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Answer all EIGHT questions from this section (5 marks each)

Economics

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Answer all EIGHT questions from this section (5 marks each).

1. Consider the strategic-form game below with two players, 1 and 2. Solve the game by iteratively eliminating dominated strategies.

Player 2

Az By Co

Ai

Player By

C;

 

2. Consider an economy with two goods, x and y with prices p, and py, respec-tively. We observe the following choices made by Rob: if py > p, he chooses to consume only y, and if py > px he chooses to consume ony x. Suggest a utility function for Rob that represents preferences consistent with the given data.

 

3. Consider a market for used cars. There are many sellers and even more buyers.

A seller values a high quality car at 800 and a low quality car at 200. For any quality, the value to buyers is m times the value to sellers, where m > 1. All agents are risk-neutral. Sellers know the quality of their own car, but buyers only know that 2/3 of the cars are low quality and the remaining 1/3 of them are high quality. For what values of m do all sellers sell their used cars?

 

4. If the price elasticity of supply is zero, a tax on suppliers will raise the market price. Is this true or false? Explain your answer.

5. Amal consumes pizzas and also consumes good O which is a composite of all other goods. His income-consumption curve is a vertical line as shown in the picture below. Pizza might be a Giffen good for Amal. Is this true or false?

Explain your answer.

Pizza

O

Income-consumption curve

6. An individual consumes two goods and her preferences satisfy non-satiation. It follows that at least one of the two goods must be a normal good. Is this true or false? Explain your answer.

7. Under first-degree price discrimination, a monopolist produces the efficient out- put. Is this true or false? Explain using an appropriate diagram.

8. Several generators pollute the environment by emitting carbon dioxide. Gener- ators have different costs of reducing carbon emissions. The government wants to put a cap on total emissions. Putting a cap on each generator is more efficient compared to issuing tradeable emissions permits to each generator. Is this true or false? Explain your answer.

SECTION B

Answer THREE questions from this section (20 marks each).

9. (a) Consider the following simultaneous move game with two players.

2

A2 B2

1 A1 3,1 1,2

B1 1,2 3,1

Consider the pure strategies of player 1. Note that A1 does not dominate

B1, and B1 does not dominate A1. Is it possible for a mixed strategy of player 1 to be a dominant strategy? Explain. [5 marks]

(Hint: For any mixed strategy of 1 to be a dominant strategy, it must dom-inate both A1 and B1. Is this possible?)

(b) For the following extensive-form game:

i. Identify the pure and mixed strategy Nash Equilibria. [5 marks]

ii. Identify all Subgame Perfect Nash equilibria. [5 marks]

B

A

1

D

(1,-1)

C

(3,2)

D

(2,3)

C

(-1,0)

2

(c) Suppose the following game is repeated infinitely. The players have a com-mon discount factor δ ∈ (0, 1). Show that for high enough values of δ, there is an equilibrium of the infinitely repeated game in which (C, C) is played

in every period. Your answer must state the strategies of the players clearly.

[5 marks]

2

C D

1 C 4,2 0,3

D 5,0 1,1
10. A seller sells a good of quality q at a price t. The cost of producing at quality

level q is given by q2/2. There is a buyer who receives a utility of θq − t by

consuming the unit of quality q at price t. If he decides not to buy, he gets a

utility of zero. θ can take two values θ1 = 1 and θ2 = 4.

(a) Suppose the seller can observe θ. Derive the profit maximizing price-quality

pairs offered when the type is θ1 = 1 and when the type is θ2 = 4.

[6 marks]

(b) Prove that the full information price-quality pairs are not incentive com-

patible if the seller cannot observe θ. [7 marks]

(c) Suppose the seller cannot observe θ, and suppose he decides to set q1 =

1/4 and q2 = 4. Calculate the optimal values of t1 and t2 such that both

types participate, type θ1 = 1 takes the contract (q1, t1) and type θ2 = 4

takes the contract (q2, t2). [7 marks]

[Hint: write down the participation constraint of type θ1 and the incentive

constraint of type θ2 and solve for t1 and t2.]

11. Suppose two firms (1 and 2) sell differentiated products and compete by setting

prices. The demand functions are

q1 = 7 − P1 +

P2

2

and

q2 = 7 − P2 +

P1

2

Firms have a zero cost of production.

(a) Find the Nash equilibrium in the simultaneous-move game. Also find the

quantities sold by each firm. [5 marks]

(b) Find the subgame-perfect equilibrium if 1 moves before 2. Also find the

quantities sold by each firm. [5 marks]

(c) Calculate the profits of the two firms for the case in part (b). Which firm

gets a higher profit, the first mover or the second mover? [5 marks]

(d) Briefly explain the intuition for the result in part (c). [5 marks]

12. A risk neutral principal hires a risk averse agent to work on a project. The agent’s utility function is

V(w, ei) =

√

w − g(ei),

where w is wage, g(ei) is the disutility associated with the effort level ei exerted on the project.

The agent can choose one of two possible effort levels, eH or eL, with associated disutility levels g(eH) = 2, and g(eL) = 1. If the agent chooses effort level eH, the project yields 20 with probability 3/4, and 0 with probability 1/4. If he chooses eL, the project yields 20 with probability 1/4 and 0 with probability 3/4.

The reservation utility of the agent is 0.

Let {wH, wL} be an output-contingent wage contract, where wH is the wage

paid if the project yields 20, and wL is the wage if the yield is 0. The agent

receives a fixed wage if wH = wL.

(a) If effort is observable, which effort level should the principal implement?

What is the best wage contract that implements this effort? [8 marks]

(b) Suppose effort is not observable. What is the optimal contract that the principal should offer the agent? What effort level does this contract im-plement? [8 marks]

(c) Explain in words why the principal’s payoff differs across the cases con- sidered in parts (a) and (b) above. [4 marks]
13. Pip consumes two goods, x and y. Pip’s utility function is given by

u(x, y) = x1/2

y1/2

The price of x is p and the price of y is 1. Pip has an income of M.

(a) Derive Pip’s demand functions for x and y. [5 marks]

(b) Suppose M = 72 and p falls from 9 to 4. Calculate the income and substi-

tution effects of the price change. [5 marks]

(c) Calculate the compensating variation of the price change. [5 marks]

(d) Calculate the price elasticity of demand for x. [5 marks]

14. Each firm belonging to a competitive industry has the following long-run cost

function

C(q) = 10q − 2q2

+ q3

where q denotes the output of a representative firm. Firms can enter and exit

the industry freely. The industry has constant costs: input prices do not change

as industry output changes. The market demand facing the industry is given by

Q = 20 − P

(a) Derive the long-run industry supply curve. [5 marks]

(b) How many firms operate in the industry? [5 marks]

(c) Suppose a regulator imposes a lump-sum tax of 8 on each firm. Does the

output produced by a firm rise or fall as a consequence of this policy? Ex-

plain. [5 marks]

(Hint: Consider the following equation:

−

8

q2

− 2 + 2q = 0

The solution to this is q = 2.)

(d) How much revenue does the tax policy in part (c) raise? [5 marks]