ECOS3005 Problem set 1

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ECOS3005

Problem set 1

1. Suppose two rms compete in the market for Glonks. Each rm is debating their pricing strategy. Firms simultaneously decide whether to set a “high” price or a “low” price. The high price is close to the monopoly price. If both rms choose this price, pro ts are high for each rm. If only one rm chooses the low price, that rm captures the entire market, earning even higher pro ts, while the other rm loses all of their customers. Finally, if both rms set the low price, pro ts are modest for both rms.

(a) Write out a game in normal form that describes this competitive situation. (b) Identify any dominant strategies in the game you have written.

(c) Identify any Nash equilibria to your game.

(d) Are there any similarities to any of the games we have studied in class Explain brie y.

2. Anne (player 1) and Betty (player 2) are the only bidders for a work of art at an auction. Anne has a valuation (willingness to pay) for the artwork of v1 and Betty has a valuation of v2 , and v1 and v2 are known to both players. The artwork is sold using a “second-price auction” . Each player simultaneously submits a bid, bi , i = 1y 2. The highest bidders wins the auction and pays the second highest bid. The payoff of the winner is the difference between her valuation and her bid, vi bi, and the payoff of the loser is zero.

(a) Is there a dominant or weakly dominant strategy for Anne and/or Betty (b) Find the Nash equilibrium to the game. Who wins the auction

3. Consider a market with the following properties. All rms have identical cost technology, summarised by the following total cost curve: C(qi) = 1000 + 10qi + 0●1qi(2), where qi is the output of the representative rm, i. The market (inverse) demand curve is given by P(Q) = 130 0●1Q, where Q = E1 qi is the total output of the n rms in the industry.

(a) Cost curves

i. Find an algebraic expression for the marginal cost curve, MC(qi), of the represen- tative rm. Also nd the rm’s average costs, average xed costs, and average variable costs.

ii. Does the representative rm enjoy economies of scale at qi = 50 At qi = 100 At qi = 200

(b) Perfect competition – Short run

i. Suppose there are 4 competitive (price-taking) rms in the market. How much does each rm produce (Hint: Each rm is identical. Therefore assume each rm produces the same quantity).

ii. What is the market price

iii. What are the pro ts (or losses) of each rm

(c) Perfect competition – Long run (harder)

i. Would we expect entry into the perfectly competitive market with 8 rms Why, or why not

ii. How many competitive rms can be sustained in this industry in the long run (Hint: let the number of rms in the market be n, solve for the competitive equilib- rium as before, and think about the long run equilibrium conditions).

iii. What about the short run If there were 20 competitive rms in the market, what would happen in the short run What about 100 competitive rms

(d) Monopoly

i. Suppose there is a single rm operating in the market. How much would the mo- nopolist produce

ii. What is the market price

iii. What are the pro ts (or losses) of the monopolist

4. The market for good X has demand

P(Q) = A Qy

where Q is market output and A is a positive constant. n rms compete by simultaneously choosing output in a single period. Each rm has costs given by

C(q) = cqy

where q is an individual rm’s output and c is a positive constant.

(a) Find the reaction function for each rm. Explain how the reaction functions depend on

market demand, marginal costs and the number of rms.

(b) Find the equilibrium output of each rm.

(c) Explain the impact on equilibrium output and the market price of

i. an increase in demand

ii. a decrease in marginal costs

iii. entry