Mechanism Design and Social Choice Part II:
Problem Set 3
Problem 1 Consider a quasi-linear environment with one agent:I = 1.The type-space is discrete Θ = {θ1, θ2, . . . , θN} where θ1 ≤ θ2 ≤ . . . ≤ θN. The set of alternatives is X= [0,1]×R with elements (q, t). The payoff of the agent with typeθ when alternative (q, t) is chosen, is given by u((q, t), θ) = θv(q) +t. Assume that v is twice continuously differentiable withv(0) = 0,v0(q)>0 andv00(q)<0 for allq∈[0,1]. For a given allocation ruleq : Θ→[0,1], let ¯v(θ) =v(q(θ)).
1. Let (q(θ), t(θ)) be an incentive compatible social choice function. Show that ¯v is non-decreasing inθ.
2. Show that all incentive compatibility constraints are satisfied if the followinglocal constraints are satisfied for all k= 1, . . . , N−1:
U(θk, θk+1) = ¯v(θk+1)θk+ ¯t(θk+1)≤U(θk), (IC(k,k+1)) U(θk+1, θk) = ¯v(θk)θk+1+ ¯t(θk)≤U(θk+1). (IC(k+1,k)) 3. Show that IC(k-1,k) is fulfilled if IC(k,k-1) holds with equality and that IC(k,k-1) is
fulfilled if IC(k-1,k) holds with equality
Now suppose thatqis the quantity of a good that is produced at marginal costcby a firm. The firm offers different quantities of the good at different prices. The agent is a consumer that can choose any of these options. In other words, the consumer faces a menu of quantity-price pairs (q1, t1),(q2, t2), . . . . Given her type θ,the consumer chooses the quantity that maximizes her utility. By the revelation principle, the firm can restrict attention to an incentive compatible menu of options (q(θ), t(θ))θ∈Θ. (Incentive compatibility here means that for a consumer of type θ it is optimal to choose (q(θ), t(θ)).)
4. Suppose that the firm has already fixed the quantities q(θ1) ≤ q(θ2) ≤ . . .. Derive a formula for U(θ) if the firm sets t(θ1) = −θ1v(θ¯ 1) and chooses t(θk) for k > 1 such that the profit of the firm is maximized subject to incentive compatibility of (q(θ), t(θ))θ∈Θ. (Hint: The formula is a discrete version of condition (ii) in the characterization of Bayes-Nash incentive compatible social choice functions).
5. Now consider the firm’s optimal choice ofq.LetN = 2,φ(θ1) =β, andφ(θ2) = 1−β.
Derive the optimal menu (q(θ1), t(θ1)),(q(θ2), t(θ2)) that maximizes the expected profit of the firm! Which of the quantity choicesq(θ1) andq(θ2) is efficient? Interpret!
(Use the result of part 3 in your interpretation!)
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