Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

Econ6025 Problem Set (due on Feb 24 - 4pm)

1. Suppose a seller wants to sell his good to a buyer by optimally pricing it.  Suppose the seller’s utility function is given by t − q and the buyer’s utility function is given by θq1/3  − t where q  ≥ 0 is the amount of good sold to the buyer in exchange of t ≥ 0 amount of transfer made to the seller.  Suppose the seller does not observe the buyer’s taste parameter θ, but it is common knowledge that this parameter can be either θ¯ = 3 with ν = 1/4 probability or θ = 1 with 3/4 probability. Suppose that the seller offers a menu of contracts (i.e. quantity-transfer pairs) after the buyer observes her type.

(a) Write down the seller’s optimization problem by clearly stating the objective func- tion and the constraints. [10 pts]

(b) Simplify this problem by identifying binding and non-binding constraints, and mak- ing necessary substitutions. [10 pts]

(c) Solve the simplified problem; that is, find an optimal contract as a quantity-transfer pair offered for each type of the buyer. [10 pts]

(d) Calculate the expected utility of the seller. Find each type of buyer’s information rent. [10 pts]

(e) Find the minimum value of ν such that the shutdown policy becomes optimal given all other parameters. [10 pts]

2. Suppose a medical company supplies vaccine to a benevolent planner.  Suppose with ν =  probability the company is efficient such that its efficiency parameter is θ = 1 and with  probability it is inefficient such that its efficiency parameter is θ = 2.  Suppose the planner can use an audit technology at a cost c(p) = p2  in order to verify the true state (i.e., the company’s efficiency parameter) with p ∈ [0, 1] probability of detection if the company is misreporting. Suppose also that, as part of its audit mechanism, the planner can charge the company up to l = 1 amount of payoff as a punishment P ≥ 0 whenever the company is caught lying about its true parameter θ ∈ {1, 2}.  Suppose, when the quantity produced is q  ≥ 0, the transfer made is t ≥ 0, the accuracy level is p ∈ [0, 1], and the induced punishment is P ≥ 0, the company’s profit is given by t − θq − pP whenever the company is misreporting, and t − θq whenever it is not, while the planner’s surplus is given by 2ln(q) − t − p2 . Finally, suppose that the seller offers a menu of contracts after the company observes its type θ .

(a) Write down the planner’s optimization problem by clearly stating the objective function and the constraints. [10 pts]

(b) Simplify this problem by noting binding and non-binding constraints, and making necessary substitutions. [10 pts]

(c) Find the efficient type company’s intended contract (U , q , p, P) as part of the opti- mal audit mechanism. [10 pts]

(d) Find the inefficient type company’s intended contract (  , , p¯, ) as part of the optimal audit mechanism. [10 pts]

(e) Find each type of company’s optimal transfer and the planner’s expected payoff. [10 pts]