A firm is considering building a public good (e.g. a swimming pool). There are n agents...

 

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A firm is considering building a public good (e.g. a swimming pool). There are n agents in the econ- omy, each with IID private value 0; € [0, 1]. Agents' valuations have density f(0) and distribution F(0₁). Assume that 1 - F(0₂) f(0₁) is increasing in ₁. The cost of the swimming pool is cn, where c > 0. MR(0₁) = 0; - First suppose the government passes a law that says the firm cannot exclude people from entering the swimming pool. A mechanism thus consists of a build decision p(0₁,...,0n) [0, 1] and a payment by each agent ti(0₁,...,n) R. The mechanism must be individually rational and incentive compatible. [Note: When showing familiar results your derivation can be heuristic.] (a) Consider an agent with type 0₁, whose utility is given by 0ip - ti Derive her utility in a Bayesian incentive compatible mechanism (p, t;). (b) Given a build decision p(), derive the firm's profits. (c) What is the firm's profit-maximizing build decision? (d) Show that E[MR(0₂)] = 0. (e) Show that as n →∞, the probability of provision goes to zero.² Next, suppose the firm can exclude agents. A mechanism now consists of a build decision p(01,...,0n) € [0, 1], a participation decision for each agent x₁(01,...,n) € [0, 1] and a payment ti(01,..., On) ≤ R. Agent i's utility is now given by 0₁x₂P - ti The cost is still given by cn, where n is the number of agents in the population. (f) Solve for the firm's optimal build decision p() and participation rule x₂(). (g) Suppose n→∞. Show there exists a cutoff c* such that the firm provides the pool with probability one if c < c*, and with probability zero if c> c*. A firm is considering building a public good (e.g. a swimming pool). There are n agents in the econ- omy, each with IID private value 0; € [0, 1]. Agents' valuations have density f(0) and distribution F(0₁). Assume that 1 - F(0₂) f(0₁) is increasing in ₁. The cost of the swimming pool is cn, where c > 0. MR(0₁) = 0; - First suppose the government passes a law that says the firm cannot exclude people from entering the swimming pool. A mechanism thus consists of a build decision p(0₁,...,0n) [0, 1] and a payment by each agent ti(0₁,...,n) R. The mechanism must be individually rational and incentive compatible. [Note: When showing familiar results your derivation can be heuristic.] (a) Consider an agent with type 0₁, whose utility is given by 0ip - ti Derive her utility in a Bayesian incentive compatible mechanism (p, t;). (b) Given a build decision p(), derive the firm's profits. (c) What is the firm's profit-maximizing build decision? (d) Show that E[MR(0₂)] = 0. (e) Show that as n →∞, the probability of provision goes to zero.² Next, suppose the firm can exclude agents. A mechanism now consists of a build decision p(01,...,0n) € [0, 1], a participation decision for each agent x₁(01,...,n) € [0, 1] and a payment ti(01,..., On) ≤ R. Agent i's utility is now given by 0₁x₂P - ti The cost is still given by cn, where n is the number of agents in the population. (f) Solve for the firm's optimal build decision p() and participation rule x₂(). (g) Suppose n→∞. Show there exists a cutoff c* such that the firm provides the pool with probability one if c < c*, and with probability zero if c> c*.

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a An agent with type i has utility given by i pt i The mechanism is individually rational because the agents utility is a function of their private va... View the full answer