研究ノート（Note）In Aoki (2007), I claimed that, under a Cournot oligopoly model with international linkage, a long-run
equilibrium exchange rate deviates, into the same direction but in a larger extent, from the purchasing
power parity (PPP) in factor prices than the PPP in product prices does, with the extent of asymmetry
in the degree of competitiveness, consumer’s preference and market volume between countries, or with
the extent of asymmetry in firms’ marginal costs between countries. In this note, I show the detailed
computational process for deriving equilibrium prices and quantities, and the equilibrium exchange rate
under the settings of the model.application/pdfdepartmental bulletin pape

青木　隆明

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Niigata Sangyo University Repository

91新潟産業大学経済学部紀要　第63号AbstractIn Aoki (2007), I claimed that, under a Cournot oligopoly model with international linkage, a long-run equilibrium exchange rate deviates, into the same direction but in a larger extent, from the purchasing power parity (PPP) in factor prices than the PPP in product prices does, with the extent of asymmetry in the degree of competitiveness, consumer’s preference and market volume between countries, or with the extent of asymmetry in firms’ marginal costs between countries. In this note, I show the detailed computational process for deriving equilibrium prices and quantities, and the equilibrium exchange rate under the settings of the model.Keywords: Cournot equilibrium, Exchange rate, Two-country modelJEL classification: F10, F181. Basic modelAs explained in Aoki (2007), we set the following model: There exist two countries (country 1 and 2) and one traded product. There are n identical firms (denoted by x) in country 1, and m identical firms (denoted by y) in country 2. Firm x has a constant marginal cost cx and produces quantity xi for country i (i=1,2). Similarly, firm y has a constant marginal cost cy and produces quantity yi for country i (i=1,2). Country i has an inverse demand function pi=αi-βi qi, where qi=nxi+myi is the quantity supplied in country i, and pi is the real price of the product in country i. n and m represent the degree of competitiveness in country 1 and 2, respectively. The larger αi implies greater consumer’s preference for the product, and the less βi implies larger (inherent) market volume. The current real exchange rate of country 2 in terms of country 1’s real price level is denoted by e. Under the above assumptions, further settings are as follows.1.  Each firm behaves as a Cournot oligopolist, and is allowed to set a differentiated price in each coun-try’s market for each good it produces.2. Each firm takes a current exchange rate as given.3. Each firm exits from the market, if the current Cournot equilibrium makes no profit.4.  A long-run equilibrium exchange rate is determined so that the account of each country’s currency is zero, that is, the demand and supply of the currency meet.5. Every price including exchange rate and cost is represented in real terms, not nominal.6.  We relax the assumption of the law of PPP (Purchasing Power Parity) in product prices, and allow firm’s price differentiation across countries. (That is, we assume that there do not occur reversible A note on the two-country Cournot model  with an exchange rate linkageTakaaki Aoki**  Niigata Sangyo University, Dept. of Economics, Karuigawa 4730, Kashiwazaki, 945-1393, Japan, Phone: +81-257-24-6655   E-mail: taoki@econ.nsu.ac.jp92A note on the two-country Cournot model with an exchange rate linkageand instantaneous flows of products across countries.)7.  The factors in each country are not traded between countries, and the factor price (represented by the real marginal cost) of each firm is exogenously given.8.  We assume the linear demand for each market, allowing to set diversified consumer’s preferences (diversified intercepts), diversified market volumes (diversified slopes) across markets, or diversified degree of competitiveness (diversified number of firms) between countries, or to set diversified mar-ginal costs between firms.In addition, we consider the following two cases: Case A: Each firm has its production line only in its mother country (country 1 for firm x and country 2 for firm y). Case B: Each firm produces the product in each country where the market exists. (That is, firm x pro-duces the quantity xi and firm y produces the quantity  yi in country i.)The long-run equilibrium free exchange rate is determined, so that the demand and supply of the cur-rency meet. Then, in case A and B, the following equations hold:In case A,  ep2 nx2=p1 my1,        (1)1and in case B,  e(p2-cx)nx2=(p1-cy)my1.       (2)Also we define the PPP ratio in factor prices (in terms of country 1) as c̃ ≡ cx/cy, and the PPP ratio in product prices as p̃≡p1/p2. Then the law of PPP can be interpreted as the following relation: p̃=e=c̃         (3)Note that, in equilibrium, generally we may have p̃≠c̃ and e≠c̃ . 2. Derivation and resultsCase AIn country i (i=1,2), the quantity supplied is qi=nxi+myi, and the price of the product is pi. Taking the current exchange rate of country 2 in terms of country 1’s price level, e, as given, we denote the Cournot equilibrium prices and quantities (supplied in and produced for country i (i=1,2)), by pi*, qi*, xi* and yi*.In country 1, in equilibrium, firm x takes other firms’ equilibrium quantities as given, and solves the fol-lowing problem in the country: x1*= arg maxx1 (p1-cx)x1,       (4) where p1=α1-β1 q1 and q1= x1+(n-1)x1*+my1*.The F.O.C. of problem (4) and setting an equilibrium condition, x1= x1*, derive (α1-cx)=β1 {(n+1)nx1*+my1*}.      (5)Similarly, firm y, in country 1, takes other firms’ equilibrium quantities as given, and solves the follow-ing problem in the country: y1*= arg maxy1 (p1-ecy)y1,       (6)  where p1=α1-β1 q1 and q1= y1+nx1*+(m-1)y1*.1　From equation (1) we have de　e =d(p1　p2 )/(p1　p2 )+d(my1　　nx2 )/(my1　　nx2 ).93新潟産業大学経済学部紀要　第63号The F.O.C. of problem (6) and setting an equilibrium condition, y1= y1*, derive (α1-ecy)=β1 {nx1*+(m+1)y1*}.      (7)Then, from (5) and (7), the equilibrium quantity produced by each firm for country 1 is calculated as a function of parameters explaining the market structure, α1, β1, n, m, cx, cy, and a current exchange rate, e, as follows. x1*= 1　　　　　β1(m+n+1) {(m+1)(α1-cx)-m(α1-ecy)}    (8) y1*= 1　　　　　β1(m+n+1) {-n(α1-cx)+(n+1)(α1-ecy)}    (9)Then the equilibrium quantity supplied for country 1 is q1*= nx1*+ my1* = 1　　　　　β1(m+n+1) {n(α1-cx)+m(α1-ecy)},   (10)and the equilibrium market price is p1*= α1- 1　　　　(m+n+1)  {n(α1-cx)+m(α1-ecy)}.     (11)Quite similarly, we obtain in country 2, x2*= 1　　　　　β2(m+n+1) {(m+1)(α2- cx　e )-m(α2-cy)},    (12) y2*= 1　　　　　β2(m+n+1) {-n(α2- cx　e )+(n+1)(α2-cy)},    (13) q2*=nx2*+my2*= 1　　　　　β2(m+n+1) {n(α2- cx　e )+m(α2-cy)},   (14) p2*=α2- 1　　　　(m+n+1)  {n(α2- cx　e )+m(α2-cy)}.     (15) By substituting (9), (11), (12), (15) into y1, p1, x2, p2 in (1), respectively, the equilibrium exchange rate, e*, is derived as e which satisfies the following simplified equation. eβ1 [α2- 1　　　　(m+n+1)  {n(α2- cx　e )+m(α2-cy)}]{(m+1)(α2- cx　e )-m(α2-cy)} =β2 [α1- 1　　　　(m+n+1)  {n(α1-cx)+m(α1-ecy)}]{-n(α1-cx)+(n+1)(α1-ecy)}          (16)Case BQuite similarly as in case A, we obtain, in place of (8) ~ (15), x1*=1　　　　　β1(m+n+1) {(m+1)(α1-cx)-m(α1-cy)},     (8’) y1*=1　　　　　β1(m+n+1) {-n(α1-cx)+(n+1)(α1-cy)},     (9’) q1*=nx1*+my1*=1　　　　　β1(m+n+1) {n(α1-cx)+m(α1-cy)},    (10’) p1*=α1- 1　　　　(m+n+1)  {n(α1-cx)+m(α1-cy)}.     (11’)94A note on the two-country Cournot model with an exchange rate linkage x2*=1　　　　　β2(m+n+1) {(m+1)(α2-cx)-m(α2-cy)},     (12’) y2*=1　　　　　β2(m+n+1) {-n(α2-cx)+(n+1)(α2-cy)},     (13’) q2*=nx2*+my2*=1　　　　　β2(m+n+1) {n(α2-cx)+m(α2-cy)},    (14’) p2*=α2- 1　　　　(m+n+1)  {n(α2-cx)+m(α2-cy)}.     (15’)By substituting (9’), (11’), (12’), (15’) into y1, p1, x2, p2 in (2), respectively, the equilibrium exchange rate, e*, is derived as e which satisfies the following simplified equation. eβ1 {(m+1)(α2-cx)-m(α2-cy)}2=β2 {-n(α1-cx)+(n+1)(α1-cy)}2   (16’) The results in Aoki (2007) can be derived, in (16) and (16’) by treating e as a function of moving pa-rameters and solving the equation.3. Final remarksIn this note, a long-run (sustainable) equilibrium exchange rate was strictly derived, according to the determination mechanism as in Aoki (2007) under an international Cournot oligopoly setting, relaxing one significant assumption of the law of single price for one good. These results prove to show how the larger deviation in equilibrium exchange rate arises, compared with the PPP in product prices, with some asymmetric elements in the international economic system.References Aoki, T. (2007) “Some Remarks on Purchasing Power Parity and Exchange Rate Determination Mechanism,” Applied Eco-nomics Letters, 14, 179-182.Krugman, P. (1980) “Scale economics, product differentiation, and the pattern of trade,” American Economic Review, 70, 950–959Mas-Colell, A., M. D. Whinston, and J. R. Green (1995) “Microeconomic Theory”, The MIT Press.Obstfeld, Maurice and Kenneth Rogoff (1996) “Foundation of International Macroeconomics”, The MIT Press.Taylor, A. M., and M. P. Taylor (2004) “The Purchasing Power Parity Debate”, Journal of Economic Perspective, 18, 135-158. 

A note on the two-country Cournot model with an exchange rate linkage

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A note on the two-country Cournot model with an exchange rate linkage

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