3. Assume that two countries, A and B, produce steel for export markets. Steel f

Question

3. Assume that two countries, A and B, produce steel for export markets. Steel from both countries is regarded by world buyers as perfectly substitutable (so that steel from each country has to sell at the same price in the world market). To make the analysis easier, assume that there is only one steel producer in each country. Assume also that the marginal cost of producing a ton of steel in country A is $10 and in country B it is $11.
(a) If the two countries’ producers play a Bertrand game in the export markets, what would be the Bertrand-Nash equilibrium price of steel?
(b) Now assume that the two countries are engaged in Cournot competition in the steel export market. With this assumption, can you calculate the world price of steel if the world (inverse) demand for steel is given by p = 100 – X, where X is the total supply of steel into the world market. [Hint: write down first order conditions for profit maximization for each steel producer. This will give you two equations in two unknowns (which are each country’s steel output). You can then solve for Cournot-Nash equilibrium output supplies.]
(c) Calculate shares of steel exports going to each country in the Cournot-Nash equilibrium. Now assume that country B decides to subsidize its country’s steel export to the amount of $1 per ton. What will be the share of world exports of steel going to country B’s manufacturer? What will happen to country A’s exports of steel? What will happen to world price of steel?

Explanation / Answer

(a) A Bertrand game is a price war. Country A will price at $10.99 and serve the entire market because A has a lower marginal cost. Country B will produce no quantity. (b) Yes, you can calculate the world price. Let's use Y to mean profit, MY to mean marginal profit, A to be the quantity that A produces, and B to be the quantity that B produces. This implies Q=A+B For A: Y=(P-10)A Substitute in the demand function Y=(100 - A - B - 10)A Set marginal profit equal to zero 100-2A-B-10=0 Solve for A A=45-B/2 This is A's best reply function. Let's do the same thing for B. Y=(100 - A - B - 11)B MY=100-A-2B-11=0 B=(89-A)/2 Now, plug B's best response into A's best response. A=45-[(89-A)/2]/2 A=45-(89-A)/4 3A/4=45-89/4 A=(4/3)*[45-89/4] A=30.33 Now, plug this into B's best response function B=(89-A)/2 B=(89-30.33)/2 B=29.34 The Cournot/Nash equilibrium is A=30.33 and B=29.34 This implies that: Q=30.33+29.34 Q=59.67 And since: P=100 - Q P=100 - 59.67 P=40.33 Notice that both firms make a positive economic profit at this price and quantity and even A is better off. (c) We already solved the Cournot/Nash equilibrium. A=30.33 and B=29.34 If B subsidizes steel by $1, then this will change B's marginal cost from $11 to $10. We'll resolve the Cournot equilibrium. A's best response function does not depend on B's costs, so it is the same. A=45-B/2 But now B has a symmetric best response function because B's costs are now identical to A's. You can check this if you like (and practice with these is good), but it is the same math we did above. B=45-A/2 Plug B's best response function into A's best response function. A=45-B/2 A=45-[45-A/2]/2 A=45 - 45/2 + A/4 (3/4)A=45 - 45/2 A=(4/3)[45 - 45/2] A=30 B=30 (you can do the math again if you like) A's share is 30, which is 50% of exports. B's share is 30, which is 50% of exports. A's exports have decreased from 30.33. Now: Q=A+B Q=30+30 Q=60 Plug this into our demand function P=100-Q P=100-60 P=40 This is a lower price than before, which makes sense. If you subsidize something, you should get a lower price.

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