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Oligopoly_sol - COMM/FRE 295...

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COMM/FRE 295 Oligopoly (Cournot/Stackelberg/Betrand) Question #1 Two large farmers, A and B, produce wheat exclusively for a local village in Saskatchewan. They are the only farmers in the local market for wheat. The demand for wheat in the small village is given as follows: P = 1,010 – Q, where Q is the total supply measured in tons and P is the price in dollars per ton. Each farmer’s MC = $10. Assume that FC = 0 for each farmer. a) Derive the reaction function for Farmer A when the two farmers play Cournot-Nash quantity competition game. If Farmer B chooses to increase production by one unit, by how much will Farmer A choose to change its optimal production? Revenues: R A = (1,010 – Q A - Q B )Q A R B = (1,010 – Q A - Q B )Q B Marginal revenues: MR A = 1,010 – 2Q A - Q B MR B = 1,010 – Q A - 2Q B Set MR A = MC A 1,010 – 2Q A - Q B = 10 Q A = 500 – 0.5Q B When B increases his quantity by one unit, A will reduce his quantity by 0.5 units (note mathematically the question is asking dQ A /dQ B = ?). b) Derive the quantities of wheat supplied by each farmer when they decide simultaneously about their quantities using Cournot-Nash quantity competition. Calculate the equilibrium price. By symmetry B’s RF is: Q B = 500 – 0.5Q A (Note if B’s MC is different, you have to solve for RF all over again as done for A) Substitute B’s RF into A’s to get Q A = 333.33. From B’s RF, Q B = 333.33. Q = Q A + Q B = 666.66. Substituting Q = 666.66 in the demand function
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P = $343.34. c) Suppose Farmer A is able to plant his wheat in early spring, which enables him to harvest it in late spring before farmer B plants his wheat. Calculate the new respective quantities supplied and the new market price. Show that farmer A is better off planting and harvesting his wheat before farmer B. Briefly explain why this is the case. Substitute B’s RF into A’s ( leader’s ) revenue function, R A = (1,010 – Q A - Q B )Q A = 1010 Q A - Q A 2 - Q A ( 500 – 0.5Q A ) = (510 Q A – 0.5Q A 2 MR A = 510 – Q A Equating MR A = MC A 510 – Q A = 10 Q A =500. Substituting Q A in B’s RF, Q B = 250 Q = Q A + Q B = 750. Substituting Q in the demand function, P = $260. Farmer A’s profit under Cournot competition is П = P* Q A – MC* Q A ( note FC = 0 ) = (343.34 - 10) * 333.33 = $111,112.22 Under Stackelberg competition, where A moves first, A’s profit is П = P* Q A – MC* Q A = (260 - 10) * 500 = $125,000 A’s profit under Stackelberg competition is higher than under Cournout competition as a result of the first mover advantage. d) Suppose now that both farmers compete using prices rather than quantities. Calculate the market price for wheat and the share of the market supplied by each firm if both farmers decide about their prices simultaneously. How would your answer change if Farmer A received as a gift a new harvester that lowered his marginal cost from 10 down to 5?
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