Question #1:
The table below displays the profits from a price game between Apple and Microsoft.
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Microsoft
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Apple
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Low Price
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High Price
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Low
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(50,50)
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(190,0)
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High
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(0,425)
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(100,150)
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Each firm can set either a high price, pH, or a low price, pL, where pH > pL > 0. Solve the following problems.
a) Write down Apple's best-response function. Also, write down Microsoft's best-response function.
b) Conclude which outcomes of the game constitute a Nash equilibrium.
c) Are there outcomes in this game which Pareto dominate the Nash equilibrium outcomes that you found in the previous problem? Prove your result.
d) Suppose Apple and Microsoft form a cartel. Which outcome maximizes joint industry profit?
Question #2
Firms A and B can choose to adopt a new technology (N) or to adhere to their old technology (O). Formally, firms' action sets are: tA ∈ {N,O} and tB ∈ {N,O}. The table below exhibits the profit made by each firm under different technology choices.
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Firm B
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Firm A
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New Technology
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Old Technology
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New
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(200,0)
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(0,200)
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Old
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(50,100)
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(100,50)
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a) Write down the best-response functions of firms A and B, tA = RA(tB) and tB = RB(tA)
b) Draw the tree of a two-stage extensive-form game in which firm A chooses its technology tA in stage I, and Firm B chooses its tB in stage II (after observing the choice made by firm A). Make sure that you indicate firms' profits at the termination points on the tree. Solve for the subgame-perfect equilibrium of this game. Provide a short proof or an explanation justifying your answer.
c) Draw the tree of a two-stage extensive-form game in which firm B chooses its technology tB in stage I and Firm A chooses its tA in stage II (after observing the choice made by firm B). Solve for the subgame-perfect equilibrium of this game. Provide a short proof or an explanation justifying your answer.
d) Compare the equilibrium firms' profit levels of the games played in (b) and in (c). Conclude under which game firm A earns a higher profit. Briefly explain your answer.
Question #3
Consider the following 2x2 game
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Player 2
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Player 1
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L
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R
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L
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(10,15)
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(0,4)
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R
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(5,0)
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(13,13)
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a) By inspection, what are the pure strategy Nash equilibria?
b) Draw the extensive form game and find the Subgame Perfect Equilibrium.
c) Find the additional mixed strategy equilibrium by using the fact that if a player is willing to mix between two or more strategies, she will be indifferent between them in equilibrium.
d) Draw the best-response correspondences. Where do they intersect?
Question #4
Consider the cost function TC(Q) ≡ F + cQ, where F,c>0.
a) Calculate and plot the TC(Q), AC(Q) and MC(Q).
b) At what output level is the average cost minimized?
c) Infer whether this technology exhibits IRS, CRS, or DRS. Explain.
Question #5
In Seattle there is only one shoe repair factory who acts as a monopoly. The inverse demand function for this service is given by P = 12 - Q/2, where P denotes the price charged per visit, and Q the quantity demanded for shoe repair.
a) Suppose the cost function of this shoe repair factory is given by C(Q) = 4 + 2Q. That is, the marginal cost is c = $2 (consisting of her value time and other "communication" expenses), and the fixed cost is F = $4 (say, monthly rent on her office space). Compute and draw the shoe factory's marginal cost and average functions, as well as the marginal revenue function.
b) Algebraically compute the shoe factory's profit-maximizing output, price, and profit.
c) Compute the price elasticity at the profit-maximizing output.
Question #6
Historically, most of the diamond mines in the world have been controlled by a few companies and governments. Through clever marketing by diamond producers, many consumers have furthermore become convinced that "diamonds are a girl's best friend" because "diamonds are forever." Suppose that the demand for diamond is given by p = 90-x and suppose further that the monopolist's marginal cost curve is given by MC = x.
a) Derive the equation for the marginal revenue curve
b) What is the profit-maximizing output level xM? What is the profit-maximizing price pM (assuming that the monopolist can only charge a single per-unit price to all consumers)?
c) In absence of recurring fixed costs, what is the monopolist's profit?
d) What is the consumer surplus and deadweight loss (assuming that demand is equal to marginal willingness to pay)?
e) What is the cost function if recurring fixed costs are sufficiently high to cause the monopolist's profit to be zero?