Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

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

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_1995-0522.pdf [ 3.48MB ]
Metadata
JSON: 831-1.0079817.json
JSON-LD: 831-1.0079817-ld.json
RDF/XML (Pretty): 831-1.0079817-rdf.xml
RDF/JSON: 831-1.0079817-rdf.json
Turtle: 831-1.0079817-turtle.txt
N-Triples: 831-1.0079817-rdf-ntriples.txt
Original Record: 831-1.0079817-source.json
Full Text
831-1.0079817-fulltext.txt
Citation
831-1.0079817.ris

Full Text

INTERNATIONAL TAX COMPETITION: T H E EFFECTS OF TRANSPORTATION COSTS  By JAMES ROBERT RITCHIE A. Sc. (Mathematics and Engineering), Queen's University, 1989  A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T 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 F A C U L T Y OF G R A D U A T E STUDIES D E P A R T M E N T 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 R I T C H I E , 1995  In  presenting  degree freely  at  this  the  thesis  in  partial  fulfilment  University  of  British  Columbia,  available for reference  copying  of  department publication  this or of  thesis by  this  for  his thesis  or  and  her  Department of  DE-6 (2/88)  I further  Columbia  requirements that the  agree  may be  representatives.  for financial  the  I agree  scholarly purposes  permission.  The University of British Vancouver, Canada  study.  of  gain shall  It not  is  that  an  advanced  Library shall  make  it  permission for extensive  granted  by the  understood be  for  allowed  that  head  of  my  copying  or  without my written  Abstract  We examine a four player stochastic game i n which two of the players are representative households and two of the agents are governments. T h e game is set i n a two country general e q u i l i b r i u m model where it is costly to transport goods between nations.  At  the beginning of each period, each country receives a random endowment of a single non-storable consumption good. Consumers determine o p t i m a l consumption levels and goods shipments based on tax rates and the amount of endowments received that period. Governments determine o p t i m a l tax rates and expenditure levels i n 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 i n each country after taxation and thus governments can manipulate the pattern of trade through tax policies. T h e choice between cooperating or acting independently i n the stage game creates a Prisoners' D i l e m m a . 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 i n order to prey on the resources of its opponent. In an application of the Folk T h e o r e m we find that the long run behaviour is cooperative and over-taxation is avoided, provided that governments and households are sufficiently patient. T h e addition of transportation costs adds another dimension to the short run behaviour of governments. In the same model without transportation costs, governments implement a single o p t i m a l policy strategy that is independent of the direction of net flow of goods and a unique pure strategy Nash E q u i l i b r i u m always exists for the stage game. W h e n transportation costs are introduced the o p t i m a l 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. T h e ability to manipulate particular directional flows of goods combined w i t h governments acting i n their own self-interests results i n the nonexistence of a pure strategy Nash equilibrium i n 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 exists. R e p e t i t i o n 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 equilibrium 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 E q u i l i b r i a for Stage Game w i t h N o Transportation Costs  17  5.2  Cooperative E q u i l i b r i a for the Stage Game w i t h N o Transportation Costs  23  5.3  E q u i l i b r i u m for Repeated Game w i t h N o Transportation Costs  28  6  Tax Policy Game with Transportation Costs  33  6.1  T h e Consumers' O p t i m i z a t i o n Problem: Solution to the C P P  33  6.2  Nash Correspondences for Stage Game w i t h Transportation Costs  6.3  Nash E q u i l i b r i a for the Stage Game w i t h Transportation Costs  51  6.4  Cooperative E q u i l i b r i a for the Stage Game w i t h Transportation Costs . .  63  iv  . . . .  37  Appendices  68  A Proofs for Chapter 5  69  B Proofs for Chapter 6  76  Bibliography  93  v  List o f Tables  5.1  Nash E q u i l i b r i a i n the Policy Stage Game without Shipping Costs  5.2  Nash Tax Rates i n the Policy Stage Game without Shipping Costs.  5.3  Cooperative E q u i l i b r i a i n the Policy Stage G a m e without Shipping Costs.  25  5.4  Cooperative Tax Rates i n the Policy Stage Game without Shipping Costs.  25  6.1  O p t i m a l Shipping and Consumption Plans for u(c) = log(c)  34  6.2  N a s h E q u i l i b r i a i n the Policy Stage G a m e w i t h Shipping Costs when (1 + s)  >  2  6.3  22 . . .  22  53  - f r  Nash Tax Rates i n the Policy Stage Game Shipping Costs when (1 + s)  2  > 54  - f r  6.4  Nash E q u i l i b r i a i n the Policy Stage Game w i t h Shipping Costs when (1 + s)  <  2  6.5  55  Nash Tax Rates i n the Policy Stage Game w i t h Shipping Costs when (l  +  * )  2  < i )  5  6  6.6  Cooperative E q u i l i b r i a i n the Policy Stage Game w i t h Shipping Costs.  .  65  6.7  Cooperative Tax Rates i n the Policy Stage Game w i t h Shipping Costs.  .  66  vi  List of Figures  5.1  Nash E q u i l i b r i u m for Policy Stage Game w i t h N o Shipping Costs  20  6.1  Partitioning of Strategy Space into Shipping Regions  36  6.2  Nash Strategy for the Home Government i n the No Shipping Region.  . .  40  6.3  O p t i m a l Regional Strategies for the Home Government w i t h ?/* > 77*. . .  44  6.4  O p t i m a l Regional Strategies for the Home Government w i t h 77* < 77*. . .  45  6.5  Home Government Nash Reaction Correspondence when (1 + s)  >  48  6.6  Home Government Nash Reaction Correspondence when (1 + s)  <  49  6.7  A N o Shipping Nash E q u i l i b r i u m i n the Policy Stage G a m e w i t h Shipping  2  2  Costs 6.8  57  Non-existence of a Pure Strategy Nash E q u i l i b r i u m i n the Policy G a m e w i t h Shipping Costs.  6.9  58  Partitioning of the Endowment Space i n the Policy game w i t h Shipping Costs when (1 + s)  2  > K(-J)  61  vii  Acknowledgement  I would like to thank both of m y supervisors, D r . C o l i n W . C l a r k of the Department of Mathematics and D r . R a m a n U p p a l of the Faculty of Commerce and Business A d m i n istration for their help and guidance i n 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 i n this paper.  Last of all I would like to thank m y fellow stu-  dents i n the Department of Mathematics for their constant support and encouragement throughout the duration of m y studies at the University of B r i t i s h C o l u m b i a .  vm  Chapter 1  Introduction  A s 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 w i l l have a marked effect on the economy of its neighbour. T h e 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 i n the design of national fiscal policies i n integrated economies. T h e importance of this study has not gone unoticed; there is a large literature that already exists i n this field. T h e standard model consists of two or more regional governments 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 i a 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 i n a competitive equilibrium, there is a lack of public goods and that government coordinated policies are Pareto i m p r o v i n g . 1  O u r 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  O u r model is based on a stochastic endowment process, while the standard fiscal competition model is based on a production economy. A s a result, the mechanism of policy transmission i n 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. T h i s difference is at the root of the contradiction i n the predictions of our model and the standard policy competition model. Where the standard model produces an e q u i l i b r i u m where there is a shortage of public goods, our economy has an excess of public goods.  Other policy models that incorporate a stochastic element i n -  clude: Frenkel and R a z i n [5] who examine a two country stochastic taxation model, i n order to assess the transmission of fiscal shocks v i a 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 i n game theoretic terms). Secondly, our model has an infinite time horizon. T h e standard fiscal competition model is static or two periods at most. T h e infinte time horizon permits richer strategic behaviour on the part of governments that cannot be realized i n models of shorter duration. Coates [3] examines tax competition i n 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. Finally, 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 w i t h costly transportation for labour. T h e geometry of their results are strikingly similar to that our results, but as they are dealing w i t h 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. T h e 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 i n t u r n leads to trade disagreements. In an application of the Folk Theorem we show that coordinated fiscal policies are more efficient and are enforcible i n our stochastic game v i a a G r i m Trigger strategy.  Chapter 2  The Model  We consider a two country general equilibrium model - an extension of of the endowment economy found i n Secru, U p p a l and van Hulle [15] which was, i n 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. T h e endowment processes are taken to be identical, but independent of one another. T h e endowed good is non-storable and can be shipped between countries, but shipping is costly and a portion of the shipment is lost i n transit. This dissipative cost is understood to be a proxy for any impediments to trade, such as time lags and informational costs as well as the physical costs incurred i n m a k i n g shipments. Representative households hold ownership claims to the endowment process i n their respective countries and use these claims to create securities for trading w i t h their peers i n order to obtain claims to foreign consumption goods. In this model, households derive utility 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 representative household belonging to their own country. We assume that this behaviour w i l l be observable regardless of the political faction that is i n power, hence we treat the government of each country as a single infinitely lived agent w i t h a "fixed" objective. In particular, governments w i l l attempt to maximize the utility of representative household of their respective countries accounting for both the utility derived from consumption,  4  Chapter 2.  The Model  5  as well as the u t i l i t y due to government expenditures. T h e government of each country also behaves strategically accounting for the tax policy of their peer government when determining their own o p t i m a l policy. T h e model is setup as a stochastic game between four players: home government, home household, foreign government and foreign household. T h e sequence of events w i t h i n a period of t i m e determines the structure of our game: / . A t the beginning of a time period t, households trade claims on endowments. T h e structure of the securities market is not specified, but it is assumed that a sufficient number of securities exist i n order to provide complete markets. II. E a c h country then receives a random endowment of the consumption good; 9 units t  i n the home country and 9* i n 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 ; each government claims a portion of its own country's endowments. 1  T h e home government sets a tax rate of T and the foreign government imposes a t  rate of r *, where 0 < r < 1 and 0 < r* < 1. t  t  IV. After taxes have been paid, households are left w i t h holdings of perishable goods at home and abroad. In a model w i t h no shipping cost each agent would simply ship home any goods she held abroad, however, i n the face of shipping costs this strategy is no longer optimal. To avoid redundant shipping, households w i l l engage in a second period of trading where consumption goods held abroad are swapped at The 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]. 1  Chapter 2.  The  Model  6  the prevailing exchange rate for local goods and securities. If, after the completion of this trading, an agent still owns goods abroad, then these goods w i l l be i m p o r t e d to the agent's home country. We denote the shipment of goods as exports: x is the t  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. F r o m the above, we see that there is a leader-follower relationship between governments and households. Households, being the last to act i n each period, take government policies as given, and ignore the effects that their behaviour might have on the determination 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. W h e n determining tax policies governments w i l l account for the reactions of households as well as the current foreign tax policy. Thus, it is necessary to solve this problem i n two stages: first the o p t i m a l shipping and consumption plans of households are determ i n e d , then governments set tax rates i n a game based on the solutions to the consumers' problem.  Chapter 3  The Consumers' Optimization Problem  Households derive utility from two sources: consumption and government expenditures. We assume that this utility function w i l l be additive over both consumption and government expenditures as well as over time. Home utility of consumption is given by the function: u : 3 £  — » 9 £ where u(-) is a strictly increasing concave C  2  +  function. Let  v : 3 £ + —> 3 £ represent the home household's utility of government expenditure, the value of public goods to consumers, v(-) is a strictly increasing C function. T h e home house2  hold's expected utility at time t is then given by the function  oo  E [J2^(u(c ) t  + v(g ))],  t  7  t  (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  {g } t  is the government expenditure process.  U t i l i t y for the foreign household is  defined i n an analogous fashion w i t h foreign variables being denoted by an asterisk. E a c h household strives to m a x i m i z e her expected discounted utility, taking government policies as given. Thus households ignore the portion of u t i l i t y due to government expenditures and attempt only to maximize utility due to consumption. C o n s u m p t i o n is financed by the real dividends obtained from the household's portfolio of securities. W i t h complete markets, household agents w i l l strive to reach an o p t i m a l risk sharing arrangement; home agents w i l l hedge against endowment risk by gaining claims to the  7  Chapter 3.  The Consumers'  Optimization  Problem  8  foreign endowment v i a security trading. T h e resulting equilibrium 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 m a x i m i z e the weighted sum of home and foreign household utilities by controlling the distribution of goods between countries. T h e weight assigned to each household is a function of the endowment process for each country as well as the i n i t i a l wealth of each household agent.  For the sake  of simplicity we assume symmetry between our countries: identical but independent endowment processes and equal i n i t i a l wealths. In this case each household receives an equal weighting i n the C P P objective function. In each period, the planner can choose to adjust the balance of consumption goods v i a shipments. These shipments represent the flow of goods due to post endowment trading between households. Let x be the amount t  of goods exported from the home to the foreign country and x* be the amount of goods i m p o r t e d to the home country (foreign exports). Shipping however, is costly; for every unit of goods leaving a country, only ^ Thus only  Y+I *  and only j^x  X  t  units arrive at the destination, where s > 0.  units of exports w i l l be available for the foreign agent's consumption of imports can be consumed at home. T h e n the C P P is  oo  oo  Max { B o E A l ^ l + ^oEWO]}, i=0  (3-2)  t=o  such that c< t  (1  < < (1  - T )9 -xt t  t  <)e;  x > 0, t  +  - x* + x* > 0.  x\ ^  + 3-  Xt  14V  (3.3) (3.4) (3.5)  Observing that goods are non-storable, endowments identically distributed i n each period, and u t i l i t y is time additive, there is no "linkage" between periods i n the C P P . See Chapter 17 of Varian [19] for more details.  1  Chapter 3.  The Consumers'  Optimization  Problem  9  T h e central planner faces the same problem i n 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 i n notation. We define rj and rj* to be the amount of consumption t  goods remaining after taxation, i n the home and foreign countries respectively. Thus we have  Vt  =  (l-r )9 ,  (3.6)  V;  =  (WW.  (3.7)  t  t  Once this substitution has been made, the form of the C P P is identical to the o p t i m i z a t i o n problem found i n [15]. T h e planner's problem is now  Max {x  t  {«(C)+ « « ) } ,  (3.8)  > 0,x* > 0} t  such that <H <  Vt-xt  c* <  >/? " ^ + i f ^ -  t  + T%-, 1+ 6  (3.9)  ( - ) 3  1 0  T h e problem is solved using a standard static optimization approach. T h e Lagrangian is  L (ct,c*,x ,x*) CPP  = u(c )  t  t  +  u(c*) + X (r] - x + -  +  A ^ - x J  Taking derivatives w i t h respect to c ,c*,x t  =  c" dc  dL  t  '(c )-X  u  t  t  t  t  t  t  + s  + ^ - c f ) .  (3.11)  and x* we find:  = 0,  (3.12)  Chapter 3.  The Consumers'  Optimization  Problem  10  =  u'(c?) - A * = 0,  -j*T =  ^ ; - A , < 0 ,  with  ^|f-,=0,  (3..4)  = ^ . - K < 0 ,  with  %F-x -  (3.15)  %  ^ uc  (3.13)  t  t  d  t  = 0.  F r o m (3.12) and (3.13), we see that X = u'(c ) > 0 and A* = u'(c*) > 0 i m p l y i n g t  t  that the constraints on consumption, (3.9) and (3.10), are binding.  These quantities  are the shadow prices of the consumption good i n the home and i n the foreign country respectively. Substituting these results into (3.14) and (3.15), we find that  V-<l + s A  and  v->—^> A 1+ s  4  (3-16)  t  hence < ^ < l . + s.  1  (3.17)  1 + s ~ u'(ct)  T h e 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]. T h e K u h n Tucker conditions also determine the size and the t i m i n g of shipments. F r o m (3.14), we see that — £ ^ < 0 ox  =>x  t  = 0,  (3.18)  t  u'(c*) —V^r < 1 + s u'{ct)  thus  =>  x = 0. t  (3.19)  If the exchange rate is sufficiently low, it w i l l not be o p t i m a l to export goods from the home country.  X t  >0  ^1^  = 0,  (3.20)  Chapter 3. The Consumers' Optimization Problem  thus  •—LiZ w (Q)  =  i  +  11  a  X t  >o.  (3.21)  If the exchange rate rises to its highest possible value, then home goods are cheap relative to foreign goods and it w i l l be o p t i m a l to export goods from the home country. T h e amount of goods shipped must satisfy the equation for the exchange rate  u'(T]  t  -  = 1 + 6.  X)  (3.22)  t  Once (3.22) has been solved, then we have a state contingent policy for exports:  x (r}*, rj ) : t  t  [0,0* ]x[0,6 ]^[0,Vt]t  t  Similar analysis can be performed for the condition i n (3.15) to determine a feedback rule for exports from the foreign country.  and x *>0 t  when  ^ p i= ——,  *  (3.24)  1 + 5'  U'(c ) t  V  7  where x* satisfies: " ' ^ " ^  -  «'(»/*+ rfc *) s  The  (3.25)  1  l+ s  '  solution to (3.25) forms a feedback rule for foreign exports:  x*(t]*,rj ) t  :  [0,0*] x  [o,^][0,17;]. Substituting these shipping policies into the C P P consumption constraints, (3.9) and (3.10), gives the o p t i m a l consumption policies:  CM,*):  [0,0,1 x[O,0 ] t  c*( ;, ):[o,e;)x[o,0 ] v  Vt  t  -  »  +  ,  (3.26)  ->  »  +  .  (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/ and 77* are t  functions of T and r*, see (3.6) and (3.7)). Each correspondence acts as a constraint for t  the optimization problem that each government faces in the policy game that follows.  Chapter 4  Government Policy Game  A s stated earlier, we assume that governments acting i n their own self-interests behave i n a benevolent fashion towards the resident of their own country and thus the objective of each government is to maximize the utility of its constituents. W h e n determining an o p t i m a l policy, each government considers the complete effects that its policies has on its resident's u t i l i t y and thus the objective function for each government includes both components of the household's utility: the portion due to consumption as well as the portion due to government expenditures. T h e objective of the home government is  Max  0 0  ^E/*'(«(c0 {0 <TT < 1}  + 7«(*))],  (4-1)  *=0  such that 9t •< TA = 9 t  V t  ,  (4.2)  c = C( l ), t  (4.3)  V m  where C(r/*, Tj ) is the reaction correspondence for the home household, denned i n (3.26). t  T h e foreign government faces a similar problem. T h e objective function is defined i n the same manner but evaluated at c* and g* and is constrained by the foreign household reaction correspondence C*(rj*,rj ) defined i n (3.26) as well as the foreign government t  budget constraint: g* < 9* — 77*. B y setting r , the home government is indirectly setting rj . T h i s suggests a change i n t  t  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 i n its own country that are available for consumption and shipping. Thus, i n period t, the home government determines a policy rj , where 0 < r) < 9 and the foreign government w i l l set a value for t  t  t  the policy r/*, where 0 <rj* < 9*. A s was the case for the C P P , there is. no linkage between decisions made i n different periods. B i n d i n g government budgets and time separable utility effectively isolate periods such that government policy only effects the current economy. T h e stochastic game, (4.1) - (4.3) is effectively a repeated game; i n each period the governments play the same one shot game w i t h 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 m a x i m i z e the sum: u(c ) + "fv(g ) subject to (4.2) and (4.3). Meanwhile, the foreign government t  t  w i l l be attempting to solve the following problem.  Max  {o < < < 9;}  {ti(c?) + 7 « ( t f ) }  (4.4)  such that 9l  <  Pt-Vl  (4-5)  cl  =  C*(r,;,r, ). t  (4.6)  A d d i t i o n a l l y , there is the issue of what solution concept should be used. Three approaches can be applied to our game: i) Nash Game: In a Nash game governments are on equal footing w i t h one another; each government has access to the same set of information and b o t h governments announce their policies simultaneously. Additionally, 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 w i l l have an incentive to deviate from an equilibrium strategy. A s 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 political or economic factor, is able to act first and impose its tax policy upon its opponent. T h i s 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  iii) Cooperation: Under this regime, governments agree to cooperate when determining tax rates. T h e governments acting as a team, attempt to m a x i m i z e a commonly agreed upon objective function. The game is replaced by a single optimization problem. T h e solution to this problem often generates unstable equilibria; i n certain scenarios one or both governments may find it advantageous to renege on the agreement. 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. T h e Stackelberg concept does not fit well w i t h our model due to the assumed 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 utility, thus for the remainder of this paper, we define A Stackelberg Equilibrium is time inconsistent if the leading player has an incentive to deviate from her selected strategy after it has been announced. 1  Chapter 4.  Government Policy  Game  16  u(c) = log(c) and v(c) = log(c) . 2  Alternatively, 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 rj , 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 — ^g 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. 2  t  2  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 i n the previous two sections w i t h 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 i n Lucas [11]. Due to similar preferences and endowment processes, each household consumes exactly one half of the total endowed goods i n each period. Thus, the o p t i m a l consumption plan for the home agent is  c = t  (5.1)  ^ ± ^ .  2  T h e foreign household has the same o p t i m a l 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 ) + l o g ( 6 / - 7 / ) 4  7  t  (5.2)  t  {0<vt< 0t} such that (5.1) is satisfied.  T h e solution to this problem is the home government's  reaction correspondence, a mapping that describes the home government's o p t i m a l replies for given a foreign government policy value. T h e foreign government faces the analogous problem defined i n 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] — » 17  3£  be the period  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  log(2Lt£)  =  + l g(0 7 O  t  -  rj ). t  (5.3)  T h e reward function for the foreign government is  R*(vhVt) = M ^ ) + 7 logW-»?;). 5  (5.4)  T h e reaction correspondences for both the home and foreign governments are described i n the following l e m m a  Lemma 1 For the policy stage game without shipping costs the Nash reaction correspondence for the home government is  n = < t  1 + 7  . 0  1 + 7  "  *  7  (5.5)  if $ > ^  The foreign Nash reaction correspondence for the same policy game is  Vt  Proof:  1+7  0  (5.6) if  vt>  Please see the A p p e n d i x .  In this economy, trade occurs as a result of imbalances i n the distribution of consumption goods between the two countries. Governments, v i a tax policies, are able to set the amount of goods available for consumption or trade and thus are able to indirectly control the terms of trade. T h e reaction correspondences (5.5) and (5.6) illustrate this fact.  Consider the o p t i m a l policy strategy of the home government. F r o m (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 i n the home country (a domestic tax increase). B y reducing rj , the home government is t  attempting to shift the imbalance of goods i n its own favour. T h e intent is to generate a shortage of goods at home i n order to draw i n goods from abroad. W h e n there are relatively 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. W e observe that the o p t i m a l 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. T h e switch i n the strategy occurs when rj* >  an extreme situation where  the relative distribution of goods is greatly skewed i n favour of the foreign country. T h e motive behind the 7/2 = 0 strategy is essentially the same; w i t h a majority of the goods in the foreign country, the home government chooses to tax away a l l the goods endowed to the home country, forcing the home resident to rely solely on imported goods for consumption. O n the other hand the foreign government realizes that it cannot influence the pattern of trade at this point and resigns itself to the policy 77* = 1  o p t i m a l policy for an autarky  0* r?  . Note that the autarky optimums, r) — t  which is the 0*  and rj* =  form the intercepts of the Nash reaction correspondences for the open economies. These policies act as "benchmarks" when determining an o p t i m a l policy i n the open economy; a best reply is the closed economy o p t i m u m adjusted downwards for the current strategic situation. T h e result is higher tax rates for open economies. To find a Nash E q u i l i b r i u m for the stage game, we must determine the intersection of the two policy rules (5.5) and (5.6). T h e rationale here is that a Nash e q u i l i b r i u m must be self-enforcing, thus it must be o p t i m a l for both parties simultaneously. For The optimum policy for an autarky is derived from the first order conditions of the problem: Max{\og(rj*) + 7 log(0* - %*)}. 1  Chapter 5. Policy Game with No Transportation  Costs  20  Figure 5.1: Nash E q u i l i b r i u m for Policy Stage G a m e w i t h N o Shipping Costs. T h e intersection of the government reaction correspondences determines the equilibrium 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 e q u i l i b r i u m i n pure strategies always exists for this stage game. A n example of a Nash e q u i l i b r i u m i n this stage game is illustrated i n 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 Proof:  the nature of the equilibrium as shown in Table 5.1.  Please see the A p p e n d i x .  B y first solving the consumers' optimization problem and then incorporating this information i n 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 equilibrium may be time inconsistent. V a n 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 i n motion. O u r model does not suffer from time inconsistency for two reasons. F i r s t , w i t h stochastic endowments and discrete time, governments never make any long t e r m policy announcements i n our model. Second, i n the sequence of events that comprise our stage game governments draw taxes from endowments before consumers can act (tax i n advance), hence the government policy decision is necessarily binding. Corollary 1 The Nash tax rates in the policy stage game without shipping functions  of the ratio of endowments,  Proof:  Please see the A p p e n d i x .  costs are  The Nash tax rates are detailed in Table 5.2.  F r o m the o p t i m a l tax rules for the home government i n Table 5.2 we note the following:  Chapter 5. Policy Game with No Transportation Costs  Table 5.1: Nash E q u i l i b r i a i n the Policy Stage Game without Shipping Costs.  Endowment Distribution  Foreign Policy  Home Policy Vt  1+7  Bt <•  7  e; -  1+7  e*  ^  ^  g* t  1+7  7  0  (1+7)0,* ~70t 1+27  (1+7)^-70,*  0  6t 1+7  e -> i+a e* ^ 7 t  1+27  Table 5.2: Nash Tax Rates i n the Policy Stage Game without Shipping Costs.  Endowment Distribution  Foreign Tax Rate  Home Tax Rate  r* t  0t  <  7  ^  1+7  ^  fit  9, ^ 9; ^  >  7 1+7  7 1+7  Of -  1+7 7  1+7. ,  7  l + 2 ^ ~ 0?' 7  1 .  1  i+7  22  Chapter 5. Policy Game with No Transportation  i)  Costs  23  > 0. T h e home government increases its tax rate as the foreign endowment increases. T h e 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. iii)  > 0. A s the relative importance of public goods i n the household u t i l i t y function increases, the home government is given license to set even larger tax rates.  T h e foreign government sets its tax rate using the same set of principles. Here, as well as i n 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. T h i s 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 jointly 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 . T h e problem 2  The 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 objective function (5.7) will result. However, it is not clear that this settlement is renegotiation proof; once 2  Chapter 5. Policy Game with No Transportation  Costs  24  that the cooperative regime faces is then  Max  log(o) + log(0* 7  {0 <  Vt) +  < 8t,0 < n < 6 } t  logCc?) + 7log(0? -  r,*)  (5.7)  t  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 5-4Proof:  Please see the A p p e n d i x .  T h e o p t i m a l policies i n Table 5.3 represent a state contingent trade agreement; government policies are set i n order to optimally distribute endowments between the two countries. Comparing the results i n Table 5.2 to the results i n Table 5.4, we see that the tax rate i n the non-cooperative equilibrium 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 i a a joint optimization process, the resulting equilibrium is Pareto O p t i m a l . T h e agreement is also very altruistic; i n extreme situations one government provides foreign aid by forgoing some of its public expenditures i n 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. ' taxes instead of the Nash amount of M  T h i s 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 E q u i l i b r i a i n the Policy Stage Game without Shipping Costs.  Endowment Distribution  Foreign Policy  < _x_ e; - 2 +  £L  7  7 ^ 2+7 ^  h. ^ 2+7 e* ^ 7 t  0* ^  v* "'  Vt  20* 2+7  0  (2+7)0*- e 2(1+7) 7  h. > 2+2  Home Policy  t  (2+7)0t- 0' 2(1+7) 7  20 2+7  0  7  t  Table 5.4: Cooperative Tax Rates i n the Policy Stage Game without Shipping Costs.  Endowment Distribution  Foreign Tax Rate  Home Tax Rate  7  1  9t < 7 e; - 2+7  7 2+7  ^ ii. ^ ^0*  0t  >  e; -  2+7  2+7 7  2+j_  7  7  2 + 2 7 V-  1  (1+°*) ^  1  2+2  7  V  1  ^  ^ * '  2+T"  '  Chapter 5. Policy Game with No Transportation  Costs  26  exacerbates the already disproportionate good distribution which i n turn creates a larger volume of net goods which flow to the home country. A l t h o u g h governments are allowed to cooperate i n 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 i n turn determine the value of their policies that are implemented i n 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 i n 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 o p t i m a l policy i n the stage game; the home government uses the function (5.5) while the foreign government uses (5.6). T h e stage game becomes a b i - m a t r i x 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 m a t r i x 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 '  J  -  < % < ^- - then V is a Prisoners' 7  9*  1+7  ii) If |jr <  Dilemma.  7  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' D i l e m m a for  < If < 1+7  6*  we must 7  demonstrate the following two facts for each government: i) T h e reward received from a cooperative equilibrium is strictly greater than the reward received from a Nash equilibrium. ii) T h e action "C" is strictly dominated i n rewards by the action " N " . T h e details of the proof are given i n the appendix. Thus we have the familiar story of the Prisoners' D i l e m m a . W h e n both governments decide to cooperate, they both receive a better reward than if they had decided to act independently of one another. B u t cooperation is a strictly dominated action and thus i n the one-shot game the Nash equilibrium, (N,N), is the only rational and stable equil i b r i u m . In the short run each government always prefers to play the non-cooperative action. F r o m the first fact of the proof of Theorem 3 we see that the cooperative e q u i l i b r i u m Pareto dominates the Nash equilibrium. Clearly the government over-spending witnessed i n the N a s h e q u i l i b r i u m is inefficient. T h e source of this inefficiency is a negative tax  Chapter 5.  Policy Game with No Transportation  Costs  28  externality that is not accounted for i n the optimization problems solved by the i n d i v i d u a l governments; increasing the domestic tax rate has the effect of lowering the amount of private goods consumed i n the foreign country.  In this scenario governments always  provide an excessive amount of public goods at the expense of the foreign consumer. T h i s prediction contradicts the results results reported by M i n t z and Tulkens [13] and by de Crombrugghe and Tulkens [4].  B o t h of these works predict that under fiscal  competition, Nash e q u i l i b r i u m tax rates are too low and, as a result, an insufficient amount of public goods are provided at each locale. T h i s discrepancy is due to the nature of the economy studied i n these papers.  In their economy household agents "invest"  labour at different locations i n 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 i n t u r n creates a lack of public goods at each location. In contrast w i t h our model, we see that there is a positive externality i n 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 equilibrium is obtained by having both governments simultaneously raise their tax rates [4], whereas i n 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 i n 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 i n the stage game for an i n d i v i d u a l government can become optimal (self-enforcing and rational) i n  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 . For infinitely repeated discounted games, the theorem states that 3  any feasible, individually rational payoff can be supported as an equilibrium of the game so long as the discount rate is sufficiently high. There are two problems with applying the Folk theorem to our model. F i r s t , the theorem has no predictive power. It does not specify what equilibrium w i l l evolve i n a repeated game, it only details what set of equilibrium payoffs are possible under such strategies.  We have already addressed this problem by assuming that the cooperative  action w i l l be a solution to the optimization problem described i n (5.7). O u r interest i n the theorem is to use it to find a Nash strategy that w i l l result i n cooperative actions being played by both governments at every stage i n 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. A g a i n , it is not the theorem that we wish to apply, only the methodology contained i n 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 government attempts to maximize the sum Max  x G{C,N} 4  0 0  EotE/PUW,*)]  (5.11)  t=o  while the foreign government attempts to maximize the analogous sum defined i n terms of £/*(•,•). Define the cooperative strategy profile to be (x ,x*) = (C,C), Wt > 0. T h e t  cooperative strategy profile can be a rational outcome of r ( 5 ) if each government adopts co  /  a G r i m Trigger strategy. There are two states i n the G r i m Trigger strategy: a cooperative The 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. 3  Chapter 5. Policy Game with No Transportation  Costs  30  state and a punishment state. B o t h governments begin the game i n the cooperative state by selecting "C" as their action i n the first stage of the repeated game. Thereafter, each government remains i n the cooperative state and continues to select the action "C" so long as its opponent has selected "C" i n the previous period. If i n the previous period its opponent h a d 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 a l l subsequent periods of play. T h e 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 o p t i m a l action at each stage for all possible pairs of endowments, thus we adopt a worst case design approach and require "C" to be the o p t i m a l action for the m a x i m u m possible gain from a one-shot deviation. L e t the cumulative probability distribution of the endowments be given by the function F(-) w i t h a m a x i m u m possible endowment value of 8. Define fi  cc  to be the expected reward from  the stage game when both governments cooperate  (5.12) and fi  NN  to be the expected reward from the stage game when both governments select  the Nash action "N". (5.13) Note that these values are the same for both governments. T h e largest possible one-stage deviation gain is then Max  A/7  {(*?,*,)€ (0,0) x(0,*)}  /7(C,N)  -U(C,C).  (5.14)  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 m a x i m u m exists for the above problem exists. Consider a potential deviation at t = T. T h e 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- i*™  >  T  tzzT  A  ^ -  U  (- ) 5  15  t=T  Note that continuation rewards are measured relative to the period i n which the decision is being made. Evaluating the sums i n (5.15) we find 1 (A*cc-M™)> A*7 „, 1-/3  (5.16)  m  hence we require  P > 1~ \ r j  ^  = P-  NN  max  Since the stage game is a Prisoners' D i l e m m a , \i  cc  > fi  (5-17) NN  and thus we have /5 < 1.  So i f (3 satisfies (5.17), then no government w i l l ever choose to deviate from cooperative strategy profile. Note that the punishments used to obtain this level of collusion are extreme. A onetime deviation from the cooperative strategy profile results i n an eternal punishment; it may be too costly for a government to carry out this threat. However, the punishment strategy is the Nash equilibrium of the stage game, thus once the punishment phase has begun, neither government has an incentive to leave i t . T h e 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 equilibrium i n 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 consumption  plans of the home and foreign households are contingent  ernment policy pair (r}*,r)t) and partition  the policy space into three disjoint regions rep-  resenting the direction of trade between the two i) No Shipping Region: ii) Domestic  NS = {(n*, rj )  Export Region:  iii) Foreign Export Region:  t  Proof:  countries:  £ [0,0,*] x [0,0,] : ^  DX = {(r/*,r/ ) £ t  FX = {(r/*,r/ ) £  The optimal shipping and consumption  upon the current gov-  t  <  < 1 + s} .  [0,0*] x [0,0,] :  > 1 + s} .  [0,0*] x [0,0,] : J§- < ^ } .  plans are stated in Table 6.1.  Please see the Appendix. 33  Chapter 6.  Tax Policy Game with Transportation  Costs  34  Table 6.1: O p t i m a l Shipping and Consumption Plans for u(c) = log(c).  State No Shipping  Domestic Export (l+s) V  T) -{l+s)r)*  J  Foreign Export  SL > _L_  >  —  Vi  —  l+s  *)t ^ if  ^  I i+»  2  0  0  *  x  0  0  '?t*-(l+s)»)t 2  Ct  2  t  t  r*  2  j)* + ( l + s)rj  2  2  t  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 i n the distributions of goods between the two countries are always corrected v i a 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 N o Shipping wedge, a region where the costs of shipping outweighs the benefits of additional consumption; imbalances i n 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 rj = 77*, the households w i l l always consume different t  amounts. T h e partitioning of the policy space is illustrated i n Figure 6.1. We define the upper boundary of the N o Shipping wedge to be  8DX  = {(77^) G  [0,0?] x [0,6 ] T  : 77,  = (1 + s)r,;}  (6.1)  and the lower bound as  = {(77;,77O G  OFX  [o,et] x [o,e]: 77, = T j ^ y t f } •  (-) 6 2  t  B y substituting the the shipping rules from L e m m a 2 into the consumption plans, we obtain an o p t i m a l consumption plan for each household as functions of the government policies 77* and 77,. For the household of the home country we find  Ct  —  0t  +  l_  ~—;—x  t  —  Xt  1 + 5"  , i-fc<i±ikr+T  T h e foreign household's consumption plan is given by  ( a ^ f i ) .  (  , ) 3  Chapter 6.  Tax Policy Game with Transportation  Costs  36  V  e  e*  Figure 6.1: P a r t i t i o n i n g of Strategy Space into Shipping Regions. W h e n (?7*,77 ) £ DX there is an excess of home goods and the C e n tr a l Planner reponds by shipping goods from the home country. W h e n (77*, n ) € 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. t  t  Chapter 6.  Tax Policy Game with Transportation  Costs  37  where the function (-) is defined by +  (y )  +  =  y  if y > 0  0  if y < 0  (6.5)  T h e presence of shipping costs T h e functions (6.3) and (6.4) form the constraints for the government optimization problem i n the policy game.  6.2  Nash Correspondences for Stage Game with Transportation Costs  W e 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 l o g ( 0 - r / O  (6.6)  t  {0 < t < 0t] V  such that  ^^(^^y.^^y.  ,  ( 7)  Substituting the constraint (6.7) into the objective function (6.6) results i n the period reward function for the home government.  R{vhVt)  =  log(c )-rjlog(6 -n ) t  t  t  (n;,n )eDX t  {vlvt)eNS ( *,n )eFX V  t  Chapter 6.  Tax Policy Game with Transportation  Costs  38  log(§fo + (1 + s) ;)) + log(^ - rj )  (rj;, 7/0 e DX  <  logfo) + log(0, - Tjt)  (77 *,  1  iog(|(i  v  7  t  7  ? t  +  ^ i  ?  ; ) )  +  7  iog(^-i  t  / t  7/0  £ Atf  (6.8)  fe^)eFi.  )  The functions R (-, •), - R ( " ' •) and R (-, •) are convenient notations for the functional DX  NS  FX  form of the period reward function i n each of the Domestic E x p o r t s , N o Shipping and Foreign E x p o r t s 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  i) R(-,-) is continuous  along the lines dDX and dFX.  ii) R(-,-) is not differentiate iii)  properties:  along the lines dDX and dFX.  i?(-, •) is not concave in n along the line dFX for all n* £  Proof:  t  [0,7/ *]. t  Please see the A p p e n d i x .  Since R(-,-) fails to be concave i n r\ for a l l possible ??*, the standard existence theorem t  for pure strategy Nash equilibria cannot be applied to our m o d e l . A s we shall see, this 1  lack of concavity creates a range of endowment ratios, | f for which no pure strategy Nash E q u i l i b r i a w i l l exist. Our approach is to first use i ? ( v ) to first calculate an o p t i m a l policy strategy for each of the trading regions and then to determine which one of these regional strategies w i l l be o p t i m a l for a given foreign policy 77*. W i t h i n each region we see that R(-,-) is C , 2  concave w i t h respect to T / and bounded for a given 9\ and 6 , thus the task of finding an 4  t  The 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. 1  Chapter 6.  Tax Policy Game with Transportation  Costs  39  o p t i m u m for the interior of a particular region poses no problems. In the following three lemmas we construct an o p t i m a l 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  1+7  1 1+s  for  77* < 77*  for  77* < 77* < n*  for  77* >  B  (6.9)  77*  where 1  (6.10)  l+sl+7 *  +  1B  Proof:  *7  (6.11)  1+  Please see the A p p e n d i x .  A n example of the o p t i m a l strategy for the home government i n the NS region is illustrated i n Figure 6.2.  T h e switching values 77* and 77* are due to the first order  conditions for o p t i m a l i t y 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. T h e o p t i m a l policy value is thus equivalent to the autarky o p t i m u m . 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+7  (  1  +  )rL-V;  s  1+7  for  r,; < 77;,  (6.12)  Chapter  6.  Tax Policy Game with Transportation  Costs  40  e  Optimal Home Policy in NS  Figure 6.2: Nash Strategy for the Home Government i n the N o Shipping Region. T h e graph illustrates the set of best policy replies for the home government when the policy pair is constrained to lie w i t h i n NS.  Chapter 6. Tax Policy Game with Transportation  Costs  41  where 1  (6.13)  1+ s l + 2 ' 7  Proof:  Please see the A p p e n d i x .  Lemma 6 In the policy stage game with shipping costs, the optimal strategy for the home government in the Foreign Export region is ife Vt  1+7i+7" ?* 7  f°  0  V* < Vt < V*  r  D  for  V  B  (6.14)  ; > n*  where (6.15) (6.16)  7  Proof:  Please see the A p p e n d i x .  T h e 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  ti=S  for  TJ  for  VA<rit<  T  <  TJ  A  n  B  (6-17)  Chapter 6.  Tax Policy Game with Transportation  Costs  42  where  (6.18)  (1 + *)  VB  ii) In the Domestic following  (6.19)  +7  T  Export region the foreign government finds it optimal to adopt the  strategy  7  1  Vt =  Vt for I+5I+7  1+7  Vt<Va  (6.20)  where 1 Vc 1 0  6*  (6.21)  l + 5l + 2 " 7  iii) In the Foreign Export region the foreign government finds it optimal  to use the  strategy  - r+ir+^*  fo  n <nt<  r  0  v  D  for  n > t  E  (6.22)  Tj , E  where a*  (1 + -) 1 +' 2 '  (6.23)  7  VE  Proof:  (1 + *)  n  7"  (6.24)  Please see the A p p e n d i x .  T h e final step i n constructing the Nash reaction correspondences is the elimination of dominated strategies.  B y plotting the three locally o p t i m a l 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 o p t i m a l response.  Figures 6.3 and 6.4 illustrate this fact.  In  particular, there are two regions where o p t i m a l 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 i n the Domestic E x p o r t region and a value derived from the strategy i n the N o Shipping region. B y choosing to use the strategy i n the DX region the government adopts a policy value that, for the given foreign policy value, generates domestic exports. Selecting the strategy which lies w i t h i n the NS region amounts to setting a policy which w i l l deter trade. In the NS — DX overlap region we have 77* € (77* ,9*) . For this range of foreign policies the home government must decide 2  between a strategy that generates foreign exports versus one that prevents trade. Notice that there are two possible configurations for these lines; i n Figure 6.3, the NS — FX policy overlap begins at a point to the right of 77* , the "kink" i n the NS strategy whereas i n Figure 6.4, the same overlap region begins to the left of 77*. T h i s difference is due to changes i n 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*. T h i s configuration may also be due to an increased importance of public goods w i t h i n the u t i l i t y function (large 7). Increasing 7 decreases the slope of the o p t i m a l strategy i n the DX region forcing 77*^ left. A N a s h reaction correspondence is the best reply function to the action of opponents. Faced w i t h a choice between two strategies, the home government never selects the strategy 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 i n the NS — DX overlap region, the home government always chooses the strategy that lies i n DX and that i n the NS — FX overlap region, there is a unique value of foreign Assuming of course that 6* > 77* . This needn't be true. If #* is too small than this overlap region simply won't exist. 2  Chapter 6.  Tax Policy Game with Transportation  Costs  44  Local Optimal Strategy (Home Government)  Figure 6.3: O p t i m a l Regional Strategies for the Home Government w i t h 77* > 77*. If 0 < 77* < 77* the home government can select a strategy i n 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: O p t i m a l Regional Strategies for the Home Government w i t h r?* < 77*. T h i s configuration of first order conditions occurs when shipping costs are reduced or by increasing the importance public goods w i t h i n the u t i l i t y function.  Chapter 6.  Tax Policy Game with Transportation  Costs  46  policy value that determines which strategy will be used. Theorem 5 Define /  c( ) = ( l + 7 X 2 ^ - 1 ) .  (6.25)  7  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  1+7  - (1 +  Nash reaction correspondence is  8)^1?  vi e [o,v*)  (1 + (6.26)  6t 1+7  Ot 1+7  ~  1  7  1+s  *  vie  1+7^*  Mill)  0 where **  = (1 + Vt 'II: When (1 + s)  2  s)(2^ -  < ^ y the home government's  Vt=<  (6.27)  m.  Nash reaction correspondence is  (1 + s)r,t Jt 1+7  1 1+ s  (6.28) 7  „*  1+7 It  where rj* = <j>6 and (f> solves t  1 2(1 + 3)^(1 - (1 + s)<t>y = ( — — ) ^ 1+7  7 7  1+7  (1 + 1+5  (6.29)  Chapter 6. Tax Policy Game with Transportation  Proof:  Costs  47  Please see the A p p e n d i x .  Figures 6.5 and 6.6 illustrate the two possible configurations of the Nash reaction correspondence for the home government. A g a i n , the results for the foreign government are derived v i a symmetry and are analogous 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) >  the foreign government's Nash reaction correspondence  2  e*  vt e [o,v )  1+7 "  c  (1 + Vt  vt  e*  e  IVCVA)  Vt € [v ,Vt] A  1+7  1+7 "  is  1 7 1+s1+7^  (6.30)  vt^lvlvl)  0  vt^W Al E  where  Vt =( l + s ) ( 2 ^ 7 -  (6.31)  and K ( ) is defined in (6.25). 7  II: When (1 + s) < 2  the foreign government's  fb-(l Vt = <  + *)TbVt  (l + s)vt J£ 1+7  1  7_  1+s 1+7  where f$ = <t>0 and (j> solves (6.29).  Vt  is  Vt£[0,Vc) Vt£[VaVt]  0 t  Nash reaction correspondence  (6.32)  v*t^[vhv* ) E  v*t£W Al E  Chapter 6.  Tax Policy Game with Transportation  v*o  v*  A  Costs  48  v*  o*  Home Government Reaction Correspondence  Figure 6.5: Home Government Nash Reaction Correspondence when (1 + s) > T h e above graph is the best policy reply correspondence for the home government when either s is large or 7 is small. 2  Chapter  6.  Tax Policy Game with Transportation  Costs  49  Home Government Reaction Correspondence  Figure 6.6: H o m e Government Nash Reaction Correspondence when (1 + s) < T h e above graph is the best policy reply correspondence for the home government when either s is small or 7 is large. 2  Chapter 6.  Tax Policy Game with Transportation  Costs  50  In contrast w i t h 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 o p t i m a l policy rules. W h e n 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*. W h e n the amount of foreign goods is small, the economy is i n a Domestic Export state. A s the amount of available foreign goods increases, the home government responds by decreasing the amount of domestic goods. T h i s action narrows the difference between the supply of goods i n each country and decreases the flow of goods that are leaving the home country. A s 77* continues to rise and n continues to drop. Eventually the difference i n the goods supplies becomes t  so small that domestic exports are no longer viable (the occurs at rj* — ??*.). Once this point has been reached, further increases i n foreign goods are met w i t h an increase i n 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 u n t i l the amount of goods at home reaches the autarky o p t i m u m . Since there is no trade i n NS there is also no strategic interaction; the home government chooses to m a i n t a i n this level of goods, i n 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 rj to a very low level i n order to generate foreign t  exports (this occurs at 77* = 77*). T h e downward j u m p 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*. T h e same type of discontinuity has been  found i n other models of fiscal competition where transportation is costly [4], [13]. T h e i m p l i c a t i o n for our model is that it does not necessarily yield an equilibrium i n pure strategies. F r o m 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 i n the case without shipping costs. E a c h government subscribes to a "Beggar thy Neighbour" strategy; by progressively consuming more and more of its own domestic goods, a government seeks to draw resources from its opponent. T h e major difference occurs i n the NS region. W h e n the amount of goods i n each country are nearly equivalent, both governments prefer autarky. T h i s preference for isolation breaks down one of the countries becomes too "wealthy".  6.3  Nash Equilibria for the Stage Game with Transportation Costs  F i n d i n g the pure strategy Nash E q u i l i b r i a of our game is a relatively straight forward task. Since the reaction correspondence of each government is composed of regional o p t i m a l 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" N o Shipping equilibria from our search. T h e following theorem summarizes these results. Theorem 6 In the policy stage game with shipping costs, the existence and the type of Nash equilibria and  is a function  of the ratio of endowments Jj-. and the model parameters s  7. Let K ( ) be as defined in (6.25) and let 7  m /. For (1 + s) > 2  i) When |t <  = 7 T - K 1 + 2 ) ( 2 ^ - 1) + 7].  1+7  7  there are three possible types of pure strategy Nash ui+)> a  a  unique Nash equilibria exists within the FX  (6.33) equilibria: Region.  Chapter 6.  Tax Policy Game with Transportation  Costs  52  ii) When ( ^ ^ < ff < (1 +5)^(7) a unique Nash equilibrium exists within the 1+  NS iii)  K  Region.  When ff > £( )(l+s), a unique Nash equilibrium exists within the DX 7  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)  there is two possible types of pure strategy Nash  Equilibria:  Jj ^ ^ a unique Nash equilibria exists in the DX  region,  ii) When ff < ^ j ^ V . ^ a unique Nash equilibria exists in the FX  region.  2  <  i) When ff >  1 + 8  where </> solves (6.27). equilibrium  27  If ff does not lie in side of either of these intervals then a Nash  in pure strategies fails to exist. These results are detailed in Table 6.4-  corresponding Proof:  1  The  tax policies are listed in Table 6.5.  Please see the A p p e n d i x .  W e 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 equilibrium for a given endowment pair, and so, if a Nash equilibrium 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  o p t i m a l 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  Table 6.2:  Game with Transportation  Location of Equilibrium  0t ^ 1 7 B* — (l+s) 1+-Y  1 7 (l + s) 1+7  1  (1+S)K(7)  a  „ 6t / ^ 6* -  0  <  (1+^(7)  +  53  Nash E q u i l i b r i a i n the Policy Stage Game w i t h Shipping Costs  Endowment Distribution  ( l  Costs  ) « (  f  "  <  7  t  1  nil  ^  J? < (1 + 6)ic( ) 7  ) < ^ < ( l + ^ (  (1 + ^ ( 7 ) <%<(l  | > ( 1  7  )  + s) -*  + ^)T  Region  F X Region  (l + s)?(7)  1  <  s'  FX  1  Foreign Policy  Home Policy  Vt  m  *?  0  1+7  (l+7)0*-(l+s)7t9f  (1+7)^-^70*  l+2  l+2  7  nil  7  nil  e* 1+7  Bt 1+7  nil  nil  nil  DX Region  (1+T)»t-1T7T * 1+27  (l+7)<?t-(l+s)70? l+2  0  0t 1+7  NS Region  DX Region  9  7  when  Chapter 6. Tax Policy Game with Transportation  Costs  54  Table 6.3: Nash T a x Rates i n the Policy Stage Game Shipping Costs when ( l + s ) > ^ - r . 2  Endowment Distribution  9t ^  (9*  Home Tax Rate  7  1  1 7 (l+s) 1+7  —  1+7  1 7 ^ Bt / 1 (1+5) 1+7 ^ 8; - (l+5)i;(7)  1  Foreign Tax Rate  c  6 t  C  1+^(1+(!+*) (ff))  1  (i+sK( ) ^ e* ^ (i+ )«( ) 7  s  7  l+s G 0 t  nil  1+7  T+7  ^  ni/  7  + s) -*  + * ) ^  +  ni/  (1 + 6 ) / C ( ) < | < ( 1 + ^ ( )  f>(l  l  7  7  < l\ < (1 + *)«(7)  (1 + ^ ( 7 ) <%<{l  l+2 (  1  1+27^  +  1+5  1  G*))  rfe(i + (i + *) (I)" ) 1  7  1+7  Chapter 6.  Tax Policy  Game with Transportation  Table 6.4:  Nash E q u i l i b r i a i n the  Endowment Distribution  55  Costs  P o l i c y Stage Game w i t h Shipping Costs  Location of Equilibrium  Foreign Policy  Home Policy  It  Bt s-  1  1 7 . ^ (1+5)1+7 ^  B sB* -  F X Region  +  < | < ( l  |>(1  1  t  d i ( 7 ) < ** <  (1 + ^(7)  7  (1+5)1+7  B* -  F X Region  (l+5)it(7)  +  s  ^  + 5)lf  + ^)T  nil  DX Region  DX Region  8*  0  1+7  (l+t)6*-(l+s)j6  t  (1+7)01-^70*  I+27  1+27  nil  nil  (1+7)0? 1+27  0  (l+7)0t-(l+s)70*  l+  2  Bt 1+7  7  when  Chapter 6.  Tax Policy Game with Transportation  Costs  Table 6.5:  Nash Tax Rates i n the Policy Stage Game w i t h Shipping Costs when  56  (i + *) <A2  Endowment Distribution  9t  ^  1  i  7  (1+s) 1 + 7  e  i (l+sK(7)  t  +  s  r . f e ( i + ( i + a) ( 1 ) )  + -) ? 1  1 + 2 7 ^  +  i+s { $ ; )  )  nil  ^  < | < ( l + 5)i^  |>(1  1  7  d s K ( 7 ) < l\ < (! +  (1 + ^(7)  7 1+7  ^ e* -  ^  Home Tax Rate  7  e; - (i+ ) i+ s  Foreign Tax Rate  1+27  ^  +  i+s (e*))  1  ^ ( 1 + (! + .) ( I ) " ) 1  7 1+7  Chapter 6.  Tax Policy Game with Transportation  Costs  57  Home Reaction Coorespondence Foreign Reaction Coorespondence  Figure 6.7: A N o Shipping Nash E q u i l i b r i u m i n the Policy Stage G a m e w i t h Shipping Costs. W h e n the reaction correspondences intersect i n the TVS', the resulting equilibrium 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 E q u i l i b r i u m i n the Policy G a m e w i t h Shipping Costs. A pure strategy Nash E q u i l i b r i a 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 E x p o r t E q u i l i b r i a Region: DX  = {(9* ,9 ) e^ :9 >(l 2  t  t  +  t  + *)£( )0?}. 7  ii) A n Upper N o Nash Region: UNN  = {(9* ,9 ) e %l : (1 + 8 ) K ( i ) P <9 <(l t  t  t  t  +  s)t(l)9;}.  iii) A N o Shipping Region: NS  = {WA)  € &  <9 <(l  :  +  t  + a)K(7)*t*} •  iv) A Lower N o Nash Region: LNN  EE {(9l9 ) t  G&  +  :  <9 < t  ^4^}.  v) A Foreign E x p o r t E q u i l i b r i a Region:  FX^{(9;A)en--0t<(^}Figure 6.9 illustrates the endowment space partitioning. W h e n the endowment pair lies in DX the home endowment is large and the resulting equilibria w i l l be characterized by domestic exports. Endowment pairs i n FX indicate a large level of foreign output; during this stage of the game goods w i l l be exported from the foreign country. For (#*, 9 ) G NS t  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 i n the NS region i n the policy space as illustrated i n Figure 6.8. Endowment pairs i n 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 i n 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. T h e w i d t h of these zones of disagreement is determined by re( ) and £(7), both functions of the 7  parameter 7. For sufficiently large 7, K(J) is a decreasing function. A s the importance of public goods to the welfare of the household increases, the angular w i d t h of NS decreases while the angular widths of UNN and LNN increases. where (1 + s)  2  =  Eventually 7 reaches a point  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 N o Nash region: WW = {(6;, 6) G &  +  : (1 + s ) £ ( ) < 0 < 7  scenario corresponds to the structure of Nash equilibria when (1 + s)  T  2  <  • This as seen i n  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 w i t h similar levels of output (eg. Japan and the U . S ) and not i n scenarios when one country has a clear edge i n output (eg. Japan and Canada). T h e presence of trade disagreements is an interesting result, but it is unsatisfactory i n that our model fails to specify the behaviour of governments when an endowment pair lies i n one of the N o Nash E q u i l i b r i u m zones. One method of addressing this short-coming is to extend the class of admissible solutions to include m i x e d strategies. Under a m i x e d strategy each government is allowed to randomize its action i n each stage game over a range of possible policy values. A mixed strategy equilibrium consists of the probability density functions that each government uses to determine its action. Since the policy space i n 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 equilibrium i n mixed strategies w i l l exist for the stage game (Glicksberg's Theorem, see [7]). To determine the o p t i m a l 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: P a r t i t i o n i n g of the Endowment Space i n the Policy game w i t h Shipping Costs when (1 + s) > ^(7). Conflicting government objectives creates trade conflict zones (UNN and LNN) that are sandwiched between the three shipping zones. W i t h i n these conflict zones, governments can not agree on which shipping state should prevail. 2  Chapter 6. Tax Policy Game with Transportation  Costs  62  the m i x e d strategy w i l l assign a probability of one to each of these policy values when the appropriate foreign policy is played. W h e n 77* = 77*, the home government is indifferent between two policy values: 77  NS  and rj , where FX  ft VNS  FX  S  ) * > ^ (6.34)  (1 + 5)77*  V  f o r ( l +  =  for  (l +  5  )2<_^,  = 7 ^ - 7 ^ 7 ^ ( 1 + « •  (6-35)  Since there is no other criteria to guide the home government i n its selection between these two values, we permit a random policy selection. W h e n 77* = 77*, let p £ [0,1] be the probability that the home government chooses the policy 77 ability that the policy 77 ^  NS  and let  1  —  p  be the prob-  is selected. Since a Nash equilibrium 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 o p t i m a l foreign policy reply to the home governments mixed strategy. T h e foreign government maximizes the expected reward  R  = p[log(77*) + l o g ( 0 * - 77;)] + (1 - p ) [ l o g ^  exp  The  7  +  +  + l o g ( 0 * - 77*)]. (6.36) 7  probability p must satisfy the first order conditions for the foreign governments  o p t i m i z a t i o n 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^) *-K e  0  (6.37)  Chapter  6.  Tax Policy Game with Transportation  7(1 + s)(2^ P =  Costs  - 1)2*+^  [l-7(2^-l)][(|r-(l +  We see that p is a function of the state | f .  63  5  (6.38)  ) ( 2 ^ - l ) ]  A s | f decreases p also decreases; as the  ratio of endowments begins to favour the foreign country, it becomes more likely that the home government w i l l pursue a policy that generates foreign exports. A g a i n we see the "Beggar T h y Neighbour" behaviour i n the optimal strategy of each government. T h e m i x e d strategy equilibrium provides us w i t h a solution, but the result is not very valuable. H a v i n g a government determine a national policy by random methods is at best, unrealistic. E v e n less convincing is the fact that the probabilities for a government's m i x e d strategy are selected i n order to stabilize its opponent's o p t i m a l reply. For these reasons we shall discontinue our discussion of the m i x e d strategy e q u i l i b r i u m 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 w i t h one another before the stage game commences.  W i t h pre-game  communication it becomes possible for the governments to coordinate their policies i n order to m a x i m i z e a common objective function. A s i n 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 w i t h 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 ( ;,ri ) c  v  t  = R( ;,r ) v  )t  +  K*( ;, ). v  Vt  (6.39)  Chapter 6.  Here  Tax Policy Game with Transportation  Costs  64  i]t) is defined by (6.8) and R*(n*, r} ) is defined i n similar fashion for the foreign t  government. Solving the first order conditions for this problem w i t h respect to 77* and rj  t  results i n 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 A p p e n d i x .  F r o m Tables 6.6 and 6.7 we see that the cooperative policy solution partitions the endowment space along the same lines as C P P ' s partitioning of the policy space (see L e m m a 2). T h e cooperative solution removes the effects of fiscal competition and restores the "natural" trading boundaries which are determined by an o p t i m a l risk sharing agreement between private agents. Note that the variable 7 plays no role i n determining when shipments w i l l occur. T h e 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 i n a cooperative equilibrium the o p t i m a l tax rates are lower than i n a Nash e q u i l i b r i u m . T h e 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 equilibrium. 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. T h e stage game becomes a b i - m a t r i x 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. A s before, the payoff functions  Chapter 6. Tax Policy Game with Transportation  Costs  65  Table 6.6: Cooperative E q u i l i b r i a i n the Policy Stage Game w i t h Shipping Costs.  Endowment Distribution  ii. < 8* -  1 (1+s)  7  i  f  (1+s)  2+7  ^  2+7 ^  9  ^  t  6*  T+7<|<  ^  1  Location of Equilibrium  1 1+s  Region  FX  Region  2  5  + *)T  Home Policy  Vt  »7t  26*  0  2+7  (2+7)B*-(\+s)i8t  (2+7)0.-1^770'  2(1+7)  2(1+7)  8*  0t  1+7  1+7  DX Region  (2+7)^-^77^  (2+ )0,-(l+s)70*  2(1+7)  2(1+7)  DX Region  0  iVS Region  +^  1+ < | < ( 1 + .) ^  |>(1  FX  Foreign Policy  7  28  t  2+7  Chapter 6. Tax Policy Game with Transportation  Costs  Table 6.7: Cooperative Tax Rates i n the Policy Stage G a m e w i t h Shipping Costs  Endowment Distribution  8t  ^  1  7  ^  e  2+7  ^  6*  t  T+7<|<  l + s < £ < ( l  Home Tax Rate  7  1  7 2+7  e* — (l+s)  i (l + s)  Foreign Tax Rate  1  2+7  „  I  ^  l+s  +  2+2 " 7  5  + )2±2 a  7  >(1 + * ) ^  l+s\9*J  7  7  1+7  1+7  d +  2+27 V  f  ^  1  ^  l + s 0* '  1  2+7  >  Chapter 6.  Tax Policy Game with Transportation  Costs  67  for this game are given by [/(•, •) (5.9) and £/(•, •) (5.10) but w i t h 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' D i l e m m a . We state the result for the case when (1 + s ) > - ^ y , the other case is analogous. 2  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 + ) / c ( ) < f f < 5  7  +  thenT is a  Prisoners'  Dilemma. 2+7" then T is degenerate in that the actions of home government  ii) If f f <  have  no effect on the outcome of the stage game. iii) If f f > (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 / ) ( ) < f f < (1 + +  K  7  governments  Proof:  S)K('J)  then T is degenerate in that the actions of both  have no effect on the outcome of the game.  Please see the A p p e n d i x .  T h e incentive to deviate from a coordinated solution only occurs i n the event of shipping, i n which case, if the game is not degenerate, then it forms a Prisoners' D i l e m m a . Cooperation is self enforcing when there is no trade only because i n this case there is no difference between coordinated and independent actions. T h e standard results for the Prisoners' D i l e m m a apply; for endowment pairs where shipping occur i n the Nash solution, the action "N" strictly dominates "C" i n rewards thus neither government w i l l ever choose to cooperate i n this scenario. Of course this type of behaviour is o p t i m a l only when governments only care about the present stage game. A s we saw previously,  Chapter 6.  Tax Policy Game with Transportation  Costs  68  repetition can lead to cooperation. T h e 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 o p t i m a l . Thus, 3  in the infinite horizon stochastic game w i t h transportation costs, trade disagreements never occur if governments and households are sufficiently patient.  Note that the mixed equilibrium solution is required in order to form expectations about future potential rewards. 3  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 o p t i m a l reaction of the home government to the foreign government policy 77* is:  m = T-r- - rr-vl 1+7  (A-2)  I+7  B u t as 77, > 0, this solution can only be applied if 77* <  For larger values of 77* we  note that:  4>±  7  -  ^<0  (A.3)  0Vt  and hence an o p t i m u m is achieved i n this region by setting 77, = 0. T h e home government reaction correspondence is then:  7  1+7  Vt = < _0  1 + 7 <"  if V* <7 * " t  if 77;  >  ^  ^  T h e foreign reaction correspondence is obtained through symmetry.  69  ^  n  Appendix  A.  Proofs for Chapter 5  70  Proof of Theorem 1 There are three ways i n which the reaction correspondences (5.5) and (5.6) can intersect one another:  ,  i) r] > 0 and n* > 0 t  To solve for the equilibrium policy of the home government we substitute (5.6) into (5.5).  7  0 7-7  t  Vt  =  *  7——Vt  1+7 Ot 1+7 Ot  1+7 7 ; 0;  ' 7 l + 7•XTTT. l + ~ 1+7 TTT.Vth 7 n* , 7 v  7  2  1+7 (1 + 7 ) (1 + ) f l , - fl? 7  l + 2  (1 + 7 )  2  7  2  (A.5)  7  Substituting (A.5) into (5.6) gives us the equilibrium policy of the foreign government:  . (i+7)»r-7ft =  '*  l + 2  ( A v  .  6 ) ;  7  T h e restrictions r/, > 0 and rj^ > 0 i m p l y that this equilibrium w i l l only occur when < f f < ^ - . A n example of this type of equilibria is illustrated i n Figure 5.1. 7  ii) r/i = 0 and n* > 0 Substituting rj = 0 into (5.6) we find rj* = t  |* ^ 1+^-  I  n  T h i s Nash equilibrium occurs when  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 i n the foreign country.  Appendix  iii)  t]  t  A.  Proofs for Chapter 5  71  > 0 and 77* = 0  Substituting 77* = 0 into (5.5) we find: 77, = if  >  ft.  This Nash equilibrium w i l l occur  that is, when the home country is endowed w i t h an excessive amount  of goods relative to the foreign country endowment. B y using the method of Backward Induction Rationality we have verified that this solution is subgame perfect [7]. W i t h a unique Nash equilibrium further equilibrium 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 equilibria are rational . 1  Subgame perfection guarantees that our equilibrium 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. F r o m (5.6) and (5.6) we form the following iterative map:  I \ Vt+i )  \  70?  1+7  (1+7)  1+7  161 (1+7)  \  /  0  (1+7)  +  /  2  .  7  \  (1+7)  2  2  (A.7)  0  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 F r o m Theorem 1 we see that the o p t i m a l government policies are stationary control laws based on the period endowments. T h e 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 f  0  c  1  (i )- (ft)+ 7  0*  1  7  l+2  I  i-L <^ _ 7 _ 0* 1+7  o r 0  for  T1 -+ -7- < 7  7  f0r* <  (A.9)  —  7  I 0  1+7  F r o m (3.6) we see that r = 1 - C (|)  7  t  hence the results i n Table 5.2 follow. T h e foreign  t  tax policy is determined i n the same fashion.  Proof of Theorem 2 T h e period reward function for the cooperative regime is:  R (vhVt) C  =  2 1 o g ( ^ I ) + log(0 7  rj ) + l o g ( 0 * -  t  F r o m the first order conditions W^- = 0 and  t  7  tf).  (A.10)  = 0 we find:  ovt  or)*  (2 + 7)^ + 7^  =  20„  (A.ll)  7r + (2 + 7)r *  =  20,*.  (A.12)  ?t  /t  For 77, > 0 and 77* > 0 the intersection of (A.11) and (A.12) yield the results: (2 + )0 7  Vt  - 0* 7  t  2(1+7) ' (2 + ) 0 * - 0 ,  *  7  Vt  2(1+7)  In (A.13), Vt > 0 implies that | f > -ft this region. For smaller values of  t  w  7  (A.13) (A.14)  and thus (A.13) and (A.14) are only feasible i n e  have  < 0 thus an o p t i m u m is obtained i n  this region by setting 77, = 0. Substituting this result into (A.12) we find the o p t i m a l  Appendix  A. Proofs for Chapter 5  foreign policy to be rj* = are n — t  73  Similarly, for ff- < 2 ± i we find that the o p t i m a l policies  and rj* = 0. Let  (A.15) where f 1  St. < 6* — 2 + 7  o r U  i  (2+7)-7(ftr  0*  2(1+7)  i  U  2+7 ^  i  for ^- >  2  I  6' ^  (A.16)  7  ? + 2  2+7  we see that the o p t i m a l home tax rate is r = 1 — (  c  t  (j^  and thus the results i n Table  5.4 follow. Results for the foreign government are derived by symmetry.  Proof of Theorem 3 F i r s t we consider the game for  < ff <  2+0  i) To show that the cooperative equilibrium Pareto Dominates the non-cooperative solution we must show that U(C,C)  > C/(N, N). T h e home government's reward  in a cooperative equilibrium is  U(C,C)  =  =  log(5*±^L)_ i (^_ ) 7  l 0 g (  2  og  )7t  2(1+7)  (  +  2(1+7) - 7log(0  -  ) }  (2+ t  )fl*-7^ , _ ) 2(1+7)  0  7  /  1  A  7(0*+ 0D  l o g ( J r 7 % ) + 7log( )• '2(1+7)' 2(1+7) : t  V  T h e home government's reward i n a non-cooperative (Nash) equilibrium is  [/(N,N)  =  log(5Lt^L)_ i (^_^) 7  og  (A.17)  Appendix  A.  Proofs for Chapter 5  Define A C ^ . ^  = U(C,C)  74  - <7(N,N).  A ^ c c - , , = (1 + 7 ) l o g ( ^ ^ ) + 7 l o g ( 2 ) . CC-NN  AU  =  0  W  H  E  L  (A.19)  1  (112-)™.? 2^=1 which is true when j - ^ = 0 (i.e. 7  = !  (A.20)  + -^—, 1+7  7 = 0) or  — 1 (i.e.  (A.21)  7 —•>•  00).  Since 2  7  is  concave up w i t h respect to j - ^ we have 7  2 ^ < 1+ - 1 — 1+7  (A.22)  for 0 < 7 < 0 0 . B u t this fact implies  (i^r'.^l which i n t u r n implies that AU _ CC  NN  (A.23)  > 0.  ii) 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 C / ( C , N ) > U(C,C)  R(n*,rjt).  Therefore,  and £ / ( N , N ) > U(N, C) and hence " N " dominates " C " i n  payoffs. Due to the symmetry i n the model, the same results hold for the foreign government. Thus we see that the structure of the payoffs i n T are equivalent to the structure of payoffs i n a Prisoners' D i l e m m a .  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] = 0 for t  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(c ) = log(c ) the first order conditions from the C P P , (3.12) - (3.15) become: t  t  dL  1  cpp  dc  = - ,  A,  c 1  t  dL  CPP  A* = T  dc*  0  = •  i ,  A; =  c  t  =  dx  —  A  ; -  A,  l + s *  t  (B.l)  Ct  t  with  < o ~  =•  < c* ~  (B.2)  t  1 +s  t  -^^-x ox  =  d  t  0,  (B.3)  t  dx*  =  —  A  l+s  with  , -  A: <  *  _  0  =•  ^ > c* ~  l+s  t  dL ^ - x * ux r  =  0.  (B.4)  t  F r o m the K u h n Tucker conditions, (B.3) and (B.4), we see that there are three possible shipping regimes: i) No Shipping:  W h e n there are no shipments x = 0 and x* = 0, thus from (6.3) t  and (6.4) c = n and c* = v*. F r o m (B.3) and (B.4), | t  < 1 + s and g- >  t  ^  hence the N o Shipping region is defined by NS  = ((tf.ifc) € [0,0?] x [O,0 ] : -l i+- s < I t  ii) Domestic Exports:  ^ < 1+ s i . n  J  t  For domestic exports we have x  t  from (6.3) and (6.4) c = n - x and c* = n* + ffc. t  t  t  t  76  > 0 and  F r o m (B.3),  (B.5) = 0, thus = 1 + s, but  Appendix B. Proofs for Chapter 6  77  r\t > c and rj* < c\ thus the E x p o r t region is t  DX = ^(rj*,r, ) G [0,0?] x t  : | > l  M  +  - } .  (B.6)  F r o m (3.22), the amount exported must satisfy  *  ,  j±_  = l + a,  (B.7)  ^*.  (B.8)  hence  x, = " ' iii) Foreign Exports:  (  1  + 2  Foreign exports occur when x = 0 and x* > 0, thus from t  (6.3) and (6.4) c = n + ft and c* = 77* - x*. F r o m (B.4) % = ft, but 77, < c t  t  t  and 77* > c* thus the E x p o r t region is FX = \(rj;,n )  I  G [0,0*] x [0,0,] : ^  t  <  .  1  + 5  (B.9)  J  F r o m (3.25), the amount shipped must satisfy Vt + T+S T]* -  X*  1+  (B.10)  S'  hence  Proof of Lemma 3 i) To see that  R(-,-)  RNS(V*,  is continuous along dDX we note that  V)  \v=(i+»)v*  = tMf) + 7 log(0t =  ^)]  n =  (i  + s ) r )  *  l o g ( ( l + s)ri*) + l o g ( 0 - (1 + 7  t  5)7/*).  (B.12)  Appendix  B.  Proofs for Chapter 6  78  Approaching dDX from w i t h i n the region DX we find lim  R (n*,n) DX  =  l o g ^ f a + (1 +  =  t  7  (B.13)  dDX.  Similarly, for the boundary dFX  which also lies w i t h i n NS  =  lim  7  l o g ( ( l +5)77*)+ l o g ( 0 « - ( l + *>/*)  hence R(-,-) is continuous along  Approaching dFX  + l o g ( 0 - V)  S)TJ*))  log(-^^) + l+s  l o g ( ^ - - ^ ^ ) . l+s  7  (B.14)  from within region FX results i n  R (n*,n) FX  =  [log fa + T^j—»*)) + 7log(0* - v)  l+s thus R(-, •) is continuous along dFX  (B.15)  l+s  as well.  ii) It is easy to see that R(-, •) is not differentiable along either dDX or dFX  as  l +s  dR  T  n + {l +  0tf  (B.16)  s) f  t  V  I l+s  (B.17)  ^+1+7^'  = 0. Thus 2gB2L ^ ^  along  and ^  (B.18) ^  a  i  o  n  g  iii) We need only show one counter example to show that FX.  Let  7?* t  = (1 + s ) ^ , then n = j^rj* t  5i? drjt  = 7  Tj  0 t  F  X  .  •) is not concave ij along t  B u t then we have  1 t  d  rjt  0  (B.19)  Appendix  79  B. Proofs for Chapter 6  and  1  dR  FX  Vt + j^n*  dm  1+7  7  0 -rj t  20,  t  < 0.  (B.20)  Thus R(-, •) is not concave i n 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 i n 77,.  Proof of Lemma 4 Let (77*577,) G NS, then the home reward function is  R (v;,Vt) = log(r ) + \og(8 -r ). NS  !t  1  t  (B.21)  1t  T h e first order conditions w i t h respect to 77, are  NS  _  9R  %  7  1  ?7i  0  t  - Vt  = 0  (B.22)  and therefore the o p t i m u m strategy is  77, = -A-. 1+7  (B.23)  B u t since we are restricted to the NS region employing this strategy implies  1  11  j^r<  ± l+s (j+V) ——— < v; l +sl+7'* V* < V*t A  l+s < i + *, <(i+* ) - ^ , ~ l + 7' < C v  ;  (B.24)  T h e solution (B.23) can only be adopted for foreign policies that lie i n the range 77* G [77*,77*].  E m p l o y i n g this solution outside of this range w i l l force  (77*,77,)  outside of NS.  Appendix B. Proofs for Chapter 6  80  To calculate the o p t i m u m for 77* £ [77*, 77* ] we treat this as a constrained o p t i m i z a t i o n problem and search for solutions along the boundaries of NS. dR 2  NS  dtf  1  7  V?  (0t - Vt)  2  < 0 and 77, < ft implies  thus Vt> ft implies  < 0  (B.25)  > 0. Hence, when 77, > ft it  is o p t i m a l to set 77, as low as possible within NS and when 77, < -ftj the largest possible 77, is o p t i m a l . For 77* > 77;,  (,;,,,)  €  N  S  _  | > _ ± _ -  „ > J i _ .  (B.26)  Therefore when 77* > 77^, the home government sets 77, as small as possible w i t h i n the region  NS,  and so 77, =  ftv*-  S i m i l a r l y for 77* < 77*,  (v*,Vt)eNS  -> ^<l + s »7t  ^<r^-1+7  ( - ) B  27  Thus when 77* < 77* the home government sets 77, as large as possible w i t h i n NS, hence  »* = ( i + "  s)v;.  In summary for (77*, Vt) £ NS the strategy (6.9) is o p t i m a l for the home government.  Proof of Lemma 5 Let (VtiVt) £ DX, then the home reward function is  Rnxiv^Vt) = \og( -(Vt + (1 + s)r,;)) +7log(0, l  T h e first order conditions w i t h respect to 77, are  - Vt)-  (B.28)  Appendix  B. Proofs for Chapter 6  dR  81  1  D X  df]t  7  Vt + (1 +  s)vt  = 0  (B.29)  &t - Vt  and thus the o p t i m u m strategy is  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~ Vt,  (B.32)  s  ^<rT7(i^-(  1  +  -)^  ?  )'  (  (^I^lMI^)'  R  (B  3  -  3  )  34)  T h e policy (B.30) can only be employed when T / * < ??*; pursuing this policy outside of this range forces  (vtiVt) into NS.  To determine the o p t i m a l export strategy for 77* > V*  d R dv! 2  D X  thus  R  DX  1 (m + (i + s)vt)  2  is decreasing i n Vt when  Vt > ^  —  w  e  n  7 ^ - vt) (1 + s)j^v*-  o  2  ^ that e  < 0,  (B.36)  Let (77*, x ) be o n the DX policy line (B.30). x = -^--{l+s)-l-r,;.  1+7  Note that x =  1+7  when 77* = 77*, thus 77* > 77* implies x <  we have ^ > l + s, thus when 77* > 77* it must be that 77, > 77* > 77* we find that  (B.37)  B u t for (77*, 77,) <E Z i Y > x. Thus, when  Appendix  B. Proofs for Chapter 6  82  and the home government w i l l set i] 6 DX as small as possible. A s DX does not contain t  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 N o Shipping policy for this range of r/*, more simply put: the home government does not find it o p t i m a l to set a policy that promotes exports when n* > rj*. m  Proof of Lemma 6 For (77*, rj ) £ FX the reward function is t  Fx  v  log(hn +  =  R ( ;, t) V  t  z  -^-ri;)) 1+ s  + \og(et- t). 7  V  (B.40)  T h e first order conditions with respect to nt are  dR  1  FX  9vt  Vt +  7  ftv*  = 0  Qt - Vt  (B.41)  and thus the o p t i m u m strategy is  Vt  0t  1 7 7——77—f/*l+sl+7  =  1+7  (B.42)  Being restricted to the FX region and enforcing the strategy i n (B.42) implies ^  <  (B.43)  r,; > (1 + s)vt,  (1^-1+11+1"')'  (B.44) (B  '  45)  Appendix  B.  Proofs for Chapter 6  ^  >  (  83  1  +  S  1T27"  )  =  ^-  (  E  U  ?  )  T h e strategy (B.42) is only feasible for 77* > 77* . Pursuing this strategy outside of this range forces (r}*,r)t) into NS.  T h e o p t i m u m for 77* < 77* can be determined using an  argument similar to the one employed i n the DX case w i t h 77* > 77* (see L e m m a 6). T h e results are also similar; when 77* < 77* the home government finds that it is more o p t i m a l to operate i n the N o Shipping region than i n the Foreign E x p o r t region. Thus we see that (6.14) is the o p t i m u m strategy for the home government i n the FX region. •  Proof of Lemma 2 A l l these results follow from the symmetry between the home government's optimizat i o n problem and the foreign government's optimization problem. T h e proof of i) follows the proof i n L e m m a 4 and the proofs for ii) and iii) follow Lemmas 6 and 5 respectively. •  Proof of Theorem 5 We begin our analysis w i t h the NS — DX overlap region.  i) NS - DX Overlap Region Let 77* 6 [0,77*). Since the DX strategy (B.30) is o p t i m a l for (77*, 77,) 6 DX  RDxint^t) |„t=Ji__(i+a)^_„.> R x{v?,Vt) k=(i+*)„ *, D  *  1+7  v  '1+7  t  (B.48)  *  but since R(-,-) is continuous along the boundary dDX we have  R-oxivhVt) \ ,={i+a)n'= RNsivhVt) k = ( i + » K v  (B.49)  Appendix  B. Proofs for Chapter 6  84  Thus  J ?*) I „ 7  t=  et__(i )^_,.> +s  RNsiVtiVt)  (B.50)  \ =(l+ ) * Vt  s  v  and hence the home government always finds it o p t i m a l to pursue a a strategy i n Domestic E x p o r t region for this range of foreign policies. ii) NS - FX Overlap Region  Let  rj*e( * ,e;}. v D  First consider the case when rj* G  Since  (Vg^*].  FX  strategy (6.14) is o p t i m a l for  ^l(r,lr, )eFX t  R (vUvt) FX  "•  \  v  t  =  j L . >  Kxivhrn)  1+7  \vt=j+-v:  ~r  1  s  •  ( - ) B  e  B u t i?(-, •) is continuous so  RpxivhVt)  U = ^ '  =  R (vhVt) NS  \  n  t  =  T  L .  v  ;  (B-52)  and therefore, the home government w i l l always select the the strategy i n the Foreign E x p o r t region (6.9) for this range of foreign policies. To determine the behaviour of the home government i n the range n* € (??*,"*], we introduce the function AR(r]*), the difference between the FX and the NS regional reward functions evaluated at the o p t i m a l regional strategy. T h e home government w i l l switch strategies whenever this function is equal to zero.  51  Appendix  B. Proofs for Chapter 6  85  l+l  1+7  - Ns(vUVt)  \nt={i+'ht  R  ARM)  l + « 1+7Vt  1+7  - Ns(Vt,Vt)  |„ £  R  = T  1+1  l+s  i+i  1+7  ( l + ) l o g ( ^ ^ ) + log( ) 7  T  7  7  -log(2)-log((1 + .)^) +  =  7  tf€(i£,»£)  log ( 0 , - ( 1 + 5 ) 7 7 * )  (l 7)losW + ^ )  , .  +  <  g  (  B  H  )  - l o g (2) - ( 1 + 7 ) l o g (Ot)  (l +7  )log(^5^) +  7  log( ) 7  0?].  - l o g (2) - l o g + iog(0 - ^7,*) 7  t  T  Differentiating w i t h respect to 77? gives  1+7  J_ i _ _ 2 i l ± £ l _  r,* a ( * r,*) n  (l + s)t9 +7)' t  AR'( ;) V  (B.54)  1+7  1+7  ^ + (l +s)7?t-„*  G  ("B'^<]-  Appendix  B.  For  Proofs for Chapter 6  86  77* G (77*,, 77*)  e  1  t  1 + 5 1 + 7  e  t  - (i + s) ; > (i - -J—)* > o,  (B.56)  t  v  1+7  thus 7  (  1  +  >0.  S )  (B.57)  Also  :;>4 (l + s)0, + tf vt 1  +  (B.58)  7  thus 1 + T  1 - : 7,? T > 0  7  ( 1 + 5 ) ^ + 7/*  and so For  AR>(r)t)  > 0 for  77*  G  (B.59)  (77*,, 77*).  77* G ( 7 7 * , T , ; )  A  ^- i X,r (  thus  AR(r)t)  > 0  +  is monotone increasing for  77*  G  {v* ,V* ]i D  '  ( R 6 0 )  D U  B  t  a  t  vt  ^  =  ^  n e  strategy  i n FX is o p t i m a l (see (B.51) and (B.52)) thus AR{r,* )>0  (B.61)  B  T h e strategy i n the  NS region  R s(v*iVt) N  policy is optimal for all  \ =(i+s)r,* Vt  > =  R (VtiVt) NS  Fx(vtiVt)  R  (vtiVt)  G  NS.  \(vt,vt)edFX  For  77* < 77*  (B.62)  \(r,i )edFx, m  but the strategy i n the FX region (6.14) intersects dFX when 77* = 77* and 77* < 77* thus Ai?(77*,)<0.  (B.63)  Appendix  B.  87  Proofs for Chapter 6  The continuity and monotonicity of AR(r,*) on the interval (77*, 77* ] coupled w i t h 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(rj *)  = 0 gives  t  (1 + 7) log (9 + - ^ 7 7 * ) - log (2) - (1 + 7) log (0 ) = 0, t  (B.64)  t  thus 77; = (1 + s){2^  - l)9 .  (B.65)  t  T h i s result is valid only if rj* > 77* " ' -  A  =1-J—-A_ + s 1 +7  v  (B.66)  '  which is equivalent to the condition (1 + * ) > 2  L j (l+ )(2 ^-l)  = -  1 +  7  L <V  (B.67)  W h e n rj* < 77* we arrive at the second possibility. In this case 77* G (ft*^ ?^) solves 7  AR(fj*)  = 0. Setting 77* = (j>6 we find that (j) solves t  2(1  +  3)^(1 - (1 + s)W = ( r - - - ) ^ 7 ( i + 7  1+7  rr-<t>) 1+1  (B.68)  1+ 3  T h i s case is characterized by the fact that (1 + s)  2  < — L j (l+ )(2 ^-l) 1 +  7  = - L . \f)  (B.69)  Note that this can only occur if 77* < 77*. In summary, for (1 + s ) the strategy i n (6.26). correspondence.  2  >  it is o p t i m a l for the home government to adopt  Figure 6 illustrates a specific example of this type of reaction  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  o p t i m a l strategy. A n example of this type of reaction correspondence is illustrated i n Figure 7.  • Proof of Theorem 6 F i r s t we consider the case when (1 + s)  > ^ y . For the DX region, we look for the  2  intersection of the home strategy i n the DX region (6.12) given i n L e m m a 5 and the foreign strategy i n the DX region (6.20). We find  -  (i +1)6; Vt  =  Vt  ^  l + 2  (i  =.  +  1  s l  e  t  B.70  7  )e -(i  +  t  l + 2  ) e;  s 7  B.71  7  w i t h the restrictions rj* < V from (6.12) and Vt > Vt- T h e first restriction together w i t h c  (B.70) i m p l y  (i + 7 ) f t - ijrrg? l + 2  _j  ej_  B.72  l + sl + 2 '  7  7  B.73 T h e second restriction combined w i t h (B.71) gives  (1+7)^(1 + ^  >  ( 1 +  |  >  |  >  r  a  )  (  * _  2  l  W  B.74  i  ^ [ ( l + 2 ) ( 2 ^ - l ) + ] ( l + 3), 7  7  B.75 B.76  £ ( 7 ) ( 1 + *)-  O n l y one of the inequalities (B.73) and (B.76) w i l l be binding. «7)  =  r  ^[(  + 7 ) ( 2 ^ - l ) + 7]  1  1  2  I  _ . „ ^rfc [(l + 2 ) 2 ^ - l 7  7  ]  B.77 B.78  1+7 =  (2 v  —)2 + 7  i +  r  i T T  -1.  B.79  Appendix  B.  Proofs for Chapter 6  89  We see that: £(0) = 1, £(7) —> 1 as 7 —*• co and £(•) has a unique critical 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 equilibrium w i l l occur w i t h i n the DX  region.  T h e procedure for deriving the Nash equilibrium i n the FX  is similar. T h e home  government strategy i n the FX region (6.14) and the foreign strategy i n the FX  region  (6.22) intersect at  *•  =  (  1  +  ^ ; ^  h  * ' .  (B.80)  (1+7)0! - T+-70,*  *" =  1+2,  •  ( B  T h e foreign strategy i n the FX region is only defined for rj > rj which implies ff- < t  c  '  8 1 )  j^.  T h e home strategy i n the FX region is only applicable when 77* > fj* which implies  h < °*  - 1+7  1  =  [(l + 2 ) ( 2 ^ - l ) + 7 ] ( l + s)  ^Xi  I  7  ( B  8 2 )  + s)'  Since £(7) > 1 for 7 > 0, (B.82) and determines the range of endowment ratios for which a pure strategy Nash equilibrium w i l l exist i m n the FX  region.  A pure strategy equilibrium i n the NS region is  =  «f  yqfy  (B.83)  w i t h the l i m i t a t i o n that 77* < 77* < rj* and rj < rj < fj . F r o m rjt < r)* we find that A  I  > (1 + s)(l + ) ( 2 * 1  7  t  7  t  - 1) = (1 + s)/c( ) 7  and from rj < rj we conclude that t  t  Ot  ( l  +  a  ) ( l  +  7  ) ( 2 ^ - l )  (1 +  ^(7)  (B.85)  Appendix  B. Proofs for Chapter 6  90  Thus we see that a pure strategy equilibrium w i l l occur i n the NS region whenever  < £ < ( ! + *M7).  1  (1 +  5  (B.87)  )K( )-0? 7  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 equilibrium for a given endowment pair, and so, if a Nash equilibrium 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. F i n a l l y , for (1 + s)  2  < K(J) we note that neither government utilizes a strategy w i t h i n  the interior of NS, thus, there can never be a pure strategy Nash equilibria that occurs i n the N o Shipping Region. For this set of parameters the Nash equilibria w i l l be either a point i n DX, a point i n FX or non-existent. T h e endowment ratio intervals for these cases are derived i n 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  ( ;, v  R (v:,ri ) c  t  V t  )eDx  = (rj;,rj )eFX t  Appendix  B.  Proofs for Chapter 6  l  o g  91  (2£±i^i)  +  log(fl -, )  7  t  t  + i o g ( ^ ^ ) + iog(0;-r/;) 7  logfat) + 7 log(0, - ft)  ivhVt)  G Atf(B.88)  + log(j/;) + 7 log(«?-»/«*) log(  ) + 7log(0,-7? ) t  ivhVt)eFX.  + l o g ( ^ ± ^ ) + 7log(0;-r;*) dR  c  For (r]*,Tit) G F X , from the first order conditions  = 0 and  ™ r  a  = 0 we have  the equations  (2 + 7X1 + 7(1 + ^  ^  + 77,*  =  2(l +  =  20*.  + (2 + 7)77*  5  )0„  (B.89) (B.90)  T h e intersection of (B.89) and (B.90) is given by (2 + 7)0; _ ( i + ) 7 0 , s  Vt = Vt =  (B.91)  2(1+7)  (2 + 7)0, -  ft e; 7  (B.92)  2(1+7)  Since FX is adjacent to the 77* axis, we must check the constraint 7/* > 0. This constraint along w i t h (B.91) i m p l y that ff < (1 + s) ^ 2  and so the equilibrium described by (B.91)  1  and (B.92) w i l l only exist for this range of endowment ratios. W h e n ff > (1 + s) ^ , 2  o p t i m a l to set if = 0 and thus from (B.90) it must be that 77, — -ft. t  it is  1  For (7/*, r, ) G NS  first order conditions yield the autarky o p t i m u m as expected. T h e case of  t  the  (v*iVt) G DX,  the results are analogous to the results for the Foreign E x p o r t region. To find the equilibrium tax rate we define C ((ff) c  v(0h0t)  '0  t  =  C  a s  (^1-0*.  (B.93)  T h e tax rate is now derived using the formula r, = 1 — £ ; the results i n Table 6.7 follow. c  Appendix  B. Proofs for Chapter 6  92  Proof of Theorem 8 For  < |  <  I  I  T  ^  M  or  (1 + ) « ( ) < | 5  7  < (1 + ) 2 ± 2 the actions " C " and 5  " N " specify different policy levels for both governemnts. T h e 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 7 ( C , C ) + < 7*(C,C), implying that ( C , C ) is not optimal for the joint objective function. 7  7  T h u s , we conclude that for these ranges of endowment ratios, the structure of payoffs i n 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  B a r a r i , M . and L a p a n , H . E . , 1993, Stochastic trade policy w i t h asset markets, Journal of International Economics 35, 317-333. Basar, T . and Olsder, G . J . , 1982, Dynamic demic Press, London).  Noncooperative  Game Theory  Coates D . , 1993, Property tax competition i n a repeated game, Regional and Urban Economics 23, 111-119.  (Aca-  Science  de Crombrugghe, A . and Tulkens, H . , 1990, O n pareto i m p r o v i n g c o m m o d i t y tax changes under fiscal competition, Journal of Public Economics 41, 335-350. Frenkel, J . A . and R a z i n , A . , 1982, Fiscal Policies Edition ( M I T Press, Cambridge).  and the World Economy,  Friedman, J . , 1971, Non-cooperative equilibria for supergames, Review of Studies 38, 1-12.  2nd  Economic  Fudenberg, D . and Tirole, J . , 1991, Game Theory ( M I T Press, Cambridge). Hirsch, M . W . , Smale, S., 1974, Diferential Equations, Algebra (Academic Press, San Diego).  Dynamic Systems and  Linear  Kehoe, P. J . , 1989, Policy cooperation among benevolent governments may be undesirable, Review of Economic Studies 56, 289-296. Lockwood, B . , 1993, C o m m o d i t y tax competition under destination and origin principles, Journal of Public Economics 52, 141-162. Lucas, R . E . , 1982, Interest rates and currency prices i n 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, C o m m o d i t y tax competition between member states of a federation: equilibrium and efficiency, Journal of Public Economics 29, 133-172. [14] R a z i n , A . , Sadka E . , 1991, International fiscal policy coordination and competition, i n International Taxation in an Integrated World ( M I T Press, Cambridge). 93  Bibliography  94  [15] Secru, P . , U p p a l , R . , and V a n Hulle, C , 1995, T h e exchange rate i n the presence of transaction costs, Journal of Finance, forthcoming. [16] V a n D a m m e , E . , 1987, E q u i l i b r i a i n non-cooperative games, i n CWI Tract 39: Surveys in Game Theory and Related Topics (Center for Mathematics and Computer Science, A m s t e r d a m ) . [17] V a n D a m m e , E . , 1991, Stability (Springer-Verlag, Berlin).  and Perfection  the Nash Equilibrium,  2nd ed.  [18] V a n der Ploeg, F . , 1988, International policy coordination i n interdependent monetary economies, Journal of International Economics 25, 1-23. [19] V a r i a n , H . R . , 1992, Microeconomic Company, New Y o r k ) .  Analysis,  3rd Edition  ( W . W . N o r t o n and  [20] W i l d a s i n , D . E . , 1988, Nash equilibria i n models of fiscal competition, Journal of Public Economics 35, 229-240. [21] W i l d a s i n , D . E . , 1991, Some rudimentary 'duopolity' theory, Regional Science Urban Economics 21, 393-421.  and  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0079817/manifest

Comment

Related Items