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PS18_Solutions

# PS18_Solutions - Intermediate Microeconomics 2008 Problem...

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Intermediate Microeconomics, 2008 Problem Set No 18: Solutions Problems 13.30 At a busy intersection on Route 309 in Quakertown, PN, the convenience store and gasoline station, Wawa, competes with the service and gasoline station, Fred’s Sunoco. In the Bertrand equilibrium with product differentiation competition for gasoline sales, the demand for Wawa’s gas is q W = 680 - 500 p W +400 p S , and the demand for Fred’s gas is q S = 680 - 500 p S +400 p W . Assume that the marginal cost of each gallon of gasoline is m = 2 . 00 . The gasoline retailers simultaneously set their prices. a. What is the Nash equilibrium? Let’s start by looking at Wawa’s problem: max ( p W - m ) q W = ( p W - 2)(680 - 500 p W + 400 p S ) ∂p W = (680 - 500 p W + 400 p S ) · 1 - 500( p W - 2) = 0 500 p W - 1000 = 680 - 500 p W + 400 p S 1000 p W = 1680 + 400 p S p W = 1680 + 400 p S 1000 p S = 1680 + 400 p W 1000 We know p S = p W because the problem is symmetric, and in symmetric games, the payoffs are always equal. So we can plug in for p S in the p W equation to get numeric values. p W = 1680 + 400 p W 1000 1000 p W = 1680 + 400 p W 600 p W = 1680 p W = 2 . 8 p S = 2 . 8 So the Nash equilibrium is( p W , p S ) = (2 . 80 , 2 . 80). b. Suppose that for each gallon of gasoline sold, Wawa earns a profit of \$ 0.25 from its sale of salty snacks to its gasoline customers. Fred sells no products that are related to the consumption of his gasoline. What is the Nash equilibrium? 1

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Now this is no longer a symmetric game. But Fred’s Sunoco’s best response function hasn’t changed, so we know p S = 1680 + 400 p W 1000 We still need to solve for Wawa’s new best response function. Now, instead of having a marginal cost m = \$2 . 00, Wawa’s effective marginal cost is m = 2 - . 25 = 1 . 75, since it earns \$0.25 from every sale of gasoline. So Wawa’s new profit function is π W = ( p W - 1 . 75)(680 - 500 p W + 400 p S ) ∂π W ∂p W = (680 - 500 p W + 400 p S ) · 1 - 500( p W - 1 . 75) 500 p W - 875 = 680 - 500 p W + 400 p S 1000 p W = 1555 + 400 p S p W = 1555 + 400 p S 1000 p W = 1555 + 400 1680+400 p W 1000 1000 1000 p W = 1555 + 2 5 (1680 + 400 p W ) 1000 p W = 1555 + 672 + 160 p W 840 p W = 2227 p W = 2227 840 = 2 . 65 p S = 1680 + 400 · 2 . 65 1000 = 2740 . 48 1000 = 2 . 74 So now, with the introduction of salty snacks, both companies lower their prices to the new Nash equilibrium ( p W , p S ) = (2 . 65 , 2 . 74). 13.38 Two firms, each in a different country, sell homogenous output in a third country. Gov- ernment 1 subsidizes its domestic firm by s per unit. The other government does not react. In the absence of government intervention, the market has a Cournot equilibrium.. Suppose demand is linear, p = 1 - q 1 - q 2 , and each firm’s marginal and average costs of production are constant at m . Government 1 maximizes net national income (it does not care about transfers between the government and the firm, so it maximizes the firm’s profit net of the transfers). Show that Govern- ment 1’s optimal s results in its firm producing the Stackelberg leader quantity and the other firm producing the Stackelberg follower quantity in equilibrium. 2
Given price p = 1 - q 1 - q 2 , firm 1 will earn revenue R 1 = pq 1 = (1 - q 1 - q 2 ) q 1 = q 1 - q 2 1 - q 1 q 2 .

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