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Consider a Cournot duopoly with two firms 1,2. Let q1, q2 be the quantities produced by firms 1,2. The price p is given by the inverse demand p = 24 ? Q where Q = q1 + q2. The unit cost of firm 1 is c1 = 2 and the unit cost of firm 2 is c2 = 14.

(a) [4 points] Drawing the best response functions in a diagram, identify Nash Equilibrium (NE) quantities of firms 1,2 and find their numerical values. (b) [4 points] Suppose firm 1 is constrained by capacity K = 8, while firm 2 has no capacity constraint. Draw the best response functions and find NE quantities of this capacity constrained duopoly. (c) [4 points] Suppose firm 2 is constrained by capacity K = 3, while firm 1 has no capacity constraint. Draw the best response functions and find NE quantities of this capacity constrained duopoly. (d) [4 points] Suppose both firms 1,2 are capacity constrained. Firm 1 is constrained by capacity K1 = 8 and firm 2 is constrained by capacity K2 = 3. Draw the best response functions and find NE quantities of this capacity-constrained duopoly.

There are two people i = 1, 2 and two alternatives & = a,b. One alternative must be chosen and will affect both people. Each individual's preference is characterized by a parmeter @, that is unifor distributed on the interval [-1, 1]. The , are statistically independent of each other. Thus the density (61, 02) is 4 everywhere on [-1, +1] x [-1, +1]. In addition to selecting the alternative the two individuals may receive (or pay, if negative) a monet transfer ti. An outcome is a triple (k, ti, to). The Bernoulli utilties for the outcomes are given by 1, (k, t. . 0.) = 0, +t, ifa =a = -0 + t, if x = b In addition to these consumers there is a government that can serve as a sink for any net negative trans that might be collected. Each 0, is i's private information. a) What is the Pivotal (or Clarke) mechanism in this example? What outcome does it select as a funct of (0,, 02)? Does it implement this outcome in dominant strategies, or in Bayesian optimal strategies? W. (5) b) Comment on the efficiency of Pivotal mechanism. (3) c) Conditional on observing 6,, what is the expected utility that i would have from playing this mec nism? (5) d) Suppose that after learning ; an individual had the option to cancel the mechanism unilater in which case the government would have no information, would select the alternative at random (50 probability), and would set both {; = 0. Show that there is no value of 0; at which the people would pr to cancel the mechanism in this way instead of going through with it? (3) Now consider a simple Voting Mechanism as follows. After observing 0, each person announces one of two alternatives, a or b. If they announce the same alternative, it is implemented. If their announceme disagree, the government chooses an alternative at random (50-50 probability). No monetary transfers