MN2028: n identical firms supply a market with demand function: Managerial Economics Assignment, UOL, Singapore

1. n identical firms supply a market with demand function p=1-Q where Q is the total quantity supplied, i.e. the sum of quantities supplied by the n firms (Q =q1 + q2 +…+qn). The firms have constant marginal cost c (0 < c < 1).

a) Find the Cournot equilibrium.

b) Show that a merger of two firms is not in the interest of the merging firms.

2. Consider a profit-maximizing firm in a perfectly competitive industry. Its production function is given by q = ALα K1-α. Show that α = wL/(pq) where w is the per-unit cost of Labour (L) and p is the per-unit output price.

3. Alex is considering auction formats to sell an antique. There are 3 potential buyers, Ben, Charlie, and David who have valuations 2, 5, and 8 respectively. The 3 bidders know their own valuation but not the other bidders’ although they and the seller know that valuations are drawn from a uniform distribution on [0,10].

a) What is Alex’s revenue if he uses an English auction? Who wins the auction?

b) What is Alex’s revenue if he uses a second price sealed bid auction? Who wins the auction?

c) Determine the bidders’ equilibrium strategies in a first-price sealed-bid auction. What is Alex’s revenue if he uses this type of auction?

4. Diane always has tea with two sugars and coffee without sugar.

a) Draw her indifference curves for tea and sugar and write down her utility function for tea and sugar.

b) Draw her indifference curves for coffee and sugar and write down her utility function for coffee and sugar.

SECTION B

5. Consider the game in the normal form below.

a) Assume x = y. For which values of x do both players have a dominant strategy? What are their dominant strategies?

b) Assume x = y. For which values of x is this game a prisoner’s dilemma?

c) Now assume x > 0 > y. Find the Nash equilibrium in pure strategies.

d) Are there any values of x and y for which this game does not have an equilibrium in pure strategies? Explain your answer.

6. A company has two plants with different production technologies to produce the same good. Plant A has production function Q = min(3L, 2K) and plant B has production function Q = L0.5 K0.5. Unit costs of labour (L) and capital (K) are w and r respectively.

a) Find the cost function for plant A.

b) Find the cost function for plant B.

c) For w = r = 10, how should production be divided between the two plants if the company wants to produce 100 units?

d) Assuming r = 10, what is the maximum value for w so that plant B will be used?

7. In Russian roulette a player spins a revolver’s 6 round cylinder, places the muzzle against his head and pulls the trigger. If the revolver contains x bullets then the chance of being killed is x/6.

a) Alexey is going to play Russian roulette. There are 5 bullets in the revolver. He could pay an amount of money p1 to reduce the number of bullets to 4. Draw his decision tree. [Hint: the outcomes are ‘dead’, ‘alive with wealth W’, ‘alive with wealth W-p1’.]

b) Suppose Alexey starts out with 1 bullet in the revolver and if he pays p2 that bullet is removed. Draw his decision tree.

c) Using Alexey’s utility function, write down the equation indicating indifference between the two decisions in a).

d) Using Alexey’s utility function, write down the equation indicating indifference between the two decisions in b).

e) Use your answers to c) and d) to derive the conclusion that Alexey is prepared to pay more to reduce the number of bullets from 5 to 4 than to reduce the number of bullets from 1 to zero.

8. What is meant by ‘pure bundling’ and ‘mixed bundling’? Explain how bundling can increase profits. Give a numerical example.

9. It is very common for lawyers to bill their clients by the hour. Explain how this creates moral hazard problems. Suggest a different billing structure that avoids these problems and discuss its advantages and disadvantages.

10. Employees typically have wage profiles that rise with age whereas the self-employed tend to have flatter earnings profiles. Discuss possible explanations for this pattern.