EC2066: Consider the Strategic-form Game below with two Players, 1 and 2 Solve the game by iteratively Eliminating Dominated strategy: Microeconomics Assignment, UOL, Singapore

SECTION A

Answer all EIGHT questions from this section

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

2. Consider an economy with two goods, x and y with prices px and py, respectively. We observe the following choices made by Rob: if px > py he chooses to consume the only y, and if py > px he chooses to consume the only 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.

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 an efficient output. Is this true or false? Explain using an appropriate diagram.

8. Several generators pollute the environment by emitting carbon dioxide. Generators 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

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

Consider the pure strategies of player 1. Note that A1 does not dominate B1 and B1 do 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 dominate 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.

(c) Suppose the following game is repeated infinitely. The players have a common 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.

10. A seller sells a good of quality q at a price t. The cost of producing at quality level q is given by q 2/2. 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.

(b) Prove that the full information price-quality pairs are not incentive compatible if the seller cannot observe θ.

(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).

[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 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.

(b) Find the subgame-perfect equilibrium if 1 move before

2. Also, find the quantities sold by each firm.

(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?

(d) Briefly explain the intuition for the result in part (c).

12. A risk-neutral principal hires a risk-averse agent to work on a project. The agent’s utility function is k(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, 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 the effort is observable, which effort level should the principal implement? What is the best wage contract that implements this effort?

(b) Suppose effort is not observable. What is the optimal contract that the principal should offer the agent? What effort level does this contract implement?

(c) Explain in words why the principal’s payoff differs across the cases considered in parts (a) and (b) above

13. Pip consumes two goods, x and y. Pip’s utility function is given by 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 substitution 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 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.

(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? Explain. (Hint: Consider the following equation:

(d) How much revenue does the tax policy in part (c) raise?