consumption and government expenditures, but taking government behaviour as given, the households only attempt to maximize their respective utilities over consumption. Governments, motivated by a desire to be re-elected, are benevolent towards the rep-resentative household belonging to their own country. We assume that this behaviour wi l l be observable regardless of the polit ical faction that is in power, hence we treat the government of each country as a single infinitely lived agent wi th a "fixed" objective. In particular, governments wi l l attempt to maximize the ut i l i ty of representative household of their respective countries accounting for both the ut i l i ty derived from consumption, 4 Chapter 2. The Model 5 as well as the ut i l i ty due to government expenditures. The government of each country also behaves strategically accounting for the tax policy of their peer government when determining their own opt imal policy. The model is setup as a stochastic game between four players: home government, home household, foreign government and foreign household. The sequence of events wi thin a period of t ime determines the structure of our game: / . A t the beginning of a t ime period t, households trade claims on endowments. The structure of the securities market is not specified, but it is assumed that a sufficient number of securities exist in order to provide complete markets. II. Each country then receives a random endowment of the consumption good; 9t units in the home country and 9* in the foreign country. We assume that endowments are strictly positive. III. Once the endowments have been realized, governments announce their tax policies and collect their taxes. Governments levy commodity taxes according to the origin principle 1 ; each government claims a portion of its own country's endowments. The home government sets a tax rate of Tt and the foreign government imposes a rate of rt*, where 0 < rt < 1 and 0 < r* < 1. IV. After taxes have been paid, households are left with holdings of perishable goods at home and abroad. In a model with no shipping cost each agent would simply ship home any goods she held abroad, however, in the face of shipping costs this strategy is no longer optimal. To avoid redundant shipping, households wi l l engage in a second period of trading where consumption goods held abroad are swapped at 1The alternative taxation principle is the destination principle which places a tax on consumption. Under this tax scheme there is no strategic interaction between governments in our model as we define the utility functions to be logarithms in the later sections of this paper. For a study of non-cooperative fiscal competition and the effects of switching between these two principles see Lockwood [10]. Chapter 2. The Model 6 the prevailing exchange rate for local goods and securities. If, after the completion of this trading, an agent st i l l owns goods abroad, then these goods wi l l be imported to the agent's home country. We denote the shipment of goods as exports: xt is the amount exported from the home country and x* is the amount of goods exported from the foreign country (i.e. imports). V. Once shipments have arrived at their destinations, government expenditures occur and consumption ensues. From the above, we see that there is a leader-follower relationship between govern-ments and households. Households, being the last to act in each period, take government policies as given, and ignore the effects that their behaviour might have on the determi-nation of tax rates (i.e. households do not exhibit strategic behaviour). Governments, by virtue of the fact that they act first, are able to impose their policies upon the households. When determining tax policies governments wi l l account for the reactions of households as well as the current foreign tax policy. Thus, it is necessary to solve this problem in two stages: first the optimal shipping and consumption plans of households are deter-mined, then governments set tax rates in a game based on the solutions to the consumers' problem. Chapter 3 The Consumers' Optimization Problem Households derive ut i l i ty from two sources: consumption and government expenditures. We assume that this ut i l i ty function wi l l be additive over both consumption and gov-ernment expenditures as well as over time. Home ut i l i ty of consumption is given by the function: u : 3 £ + — » 9 £ where u(-) is a strictly increasing concave C2 function. Let v : 3 £ + —> 3 £ represent the home household's ut i l i ty of government expenditure, the value of public goods to consumers, v(-) is a strictly increasing C2 function. The home house-hold's expected ut i l i ty at t ime t is then given by the function oo Et[J2^(u(ct) + 7v(gt))], (3.1) i=0 where 0 < /3 < 1 and 7 > 0. The constant (3 is the household's discount factor and the parameter, 7 is a measure of the relative importance of government expenditures for the well-being of the household and {gt} is the government expenditure process. U t i l i t y for the foreign household is defined in an analogous fashion with foreign variables being denoted by an asterisk. Each household strives to maximize her expected discounted util i ty, taking govern-ment policies as given. Thus households ignore the portion of ut i l i ty due to government expenditures and attempt only to maximize ut i l i ty due to consumption. Consumption is financed by the real dividends obtained from the household's portfolio of securities. W i t h complete markets, household agents wi l l strive to reach an optimal risk sharing arrangement; home agents wi l l hedge against endowment risk by gaining claims to the 7 Chapter 3. The Consumers' Optimization Problem 8 foreign endowment v ia security trading. The resulting equil ibrium can be replicated by a Central Planning Problem ( C P P ) 1 . In the C P P , we introduce a planner whose objective is to maximize the weighted sum of home and foreign household utilities by controlling the distribution of goods between countries. The weight assigned to each household is a function of the endowment process for each country as well as the ini t ia l wealth of each household agent. For the sake of s implici ty we assume symmetry between our countries: identical but independent endowment processes and equal ini t ia l wealths. In this case each household receives an equal weighting in the C P P objective function. In each period, the planner can choose to adjust the balance of consumption goods via shipments. These shipments represent the flow of goods due to post endowment trading between households. Let xt be the amount of goods exported from the home to the foreign country and x* be the amount of goods imported to the home country (foreign exports). Shipping however, is costly; for every unit of goods leaving a country, only ^ units arrive at the destination, where s > 0. Thus only Y+IX* units of exports wi l l be available for the foreign agent's consumption and only j^xt of imports can be consumed at home. Then the C P P is oo oo Max {BoEAl^ l + ^ oEWO]}, (3-2) t=o such that ct< (1 << (1 xt > 0, x* > 0. (3.5) Observing that goods are non-storable, endowments identically distributed i n each period, and ut i l i ty is t ime additive, there is no "linkage" between periods in the C P P . 1See Chapter 17 of Varian [19] for more details. i=0 - Tt)9t -xt + ^ <)e; - x* + x\ + 3-Xt 1 4 V (3.3) (3.4) Chapter 3. The Consumers' Optimization Problem 9 The central planner faces the same problem in every period and thus the above problem can be replaced by a single period static optimization problem. Before proceeding, we shall also make a change in notation. We define rjt and rj* to be the amount of consumption goods remaining after taxation, in the home and foreign countries respectively. Thus we have Vt = (l-rt)9t, (3.6) V; = ( W W . (3.7) Once this substitution has been made, the form of the C P P is identical to the optimizat ion problem found in [15]. The planner's problem is now Max { « ( C ) + « « ) } , (3.8) {xt > 0,x*t > 0} such that <H < Vt-xt + T%-, (3.9) 1 + 6 c*t < >/? " ^ + i f ^ - ( 3 - 1 0 ) The problem is solved using a standard static optimization approach. The Lagrangian is LCPP(ct,c*,xt,x*) = u(ct) + u(c*) + Xt(r]t - xt + - + s + A ^ - x J + ^ - c f ) . (3.11) Taking derivatives with respect to ct,c*,xt and x* we find: dLc" = u'(ct)-Xt = 0, (3.12) dct Chapter 3. The Consumers' Optimization Problem 10 % ^ = u'(c?) - At* = 0, (3.13) uct d-j*T = ^ ; - A , < 0 , with ^ | f - , = 0 , (3..4) = ^ . - K < 0 , wi th % F - x t - = 0. (3.15) From (3.12) and (3.13), we see that Xt = u'(ct) > 0 and A* = u'(c*) > 0 implying that the constraints on consumption, (3.9) and (3.10), are binding. These quantities are the shadow prices of the consumption good in the home and in the foreign country respectively. Substituting these results into (3.14) and (3.15), we find that V - < l + s and v - > — ^ > (3-16) A 4 At 1 + s hence 1 < ^ < l . + s. (3.17) 1 + s ~ u'(ct) The ratio of shadow prices, ^r^y , can be interpreted as an exchange rate (units of foreign goods per unit of home goods). We see that the exchange rate is bounded above and below by 1 + s and 1/(1 + s) respectively. These same bounds exist for the same economy without government agents [15]. The K u h n Tucker conditions also determine the size and the t iming of shipments. From (3.14), we see that — £ ^ < 0 =>xt = 0, (3.18) oxt u'(c*) thus —V^r < 1 + s => xt = 0. (3.19) u'{ct) If the exchange rate is sufficiently low, it wi l l not be optimal to export goods from the home country. X t > 0 1^^  = 0, (3.20) Chapter 3. The Consumers' Optimization Problem 11 thus • — L i Z = i + a X t > o . (3.21) w (Q) If the exchange rate rises to its highest possible value, then home goods are cheap relative to foreign goods and it wi l l be optimal to export goods from the home country. The amount of goods shipped must satisfy the equation for the exchange rate = 1 + 6. (3.22) u'(T]t - Xt) Once (3.22) has been solved, then we have a state contingent policy for exports: xt(r}*, rjt) : [0,0*t]x[0,6t]^[0,Vt]-Similar analysis can be performed for the condition in (3.15) to determine a feedback rule for exports from the foreign country. and where x* satisfies: x t * > 0 when ^ p i = — — , (3.24) * U'(ct) 1 + 5' V 7 " ' ^ " ^ - 1 (3.25) «'(»/*+ rfcs*) l + s' The solution to (3.25) forms a feedback rule for foreign exports: x*(t]*,rjt) : [0,0*] x [o ,^] [0 ,17; ] . Substituting these shipping policies into the C P P consumption constraints, (3.9) and (3.10), gives the optimal consumption policies: CM,*): [0,0,1 x[O ,0 t ] - » + , (3.26) c*(v;,Vt):[o,e;)x[o,0t] -> » + . (3.27) Chapter 3. The Consumers' Optimization Problem 12 Although household behaviour is not strategic, these functions can be interpreted as the household reaction correspondences to government tax policies (recall that r/t and 77* are functions of Tt and r*, see (3.6) and (3.7)). Each correspondence acts as a constraint for the optimization problem that each government faces in the policy game that follows. Chapter 4 Government Policy Game As stated earlier, we assume that governments acting in their own self-interests behave in a benevolent fashion towards the resident of their own country and thus the objective of each government is to maximize the ut i l i ty of its constituents. When determining an opt imal policy, each government considers the complete effects that its policies has on its resident's ut i l i ty and thus the objective function for each government includes both components of the household's ut i l i ty: the portion due to consumption as well as the portion due to government expenditures. The objective of the home government is Max 0 0 ^ E / * ' ( « ( c 0 + 7«(*))] , (4-1) {0 <TT < 1} *=0 such that 9t •< TA = 9t - V t , (4.2) ct = C(Vlm), (4.3) where C(r/*, Tjt) is the reaction correspondence for the home household, denned in (3.26). The foreign government faces a similar problem. The objective function is defined in the same manner but evaluated at c* and g* and is constrained by the foreign household reaction correspondence C*(rj*,rjt) defined in (3.26) as well as the foreign government budget constraint: g* < 9* — 77*. B y setting r t , the home government is indirectly setting rjt. This suggests a change in how we interpret this control problem; instead of having governments set tax rates, we 13 Chapter 4. Government Policy Game 14 shall define the government policy variable to be the amount of goods in its own country that are available for consumption and shipping. Thus, in period t, the home government determines a policy rjt, where 0 < r)t < 9t and the foreign government wi l l set a value for the policy r/*, where 0 <rj* < 9*. As was the case for the C P P , there is. no linkage between decisions made in different periods. Binding government budgets and time separable ut i l i ty effectively isolate periods such that government policy only effects the current economy. The stochastic game, (4.1) - (4.3) is effectively a repeated game; in each period the governments play the same one shot game wi th a new pair of endowments. To solve this problem, we need only consider the single period stage game. In period t, the home government attempts to maximize the sum: u(ct) + "fv(gt) subject to (4.2) and (4.3). Meanwhile, the foreign government wi l l be attempting to solve the following problem. Max {ti(c?) + 7 « ( t f ) } (4.4) {o < < < 9;} such that 9l < Pt-Vl (4-5) cl = C*(r,;,r,t). (4.6) Addi t ional ly , there is the issue of what solution concept should be used. Three ap-proaches can be applied to our game: i) Nash Game: In a Nash game governments are on equal footing wi th one an-other; each government has access to the same set of information and both gov-ernments announce their policies simultaneously. Addit ionally, governments are not allowed to communicate before the game commences and thus the possibility of a coordinated solution is eliminated. Nash equilibria are characterized by their Chapter 4. Government Policy Game 15 self-enforcing nature; neither government wi l l have an incentive to deviate from an equil ibrium strategy. As a result, governments treat the policy of their opponents as fixed when determining their own best policy response. ii) Stackelberg Game: In this scenario there is a dominant government which, due to some external poli t ical or economic factor, is able to act first and impose its tax policy upon its opponent. This situation is modeled by having the dominant government calculate and announce its optimal tax policy before its peer does. Stackelberg equilibria are not necessarily self-enforcing, and often suffer from the problem of time consistency 1 . i i i) Cooperation: Under this regime, governments agree to cooperate when determin-ing tax rates. The governments acting as a team, attempt to maximize a commonly agreed upon objective function. The game is replaced by a single optimization prob-lem. The solution to this problem often generates unstable equilibria; in certain scenarios one or both governments may find it advantageous to renege on the agree-ment. However, according to the Folk theorem, given sufficiently patient players an appropriate strategy, the cooperative solution can be made to be a stable and rational outcome of the game. In the sequel, we examine both the Nash and cooperative solutions to the policy stage game. The Stackelberg concept does not fit well wi th our model due to the as-sumed similarities between the two countries. We shall also examine the solution of the corresponding repeated game by applying a version of the Folk Theorem that has been altered for our stochastic game. For the sake of mathematical tractability, we restrict our study to the case of logarithmic util i ty, thus for the remainder of this paper, we define 1 A Stackelberg Equilibrium is time inconsistent if the leading player has an incentive to deviate from her selected strategy after it has been announced. Chapter 4. Government Policy Game u(c) = log(c) and v(c) = log(c) 2 . 16 2Alternatively, one could use power utility: u(c) = j ^ - , p > 0. For this case, the first order conditions for the CPP also yield reaction curves that are linear in r}* and rjt, hence power utility yields the same "qualitative" results as logarithmic utility. Other variations include changing the utility of government expenditures. Linear utility, v(g) = Ag fails to generate Nash Equilibria in the Domestic or the Foreign Export regions. Quadratic utility, v(g) = g — ^ g2 has the advantage of capturing public displeasure with excessive government expenditures but does not yield a closed form solution when combined with a HARA utility function. Chapter 5 Policy Game with No Transportation Costs First we shall examine an economy with no transportation costs, a special case of the economy described in the previous two sections with s = 0. W i t h this modification, the solution to the C P P is greatly simplified; households, who are risk averse, are now able to perfectly pool their endowment risks as in Lucas [11]. Due to similar preferences and endowment processes, each household consumes exactly one half of the total endowed goods in each period. Thus, the optimal consumption plan for the home agent is ct = ^ ± ^ . 2 The foreign household has the same optimal consumption scheme 5.1 Nash Equilibria for Stage Game with No Transportation Costs W i t h perfect pooling, the home government faces the following problem Max log(c 4) + 7 l o g ( 6 / t - 7 / t ) (5.2) {0<vt< 0t} such that (5.1) is satisfied. The solution to this problem is the home government's reaction correspondence, a mapping that describes the home government's opt imal replies for given a foreign government policy value. The foreign government faces the analogous problem defined in terms of c*, 6* and rj*. To solve this problem, we introduce a period reward function for each government. Let R(r}*,T]t) : [0,0*] x [0,6t] — » 3 £ be the period 17 (5.1) Chapter 5. Policy Game with No Transportation Costs 18 reward function for the home government. W i t h pooled consumption the reward function is R ( V ; , m ) = l o g ( 2 L t £ ) + 7lOg(0t - rjt). (5.3) The reward function for the foreign government is R*(vhVt) = M 5 ^ ) + 7 logW-»?;). (5.4) The reaction correspondences for both the home and foreign governments are described in the following lemma Lemma 1 For the policy stage game without shipping costs the Nash reaction correspon-dence for the home government is nt = < 1 + 7 1 + 7 " * 7 (5.5) . 0 if $ > ^ The foreign Nash reaction correspondence for the same policy game is Vt 1+7 (5.6) 0 if vt> Proof: Please see the Appendix. In this economy, trade occurs as a result of imbalances in the distribution of con-sumption goods between the two countries. Governments, v ia tax policies, are able to set the amount of goods available for consumption or trade and thus are able to indi-rectly control the terms of trade. The reaction correspondences (5.5) and (5.6) illustrate this fact. Consider the optimal policy strategy of the home government. From (5.5), Chapter 5. Policy Game with No Transportation Costs 19 < 0, thus, as the amount of available foreign goods increases (i.e. as the foreign tax rate drops), the home government responds by decreasing the amount of available goods in the home country (a domestic tax increase). B y reducing rjt, the home government is attempting to shift the imbalance of goods in its own favour. The intent is to generate a shortage of goods at home in order to draw in goods from abroad. When there are rela-tively few domestic goods this strategy has the effect of increasing the net imports and when goods are abundant at home, the effect is to decrease the amount of net exports. We observe that the optimal strategy for each government does not discriminate between these two situations; the optimal policy rule is independent of the direction of net flow of goods. The switch in the strategy occurs when rj* > an extreme situation where the relative distribution of goods is greatly skewed in favour of the foreign country. The motive behind the 7/2 = 0 strategy is essentially the same; with a majority of the goods in the foreign country, the home government chooses to tax away all the goods endowed to the home country, forcing the home resident to rely solely on imported goods for con-sumption. O n the other hand the foreign government realizes that it cannot influence 0* the pattern of trade at this point and resigns itself to the policy 77* = which is the 1 r? 0* optimal policy for an autarky . Note that the autarky optimums, r)t — and rj* = form the intercepts of the Nash reaction correspondences for the open economies. These policies act as "benchmarks" when determining an optimal policy in the open economy; a best reply is the closed economy opt imum adjusted downwards for the current strategic situation. The result is higher tax rates for open economies. To find a Nash Equi l ib r ium for the stage game, we must determine the intersection of the two policy rules (5.5) and (5.6). The rationale here is that a Nash equil ibr ium must be self-enforcing, thus it must be optimal for both parties simultaneously. For 1The optimum policy for an autarky is derived from the first order conditions of the problem: Max{\og(rj*) + 7 log(0* - %*)}. Chapter 5. Policy Game with No Transportation Costs 20 Figure 5.1: Nash Equi l ib r ium for Policy Stage Game with No Shipping Costs. The inter-section of the government reaction correspondences determines the equil ibrium policies of each government as functions of the period endowments. Chapter 5. Policy Game with No Transportation Costs 21 this economy, the reaction correspondences always intersect at a unique point; a unique equil ibr ium in pure strategies always exists for this stage game. A n example of a Nash equil ibr ium in this stage game is illustrated in Figure 5.1. Theorem 1 The policy stage game without shipping costs has a unique subgame perfect equilibrium in pure strategies for all possible endowment ratios IS-. The ratio of endow-"t ment determines the nature of the equilibrium as shown in Table 5.1. Proof: Please see the Appendix. B y first solving the consumers' optimization problem and then incorporating this information in the optimization problems faced by each government we have employed the technique of Backwards Induction Rationality. This technique guarantees that our solution is subgame perfect, but it also introduces the possibility that this equil ibrium may be time inconsistent. Van de Ploeg found this to be the case for the continuous time version of this model [18]. In his model governments announce a policy a the beginning of the planning horizon and then renege on this announcement as soon as the model is set in motion. Our model does not suffer from time inconsistency for two reasons. Firs t , wi th stochastic endowments and discrete time, governments never make any long term policy announcements in our model. Second, in the sequence of events that comprise our stage game governments draw taxes from endowments before consumers can act (tax in advance), hence the government policy decision is necessarily binding. Corollary 1 The Nash tax rates in the policy stage game without shipping costs are functions of the ratio of endowments, The Nash tax rates are detailed in Table 5.2. Proof: Please see the Appendix. From the optimal tax rules for the home government in Table 5.2 we note the following: Chapter 5. Policy Game with No Transportation Costs 22 Table 5.1: Nash Equi l ibr ia in the Policy Stage Game without Shipping Costs. Endowment Distr ibution Foreign Policy Home Policy Vt Bt <• 7 e; - 1 +7 gt* 1+7 0 1 + 7 ^ e* ^ 7 (1+7)0,* ~70t 1+27 (1+7)^-70,* 1+27 et -> i+a e* ^ 7 0 6t 1+7 Table 5.2: Nash Tax Rates in the Policy Stage Game without Shipping Costs. Endowment Distr ibution Foreign Tax Rate Home Tax Rate rt* 0t < 7 Of - 1+7 7 1+7 1 7 ^ 9, ^ 1+7 1 + 7 ^ 9; ^ 7 l+2 7 ^ ~ 0?' fit > 1+7. , 7 . 1 i + 7 Chapter 5. Policy Game with No Transportation Costs 23 i) > 0. The home government increases its tax rate as the foreign endowment increases. The intent is to take advantage of the foreign wealth by inducing imports through over-taxation. ii) | ^ < 0. W i t h a larger local endowment, the home government feels less pressure to rely on trade to provide consumption goods for its constituents, and relents on its trade manipulation strategy of over-taxation. i i i) > 0. As the relative importance of public goods in the household ut i l i ty function increases, the home government is given license to set even larger tax rates. The foreign government sets its tax rate using the same set of principles. Here, as well as in the Nash reaction correspondences, we see that each government adopts a "beggar thy neighbour" policy; each government tries to better its domestic situation by preying on its neighbour's resources. In the resulting equilibrium, endowments are over-taxed and the final distribution of consumption goods is inefficient. This suboptimal outcome might be avoided if governments were able to coordinate their tax policies, but that scenario comprises a different game. 5.2 Cooperative Equilibria for the Stage Game with No Transportation Costs We now alter the rules of the stage game and allow governments to communicate before play commences. W i t h communication, cooperation can occur, but before governments can cooperate, they must first agree upon a common objective function to be joint ly optimized. Due to the symmetry of our model, a reasonable candidate for this function would be the equally weighted sum of the individual objective functions 2 . The problem 2The objective function of the cooperative regime will be arrived at via a bargaining process between the two governments. If this function is renegotiated at the beginning of each new period then the ob-jective function (5.7) will result. However, it is not clear that this settlement is renegotiation proof; once Chapter 5. Policy Game with No Transportation Costs 24 that the cooperative regime faces is then Max log(o) + 7log(0* - Vt) + logCc?) + 7log(0? - r,*) (5.7) {0 < < 8t,0 < nt < 6t} such that c = 4 = *±* . M) Theorem 2 T%e cooperative solution to the policy stage game without shipping costs is as stated in Table 5.3. Under the cooperative regime the optimal tax policies are functions of the ratio of endowments, fj-. These policies are detailed in Table 5-4-Proof: Please see the Appendix. The opt imal policies in Table 5.3 represent a state contingent trade agreement; gov-ernment policies are set in order to optimally distribute endowments between the two countries. Comparing the results in Table 5.2 to the results in Table 5.4, we see that the tax rate in the non-cooperative equil ibrium is always greater than or equal to the tax rate under a cooperative regime. Coordinating policies eliminates the excessive spending of each government that was designed to generate imports. Since the agreement was arrived at v ia a joint optimization process, the resulting equil ibrium is Pareto Opt imal . The agreement is also very altruistic; in extreme situations one government provides for-eign aid by forgoing some of its public expenditures in order to boost exports to the less fortunate country. For example, if Jj- < then, as an act of good w i l l , the foreign government only claims J^L. 'M taxes instead of the Nash amount of This action endowments have been realized, the government having access to the greater supply of endowed goods may attempt to renegotiate the settlement in its favour. We shall assume that no such resettlements occur allowing negotiations to occur only at the beginning of each period. Chapter 5. Policy Game with No Transportation Costs 25 Table 5.3: Cooperative Equi l ibr ia in the Policy Stage Game without Shipping Costs. Endowment Foreign Home Distr ibution Policy Policy v* "' Vt £L < _x_ e; - 2+ 7 20* 2+7 0 7 ^ h. ^ 2+7 2+7 ^ e*t ^ 7 (2+7)0*-7e t 2(1+7) (2+7)0t-70' 2(1+7) h. > 2+2 0* ^  7 0 20t 2+7 Table 5.4: Cooperative Tax Rates in the Policy Stage Game without Shipping Costs. Endowment Distr ibution Foreign Tax Rate Home Tax Rate 9t < 7 e; - 2+7 7 2+7 1 7 ^ ii. ^ 2+7 2+7 ^ 0 * 7 7 (1+°*) 2+27 V- 1 ^ 2 + 2 7 V 1 ^ ^ * ' ' 0t > 2+j_ e; - 7 1 2+T" Chapter 5. Policy Game with No Transportation Costs 26 exacerbates the already disproportionate good distribution which in turn creates a larger volume of net goods which flow to the home country. Al though governments are allowed to cooperate in this game there is nothing that binds them to any agreements. W i t h i n the game the choice of policy is now superseded by the choice of mode of behaviour. Each government must now select one of two possible actions: to cooperate or not to cooperate. Their actions in turn determine the value of their policies that are implemented in the stage game. Let "C" indicate a choice to cooperate when determining optimal policies. A government that selects the action "C" chooses to optimize the joint objective function (5.7) and adopts a policy according to the rules prescribed in Table 5.3 of Theorem 2. Let "N" denote a choice to not cooperate. If a government chooses "N" then that government uses its Nash reaction correspondence to determine its optimal policy in the stage game; the home government uses the function (5.5) while the foreign government uses (5.6). The stage game becomes a bi-matr ix game defined over the action space {C, N} x {C, N}. To complete the description of this game we must define the reward function over this action space. Let r/(x*,x) and r/*(x*,x) be the value of the policies adopted by the home and foreign governments respectively when the home government selects the action x £ {C,N} and the foreign government selects the action x* £ {C,N}. Define t/(x*,x) = i?(r/*(x*,x),r?(x*,x)) (5.9) to be the reward function for the home government when the action pair (x*, x) is played and define t/*(x*,x) = ir(i/*(x*,x),i/(x*,x)) (5.10) to be reward the foreign government receives for the same action pair. We now have a properly defined matrix game with some curious properties. Chapter 5. Policy Game with No Transportation Costs 27 Theorem 3 Let T be the 2 player bi-matrix game defined by the reward functions £/(•, •) and £/*(•,•). i) If r r - < % < ^-7- then V is a Prisoners' Dilemma. ' J 1 + 7 9* 7 ii) If |jr < then T is degenerate in that the actions of home government have no effect on the outcome of the stage game. in) If$r> — then T is degenerate in that the actions of foreign government have no effect on the outcome of the stage game. Proof: In order to show that T is a Prisoners' Di lemma for < If < we must 1 + 7 6* 7 demonstrate the following two facts for each government: i) The reward received from a cooperative equil ibrium is strictly greater than the reward received from a Nash equilibrium. ii) The action "C" is strictly dominated in rewards by the action "N" . The details of the proof are given in the appendix. Thus we have the familiar story of the Prisoners' Di lemma. When both governments decide to cooperate, they both receive a better reward than if they had decided to act independently of one another. But cooperation is a strictly dominated action and thus in the one-shot game the Nash equilibrium, (N,N), is the only rational and stable equi-l ib r ium. In the short run each government always prefers to play the non-cooperative action. F rom the first fact of the proof of Theorem 3 we see that the cooperative equil ibr ium Pareto dominates the Nash equilibrium. Clearly the government over-spending witnessed in the Nash equil ibrium is inefficient. The source of this inefficiency is a negative tax Chapter 5. Policy Game with No Transportation Costs 28 externality that is not accounted for in the optimization problems solved by the individual governments; increasing the domestic tax rate has the effect of lowering the amount of private goods consumed in the foreign country. In this scenario governments always provide an excessive amount of public goods at the expense of the foreign consumer. This prediction contradicts the results results reported by M i n t z and Tulkens [13] and by de Crombrugghe and Tulkens [4]. Both of these works predict that under fiscal competition, Nash equil ibrium tax rates are too low and, as a result, an insufficient amount of public goods are provided at each locale. This discrepancy is due to the nature of the economy studied in these papers. In their economy household agents "invest" labour at different locations in exchange for consumption goods which are subject to a local tax. Increasing the local tax rate causes more labour to be invested at the remote site thereby increasing the tax base there. To avoid the flight of labour capital, government set tax rates that are lower than the socially optimal rate which in turn creates a lack of public goods at each location. In contrast with our model, we see that there is a positive externality in this economy; the tax rate of one government has a positive affect on the tax base of its opponent and thereby increases the amount of public goods provided at the remote site. In this scenario, a socially efficient equil ibrium is obtained by having both governments simultaneously raise their tax rates [4], whereas in our economy, social efficiency is achieved a simultaneous tax cut. 5 .3 Equilibrium for Repeated Game with No Transportation Costs Since each government can observe the past actions of its opponent, this information can be used in determining its current actions. W i t h history dependent strategies, the current actions of a government effect its future gains; actions that are sub-optimal in the stage game for an individual government can become optimal (self-enforcing and rational) in Chapter 5. Policy Game with No Transportation Costs 29 the repeated game due to the existence of punishment strategies. This is the essence of the Folk theorem 3 . For infinitely repeated discounted games, the theorem states that any feasible, individually rational payoff can be supported as an equil ibrium of the game so long as the discount rate is sufficiently high. There are two problems with applying the Folk theorem to our model. Firs t , the theorem has no predictive power. It does not specify what equil ibrium wi l l evolve in a repeated game, it only details what set of equil ibrium payoffs are possible under such strategies. We have already addressed this problem by assuming that the cooperative action wi l l be a solution to the optimization problem described in (5.7). Our interest in the theorem is to use it to find a Nash strategy that wi l l result in cooperative actions being played by both governments at every stage in the repeated game. Second, the Folk theorem is not designed for stochastic games; the result relies upon the fact that the stage game is stationary. Again , it is not the theorem that we wish to apply, only the methodology contained in its proof. B y building a punishment strategy that works for a worst case scenario, we can ensure that deviations from cooperation never dominate the future potential rewards for cooperating regardless of the realization of endowments. Let r°°(/3) represent the infinitely repeated discounted game where the home govern-ment attempts to maximize the sum Max 0 0 EotE/PUW,*)] (5.11) x 4 G { C , N } t=o while the foreign government attempts to maximize the analogous sum defined in terms of £/*(•,•). Define the cooperative strategy profile to be (xt,x*) = (C,C), Wt > 0. The cooperative strategy profile can be a rational outcome of r c o ( / 5) if each government adopts a G r i m Trigger strategy. There are two states in the G r i m Trigger strategy: a cooperative 3The Folk theorem has a long tradition in game theory. See van Damme [17] for a summary of the Folk theorem and all of it's variants. McMillan [12] discusses some applications of the Folk theorem in international economics. Chapter 5. Policy Game with No Transportation Costs 30 state and a punishment state. Both governments begin the game in the cooperative state by selecting "C" as their action in the first stage of the repeated game. Thereafter, each government remains in the cooperative state and continues to select the action "C" so long as its opponent has selected "C" in the previous period. If in the previous period its opponent had selected the non-cooperative action "N", then the government responds by entering the punishment state and playing the action "N". Once a government has entered the punishment state, it remains there for al l subsequent periods of play. The cooperative strategy profile is the rational outcome of r°°(/?) if neither government can gain by playing "N" at any stage of the game. In order to make cooperation the optimal action at each stage for all possible pairs of endowments, thus we adopt a worst case design approach and require "C" to be the opt imal action for the maximum possible gain from a one-shot deviation. Let the cumulative probability distribution of the endowments be given by the function F(-) wi th a max imum possible endowment value of 8. Define ficc to be the expected reward from the stage game when both governments cooperate and fiNN to be the expected reward from the stage game when both governments select Note that these values are the same for both governments. The largest possible one-stage deviation gain is then (5.12) the Nash action "N". (5.13) Max A/7 /7(C,N) -U(C,C). (5.14) {(*?,*,)€ (0,0) x(0,*)} Chapter 5. Policy Game with No Transportation Costs 31 Since R(-,-) and #*(•,•) are continuous on the closed and bounded set [0, rjt] x [0, TJ*\ C [0,6t] x [0,#*], it follows that a maximum exists for the above problem exists. Consider a potential deviation at t = T. The long term expected benefits of cooperation (the difference of expected continuation rewards for cooperation less the expected continuation rewards for punishment), must be greater than the largest possible one stage deviation gain. OO OO £ ^ V c - E P-Ti*™ > A U ^ - (5-15) tzzT t=T Note that continuation rewards are measured relative to the period in which the decision is being made. Evaluating the sums in (5.15) we find 1 1-/3 hence we require ( A * c c - M ™ ) > A * 7 m „ , (5.16) P > 1 ~ \ r j ^ NN = P- (5-17) max Since the stage game is a Prisoners' Di lemma, \icc > fiNN and thus we have /5 < 1. So if (3 satisfies (5.17), then no government wi l l ever choose to deviate from cooperative strategy profile. Note that the punishments used to obtain this level of collusion are extreme. A one-time deviation from the cooperative strategy profile results in an eternal punishment; it may be too costly for a government to carry out this threat. However, the punishment strategy is the Nash equil ibrium of the stage game, thus once the punishment phase has begun, neither government has an incentive to leave i t . The threat of punishment is therefore credible and the G r i m Trigger strategy is self-enforcing. Furthermore, the punishment strategy is subgame perfect; it generates a subgame perfect equil ibr ium in all subsequent stages of the game, thus the G r i m Trigger strategy is subgame perfect as well . This result, known as the "Nash-threats" Folk Theorem, is originally due to Chapter 5. Policy Game with No Transportation Costs 32 Friedman [6]. As a result of this theorem, collusion is made enforcible without the use of binding contracts. In our case it leads to the following result: Theorem 4 If f3 £ (/?, 1) then the realization of the Grim Trigger strategy is equivalent to the cooperative strategy profile, (x<,x*) = (C,C), Vt > 0. We see that if governments and households are sufficiently patient, then cooperation is optimal at every stage of the game and a Pareto Optimal distribution of endowments is achievable. With lower discount factors, cooperation may be sustainable for a period of time, but eventually an endowment distribution will occur where it is optimal to deviate from the cooperative strategy profile. Once the cooperative agreement is violated each government reverts to a "Beggar thy Neighbour" strategy thereafter. Chapter 6 Tax Policy Game with Transportation Costs We shall now examine the same economy in the presence of shipping costs (s > 0). This economy is unlike the previous one in that the policy space, rj* x % = [0,0*] x [0,0,], is partitioned into three regions representing the direction of trade generated by various policy combinations. This partitioning is due to the actions of the households and their attempts to hedge against endowment risks. 6.1 The Consumers' Optimization Problem: Solution to the C P P Under the assumption of complete markets we invoke a C P P to solve the optimization problem that consumers face. The results are stated in the following lemma. Lemma 2 In the policy stage game with shipping costs, the optimal shipping and con-sumption plans of the home and foreign households are contingent upon the current gov-ernment policy pair (r}*,r)t) and partition the policy space into three disjoint regions rep-resenting the direction of trade between the two countries: i) No Shipping Region: NS = {(n*, rjt) £ [0,0,*] x [0,0,] : ^  < < 1 + s} . ii) Domestic Export Region: DX = {(r/*,r/t) £ [0,0*] x [0,0,] : > 1 + s} . iii) Foreign Export Region: FX = {(r/*,r/t) £ [0,0*] x [0,0,] : J§- < ^ } . The optimal shipping and consumption plans are stated in Table 6.1. Proof: Please see the Appendix. 33 Chapter 6. Tax Policy Game with Transportation Costs 3 4 Table 6.1: Opt imal Shipping and Consumption Plans for u(c) = log(c). State Domestic Export No Shipping (l+s) > SL > _L_ V  J — Vi — l + s Foreign Export *)t ^ I i f ^ i + » T)t-{l+s)r)* 2 0 0 * xt 0 0 '?t*-(l+s)»)t 2 Ct 2 2 r* 2 j)* + ( l + s)rj t 2 Chapter 6. Tax Policy Game with Transportation Costs 35 W i t h no shipping costs, consumers are able to perfectly pool risks; after taxes have been extracted, imbalances in the distributions of goods between the two countries are always corrected v ia shipments and each household consumes an equal portion of goods (ct = c*). W i t h shipping costs a balanced consumption plan is unobtainable. The shipping costs create a No Shipping wedge, a region where the costs of shipping outweighs the benefits of additional consumption; imbalances in consumption levels are allowed to persist. Outside of this wedge the amount to be shipped is determined by the relative distribution of goods, but unless rjt = 77*, the households wi l l always consume different amounts. The partitioning of the policy space is illustrated in Figure 6.1. We define the upper boundary of the No Shipping wedge to be 8DX = { ( 7 7 ^ ) G [0,0?] x [0,6T] : 77, = (1 + s)r,;} (6.1) and the lower bound as OFX = {(77;,77O G [o,et] x [o,et]: 77, = T j^y t f} • (6-2) B y substituting the the shipping rules from Lemma 2 into the consumption plans, we obtain an opt imal consumption plan for each household as functions of the government policies 77* and 77,. For the household of the home country we find l_ 1 + 5" Ct — 0t + ~ — ; — x t — Xt , + T i - f c< i± ikr - ( a ^ f i ) . ( , 3 ) The foreign household's consumption plan is given by Chapter 6. Tax Policy Game with Transportation Costs 36 V e e* Figure 6.1: Part i t ioning of Strategy Space into Shipping Regions. When (?7*,77t) £ DX there is an excess of home goods and the Central Planner reponds by shipping goods from the home country. When (77*, nt) € FX the excess of foreign goods generates foreign exports. For (r)*,r]t) E NS the cost of shipping is greater than the additional benefits received due to the consumption of shipped goods. In this scenario no goods are shipped. Chapter 6. Tax Policy Game with Transportation Costs 37 where the function (-) + is defined by y if y > 0 ( y ) + = (6.5) 0 if y < 0 The presence of shipping costs The functions (6.3) and (6.4) form the constraints for the government optimization problem in the policy game. 6.2 Nash Correspondences for Stage Game with Transportation Costs We begin by solving for the Nash reaction correspondence for the home government. W i t h the shipping rules (6.3) and (6.4) the home government's optimization problem becomes Max log(Q) + 7 log(0 t - r /O (6.6) {0 < Vt < 0t] such that ^^(^^y.^^y. (,7) Substituting the constraint (6.7) into the objective function (6.6) results in the period reward function for the home government. R{vhVt) = log(ct)-rjlog(6t-nt) (n;,nt)eDX {vlvt)eNS (V*,nt)eFX Chapter 6. Tax Policy Game with Transportation Costs 38 < log(§fo + (1 + s)v;)) + 7log(^ - rjt) (rj;, 7/0 e DX logfo) + 7log(0, - Tjt) (77t*, 7/0 £ Atf (6.8) 1 iog( | ( i ? t + ^ i ? ; ) ) + 7 i o g ( ^ - i / t ) fe^)eFi. The functions RDX(-, •), -R N S (" ' •) and RFX(-, •) are convenient notations for the functional form of the period reward function in each of the Domestic Exports , No Shipping and Foreign Exports regions respectively. Before proceeding, we should examine some of the properties of •). Lemma 3 In the policy stage game with shipping costs, the period reward function for the home government defined in (6.8) has the following properties: i) R(-,-) is continuous along the lines dDX and dFX. ii) R(-,-) is not differentiate along the lines dDX and dFX. iii) i?(-, •) is not concave in nt along the line dFX for all n* £ [0,7/t*]. P r o o f : Please see the Appendix. Since R(-,-) fails to be concave in r\t for al l possible ??*, the standard existence theorem for pure strategy Nash equilibria cannot be applied to our m o d e l 1 . As we shall see, this lack of concavity creates a range of endowment ratios, | f for which no pure strategy Nash Equi l ib r ia w i l l exist. Our approach is to first use i ? ( v ) to first calculate an opt imal policy strategy for each of the trading regions and then to determine which one of these regional strategies w i l l be opt imal for a given foreign policy 77*. W i t h i n each region we see that R(-,-) is C 2 , concave wi th respect to T / 4 and bounded for a given 9\ and 6t, thus the task of finding an 1The standard existence theorem states that if the reward functional is jointly continuous in each of its arguments (player actions, policies in our model) and strictly convex in each action variable then the game will admit a Nash equilibrium in pure strategies. See [2] or [7] for more details. Chapter 6. Tax Policy Game with Transportation Costs 39 opt imum for the interior of a particular region poses no problems. In the following three lemmas we construct an optimal strategy for the home government for each of the three regions. Lemma 4 In the policy stage game with shipping costs, the optimal strategy for the home government in the No Shipping Region is where 1+7 1 1+s * 1B for 77* < 77* for 77* < 77* < n*B for 77* > 77* (6.9) 1 l + s l + 7 + * 1 + 7 (6.10) (6.11) Proof: Please see the Appendix. A n example of the optimal strategy for the home government in the NS region is il lustrated in Figure 6.2. The switching values 77* and 77* are due to the first order conditions for optimali ty intersecting the edges of the NS region. Note that the policy between 77* and 77* is constant. W i t h i n the NS region there is no shipments hence there is no strategic interaction between the two governments. The optimal policy value is thus equivalent to the autarky opt imum. Lemma 5 In the policy stage game with shipping costs, the optimal strategy for the home government in the Domestic Export region is % = T~7 ( 1 + s)rL-V; for r,; < 77;, (6.12) 1+7 1+7 Chapter 6. Tax Policy Game with Transportation Costs 40 e Optimal Home Policy in NS Figure 6.2: Nash Strategy for the Home Government in the No Shipping Region. The graph illustrates the set of best policy replies for the home government when the policy pair is constrained to lie within NS. Chapter 6. Tax Policy Game with Transportation Costs 41 where 1 1 + s l + 2 7 ' (6.13) Proof: Please see the Appendix. Lemma 6 In the policy stage game with shipping costs, the optimal strategy for the home government in the Foreign Export region is where Vt i f e - 1+7i+7" 7?* f ° r V*D < Vt < V*B 0 for V ; > n* 7 (6.14) (6.15) (6.16) Proof: Please see the Appendix. The results for the foreign government are obtained through symmetry and mirror the results for the home government. Corollary 2 The optimal regional strategies in the policy stage game with shipping costs for the foreign government are as follows: i) In the No Shipping region the foreign government finds it optimal to employ the strategy (1 + s)j]t for TJT < TJA ti=S for VA<rit< nB (6-17) Chapter 6. Tax Policy Game with Transportation Costs 42 where VB (1 + *)T + 7 (6.18) (6.19) ii) In the Domestic Export region the foreign government finds it optimal to adopt the following strategy 1 7 Vt = 1+7 I + 5 I + 7 Vt for Vt<Va where Vc 1 6* (6.20) (6.21) 1 0 l + 5 l + 2 7 " iii) In the Foreign Export region the foreign government finds it optimal to use the strategy - r+ir+^* for nD<nt< vE 0 for nt > TjE, (6.22) where VE a* (1 + -) ' (1 + *) 1 + 2 7 ' n 7 " (6.23) (6.24) P r o o f : Please see the Appendix. The final step in constructing the Nash reaction correspondences is the elimination of dominated strategies. B y plotting the three locally optimal strategies on the same graph, we find that for certain ranges of foreign policies the home government w i l l have more than one locally optimal response. Figures 6.3 and 6.4 illustrate this fact. In particular, there are two regions where optimal regional strategies "overlap" one another: Chapter 6. Tax Policy Game with Transportation Costs 43 the NS — DX overlap region and the NS — FX overlap region. In the NS — DX, defined as the range 77* £ [0,??*), the home government has a choice between two policy values: a value formulated according to the strategy in the Domestic Export region and a value derived from the strategy in the No Shipping region. B y choosing to use the strategy in the DX region the government adopts a policy value that, for the given foreign policy value, generates domestic exports. Selecting the strategy which lies wi thin the NS region amounts to setting a policy which wi l l deter trade. In the NS — DX overlap region we have 77* € (77* ,9*) 2 . For this range of foreign policies the home government must decide between a strategy that generates foreign exports versus one that prevents trade. Notice that there are two possible configurations for these lines; in Figure 6.3, the NS — FX policy overlap begins at a point to the right of 77* , the "kink" in the NS strategy whereas in Figure 6.4, the same overlap region begins to the left of 77*. This difference is due to changes in the model parameters. W i t h lower shipping costs (small s), the NS wedge shrinks drawing its borders closer to each other, eventually forcing 77* to the left of 77*. This configuration may also be due to an increased importance of public goods wi th in the ut i l i ty function (large 7). Increasing 7 decreases the slope of the optimal strategy in the DX region forcing 77*^  left. A Nash reaction correspondence is the best reply function to the action of opponents. Faced wi th a choice between two strategies, the home government never selects the strat-egy which yields the lower reward, hence payoff dominated strategies must be removed from the set of replies. Once dominated strategies have been removed, the remaining strategies form the Nash reaction correspondence. In the following theorem we find that in the NS — DX overlap region, the home government always chooses the strategy that lies in DX and that in the NS — FX overlap region, there is a unique value of foreign 2 Assuming of course that 6* > 77* . This needn't be true. If #* is too small than this overlap region simply won't exist. Chapter 6. Tax Policy Game with Transportation Costs 44 Local Optimal Strategy (Home Government) Figure 6.3: Op t ima l Regional Strategies for the Home Government with 77* > 77*. If 0 < 77* < 77* the home government can select a strategy in either the DX region or one that lies on the border of the NS region. For 77* < 77* < 9* the home government can choose a strategy that lies inside of the FX region or a strategy that is inside of the NS region. Chapter 6. Tax Policy Game with Transportation Costs 45 Local Optimal Strategy (Home Government) Figure 6.4: Opt imal Regional Strategies for the Home Government wi th r?* < 77*. This configuration of first order conditions occurs when shipping costs are reduced or by increasing the importance public goods within the ut i l i ty function. Chapter 6. Tax Policy Game with Transportation Costs 46 policy value that determines which strategy will be used. Theorem 5 Define / c( 7 ) = ( l + 7 X 2 ^ - 1 ) . (6.25) In the policy stage game with shipping costs, there are two possible forms for the home government's Nash reaction correspondence I: When (1 + s)2 > ^ y the home government's Nash reaction correspondence is where 1+7 - (1 + 8)^1? vi e [o,v*) (1 + 6t 1+7 Ot 1+7 1 7 * ~ 1+s 1+7^* vie Mill) 0 * * Vt '-= (1 + s)(2^ - m . (6.26) (6.27) II: When (1 + s)2 < ^ y the home government's Nash reaction correspondence is Vt=< (1 + s)r,t Jt 1 7 „ * 1+7 1 + s 1+7 It (6.28) where rj* = <j>6t and (f> solves 1 2(1 + 3)^(1 - (1 + s)<t>y = (—— ) ^ 7 7 ( 1 + 1 + 7 1 + 5 1+7 (6.29) Chapter 6. Tax Policy Game with Transportation Costs 47 Proof: Please see the Appendix. Figures 6.5 and 6.6 illustrate the two possible configurations of the Nash reaction corre-spondence for the home government. Again , the results for the foreign government are derived v ia symmetry and are anal-ogous to the results for the home government. Corollary 3 The foreign government's Nash reaction correspondence for the policy stage game with shipping costs has two possible forms: I: When (1 + s)2 > the foreign government's Nash reaction correspondence is Vt where e* 1+7 " vt e [o,vc) (1 + vt e IVCVA) 1+7 Vt € [vA,Vt] e* 1+7 " 1 7 1 + s 1 + 7 ^ vt^lvlvl) 0 vt^WEAl Vt = ( l + s ) ( 2 ^ 7 -(6.30) (6.31) and K ( 7 ) is defined in (6.25). II: When (1 + s)2 < the foreign government's Nash reaction correspondence is Vt = < f b - ( l + *)TbVt Vt£[0,Vc) (l + s)vt Vt£[VaVt] v*t^[vhv*E) v*t£WEAl where f$ = <t>0t and (j> solves (6.29). J £ 1 7 _ 1+7 1+s 1+7 Vt 0 (6.32) Chapter 6. Tax Policy Game with Transportation Costs 48 v*o v*A v* o* Home Government Reaction Correspondence Figure 6.5: Home Government Nash Reaction Correspondence when (1 + s)2 > The above graph is the best policy reply correspondence for the home government when either s is large or 7 is small. Chapter 6. Tax Policy Game with Transportation Costs 49 Home Government Reaction Correspondence Figure 6.6: Home Government Nash Reaction Correspondence when (1 + s)2 < The above graph is the best policy reply correspondence for the home government when either s is small or 7 is large. Chapter 6. Tax Policy Game with Transportation Costs 50 In contrast wi th the economy having no shipping costs, governments now are able to discern the direction of flow of goods between the two countries and encorporate this information into their optimal policy rules. When shipping is costly, governments actively discriminate against particular flows of goods. Consider the correspondence of the home government, as 77* is increased for 0 to 9*. When the amount of foreign goods is small , the economy is in a Domestic Export state. As the amount of available foreign goods increases, the home government responds by decreasing the amount of domestic goods. This action narrows the difference between the supply of goods in each country and decreases the flow of goods that are leaving the home country. As 77* continues to rise and nt continues to drop. Eventually the difference in the goods supplies becomes so small that domestic exports are no longer viable (the occurs at rj* — ??*.). Once this point has been reached, further increases in foreign goods are met with an increase in domestic goods; the home government increases the amount of goods at home by just the amount needed to avoid loosing output to the foreign country. This trend continues unt i l the amount of goods at home reaches the autarky opt imum. Since there is no trade in NS there is also no strategic interaction; the home government chooses to maintain this level of goods, in spite of the increasing amount of foreign goods. Eventually the amount of foreign goods becomes too great of a temptation; the home government takes advantage of the situation by reducing rjt to a very low level in order to generate foreign exports (this occurs at 77* = 77*). The downward jump of the reaction correspondence is due to the concavity of the period reward function; the surface has a "valley" along dFX extending from 77* = 77* to 77* = 77*. The same type of discontinuity has been found in other models of fiscal competition where transportation is costly [4], [13]. The implicat ion for our model is that it does not necessarily yield an equil ibrium in pure strategies. From this point onward, 77, decreases as 77* increases; to encourage imports, the home government pursues a policy that widens the difference between the supply of Chapter 6. Tax Policy Game with Transportation Costs 51 goods at home and abroad. We see that the underlying philosophy of this strategy is the same as in the case without shipping costs. Each government subscribes to a "Beggar thy Neighbour" strat-egy; by progressively consuming more and more of its own domestic goods, a government seeks to draw resources from its opponent. The major difference occurs in the NS region. When the amount of goods in each country are nearly equivalent, both governments pre-fer autarky. This preference for isolation breaks down one of the countries becomes too "wealthy". 6.3 Nash Equilibria for the Stage Game with Transportation Costs Finding the pure strategy Nash Equi l ibr ia of our game is a relatively straight forward task. Since the reaction correspondence of each government is composed of regional optimal strategies, we need only look for where those regional strategies intersect. We are further aided by the observation that the home government reaction correspondence does not intersect dFX nor does the foreign correspondence intersect dDX, thus, we may exclude "borderline" No Shipping equilibria from our search. The following theorem summarizes these results. Theorem 6 In the policy stage game with shipping costs, the existence and the type of Nash equilibria is a function of the ratio of endowments Jj-. and the model parameters s and 7. Let K ( 7 ) be as defined in (6.25) and let m = 7 T - K 1 + 2 7 )(2^ - 1) + 7]. (6.33) 1+7 /. For (1 + s)2 > there are three possible types of pure strategy Nash equilibria: i) When |t < ui+a)> a unique Nash equilibria exists within the FX Region. Chapter 6. Tax Policy Game with Transportation Costs 52 ii) When (1+^K^ < ff < (1 +5)^(7) a unique Nash equilibrium exists within the NS Region. iii) When ff > £(7)(l+s), a unique Nash equilibrium exists within the DX Region. If ff does not lie inside of any of these intervals then a Nash equilibrium in pure strategies fails to exist. These results for (1 + s)2 > are detailed in Table 6.2. The corresponding tax policies are listed in Table 6.3. II. For (1 + s)2 < there is two possible types of pure strategy Nash Equilibria: i) When ff > 1 + 8 Jj 1 ^ 2 7 ^ a unique Nash equilibria exists in the DX region, ii) When ff < ^ j ^ V . ^ a unique Nash equilibria exists in the FX region. where </> solves (6.27). If ff does not lie in side of either of these intervals then a Nash equilibrium in pure strategies fails to exist. These results are detailed in Table 6.4- The corresponding tax policies are listed in Table 6.5. P r o o f : Please see the Appendix. We see that the intervals for the various Nash equilibria are disjoint from one another. Since these ranges do not overlap, it is impossible to have any more that one pure strategy Nash equil ibrium for a given endowment pair, and so, if a Nash equil ibrium exists, it must be unique. If an endowment ratio falls outside of these intervals then there is no pure strategy Nash equilibria. Just as the behaviour of households partitioned the policy space into different shipping regions, the actions of government creates a partioning of the endowment space The opt imal policy plans of the two governments divide the endowment space into wedges. For (1 + s)2 > ^ - y there are five regions: Chapter 6. Tax Policy Game with Transportation Costs 53 Table 6.2: Nash Equi l ibr ia in the Policy Stage Game wi th Shipping Costs when Endowment Distr ibution Location of Equi l ibr ium Foreign Pol icy Vt Home Policy m 0t ^ 1 7 B* — (l+s) 1+-Y FX Region *? 1+7 0 1 7 „ 6t / 1 F X Region (l+7)0*-(l+s)7t9f ( 1 + 7 ) ^ - ^ 7 0 * (l + s) 1+7 ^ 6* - (l + s)?(7) l + 2 7 l + 2 7 1 < 0f < 1 ( 1 + ^ ( 7 ) " s't ^ nil nil nil (1+S)K(7) < J? < (1 + 6)ic(7) NS Region e* 1+7 Bt 1+7 ( l + a ) « ( 7 ) < ^ < ( l + ^ ( 7 ) nil nil nil (1 + ^ ( 7 ) <%<(l + s)1-* DX Region ( 1 + T ) » t - 1 T 7 T9 * 1+27 (l+7)<?t-(l+s)70? l + 2 7 | > ( 1 + ^ )T DX Region 0 0t 1+7 Chapter 6. Tax Policy Game with Transportation Costs 54 Table 6.3: Nash Tax Rates in the Policy Stage Game Shipping Costs when ( l + s ) 2 > ^ - r . Endowment Distr ibution Foreign Tax Rate Home Tax Rate 9t ^  1 7 (9* — (l+s) 1+7 7 1+7 1 1 7 ^ Bt / 1 (1+5) 1+7 ^ 8; - (l+5)i;(7) 1+^(1+(!+*) ( f f ) ) l+2 7 (l + l+s Gt0 ^ 1 c 6 t C 1 (i+sK(7) ^ e* ^ (i+s)«(7) n i / nil < l\ < (1 + *)«(7) 1+7 T+7 (1 + 6 ) / C ( 7 ) < | < ( 1 + ^ ( 7 ) n i / (1 + ^(7) <%<{l + s)1-* 1+27^ + 1+5 G*)) rfe(i + (i + *) (I)"1) f > ( l + * ) ^ 1 7 1+7 Chapter 6. Tax Policy Game with Transportation Costs 55 Table 6.4: Nash Equi l ibr ia in the Policy Stage Game wi th Shipping Costs when Endowment Distr ibution Location of Equi l ib r ium Foreign Pol icy It Home Policy Bt s- 1 7 B* - ( 1 +5 ) 1 +7 F X Region 8* 1+7 0 1 7 . ^ Bt s- 1 F X Region (l+t)6*-(l+s)j6t ( 1 + 7 ) 0 1 - ^ 7 0 * ( 1 +5 ) 1 +7 ^ B* - (l+5)it(7) I + 2 7 1+27 d + i ( 7 ) < ** < + s ^ nil nil nil (1 + ^(7) < | < ( l + 5 ) l f DX Region (1+7)0? 1 + 2 7 (l+7)0t-(l+s)70* l + 2 7 | > ( 1 + ^)T DX Region 0 Bt 1+7 Chapter 6. Tax Policy Game with Transportation Costs 56 Table 6.5: Nash Tax Rates in the Policy Stage Game wi th Shipping Costs when (i + *) 2<A-Endowment Distr ibution Foreign Tax Rate Home Tax Rate 9t ^ 1 7 e; - (i+s) i+7 7 1 + 7 1 i 7 ^ et i (1+s) 1 + 7 ^ e* - (l+sK(7) r . f e ( i + ( i + a) ( 1 ) ) 1 + 2 7 ^ + i+s {$ ; ) ) d + s K ( 7 ) < l\ < (! + s ^ nil (1 + ^(7) < | < ( l + 5 ) i ^ 1 + 2 7 ^ + i+s (e*)) ^ ( 1 + (! + .) ( I ) "1 ) | > ( 1 + - ) 1 ? 1 7 1 + 7 Chapter 6. Tax Policy Game with Transportation Costs 57 Home Reaction Coorespondence Foreign Reaction Coorespondence Figure 6.7: A No Shipping Nash Equi l ib r ium in the Policy Stage Game wi th Shipping Costs. When the reaction correspondences intersect in the TVS', the resulting equil ibrium is characterized by a lack of shipments between the two countries. Chapter 6. Tax Policy Game with Transportation Costs 58 Figure 6.8: Non-existence of a Pure Strategy Nash Equi l ib r ium in the Pol icy Game wi th Shipping Costs. A pure strategy Nash Equi l ibr ia fails to exist when the policy reaction correspondences don not intersect. This scenario can be interpreted as a trade disagreement; a situation where neither government can agree on what pattern of trade should prevail. Chapter 6. Tax Policy Game with Transportation Costs 59 i) A n Domestic Export Equi l ibr ia Region: DX = {(9*t,9t) e^2+:9t>(l + * ) £ ( 7 ) 0 ? } . ii) A n Upper No Nash Region: UNN = {(9*t,9t) e %l : (1 + 8 ) K ( i ) P t <9t<(l + s)t(l)9;}. i i i) A No Shipping Region: NS = {WA) € &+ : <9t<(l + a)K(7)*t*} • iv) A Lower No Nash Region: LNN EE {(9l9t) G &+ : <9t< ^4^}. v) A Foreign Export Equi l ibr ia Region: FX^{(9;A)en--0t<(^}-Figure 6.9 illustrates the endowment space partitioning. When the endowment pair lies in DX the home endowment is large and the resulting equilibria wi l l be characterized by domestic exports. Endowment pairs in FX indicate a large level of foreign output; during this stage of the game goods wi l l be exported from the foreign country. For (#*, 9t) G NS the amounts of goods endowed to each country do not differ enough to merit any shipping. In this case reaction correspondences of the two countries intersect in the NS region in the policy space as illustrated in Figure 6.8. Endowment pairs in UNN and LNN represent relative distributions of national endowments that cause trade disputes to occur between the two countries. W i t h i n these zones of disagreement one government utilizes a strategy that promotes trade while the second adopts an isolationist strategy; the two cannot agree on what the prevailing trading state should be for the given endowment distribution. Graphically, the lack of a pure strategy Nash equilibria occurs when the two policy correspondences fail to intersect as in Figure 6.9. Note that UNN and LNN separate Chapter 6. Tax Policy Game with Transportation Costs 60 the other three zones from one another creating uncertain trade boundaries. The width of these zones of disagreement is determined by re(7) and £(7), both functions of the parameter 7. For sufficiently large 7, K(J) is a decreasing function. As the importance of public goods to the welfare of the household increases, the angular width of NS decreases while the angular widths of UNN and LNN increases. Eventually 7 reaches a point where (1 + s)2 = and the two zones of disagreement meet, eliminating the TVS' zone altogether. For (1 + s)2 < the center three regions: UNN, NS and L A W are merged into a single No Nash region: WW = {(6;, 6) G & + : (1 + s ) £ ( 7 ) < 0T < • This scenario corresponds to the structure of Nash equilibria when (1 + s)2 < as seen in Table 6.4. We see that disagreements only occur when countries have nearly the same amount of endowments. Thus, this model would predict that trade disagreements would only occur between countries with similar levels of output (eg. Japan and the U.S) and not in scenarios when one country has a clear edge in output (eg. Japan and Canada). The presence of trade disagreements is an interesting result, but it is unsatisfactory in that our model fails to specify the behaviour of governments when an endowment pair lies in one of the No Nash Equi l ib r ium zones. One method of addressing this short-coming is to extend the class of admissible solutions to include mixed strategies. Under a mixed strategy each government is allowed to randomize its action in each stage game over a range of possible policy values. A mixed strategy equil ibrium consists of the probabili ty density functions that each government uses to determine its action. Since the policy space in our game, a closed and bounded subset of is a non-empty compact space and since the period reward functions for each government is continuous over this space, we are guaranteed that a Nash equil ibrium in mixed strategies w i l l exist for the stage game (Glicksberg's Theorem, see [7]). To determine the optimal mixed strategy for the home government, we first note that for a l l 77* ^  rj* there is a unique policy value that the home government prefers to play; Chapter 6. Tax Policy Game with Transportation Costs 61 Figure 6.9: Part i t ioning of the Endowment Space in the Pol icy game wi th Shipping Costs when (1 + s)2 > ^(7). Conflicting government objectives creates trade conflict zones (UNN and LNN) that are sandwiched between the three shipping zones. W i t h in these conflict zones, governments can not agree on which shipping state should prevail. Chapter 6. Tax Policy Game with Transportation Costs 62 the mixed strategy wi l l assign a probability of one to each of these policy values when the appropriate foreign policy is played. When 77* = 77*, the home government is indifferent between two policy values: 77NS and rjFX, where VNS = ft f o r ( l + S ) * > ^ (6.34) (1 + 5)77* for ( l + 5 ) 2 < _ ^ , VFX = 7 ^ - 7 ^ 7 ^ ( 1 + « • (6-35) Since there is no other criteria to guide the home government in its selection between these two values, we permit a random policy selection. When 77* = 77*, let p £ [0,1] be the probabili ty that the home government chooses the policy 77NS and let 1 — p be the prob-abil i ty that the policy 77 ^  is selected. Since a Nash equil ibrium must be self-enforcing, the random actions of the home government must not cause the foreign government to deviate from its policy rj*; the value of p must be set such that rj* is the opt imal foreign policy reply to the home governments mixed strategy. The foreign government maximizes the expected reward Rexp = p[log(77*) + 7 l o g ( 0 * - 77;)] + (1 -p ) [ log ^ + + + 7 l o g ( 0 * - 77*)]. (6.36) The probabili ty p must satisfy the first order conditions for the foreign governments optimizat ion problem evaluated at 77* = 77*and the appropriate home government policies, thus P 1 7 et - vt which leads us to conclude that 7 «? + (i + *)(iSr-rfcifc^ )  e*-K 0 (6.37) Chapter 6. Tax Policy Game with Transportation Costs 63 P = 7(1 + s)(2^ - 1)2*+^  (6.38) [l-7(2^-l)][(|r-(l + 5 ) ( 2 ^ - l ) ] We see that p is a function of the state | f . As | f decreases p also decreases; as the ratio of endowments begins to favour the foreign country, it becomes more likely that the home government wi l l pursue a policy that generates foreign exports. Again we see the "Beggar Thy Neighbour" behaviour in the optimal strategy of each government. The mixed strategy equil ibrium provides us with a solution, but the result is not very valuable. Having a government determine a national policy by random methods is at best, unrealistic. Even less convincing is the fact that the probabilities for a government's mixed strategy are selected in order to stabilize its opponent's optimal reply. For these reasons we shall discontinue our discussion of the mixed strategy equil ibrium and assume that some unspecified negotiation process occurs when a trade dispute arises. 6.4 Cooperative Equilibria for the Stage Game with Transportation Costs A n obvious way of solving the dilemma of trade disagreements is to allow governments to communicate wi th one another before the stage game commences. W i t h pre-game communication it becomes possible for the governments to coordinate their policies in order to maximize a common objective function. As in the case without shipping costs, we adopt an equally weighted sum of the individual objective functions as the objective function for the cooperative regime, thus, the regime maximizes (5.7) subject to the consumption constraints from the economy with shipping costs (6.3) and (6.4). To solve this problem we create a period reward function for the regime; the sum of the period reward functions for the individual governments R c(v;,rit) = R(v;,r)t) + K*(v;,Vt). (6.39) Chapter 6. Tax Policy Game with Transportation Costs 64 Here i]t) is defined by (6.8) and R*(n*, r}t) is defined in similar fashion for the foreign government. Solving the first order conditions for this problem with respect to 77* and rjt results in the cooperative policy equilibrium. Theorem 7 The cooperative solution to the policy stage game with shipping costs is as stated in Table 6.6. Under the cooperative regime the optimal tax policies are functions of the ratio of endowments, If. These policies are detailed in Table 6.7. Proof: Please see the Appendix. From Tables 6.6 and 6.7 we see that the cooperative policy solution partitions the en-dowment space along the same lines as C P P ' s partitioning of the policy space (see L e m m a 2). The cooperative solution removes the effects of fiscal competition and restores the "natural" trading boundaries which are determined by an optimal risk sharing agree-ment between private agents. Note that the variable 7 plays no role in determining when shipments w i l l occur. The coordination of fiscal policies prevents individual governments from manipulating the balance of trade so that trade conflicts never occur; the fiscal agreement acts like a trade treaty between the two governments. We also observe that once again in a cooperative equil ibrium the optimal tax rates are lower than in a Nash equil ibr ium. The cooperative objective function accounts for the negative consumption externality that government tax policies have on the household of the opposing country and avoids the unnecessarily high tax rates of the Nash equil ibrium. Now that collusion is possible, at the beginning of each stage game each government must decide if it shall abide by the cooperative agreement or not. The stage game becomes a bi-matr ix game defined over the action space { C , N } x { C , N } , where " C " indicates a choice to cooperate and maximize the objective function (6.39) and " N " represents a choice to act independently and employ a Nash policy. As before, the payoff functions Chapter 6. Tax Policy Game with Transportation Costs 65 Table 6.6: Cooperative Equi l ibr ia in the Policy Stage Game with Shipping Costs. Endowment Distr ibution Location of Equi l ib r ium Foreign Policy Vt Home Policy »7t ii. < i f 8* - (1+s) 2+7 FX Region 26* 2+7 0 1 7 ^ 9t ^ 1 FX Region (2+7)B*-(\+s)i8t ( 2 + 7 ) 0 . - 1 ^ 7 7 0 ' (1+s) 2+7 ^ 6* ^ 1+s 2(1+7) 2(1+7) T + 7 < | < 1 + ^ iVS Region 8* 1+7 0t 1+7 1 + 5 < | < ( 1 + . ) 2 ^ DX Region ( 2 + 7 ) ^ - ^ 7 7 ^ 2(1+7) (2+ 7 )0 , -( l+s)70* 2(1+7) | > ( 1 + *)T DX Region 0 28t 2+7 Chapter 6. Tax Policy Game with Transportation Costs Table 6.7: Cooperative Tax Rates in the Policy Stage Game wi th Shipping Costs Endowment Distr ibution Foreign Tax Rate Home Tax Rate 8t ^ 1 7 e* — (l+s) 2+7 7 2+7 1 i 7 ^ et „ I (l + s) 2+7 ^ 6* ^ l + s 2 + 2 7 " ^ l+s\9*J > T + 7 < | < 1 + 5 7 1+7 7 1+7 l + s < £ < ( l + a ) 2 ± 2 7 d + 1 ^ 2+27 V l + s 0* ' f > ( 1 + * ) ^ 1 2+7 Chapter 6. Tax Policy Game with Transportation Costs 67 for this game are given by [/(•, •) (5.9) and £/(•, •) (5.10) but with R(-, •) and R*(-, •) being defined by the period reward function when there is shipping costs (6.8). We again find that for certain endowment ratios the game becomes a Prisoners' Di lemma. We state the result for the case when (1 + s ) 2 > - ^ y , the other case is analogous. Theorem 8 Let T be the 2 player bi-matrix game defined by the reward functions U(-, •) and U*(-, •) based on the stage game with shipping costs. i) # (1+7)2+^ < ff < ( i + f e j ° r i / ( l + 5 ) /c ( 7 ) < ff < + thenT is a Prisoners' Dilemma. ii) If ff < 2+7" then T is degenerate in that the actions of home government have no effect on the outcome of the stage game. iii) If ff > (1 + s)^1 then T is degenerate in that the actions of foreign government have no effect on the outcome of the stage game. iv) If ( i + / ) K ( 7 ) < ff < (1 + S)K('J) then T is degenerate in that the actions of both governments have no effect on the outcome of the game. Proof: Please see the Appendix. The incentive to deviate from a coordinated solution only occurs in the event of shipping, in which case, if the game is not degenerate, then it forms a Prisoners' Di lemma. Cooperation is self enforcing when there is no trade only because in this case there is no difference between coordinated and independent actions. The standard results for the Prisoners' Di lemma apply; for endowment pairs where shipping occur in the Nash solution, the action "N" strictly dominates "C" in rewards thus neither government w i l l ever choose to cooperate in this scenario. Of course this type of behaviour is opt imal only when governments only care about the present stage game. As we saw previously, Chapter 6. Tax Policy Game with Transportation Costs 68 repetition can lead to cooperation. The Folk Theorem results for the economy without shipping cost are easily extended to this economy. B y having each government adopt a G r i m Trigger strategy that is designed to prevent even the most profitable possible deviation, one finds that if 8 is large enough, then cooperation is made opt imal 3 . Thus, in the infinite horizon stochastic game with transportation costs, trade disagreements never occur if governments and households are sufficiently patient. 3Note that the mixed equilibrium solution is required in order to form expectations about future potential rewards. Appendix A Proofs for Chapter 5 Proof of Lemma 1 We consider the problem for the home government. Taking the derivative of (5.3) gives the first order condition hence the opt imal reaction of the home government to the foreign government policy 77* is: m = T - r - - r r - v l (A-2) 1 + 7 I + 7 But as 77, > 0, this solution can only be applied if 77* < For larger values of 77* we note that: 4>± - ^ < 0 (A.3) 7 0Vt and hence an opt imum is achieved in this region by setting 77, = 0. The home government reaction correspondence is then: Vt = < 7 < if V*t < * 1+7 1 + 7 " " 7 ^ ^ _ 0 if 77; > ^ The foreign reaction correspondence is obtained through symmetry. n 69 Appendix A. Proofs for Chapter 5 70 Proof of Theorem 1 There are three ways in which the reaction correspondences (5.5) and (5.6) can inter-sect one another: , i) r]t > 0 and n* > 0 To solve for the equil ibrium policy of the home government we substitute (5.6) into (5.5). 0t 7 * Vt = 7-7 7——Vt 1+7 1+7 Ot 7 ; 0; ' 7 •XTTT. ~ TTT.Vth 1+7 l + 7 v l + 7 1+7 Ot 7 n* , 72 1 + 7 (1 + 7 ) 2 (1 + 7 ) 2 (1 + 7 ) f l , - 7 f l? l + 2 7 (A.5) Substituting (A.5) into (5.6) gives us the equil ibrium policy of the foreign govern-ment: . = (i+7)»r-7ft ( A . 6 ) '* l + 2 7 v ; The restrictions r/, > 0 and rj^ > 0 imply that this equil ibrium wi l l only occur when < f f < ^ 7 - . A n example of this type of equilibria is illustrated in Figure 5.1. ii) r/i = 0 and n* > 0 Substituting rjt = 0 into (5.6) we find rj* = This Nash equil ibrium occurs when |* ^ 1+^ - I n this scenario, the r?* intercept of the home reaction function lies to the left of the rj* intercept of the foreign reaction curve, representing the fact that the majority of endowed goods are in the foreign country. Appendix A. Proofs for Chapter 5 71 i i i ) t]t > 0 and 77* = 0 Substituting 77* = 0 into (5.5) we find: 77, = ft. This Nash equil ibrium wi l l occur if > that is, when the home country is endowed wi th an excessive amount of goods relative to the foreign country endowment. B y using the method of Backward Induction Rationali ty we have verified that this solution is subgame perfect [7]. W i t h a unique Nash equil ibrium further equil ibrium refinements are unnecessary; the primary purposes of Nash refinements are to eliminate "weak" equilibria when more than one equilibria exists and to insure that the Nash equil ibria are rational 1 . Subgame perfection guarantees that our equil ibrium is both rational and stable. Basar and Olsder [2] makes mention of another form of stability for infinite game equilibria where one treats the best reply correspondences as iterative maps and then verifies that a perturbations about this equilibria vanish over successive iterations. From (5.6) and (5.6) we form the following iterative map: \ Vt+i ) I 1 + 7 70? \ / + (1+7) 161 \ 1+7 (1+7) 2 / 0 . 7 \ ( 1 + 7 ) 2 ( 1 + 7 ) 2 0 (A.7) 2 where k is a positive integer. Since (fty < 1 the eigenvalues of this mapping have magnitude less than 1. Thus the map is a linear contraction and has a fixed point [8] which is the Nash equilibria. Proof of Corollary 1 From Theorem 1 we see that the optimal government policies are stationary control laws based on the period endowments. The home government's policy can be re-written as: "^Most refinements only apply to finite games [17]. A comprehensive survey of these refinements can be found in [16]. Appendix A. Proofs for Chapter 5 72 (A.8) where c 0* 0 (i + 7 ) - 7 ( f t ) -l + 2 7 f o r i-L <^ _ 7 _ 1 0 1 0* - 1 + 7 for T - 7 - - < f r < — 1 + 7 0* 7 (A.9) I I 1 + 7 0 t 7 From (3.6) we see that rt = 1 - C (|) hence the results in Table 5.2 follow. The foreign tax policy is determined in the same fashion. Proof of Theorem 2 The period reward function for the cooperative regime is: RC(vhVt) = 2 1 o g ( ^ I ) + 7log(0 t - rjt) + 7 l o g ( 0 * - tf). F rom the first order conditions W^- = 0 and = 0 we find: ovt or)* (2 + 7)^ + 7^ = 20„ 7r ? t + (2 + 7)r/t* = 20,*. For 77, > 0 and 77* > 0 the intersection of (A.11) and (A.12) yield the results: (2 + 7)0 t - 7 0 t * Vt * Vt 2 ( 1 + 7 ) ' (2 + 7 )0* - 7 0 , (A.10) ( A . l l ) (A.12) (A.13) (A.14) 2 ( 1 + 7 ) In (A.13), Vt > 0 implies that | f > -ft and thus (A.13) and (A.14) are only feasible in this region. For smaller values of w e have < 0 thus an opt imum is obtained in this region by setting 77, = 0. Substituting this result into (A.12) we find the opt imal Appendix A. Proofs for Chapter 5 73 foreign policy to be rj* = Similarly, for ff- < 2±i we find that the opt imal policies are nt — and rj* = 0. Let where 0* (2+7) -7 ( f t r 2 (1+7) 2 I 2+7 f o r St. < 1 U i 6* — 2 + 7 i U i 2 + 7 ^ 6' ^ 7 for ^- > ? + 2 (A.15) (A.16) we see that the optimal home tax rate is rt = 1 — (c (j^ and thus the results in Table 5.4 follow. Results for the foreign government are derived by symmetry. Proof of Theorem 3 Firs t we consider the game for < ff < 2 + 0 i) To show that the cooperative equil ibrium Pareto Dominates the non-cooperative solution we must show that U(C,C) > C/(N, N). The home government's reward in a cooperative equil ibrium is U(C,C) = l og (5*±^L)_ 7 i o g (^_ ) 7 t ) = l 0 g ( 2 ( 2 ( 1 + 7 ) + 2 ( 1 + 7 ) ) } ( 2+ 7 ) f l * - 7 ^ - 7 log(0 t 0 / 1 , _ A ) 2 ( 1 + 7 ) - l o g ( J r : t 7 % ) + 7log( 7(0*+ 0D )• '2(1+7)' V 2 ( 1 + 7 ) The home government's reward in a non-cooperative (Nash) equil ibrium is (A.17) [/(N,N) = log (5Lt^L)_ 7 i o g (^_^) Appendix A. Proofs for Chapter 5 74 Define A C ^ . ^ = U(C,C) - <7(N,N). A ^ c c - , , = (1 + 7 ) l o g ( ^ ^ ) + 7log(2). (A.19) AUCC-NN = 0 W H E L 1 (112-)™.? = ! (A.20) 2 ^ = 1 + - ^ — , (A.21) 1 + 7 which is true when j - 7 ^ = 0 (i.e. 7 = 0) or — 1 (i.e. 7 —•>• 0 0 ) . Since 2 7 is concave up with respect to j - 7 ^ we have 2 ^ < 1 + - 1 — (A.22) 1 + 7 for 0 < 7 < 0 0 . But this fact implies ( i ^ r ' . ^ l (A.23) which in turn implies that AUCC_NN > 0. i i) To see that " C " is a strictly dominated strategy we note that from L e m m a 1, playing " N " satisfies the first order conditions for optimizing R(n*,rjt). Therefore, C / ( C , N ) > U(C,C) and £ / ( N , N ) > U(N, C) and hence " N " dominates " C " in payoffs. Due to the symmetry in the model, the same results hold for the foreign government. Thus we see that the structure of the payoffs in T are equivalent to the structure of payoffs in a Prisoners' Di lemma. Appendix A. Proofs for Chapter 5 75 To see that T is degenerate when ff < j^. we note that the both the cooperative strategy (Table 5.3) and the non-cooperative strategy (Table 5.1) specify that r]t = 0 for this range of ff hence the actions "C" and "N" are indistinguishable in this case and the home government has but one action to choose from. The same argument holds for the foreign government when ff > Appendix B Proofs for Chapter 6 Proof of Lemma 2 Setting u(ct) = log(c t) the first order conditions from the CPP, (3.12) - (3.15) become: A , = - , (B. l) dLcpp 1 dct ct dLCPP 1 dc*t Ct AT* = 0 = • A ; = i , (B.2) c t = — A ; - A , < o = • < 1 + s dxt l + s * ~ c*t ~ with d-^^-xt = 0, (B.3) oxt = — A , - A : < 0 = • ^ > dx* l + s * _ c*t ~ l+s dLr with ^ - x * = 0. (B.4) uxt From the K u h n Tucker conditions, (B.3) and (B.4), we see that there are three possible shipping regimes: i) No Shipping: When there are no shipments xt = 0 and x* = 0, thus from (6.3) and (6.4) ct = nt and c* = v*. From (B.3) and (B.4), | < 1 + s and g- > ^ hence the No Shipping region is defined by NS = ((tf.ifc) € [0,0?] x [O,0t] : - i - < ^ < 1 + s i . (B.5) I l + s nt J ii) Domestic Exports: For domestic exports we have xt > 0 and = 0, thus from (6.3) and (6.4) ct = nt - xt and c*t = n* + ffc. From (B.3), = 1 + s, but 76 Appendix B. Proofs for Chapter 6 77 r\t > ct and rj* < c\ thus the Export region is DX = ^(rj*,r,t) G [0,0?] x M : | > l + - } . (B.6) From (3.22), the amount exported must satisfy * , j±_ = l + a, (B.7) hence x, = " ' - ( 1 2 + ^ * . (B.8) i i i ) Foreign Exports: Foreign exports occur when xt = 0 and x* > 0, thus from (6.3) and (6.4) ct = nt + ft and c* = 77* - x*. From (B.4) % = ft, but 77, < ct and 77* > c* thus the Export region is FX = \(rj;,nt) G [0,0*] x [0,0,] : ^ < . (B.9) I 1 + 5 J From (3.25), the amount shipped must satisfy Vt + T+S T]* - X* 1 + S' (B.10) hence Proof of Lemma 3 i) To see that R(-,-) is continuous along dDX we note that RNS(V*, V) \v=(i+»)v* = tMf) + 7 log(0t - ^ ) ] n = ( i + s ) r ) * = log(( l + s)ri*) + 7 l o g ( 0 t - (1 + 5)7/*). (B.12) Appendix B. Proofs for Chapter 6 78 Approaching dDX from within the region DX we find l i m RDX(n*,n) = l o g ^ f a + (1 + S)TJ*)) + 7 l og (0 t - V) = log(( l +5)77*)+ 7 l o g ( 0 « - ( l + *>/*) (B.13) hence R(-,-) is continuous along dDX. Similarly, for the boundary dFX which also lies within NS = l o g ( - ^ ^ ) + 7 l o g ( ^ - - ^ ^ ) . (B.14) l + s l + s Approaching dFX from within region FX results in l i m RFX(n*,n) = [log fa + T^j—»*)) + 7log(0* - v) l + s l + s thus R(-, •) is continuous along dFX as well. (B.15) ii) It is easy to see that R(-, •) is not differentiable along either dDX or dFX as dRT l + s 0tf nt + {l + s)Vf I l + s ^+1+7^' = 0. (B.16) (B.17) (B.18) Thus 2gB2L ^ ^ along and ^ ^ a i o n g d F X . i i i) We need only show one counter example to show that •) is not concave ijt along FX. Let 7?t* = (1 + s ) ^ , then nt = j^rj* = But then we have 7 5i? 1 drjt Tjt 0t - rjt 0 (B.19) Appendix B. Proofs for Chapter 6 79 and dRFX 1 7 1 + 7 < 0. (B.20) dm Vt + j^n* 0t-rjt 20, Thus R(-, •) is not concave in T / , on FX for this value of 77*. We shall discover later that there is a whole range of 77* values for which R(-,-) is not concave in 77,. Proof of Lemma 4 Let (77*577,) G NS, then the home reward function is RNS(v;,Vt) = log(r!t) + 1\og(8t-r1t). (B.21) The first order conditions wi th respect to 77, are 9RNS _ 1 7 % ?7i 0t - Vt and therefore the opt imum strategy is = 0 (B.22) 77, = -A-. (B.23) 1 + 7 B u t since we are restricted to the NS region employing this strategy implies 111 l + s j^r< ± < i + *, l + s (j+V) ——— < v; < ( i + * ) - ^ , l + s l + 7 - '* ~ v ; l + 7' V*A< V*t < C (B.24) The solution (B.23) can only be adopted for foreign policies that lie in the range 77* G [77*,77*]. Employing this solution outside of this range wi l l force (77*,77,) outside of NS. Appendix B. Proofs for Chapter 6 80 To calculate the opt imum for 77* £ [77*, 77* ] we treat this as a constrained optimization problem and search for solutions along the boundaries of NS. d2RNS 1 7 dtf V? (0t - Vt)2 < 0 (B.25) thus Vt> ft implies < 0 and 77, < ft implies > 0. Hence, when 77, > ft it is opt imal to set 77, as low as possible within NS and when 77, < -ftj the largest possible 77, is opt imal . For 77* > 77;, (,;,,,) € N S _ | > _ ± _ - „ > J i _ . (B.26) Therefore when 77* > 77^ , the home government sets 77, as small as possible wi thin the region NS, and so 77, = ftv*-Similar ly for 77* < 77*, (v*,Vt)eNS -> ^ <l + s - ^ < r ^ - - ( B - 2 7 ) »7t 1 + 7 Thus when 77* < 77* the home government sets 77, as large as possible wi th in NS, hence »* = ( i + s)v;. " In summary for (77*, Vt) £ NS the strategy (6.9) is optimal for the home government. Proof of Lemma 5 Let (VtiVt) £ DX, then the home reward function is Rnxiv^Vt) = \og(l-(Vt + (1 + s)r,;)) + 7log(0, - Vt)- (B.28) The first order conditions wi th respect to 77, are Appendix B. Proofs for Chapter 6 81 d R D X 1 7 df]t Vt + (1 + s)vt &t - Vt and thus the opt imum strategy is = 0 (B.29) V ^ ^ - i l + s ^ v l (B.30) Since we are restricted to the DX region employing this strategy implies ^>l + s, (B.31) Vt < TT~sVt, (B.32) ^ < r T 7 ( i ^ - ( 1 + - ) ^ ? ) ' ( R 3 3 ) ( ^ I ^ l M I ^ ) ' (B-34) The policy (B.30) can only be employed when T / * < ??*; pursuing this policy outside of this range forces (vtiVt) into NS. To determine the optimal export strategy for 77* > V* w e n o ^ e that d 2 R D X 1 7 < 0, (B.36) dv! (m + (i + s)vt)2 ^ - vt)2 thus RDX is decreasing in Vt when Vt > ^ — (1 + s)j^v*-Let (77*, x) be on the DX policy line (B.30). x = - ^ - - { l + s ) - l - r , ; . (B.37) 1 + 7 1 + 7 Note that x = when 77* = 77*, thus 77* > 77* implies x < But for (77*, 77,) <E Z i Y we have ^  > l + s, thus when 77* > 77* it must be that 77, > > x. Thus, when 77* > 77* we find that Appendix B. Proofs for Chapter 6 82 and the home government wi l l set i]t 6 DX as small as possible. As DX does not contain its own boundaries, the government must adopt the following e-strategy It = (1 + 3)$ + e, where e > 0. (B.39) Since R(-,-) is continuous along this boundary, letting e —> 0 is equivalent to selecting a No Shipping policy for this range of r/*, more simply put: the home government does not find it opt imal to set a policy that promotes exports when n* > rj*. m Proof of Lemma 6 For (77*, rjt) £ FX the reward function is RFx(v;,Vt) = log(hnt + -^-ri;)) + 7\og(et-Vt). (B.40) z 1 + s The first order conditions with respect to nt are dRFX 1 7 9vt Vt + ftv* Qt - Vt and thus the opt imum strategy is = 0 (B.41) 0t 1 7 Vt = 7——77—f/*- (B.42) 1+7 l + s l + 7 Being restricted to the FX region and enforcing the strategy in (B.42) implies ^ < (B.43) r,; > (1 + s)vt, (B.44) (1^-1+11+1"')' ( B' 4 5 ) Appendix B. Proofs for Chapter 6 83 ^ > ( 1 + S ) 1 T 2 7 " = ^ - ( E U ? ) The strategy (B.42) is only feasible for 77* > 77* . Pursuing this strategy outside of this range forces (r}*,r)t) into NS. The opt imum for 77* < 77* can be determined using an argument similar to the one employed in the DX case with 77* > 77* (see L e m m a 6). The results are also similar; when 77* < 77* the home government finds that it is more opt imal to operate in the No Shipping region than in the Foreign Export region. Thus we see that (6.14) is the opt imum strategy for the home government in the FX region. • Proof of Lemma 2 A l l these results follow from the symmetry between the home government's optimiza-t ion problem and the foreign government's optimization problem. The proof of i) follows the proof in L e m m a 4 and the proofs for ii) and i i i) follow Lemmas 6 and 5 respectively. • Proof of Theorem 5 We begin our analysis wi th the NS — DX overlap region. i) NS - DX Overlap Region Let 77* 6 [0,77*). Since the DX strategy (B.30) is optimal for (77*, 77,) 6 DX RDxint^t) |„t=Ji__(i+a)^_„.> RDx{v?,Vt) k=(i+*)„ t*, (B.48) * 1+7 v ' 1 + 7 * but since R(-,-) is continuous along the boundary dDX we have R-oxivhVt) \v,={i+a)n'= RNsivhVt) k = ( i + » K (B.49) Appendix B. Proofs for Chapter 6 84 Thus J7?*) I „ t = et__(i + s)^_,.> RNsiVtiVt) \Vt=(l+s)v* (B.50) and hence the home government always finds it optimal to pursue a a strategy in Domestic Export region for this range of foreign policies. i i) NS - FX Overlap Region Let rj*e(v*D,e;}. First consider the case when rj* G (Vg^*]. Since FX strategy (6.14) is opt imal for ^l(r,lr,t)eFX RFX(vUvt) \ v t = j L . > Kxivhrn) \vt=j+-v: • (B-51) "• 1 + 7 1~rs e But i?(-, •) is continuous so RpxivhVt) U = ^ ' = RNS(vhVt) \ n t = T L . v ; (B-52) and therefore, the home government wi l l always select the the strategy in the For-eign Export region (6.9) for this range of foreign policies. To determine the behaviour of the home government in the range n* € (??*,"*], we introduce the function AR(r]*), the difference between the FX and the NS regional reward functions evaluated at the optimal regional strategy. The home government wi l l switch strategies whenever this function is equal to zero. Appendix B. Proofs for Chapter 6 85 ARM) = < 1+7 l+l - RNs(vUVt) \nt={i+'ht 1 + 7 l + « 1+7 Vt -RNs(Vt,Vt) |„ = T£ i+i 1 + 1 l + s 1 + 7 ( l + 7 ) l o g ( ^ T ^ ) + 7 log( 7 ) - l o g ( 2 ) - l o g ( ( 1 + . ) ^ ) tf€(i£,»£) + 7 log ( 0 , - (1+5)77*) (l +7)losW + ^ ) , . g ( B H ) - l o g (2) - ( 1 + 7 ) l o g (Ot) ( l + 7 ) l o g ( ^ 5 ^ ) + 7 l o g ( 7 ) - l o g (2) - l o g + 7 i o g ( 0 t - T ^ 7 , * ) 0?]. Differentiating with respect to 77? gives A R ' ( V ; ) 1+7 (l + s)t9t+7)' 1+7 1+7 J_ i __2 i l±£ l_ r,* a (n* r,*) ^ + (l + s)7?t-„* G ( " B ' ^ < ] -(B.54) Appendix B. Proofs for Chapter 6 86 For 77* G (77*,, 77*) 1 et thus Also thus 1 + 5 1 + 7 et - (i + s)v; > (i - -J—)*t > o, (B.56) 1 + 7 7 ( 1 + S ) > 0 . (B.57) 1 + 7 : ;>4 (B.58) ( l + s)0, + tf vt 1 + T 1 7 - : T > 0 (B.59) ( 1 + 5 ) ^ + 7/* 7,? and so AR>(r)t) > 0 for 77* G (77*,, 77*). For 77* G (77*,T, ; ) A ^ - ( i + X , r > 0 ' ( R 6 0 ) thus AR(r)t) is monotone increasing for 77* G {v*D,V*B]i D U t a t vt = ^ ^ n e strategy in FX is optimal (see (B.51) and (B.52)) thus AR{r,*B)>0 (B.61) The strategy in the NS region policy is optimal for al l (vtiVt) G NS. For 77* < 77* RNs(v*iVt) \Vt=(i+s)r,* > RNS(VtiVt) \(vt,vt)edFX (B.62) = RFx(vtiVt) \(r,im)edFx, but the strategy in the FX region (6.14) intersects dFX when 77* = 77* and 77* < 77* thus Ai?(77*,)<0. (B.63) Appendix B. Proofs for Chapter 6 87 The continuity and monotonicity of AR(r,*) on the interval (77*, 77* ] coupled wi th the conditions (B.61) and (B.63) guarantee the existence of a unique value 77* G (v^iVg] such that AR(rj*) = 0. There are two possibilities. For rj* G [??*,?7*], AR(rjt*) = 0 gives (1 + 7) log (9t + - ^ 7 7 * ) - log (2) - (1 + 7) log (0t) = 0, (B.64) thus 77; = (1 + s){2^ - l)9t. (B.65) This result is valid only if rj* > 77* = -J—-A_ (B.66) " - ' A 1 + s 1 +7 v ' which is equivalent to the condition (1 + * ) 2 > L j - = - L (B.67) ( l + 7 ) ( 2 1 + ^ - l ) <V When rj* < 77* we arrive at the second possibility. In this case 77* G (ft*^7?^) solves AR(fj*) = 0. Setting 77* = (j>6t we find that (j) solves 2(1 + 3)^(1 - (1 + s)W = (r---)^7 7 ( i + rr-<t>)1+1- (B.68) 1 + 7 1 + 3 This case is characterized by the fact that (1 + s)2 < — L j - = - L . (B.69) ( l + 7 ) ( 2 1 + ^ - l ) \f) Note that this can only occur if 77* < 77*. In summary, for (1 + s ) 2 > it is optimal for the home government to adopt the strategy in (6.26). Figure 6 illustrates a specific example of this type of reaction correspondence. If (1 + s)2 < -^-y then the home government w i l l use (6.28) as its Appendix B. Proofs for Chapter 6 88 opt imal strategy. A n example of this type of reaction correspondence is illustrated in Figure 7. • Proof of Theorem 6 Firs t we consider the case when (1 + s)2 > ^ y . For the DX region, we look for the intersection of the home strategy in the DX region (6.12) given in L e m m a 5 and the foreign strategy in the DX region (6.20). We find (i +1)6; - ^ s l e t Vt = Vt =. l + 2 7 (i + 1)et-(i + s)7e; l + 2 7 with the restrictions rj* < Vc from (6.12) and Vt > Vt- The first restriction together wi th (B.70) imply (i + 7 ) f t - ijrrg? _ j ej_ l + 2 7 l + s l + 2 7 ' The second restriction combined with (B.71) gives ( 1 + 7 ) ^ ( 1 + ^ > ( 1 + a ) ( 2 * _ l W i | > r ^ [ ( l + 2 7 ) ( 2^-l) + 7 ] ( l + 3), | > £(7) (1 + *)-Only one of the inequalities (B.73) and (B.76) wi l l be binding. « 7 ) = r ^ [ ( 1 + 2 7 ) ( 2 I ^ - l ) + 7] 1 _ . „ ^ r f c [(l + 2 7 ) 2 ^ - l - 7 ] 1 + 7 + 7 = (2 — ) 2 i T T - 1 . v i + r B.70 B.71 B.72 B.73 B.74 B.75 B.76 B.77 B.78 B.79 Appendix B. Proofs for Chapter 6 89 We see that: £(0) = 1, £(7) —> 1 as 7 —*• co and £(•) has a unique crit ical point at 1 + 7 = 2 — 1^2), and hence for 7 > 0 we have £(7) > 1. Thus the inequality (B.76) is binding and determines the range of states for which a Nash equil ibrium wi l l occur wi th in the DX region. The procedure for deriving the Nash equil ibrium in the FX is similar. The home government strategy in the FX region (6.14) and the foreign strategy in the FX region (6.22) intersect at *• = ( 1 + ^ ; ^ h * ' . (B.80) (1+7)0! - T+-70,* *" = 1+2, • ( B ' 8 1 ) The foreign strategy in the FX region is only defined for rjt > rjc which implies ff- < j ^ . The home strategy in the FX region is only applicable when 77* > fj* which implies h < - 1 + 7 = 1 ( B 8 2 ) °* [(l + 2 7 ) ( 2 I ^ - l ) + 7 ] ( l + s) ^ X i + s ) ' Since £(7) > 1 for 7 > 0, (B.82) and determines the range of endowment ratios for which a pure strategy Nash equil ibrium wi l l exist imn the FX region. A pure strategy equil ibrium in the NS region is «f = y q f y (B.83) with the l imita t ion that 77* < 77* < rj* and rjA < rjt < fjt. From rjt < r)* we find that I > (1 + s)(l + 7 ) ( 2 1 * 7 - 1) = (1 + s)/c( 7) (B.85) and from rjt < rjt we conclude that Ot ( l + a ) ( l + 7 ) ( 2 ^ - l ) (1 + ^(7) Appendix B. Proofs for Chapter 6 90 Thus we see that a pure strategy equil ibrium wi l l occur in the NS region whenever 1 < £ < ( ! + *M7) . (B.87) (1 + 5 ) K ( 7 ) - 0 ? We see that the intervals for the various Nash equilibria are disjoint from one another. Since these ranges do not overlap, it is impossible to have any more that one pure strategy Nash equil ibrium for a given endowment pair, and so, if a Nash equil ibrium exists, it must be unique. If an endowment ratio falls outside of these intervals then there is no pure strategy Nash equilibria. This occurs when the two reaction curves fail to intersect. Figure 6.8 illustrates an example of this situation. Final ly , for (1 + s)2 < K(J) we note that neither government utilizes a strategy within the interior of NS, thus, there can never be a pure strategy Nash equilibria that occurs in the No Shipping Region. For this set of parameters the Nash equilibria w i l l be either a point in DX, a point in FX or non-existent. The endowment ratio intervals for these cases are derived in the same way as the intervals for the previous set of parameters. • Proof of Theorem 7 The period reward function for the cooperative regime is Rc(v:,rit) = ( v ; , V t ) e D x (rj;,rjt)eFX Appendix B. Proofs for Chapter 6 91 l o g ( 2 £ ± i ^ i ) + 7 l o g ( f l t - , t ) + i o g ( ^ ^ ) + 7 i o g ( 0 ; - r / ; ) logfat) + 7 log(0, - ft) + log(j/;) + 7 log(«?-»/«*) log( ) + 7 log (0 , - 7? t ) + l o g ( ^ ± ^ ) + 7 l o g ( 0 ; - r ; * ) ivhVt) G Atf(B.88) ivhVt)eFX. dRcr For (r]*,Tit) G F X , from the first order conditions = 0 and a™ = 0 we have the equations (2 + 7X1 + ^ + 77,* = 2(l + 5 ) 0 „ 7(1 + ^ + (2 + 7)77* = 20*. The intersection of (B.89) and (B.90) is given by (2 + 7)0; _ ( i + s )70 , (B.89) (B.90) Vt = Vt = 2 ( 1 + 7 ) (2 + 7)0, - ft7e; (B.91) (B.92) 2 ( 1 + 7 ) Since FX is adjacent to the 77* axis, we must check the constraint 7/* > 0. This constraint along wi th (B.91) imply that ff < (1 + s)2^1 and so the equil ibrium described by (B.91) and (B.92) w i l l only exist for this range of endowment ratios. When ff > (1 + s)2^1, it is opt imal to set ift = 0 and thus from (B.90) it must be that 77, — -ft. For (7/*, r,t) G NS the first order conditions yield the autarky opt imum as expected. The case of (v*iVt) G DX, the results are analogous to the results for the Foreign Export region. To find the equil ibrium tax rate we define C c((ff) a s '0t v(0h0t) = C ( ^ 1 - 0 * . (B.93) The tax rate is now derived using the formula r, = 1 — £ c ; the results in Table 6.7 follow. Appendix B. Proofs for Chapter 6 92 Proof of Theorem 8 For < | < I I T ^ M or (1 + 5 ) « ( 7 ) < | < (1 + 5 ) 2 ± 2 the actions " C " and " N " specify different policy levels for both governemnts. The Nash solution maximizes the individual reward functions, thus the cooperative solution is clearly sub-optimal for both governments regardless of the action of its opponent. For each government, the action " C " is strictly dominated by the action " N " . We also know that £/(C,C) > t / ( N , N ) ' a n d C / * ( C , C ) > t / * ( N , N ) . If this were not so then C/(N,N) + t / * ( N , N ) > i 77(C,C) + <77*(C,C), implying that ( C , C ) is not optimal for the joint objective function. Thus, we conclude that for these ranges of endowment ratios, the structure of payoffs in T is that of a Prisoner's Dilemma. For all other ranges of ft the actions " C " and " N " do not specify different policy values for either one of the governments or possibly both. In any of these cases, the game is degenerate. Bibliography Barar i , M . and Lapan, H . E . , 1993, Stochastic trade policy with asset markets, Journal of International Economics 35, 317-333. Basar, T . and Olsder, G . J . , 1982, Dynamic Noncooperative Game Theory (Aca-demic Press, London). Coates D . , 1993, Property tax competition in a repeated game, Regional Science and Urban Economics 23, 111-119. de Crombrugghe, A . and Tulkens, H . , 1990, O n pareto improving commodity tax changes under fiscal competition, Journal of Public Economics 41, 335-350. Frenkel, J . A . and Razin , A . , 1982, Fiscal Policies and the World Economy, 2nd Edition ( M I T Press, Cambridge). Friedman, J . , 1971, Non-cooperative equilibria for supergames, Review of Economic Studies 38, 1-12. Fudenberg, D . and Tirole, J . , 1991, Game Theory ( M I T Press, Cambridge). Hirsch, M . W . , Smale, S., 1974, Diferential Equations, Dynamic Systems and Linear Algebra (Academic Press, San Diego). Kehoe, P. J . , 1989, Pol icy cooperation among benevolent governments may be undesirable, Review of Economic Studies 56, 289-296. Lockwood, B . , 1993, Commodity tax competition under destination and origin principles, Journal of Public Economics 52, 141-162. Lucas, R . E . , 1982, Interest rates and currency prices in a two-country world, Journal of Monetary Economics 10, 335-359. M c M i l l a n , J . , 1986, Game Theory in International Economics (Harwood, London). M i n t z , J . and Tulkens, H . , 1986, Commodity tax competition between member states of a federation: equil ibrium and efficiency, Journal of Public Economics 29, 133-172. [14] Raz in , A . , Sadka E . , 1991, International fiscal policy coordination and competition, in International Taxation in an Integrated World ( M I T Press, Cambridge). 93 Bibliography 94 [15] Secru, P. , Uppal , R . , and Van Hulle, C , 1995, The exchange rate in the presence of transaction costs, Journal of Finance, forthcoming. [16] Van Damme, E . , 1987, Equi l ibr ia in non-cooperative games, in CWI Tract 39: Surveys in Game Theory and Related Topics (Center for Mathematics and Computer Science, Amsterdam). [17] Van Damme, E . , 1991, Stability and Perfection the Nash Equilibrium, 2nd ed. (Springer-Verlag, Berl in) . [18] Van der Ploeg, F . , 1988, International policy coordination in interdependent mone-tary economies, Journal of International Economics 25, 1-23. [19] Var ian , H . R . , 1992, Microeconomic Analysis, 3rd Edition ( W . W . Norton and Company, New York) . [20] Wi ldas in , D . E . , 1988, Nash equilibria in models of fiscal competition, Journal of Public Economics 35, 229-240. [21] Wi ldas in , D . E . , 1991, Some rudimentary 'duopolity' theory, Regional Science and Urban Economics 21, 393-421. 

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