EFIM20033 INTERMEDIATE MICROECONOMICS 2024Matlab

Java Python EFIM20033

INTERMEDIATE MICROECONOMICS

May 2024

SECTION A – ANSWER ALL QUESTIONS

Write all your answers in your Answer Booklet

QA1 Imagine a group of five hunters. Each hunter decides whether to pursue a big stag or a hare. If three hunters or more pursue the stag, they catch it and share the meat, each one will get the payoff of 35/x where x is the number of hunters that pursue the  stag. If less than three hunters pursue the stag, they won’t catch it and each one who pursues the stag will get a payoff of zero.  If a hunter pursues a hare, she catches it and gets a payoff of 6.

In your Answer Booklet, enter your correct answer (you may choose more than one answer and to receive credit, you must write down every correct answer and no incorrect answer)

a)  All five hunters going for the stag is a Nash equilibrium

b)   Four hunters going for the stag and one hunter going for the hare is a Nash equilibrium

c)   Three hunters going for the stag and two hunters going for the hare is a Nash equilibrium

d)  Two hunters going for the stag and three hunters going for the hare is a Nash equilibrium

e)  All five hunters going for the hare is a Nash equilibrium (10 marks)

QA2 Consider n=3 agents that consume a natural resource over two periods. Denote by yi

the number of units that player i consumes in the first period and Y  = 1yi   is the

total consumption in the first period. The initial stock of the resource is Y0   = 3 units. In the first period, each player can either consume nothing or just one unit, that is, either yi  = 0  or yi  = 1.

In the second period, the resource renews, and the remaining units will be divided

evenly between the players. The renewal function is     F(Y0  − Y) = 2(Y0  − Y)2 .

Each player’s payoff is their total consumption in both periods,

In your Answer Booklet, enter your correct answer (you may choose more than one answer and to receive credit, you must write down every correct answer and no incorrect answer)

a) The strategy profile y1  = y2  = y3  = 0 is a Nash equilibrium (that is, in the first period all players consume zero units)

b)  The strategy profile  y1   = y2  = 0 and y3  = 1 is a Nash equilibrium (that is, in    the first period two players consume 0 units, and the other player consumes 1 unit)

c)   The strategy profile  y1   = 0 and y2  = y3  = 1 is a Nash equilibrium (that is, in     the first period one player consumes 0 units, and the other players consume 1 unit each)

d)  The strategy profile y1  = y2  = y3  = 1 is a Nash equilibrium (that is, in the first period all players consume one unit)

e)   None of the above strategies are a Nash equilibrium (10 marks)

QA3 Consider a normal-form game with two firms. Each firm has three actions:

•    export to country A

•    export to country B

• don’t export

If both firms export to the same country (A or B), each one will get -10. If a firm is the only one to export to country A, it will receive a payoff of 10. If a firm is the only one   to export to country B, it will receive a payoff of 20. A firm that does not export will receive a payoff of zero no matter what the other firm does. Assume that these are the VnM payoffs.

a)   Find all the pure strategy Nash equilibria of the game. Explain your answer. (5 marks)

b)   Find all the symmetric mixed strategy Nash equilibria (where both players use the same strategy). Explain your answer. (5 marks)

QA4 Consider the Cournot model we studied in class: two firms compete by setting

quantities and prices are determined on the market by demand.  The firms have the same cost function TC(q) =cq where q is the output and c is the marginal cost.

Assume that the marginal cost c=4, and the inverse demand function is Q(p)=16-p.

a)  Write the best response function of each firm. Explain your answer. (5 marks)

b)  Solve for the Nash equilibrium. Calculate the profit of each firm.  Explain your answer. (5 marks)

c)   Assume now t EFIM20033 INTERMEDIATE MICROECONOMICS 2024Matlab hat each firm must pay an additional tax of F>0 to enter the market. If a firm produces nothing, it does not pay the tax. The new cost

function is TC(q)=cq+F if q>0 ,and TC(q)=0 if q=0.  If the tax F=4, solve for the Nash equilibrium (equilibria). Explain your answer. (5 marks)

•    Hint: Is there an equilibrium where both firms enter the market? If yes,  find the price; if no, explain why not. Is there an equilibrium where only one firm enters the market? If yes, find the price; if no, explain why not. Is there an equilibrium where no firm enters the market?

d)  Assume now that the firms move sequentially: Firm 1 first chooses the output level q1; then Firm 2 observes q1  and chooses the output level q2 . Solve for    the subgame perfect equilibrium of the game. Firms still incur the fixed cost    F=4, and so the cost functions are as in c). Explain your answer. (5 marks)

•    Hint: Use backward induction, first determine when firm 2 will not enter.

SECTION B – ANSWER ALL QUESTIONS

Write all your answers in your Answer Booklet

QB1 Consider a seller who sells an object to two potential buyers. The buyers’ valuations are independently drawn from a uniform. distribution over [0, 3].

a)  Characterize a Bayes-Nash equilibrium under the first-price auction. Explain your answer. (5 marks)

b)  Characterize a Bayes-Nash equilibrium under the second-price auction. Explain your answer. (5 marks)

c)   Calculate the seller’s expected revenue under the first-price auction. Explain your answer. (5 marks)

d)  Calculate the seller’s expected revenue under the second-price auction. Explain your answer. (4 marks)

e)  Which auction rule should the seller choose who wants to maximize his revenue? Explain your answer. (1 mark)

QB2 Consider a job market signalling model. The payoff to a firm from hiring a worker of

type θ with education e at wage w  is

-(w − θ)2 .

The utility of a worker of type θi  with education e  receiving a wage w  is w − Cie.

Suppose that the worker’s type can be either low θL  with probability  or high θH   = 3  with probability   , each with cost CL   =  10 and CH   =  1 . Let us restrict our attention to a wage function of the form.

w (e) = {w(w)2(1),,

e ≥ ẽ  e < ẽ .

a)  Characterize all pooling weak perfect Bayesian equilibria. Identify the most efficient pooling equilibrium. Explain your answer. (8 marks)

b)  Characterize all separating weak perfect Bayesian equilibria. Identify the most efficient separating equilibrium. Explain your answer. (7 marks)

QB3 Consider a public goods provision game, with 2 individuals. Each individual must

choose whether or not to contribute to the public good, and the public good is

provided if and only if at least one individual contributes. The cost of the contribution  is ci  to the individual i. The cost ci   is independently and identically distributed across individuals, and is uniformly distributed on [0,2] . The total payoff to an individual is     the value of the good v  =  1 (if provided) minus the cost of provision (which is ci   if the individual provides the good, and zero otherwise). Solve for a Bayesian Nash

equilibrium of this game where each individual provides the good if and only if ci subceeds a critical threshold c ∗ .

Find the threshold. In your Answer Booklet, enter your correct answer that corresponds to the threshold. Do not explain your answer.

1)  2/1

2)  3/2

3)  4/3

4) 5/4 (5 marks)

QB4 Consider a Cournot game with two firms and an inverse demand function

P(Q) = 9 − Q. Let ci   be Firm i's marginal cost. Suppose that c2   is 2 with probability and 4 with probability , whereas c1  is 3 with probability 1. Suppose c2   is Firm 2’s      private information.

In your Answer Booklet, write down and answer the following: The type spaces are

T1  = ,

T2  = .

In your Answer Booklet, write down and answer the following:

In a Bayes-Nash equilibrium under which all types produce positive quantities,

S1  = ,

S2(2) = ,

S2(4) = .

Do not explain your answer