Bilateral Tariffs Under International Competition
BBilateral Tariffs Under InternationalCompetition
T. Kutalia and R. Tevzadze Institute of Cybernetics, 5 Euli str., 0186, Tbilisi, Georgia andGeorgian-American University, 8 Aleksidze Str., Tbilisi 0193, Georgia,(e-mail: [email protected]) Georgian-American University, 8 Aleksidze Str., Tbilisi 0193, Georgia,Georgian Technical Univercity, 77 Kostava str., 0175, Institute ofCybernetics, 5 Euli str., 0186, Tbilisi, Georgia(e-mail: [email protected])
Abstract
This paper explores the gain maximization problem of two nationsengaging in non-cooperative bilateral trade. Probabilistic model of anexchange of commodities under different price systems is considered.Volume of commodities exchanged determines the demand each nationhas over the counter party’s currency. However, each nation can ma-nipulate this quantity by imposing a tariff on imported commodities.As long as the gain from trade is determined by the balance betweenimported and exported commodities, such a scenario results in a twoparty game where Nash equilibrium tariffs are determined for variousforeign currency demand functions and ultimately, the exchange ratebased on optimal tariffs is obtained. : The Nash equilibrium, tariff game, exchange rate1 a r X i v : . [ ec on . T H ] J a n Introduction
Scientists have studied the trade gain maximization problem from differentperspectives. R. Gibbons [1] considered a game model in which total welfareof a country consists of an economic surplus enjoyed by consumers, profitearned by firms within a given country and the tariff revenue collected fromthe imports. Maximization of the total welfare from trade leads to optimaltariff countries involved in trade should impose.In [2], a closed economy model is considered in which the country con-sists of a fixed number of households having preferences as a function ofconsumption and leisure. Within this model, consumption goods consist ofintermediate goods that can be produced by units of labor. Under the closedeconomy model, quantities of each intermediate good and the tariff a givencountry imposes on imports are optimized.Due to different circumstances of production, two nations can producesimilar goods and services at different prices. They can both benefit bygetting involved in international trade to import commodities, which undertheir own price system is of relatively low price than domestically producedcommodities, which under the same price system is of relatively high price.The volume of commodities imported determines one nation’s demand foranother nation’s currency. Balance of demands of two nations for foreigncurrency determines an exchange rate. J. T. Schwartz [5] considered a modelof gain maximization where the commodities produced and the prices forthose commodities are static. In addition, gain from trade is determined tobe the difference between the values of imported and exported commoditiesmeasured in national currency. Since importing those commodities whichcost less under the national price system is regarded as a benefit for bothnations, gain from competitive trade for a given nation is considered to bethe difference between the advantage it took over the competitor and theadvantage the competitor took over it, thus the difference between importsand exports measured at national currency. The Schwartz’s model solves thetariff optimization problem for two nations which are said to be economicallysymmetric, meaning they have equal demands for each other’s currency undera given exchange rate.The novelty of the approach examined in this paper is to make the com-modities and their prices random and solve the gain maximization problemunder Nash’s sense. In addition, non-cooperative bilateral trade model isgeneralized for asymmetric case and a more realistic problem where two na-2ions have different demands for foreign currency is solved.Greatest mutual benefit is achieved when nations cooperate and pursuea free trade policy. Here we assume the non-cooperative game, so they de-termine the optimal tariffs which results in greatest benefit for them underthe Nash’s sense.
Let us assume two nations exchange N different commodities for which thedemand and prices are known. For the domestic and foreign nations, an-nual demand and corresponding prices measured in national currency are d , ..., d N , p , ..., p N and d ∗ , ..., d ∗ N , p ∗ , ..., p ∗ N respectively. If we take x as anexchange rate of a unit of foreign currency in terms of domestic currencyunits, then the domestic and foreign nations’ demand for foreign currencyare given by D ( x ) := 1 C N N (cid:88) k =1 ¯ E (cid:18) p ∗ k d k , p k p ∗ k > x (cid:19) (1)and D ∗ ( x ) := 1 C ∗ N N (cid:88) k =1 ¯ E (cid:18) p k d ∗ k , p k p ∗ k < x (cid:19) (2)respectively, where C N = (cid:80) Nk =1 ¯ E ( p ∗ k d k ) , C ∗ N = (cid:80) Nk =1 ¯ E ( p k d ∗ k ) and ¯ E is themathematical expectation under ¯ P on a probability space ( ¯Ω , ¯ F , ¯Ω). If weintroduce the extended probability space (Ω , F, P ), whereΩ = ¯Ω × { , ..., N } , P ( A, k ) = 1 N ¯ P ( A ) , A ∈ ¯ F and define random variables p, p ∗ , d, d ∗ by p ( ω, k ) = p k ( ω ) , p ∗ ( ω, k ) = p ∗ k ( ω ) ,d ( ω, k ) = d k ( ω ) , d ∗ ( ω, k ) = d ∗ k ( ω ) , then (1),(2) can be rewritten as probability distribution functions D ( x ) = E (cid:18) p ∗ d, pp ∗ > x (cid:19) , D ∗ ( x ) = E (cid:18) pd ∗ , pp ∗ < x (cid:19) . (3)3hich indicate that the domestic nation will import the commodity if pp ∗ > x and the foreign nation will import if pp ∗ < x . Since x is the value of a unitof foreign currency in terms of the domestic currency units, increasing theexchange rate makes foreign commodities more expensive for the domesticnation and the domestic commodities less expensive for the foreign nation.Therefore, D is a decreasing function of x and D ∗ is an increasing functionof x . These functions have the following properties D (0) = 1 , D ( ∞ ) = 0 , D ∗ (0) = 0 , D ∗ ( ∞ ) = 1 . Figure 1: D ( x ) and D ∗ ( x ) Figure 2: D ( x ) and D ∗ (1 /x )For an exchange rate x , solving the equation xD ( x ) = D ∗ ( x ) (4)for x yields the equilibrium rate x = e . This equation determines the equi-librium exchange rate when both nations practice an unrestricted free tradepolicy. Left side of the equation is the foreign currency demand of a domesticnation and the right side is the foreign currency demand of a foreign nation,both measured in domestic currency units.Now suppose the domestic and foreign governments impose the followingtariffs on imported commodities: 1 − θ and 1 − θ ∗ . Then the domestic nationwill import the commodity if pθp ∗ > x , and the foreign nation will import if p ∗ θ ∗ p > x . Taking tariffs into account, the demand functions (3) now become D ( xθ ) = E (cid:0) p ∗ d { θp>xp ∗ } (cid:1) , D ∗ ( xθ ∗ ) = E (cid:0) pd ∗ { θ ∗ p ∗ x>p } (cid:1) . So the relation (4) is rewritten as xD ( xθ ) = D ∗ ( θ ∗ x ) , (5)4rom which it is clear that the equilibrium exchange rate x = e now dependson θ and θ ∗ . Equation (5) always has the solution e = 0 , e = 0, or θ = θ ∗ = 0,which do not carry any useful economic sense. Such conditions would restrictthe involvement of both nations in trade. To rule out these possibilities, weclaim M ≤ e ≤ M , for some large number M and M ≤ θ ≤ , M ≤ θ ∗ ≤ θ, ˆ θ ∗ ). The gain functions of each nation are givenby G ( e, θ, θ ∗ ) = E (cid:18) pd, p ∗ p < θe (cid:19) − E (cid:18) pd ∗ , p ∗ p > eθ ∗ (cid:19) (6)= E (cid:18) pp ∗ ( pp ∗ > eθ ) p ∗ d (cid:19) − E (cid:16) pd ∗ ( pp ∗ > eθ ) (cid:17) = − (cid:90) ∞ e/θ yD (cid:48) ( y ) dy − D ∗ ( θ ∗ e ) , and G ∗ ( e, θ, θ ∗ ) = E ( p ∗ d ∗ , pp ∗ < θ ∗ e ) − E ( p ∗ d, pp ∗ > eθ ) (7)= (cid:90) ∞ θ ∗ e y D ∗(cid:48) (cid:18) y (cid:19) dy − D (cid:16) eθ (cid:17) , respectively. Since the equilibrium exchange rate is the function of tariffs, wehave e = e ( θ, θ ∗ ). Our goal is to find the Nash equilibrium for the nations,i.e. such pair (ˆ θ, ˆ θ ∗ ) that relationsmax θ G ( e ( θ, ˆ θ ∗ ) , θ, ˆ θ ∗ ) = G ( e (ˆ θ, ˆ θ ∗ ) , ˆ θ, ˆ θ ∗ ) , max θ ∗ G ∗ ( e (ˆ θ, θ ∗ ) , θ, ˆ θ ∗ ) = G ∗ ( e (ˆ θ, ˆ θ ∗ ) , ˆ θ, ˆ θ ∗ )hold. The Nash pair is found from the system of equations ∂∂θ G ( e, θ, θ ∗ ) = 0 , (8) ∂∂θ ∗ G ∗ ( e, θ, θ ∗ ) = 0 (9)5iven the currency demand functions D ( x ) and D ∗ ( x ), solution to thesystem of equations (8),(9) leads to yet another system of equations (Ap-pendix A) D ( eθ ) = θ ∗ (1 − θ ) D ∗(cid:48) ( θ ∗ e ) , (10) D ( eθ ) = eθ ( θ ∗ − D (cid:48) ( eθ ) (11) Remark.
According to (5), D ( eθ ) = D ∗ ( θ ∗ e ) e . Then (10) can be rewrittenas D ∗ ( θ ∗ e ) = eθ ∗ (1 − θ ) D ∗(cid:48) ( θ ∗ e ) (12)Denoting ˜ e = e , ˜ D ( x ) = D ∗ ( x ), (12) now becomes˜ D ( ˜ eθ ∗ ) = ˜ eθ ∗ ( θ −
1) ˜ D (cid:48) ( ˜ eθ ∗ ) , which is similar to (11).At this point, if the demand functions for foreign currency of each nationare known, from (10) and (11) the Nash equilibrium pair (ˆ θ, ˆ θ ∗ ) can be found.Ultimately putting these values in (5) and solving for x will result in theequilibrium triple (ˆ e, ˆ θ, ˆ θ ∗ ) = ( e (ˆ θ, ˆ θ ∗ ) , ˆ θ, ˆ θ ∗ ). Hence the triple satisfyˆ eD ( ˆ e ˆ θ ) = D ∗ (ˆ θ ∗ ˆ e ) , (13) D ( ˆ e ˆ θ ) = ˆ θ ∗ (1 − ˆ θ ) D ∗(cid:48) (ˆ θ ∗ ˆ e ) , (14) D ( ˆ e ˆ θ ) = ˆ e ˆ θ (ˆ θ ∗ − D (cid:48) ( ˆ e ˆ θ ) . (15)Obviously, one should check whether the extremum points given by (10)and (11) are really maximums. Differentiating the derivatives of the gainfunctions once again and checking the signs for the equilibrium points servethis purpose. So the following inequalities must hold ∂ ∂θ G (ˆ e, ˆ θ, ˆ θ ∗ ) < , ∂θ ∗ G ∗ (ˆ e, ˆ θ, ˆ θ ∗ ) < θ ∗ (1 − ˆ θ ) e ˆ θ D ∗(cid:48)(cid:48) (ˆ θ ∗ ˆ e ) − ˆ θ ∗ D ∗(cid:48) (ˆ θ ∗ ˆ e ) − e ˆ θ ˆ θ − ˆ e ˆ θ D (cid:48) ( ˆ e ˆ θ ) < , (16)ˆ θ (ˆ θ ∗ e ˆ θ ∗ + ˆ e ) D (cid:48) ( ˆ e ˆ θ ) − (1 − ˆ θ ∗ ) e ˆ θ ∗ ˆ eD (cid:48)(cid:48) ( ˆ e ˆ θ ) > . (17)Hence we can formulate our main result: If pair (ˆ θ, ˆ θ ∗ ) ∈ ( M , is a uniquesolution of (13), (14), (15), (16), (17), then it is the Nash equilibrium of thegame. Demand functions differ from nation to nation. Specifically, two nations aresaid to be economically symmetric if D ( x ) = D ∗ (cid:18) x (cid:19) , (18)which implies that their demand for each other’s currency are equal under anygiven exchange rate. Economically, this means that on average they produceand exchange commodities of equal value. In case of symmetric nations, from(5) we can simply conclude that e ( θ ∗ , θ ) = e ( θ,θ ∗ ) . Thus e ( θ, θ ) = 1, whichmakes perfect sense. Since two nations have equal demand for each other’scurrency, neither is able to employ dominant economic power over the counterparty, so the Nash equilibrium will occur at equal tariffs and a unit exchangerate. More rigorously, since G ∗ ( e ( θ,θ ∗ ) , θ ∗ , θ ) = G ( e ( θ, θ ∗ ) , θ, θ ∗ ), from theresult of Game Theory ([3],p.134) follows that ˆ θ = ˆ θ ∗ for Nash point (ˆ θ, ˆ θ ∗ ).This fact simplifies the computations above. Specifically, taking θ = θ ∗ and e = 1, (11) becomes θD ( 1 θ ) = ( θ − D (cid:48) ( 1 θ ) (19)7iven the function D ( x ), the equilibrium pair (ˆ θ, ˆ θ ∗ ) is found.However, more realistic case is economically asymmetric nations havingdifferent demands for each other’s currency. In this case, the equality (18) nolonger holds. So the nations will have different tariffs imposed on importedcommodities.In [5], the following demand functions for symmetric nations were con-sidered: D ( x ) = D ∗ ( x ) = (1 + x ) − . Since two nations are economicallysymmetric, we have ˆ θ = ˆ θ ∗ , ˆ e = 1. Use (19) to solve the equation for θ .Given D ( x ) = D ∗ ( 1 x ) = (1 + x ) − the derivative of the function is D (cid:48) ( x ) = − x ) putting it into (19) gives θ (1 + θ ) = ( θ − − θ ) )solving for θ yields − θ θ ( θ −
1) = 1 θ = 13 = θ ∗ The solution (ˆ e, ˆ θ, ˆ θ ∗ ) = (1 , , ) agrees with Schwartz’s results. So incase of symmetric nations with the foreign currency demand functions givenby D ( x ) and D ∗ ( x ), we obtained the Nash equilibrium point at equal tariffsto be imposed that maximize the gain for both nations from trade. However,neither is able to tax the competitor by a greater amount than itself beingtaxed by.In addition, we consider two more examples. Symmetric case .Here we consider one more symmetric case. Suppose D ( x ) = D ∗ ( x ) = (1 − αx ) + , α <
1. Similarly applying (19) leads to the following solution. Given D ( x ) = D ∗ ( 1 x ) = (1 − αx ) + D (cid:48) ( x ) = − α putting it in (19) gives θ (1 − αθ ) = ( θ − − α )solving for θ θ − α = α − αθθ = 2 α α = θ ∗ So the Nash equilibrium point is (ˆ e, ˆ θ, ˆ θ ∗ ) = (1 , α α , α α ). Similarly,given any value α <
1, which defines the shapes of the demand functions,the equilibrium point will occur at the same tariffs for both nations.
Asymmetric case .Now we generalize the problem to a more common asymmetric case.Suppose D ( x ) = exp( − δx ) , D ∗ ( x ) = ( αx exp( βx )) ∧
1. Then solving (5) yields theequilibrium exchange rate e = − θ ln( αθ ∗ ) θθ ∗ β + δ The Nash equilibrium condition (12),(13) gives θ = δ ( θ ∗ −
1) ln( αθ ∗ ) − δθ ∗ β and (Appendix B) βθ ∗ ( θ ∗ −
1) = ( θ ∗ β − δ ( θ ∗ −
1) ln( αθ ∗ ) + δ )( θ ∗ − ( θ ∗ −
1) ln( αθ ∗ )) . Specifically, if α = 0 . , β = 2 , δ = 2 . e, ˆ θ, ˆ θ ∗ ) = (0 . , . , . D ( x ) (cid:54) = D ∗ (1 /x ). Solution to the system of equations (10), (11) leads to a domesticnation imposing greater tariff than the foreign nation. Initially, based onlyon the shapes of the functions D ( x ) and D ∗ ( x ), it is impossible to identifywhich nation is ”economically stronger” and therefore will have a greateroptimal tariff. It can be concluded that the non-cooperative trade game results in a problemof optimizing tariffs on imported commodities. Optimal tariffs to be imposedare found at the Nash equilibrium point which are the solutions to the systemof equations (13),(14),(15). The shapes of demand functions determine theeconomic power one nation has over another. However, from the demandfunctions alone, it is impossible to predict which nation will have a greateroptimal tariff to be imposed. It is assumed that the distributions of commodi-ties exchanged and the prices for those commodities are known. Examplesabove are intended to illustrate the typical cases of economically symmetricand asymmetric nations involved in non-cooperative bilateral trade game.Obviously, in real world scenario, the demand functions are not predeter-mined. They are derived from (1) and (2). Finally, as an important note, itmust be stressed that the vast improvement of the probabilistic model is thatit does not restrict itself within the assumption of static prices and demandfor commodities. Availability of imports and the tariffs affect the prices and10emand for commodities. As long as they are made to be random, the modeladequately responds to these changing conditions by considering differentprices and demands on commodities.
Appendix A
The following system of equations determine the tariffs each nation hasto set in order to obtain the maximum gains from trade ∂∂θ G ( e, θ, θ ∗ ) = G θ ( e, θ, θ ∗ ) + G e ( e, θ, θ ∗ ) e θ = 0 , (20) ∂∂θ G ∗ ( e, θ, θ ∗ ) = G ∗ θ ∗ ( e, θ, θ ∗ ) + G ∗ e ( e, θ, θ ∗ ) e θ = 0 , (21)where the index denotes the partial derivative of a given function with respectto a given variable.If we define g ( e, θ ) = eθ and g ( e, θ ∗ ) = θ ∗ e , the individual components of(20) and (21) become G θ ( e, θ, θ ∗ ) = ∂∂g [ − (cid:90) ∞ g yD (cid:48) ( y ) dy ] ∂g∂θ − ∂∂θ D ∗ ( θ ∗ e ) (22)= gD (cid:48) ( g ) g (cid:48) ( θ )= eθ D (cid:48) ( eθ )( − eθ ) = − e θ D (cid:48) ( eθ ) ,G e ( e, θ, θ ∗ ) = ∂∂g [ − (cid:90) ∞ g yD (cid:48) ( y ) dy ] ∂g∂e − ∂∂e D ∗ ( θ ∗ e ) (23)= gD (cid:48) ( g ) g (cid:48) ( e ) − θ ∗ D ∗(cid:48) ( θ ∗ e ) = eθ D (cid:48) ( eθ ) − θ ∗ D ∗(cid:48) ( θ ∗ e ) ,G ∗ θ ∗ ( e, θ, θ ∗ ) = ∂∂g [ (cid:90) ∞ g y D ∗(cid:48) (cid:18) y (cid:19) dy ] ∂g∂θ ∗ − ∂∂θ ∗ D (cid:16) eθ (cid:17) (24)= − θ ∗ eD ∗(cid:48) ( θ ∗ e )( − θ ∗ e ) = 1 θ ∗ D ∗(cid:48) ( θ ∗ e ) , ∗ e ( e, θ, θ ∗ ) = ∂∂g [ (cid:90) ∞ g y D ∗(cid:48) (cid:18) y (cid:19) dy ] ∂g∂e − ∂∂e D (cid:16) eθ (cid:17) (25)= − θ ∗ eD ∗(cid:48) ( θ ∗ e )( − θ ∗ e ) − θ D (cid:48) ( eθ ) = 1 e D ∗(cid:48) ( θ ∗ e ) − θ D (cid:48) ( eθ ) e θ and e θ ∗ can be found from (5) as follows. Let us define F ( e, θ, θ ∗ ) = eD ( eθ ) − D ∗ ( θ ∗ e ) . (26)Differentiating (26) with respect to θ and θ ∗ separately and equating themto zero yields the system of equations F θ ( e, θ, θ ∗ ) + F e ( e, θ, θ ∗ ) e θ = 0 ,F θ ∗ ( e, θ, θ ∗ ) + F e ( e, θ, θ ∗ ) e ∗ θ = 0 , from which solving for e θ and e θ ∗ gives e θ = − F θ ( e, θ, θ ∗ ) F e ( e, θ, θ ∗ ) = e θ D (cid:48) ( eθ ) D ( eθ ) + eθ D (cid:48) ( eθ ) − θ ∗ D ∗(cid:48) ( θ ∗ e ) , (27) e θ ∗ = − F θ ∗ ( e, θ, θ ∗ ) F e ( e, θ, θ ∗ ) = eD ∗(cid:48) ( θ ∗ e ) D ( eθ ) + eθ D (cid:48) ( eθ ) − θ ∗ D ∗(cid:48) ( θ ∗ e ) (28)Putting these solutions into the system of equations (20)(21) yields the fol-lowing results. From (20) we have ∂∂θ G ( e ( θ, θ ∗ ) , θ, θ ∗ ) (29)= − e θ D (cid:48) ( eθ ) + [ eθ D (cid:48) ( eθ ) − θ ∗ D ∗(cid:48) ( θ ∗ e )] (cid:34) e θ D (cid:48) ( eθ ) D ( eθ ) + eθ D (cid:48) ( eθ ) − θ ∗ D ∗(cid:48) ( θ ∗ e ) (cid:35) = e θ D (cid:48) ( eθ ) D ( eθ ) + eθ D (cid:48) ( eθ ) − θ ∗ D ∗(cid:48) ( θ ∗ e ) × (cid:20) eθ D (cid:48) ( eθ ) − θ ∗ D ∗(cid:48) ( θ ∗ e ) − θ (cid:104) D ( eθ ) + eθ D (cid:48) ( eθ ) − θ ∗ D ∗(cid:48) ( θ ∗ e ) (cid:105)(cid:21) = e θ θ (cid:104) eD (cid:48) ( eθ ) − θ θ ∗ D ∗(cid:48) ( θ ∗ e ) − θD ( eθ ) − eD (cid:48) ( eθ ) + θθ ∗ D ∗(cid:48) ( θ ∗ e ) (cid:105) = e θ θ [ − D ( eθ ) + θ ∗ (1 − θ ) D ∗(cid:48) ( θ ∗ e )] = 012nd from (9) we have ∂∂θ ∗ G ∗ ( e ( θ, θ ∗ ) , θ, θ ∗ ) (30)= 1 θ ∗ D ∗(cid:48) ( θ ∗ e ) + [ 1 e D ∗(cid:48) ( θ ∗ e ) − θ D (cid:48) ( eθ )][ eD ∗(cid:48) ( θ ∗ e ) D ( eθ ) + eθ D (cid:48) ( eθ ) − θ ∗ D ∗(cid:48) ( θ ∗ e ) ]= eD ∗(cid:48) ( θ ∗ e ) D ( eθ ) + eθ D (cid:48) ( eθ ) − θ ∗ D ∗(cid:48) ( θ ∗ e ) × (cid:20) eθ ∗ [ D ( eθ ) + eθ D (cid:48) ( eθ ) − θ ∗ D ∗(cid:48) ( θ ∗ e )] + [ 1 e D ∗(cid:48) ( θ ∗ e ) − θ D (cid:48) ( eθ )] (cid:21) = e θ ∗ e (cid:20) θ ∗ D ( eθ ) + eθ ∗ θ D (cid:48) ( eθ ) − D ∗(cid:48) ( θ ∗ e ) + D ∗(cid:48) ( θ ∗ e ) − eθ D (cid:48) ( eθ ) (cid:21) = e θ ∗ θ ∗ e [ D ( eθ ) − eθ ( θ ∗ − D (cid:48) ( eθ )] = 0Hence D ( eθ ) = θ ∗ (1 − θ ) D ∗(cid:48) ( θ ∗ e ) ,D ( eθ ) = eθ ( θ ∗ − D (cid:48) ( eθ ) . For the second derivatives we have ∂ ∂θ G ( e ( θ, θ ∗ ) , θ, θ ∗ )= θe θθ − e θ θ [ θ ∗ (1 − θ ) D ∗(cid:48) ( θ ∗ e ) − D ( eθ )]+ e θ θ (cid:20) θ ∗ (1 − θ ) e θ D ∗(cid:48)(cid:48) ( θ ∗ e ) − θ ∗ D ∗(cid:48) ( θ ∗ e ) − e θ θ − eθ D (cid:48) ( eθ ) (cid:21) , (31) ∂ ∂θ ∗ G ∗ ( e ( θ, θ ∗ ) , θ, θ ∗ )= eθ ∗ e θ ∗ θ ∗ − θ ∗ e θ ∗ − ee θ ∗ ( eθ ∗ ) [ D ( eθ ) − eθ ( θ ∗ − D (cid:48) ( eθ )]+ e θ ∗ eθ ∗ (cid:20) e θ ∗ θ D (cid:48) ( eθ ) − (1 − θ ∗ ) e θ ∗ − eθ D (cid:48) ( eθ ) − (1 − θ ∗ ) e θ ∗ eθ D (cid:48)(cid:48) ( eθ ) (cid:21) = eθ ∗ e θ ∗ θ ∗ − θ ∗ e θ ∗ − ee θ ∗ ( eθ ∗ ) [ D ( eθ ) − eθ ( θ ∗ − D (cid:48) ( eθ )]+ e θ ∗ eθ ∗ (cid:20) θ ∗ e θ ∗ + eθ D (cid:48) ( eθ ) − (1 − θ ∗ ) e θ ∗ eθ D (cid:48)(cid:48) ( eθ ) (cid:21) . (32)13ssuming that the system (10), (11) has a unique solution and xD ( x ) → , as x → ∞ , we get xD ( xθ ) − D ∗ ( θ ∗ x ) > , if x < e,xD ( xθ ) − D ∗ ( θ ∗ x ) = 0 , if x = e,xD ( xθ ) − D ∗ ( θ ∗ x ) < , if x > e. Then ddx (cid:12)(cid:12)(cid:12)(cid:12) x = e ( xD ( xθ ) − D ∗ ( θ ∗ x )) = D ( eθ ) + eθ D (cid:48) ( eθ ) − θ ∗ D ∗(cid:48) ( θ ∗ e ) < e θ > , e θ ∗ <
0. Since first summands of(31),(32) are zeros, the conditions ∂ ∂θ G ( e ( θ, θ ∗ ) , θ, θ ∗ ) < , ∂ ∂θ ∗ G ∗ ( e ( θ, θ ∗ ) , θ, θ ∗ ) < Appendix B
Differentiating the given demand functions D ( x ) = exp( − δx ) , D ∗ ( x ) = αx exp( βx ) (33)gives D (cid:48) ( x ) = − δ exp( − δx ) , D ∗(cid:48) ( x ) = ( αβx + α ) exp( βx ) , (34) D (cid:48)(cid:48) ( x ) = δ exp( δx ) , D ∗(cid:48)(cid:48) ( x ) = ( αβ x + 2 αβ ) exp( βx ) (35)The equilibrium exchange rate is found from (5) as follows e exp( − δ eθ ) = αθ ∗ e exp( βθ ∗ e ) , (36) − δ eθ = ln( αθ ∗ ) + βθ ∗ e,e ( θθ ∗ β + δ ) = − θ ln( αθ ∗ ) ,e = − θ ln( αθ ∗ ) θθ ∗ β + δ . (37)From (11) we find θ − δ eθ ) = eθ ( θ ∗ − − δ exp( − δ eθ )) , eθ (1 − θ ∗ ) δ, θ ∗ − δ ln( αθ ∗ ) θθ ∗ β + δ ,θ = ( θ ∗ − δ ln( αθ ∗ ) − δθ ∗ β . (38)Putting (37),(38) into (10) leads to the solution of θ ∗ . Specifically, redefining(10) in terms of (33) givesexp( − δ eθ ) = θ ∗ (1 − θ ) α exp( βθ ∗ e )( βθ ∗ e + 1) , putting (37) into this equation results in the following expressionexp( δ ln( αθ ∗ ) θθ ∗ β + δ ) = θ ∗ (1 − θ ) α exp( βθ ∗ [ − θ ln( αθ ∗ ) θθ ∗ β + δ ])( βθ ∗ [ − θ ln( αθ ∗ ) θθ ∗ β + δ ] + 1) , replacing θ with its definition from (38) exp( δ ln( αθ ∗ )( θ ∗ − δ ln( αθ ∗ ) − δ + δ )= θ ∗ θ ∗ β − ( θ ∗ − δ ln( αθ ∗ ) + δθ ∗ β α exp( 1 − ( θ ∗ −
1) ln( αθ ∗ ) θ ∗ − − ( θ ∗ −
1) ln( αθ ∗ ) θ ∗ − , eliminating and rearranging some terms gives a simplified equationexp( 1 θ ∗ − θ ∗ β − ( θ ∗ − δ ln( αθ ∗ ) + δβ α exp( 1 − ( θ ∗ −
1) ln( αθ ∗ ) θ ∗ − θ ∗ − ( θ ∗ −
1) ln( αθ ∗ ) θ ∗ − , combining the exponents givesexp( 1 θ ∗ − − − ( θ ∗ −
1) ln( αθ ∗ ) θ ∗ − θ ∗ β − ( θ ∗ − δ ln( αθ ∗ ) + δβ α θ ∗ − ( θ ∗ −
1) ln( αθ ∗ ) θ ∗ − , αθ ∗ = θ ∗ β − ( θ ∗ − δ ln( αθ ∗ ) + δβ α θ ∗ − ( θ ∗ −
1) ln( αθ ∗ ) θ ∗ − , finally, we obtain the equation involving only θ ∗ to solve for βθ ∗ ( θ ∗ −
1) = ( θ ∗ β − ( θ ∗ − δ ln( αθ ∗ ) + δ )( θ ∗ − ( θ ∗ −
1) ln( αθ ∗ )) . (39)This equation cannot be explicitly solved for θ ∗ but it can be computedapproximately. Putting α = 0 . , β = 2 , δ = 2 . θ ∗ = 0 . θ = 0 .