Econ 500 problem set questions consider the all-pay auction

Question

ECON Problem Set Questions

Question 1) Consider an economy with production with one consumer and one firm. The consumer, who has endowment of one hour of labour, has preferences over consumption x and labor L given by u (x, L) = xα(1 - L)^1-α while the firm has a constant-returns-to-scale production function x = bL, for constant b > 0. Find competitive equilibrium of this economy (What should prices be for the firm to have positive optimal production level?).

Question 2) Let's consider an economy with uncertainty and two agents. There are two possible states of the world, named as 1 and 2. Hence a consumption bundle is (x₁; x₂) ∈ R²₊ representing consumption contingent in each state. State i occurs with probability π_i ∈ (0, 1) (satisfying π₁ + π₂ = 1). There are two agents, A and B, which maximize expected utility with Bernoulli u(x) = log x (this agents are risk averse). Suppose the endowments e^A, e^B ∈ R²₊₊ (i.e., both agents have a strictly positive endowment of consumption in each stage) satisfy

e₁^A + e₁^B = 2(e₂^A + e₂^B),

which means that the economy as a whole has more resources in state 1.

a) Find the competitive equilibrium of this economy. Notice that the consumption of agents A and B "co-move", which means that the aggregate shock (which doubles total endowment in the economy) affects the consumption of both agents by the same factor.

b) Now suppose that we introduce a third agent, named C, in this economy. This agent also maximizes expected utility, but his Bernoulli is u^~(x) = x (this agent is risk neutral) with endowment e^C satisfying (this guarantees that an interior equilibrium exists):

e₂^C > π₁(e₁^A - e₂^A + e₁^B - e₂^B).

Find an interior competitive equilibrium (where the all consumers choose an interior consumption bundle). What happened to the risk held by agents 1 and 2?

Question 3) Consider an economy with uncertainty (with a single good, n = 1, and with M states of the world). Consider two agents, named i and j, that inhabit this economy, maximize expected utility and have Bernoulli

u_i(x) =  e^-ρ_ix/ρ_i,

with ρ_i > ρ_j.

The notes discuss a model in which two agents have CRRA Bernoulli's. Following the steps, show that in any interior competitive equilibrium of this economy agent j bears more risk than agent i, i.e., the final consumption (given by x^j_ω in state ω) of agent j varies more with the state of the economy than agent i's. What is the justification for this result?

Question 4) Consider the first-price auction discussed in class. There are 2 bidders i = 1, 2 that value a single good at v > 0. Both can simultaneously decide their bids b_i ≥ 0, with the highest bidder winning the auction and having to pay their own bid (in case of a tie, each bidder wins with probability ½).

a) Plot the best-response correspondence of a bidder in this auction.

b) Find all Nash-equilibria (Are there multiple)?

c) Now suppose there are three bidders i = 1, 2, 3, each valuing the good at the same value v > 0. The rules of the auction are kept the same, with the additional rule that in case of a tie among n agents each one wins the auction with probability 1/n (we can now have a two or three-way tie). Show that in any Nash equilibrium the good is sold at the same price as in (b).

d) Find at least two Nash equilibria of the three-bidder model (following (c), in both equilibria the good ends up being sold at the same price).

Question 5) Consider the model of Cournot competition discussed in class with inverse demand function

P (Q) = (A - Q)⁺.

Suppose firms 1 and 2 have marginal costs c₁ ∈ (0, A) and c₂ ∈ (0, A), respectively (Just as in class, we assume that marginal costs are constant and there are no fixed costs). Find a Nash equilibrium of this model (Are there multiple?).

Question 6) Find all Nash equilibria (Pure and Mixed) of the following two player game:

	l	m	r
U	5, 5	0, 3	8, 6
D	6, 5	2, 9	5, 1
Table 1:

Question 7) Consider the All-pay auction model considered in class, but with N bidders. All bidders value the good equally at v > 0.

a) Suppose every bidder i = 1, . . . , N chooses bids bi randomly, according to distribution F(·) with support on [0, v]. Find U^-(b), the expected utility of a bidder bidding b ≥ 0.

(Notice that each bidder only wins the auction if every other bid - there are N - 1 of them - ends up being below b).

b) Find what CDF F (·) should be so that each bidder is indifferent between biding all levels in [0, v].

c) Show that it is a (mixed) Nash equilibrium of this auction for all bidder to randomize their bids according to CDF F (·) found in (b).

d) Calculate the expected revenue obtained in this equilibrium of the auction (Hint: if there are N random variables α₁, . . . , α_N, each with CDF G(·), then P (α₁ ≤ z, . . . , α_N ≤ z) = G(z)^N for any z ≥ 0).

e) What is the expected utility of each bidder in this auction?

Attachment:- Assignment Files.rar