economics

Econ 7030: Exercise 2 Due: 12 noon, June 10

Value: 30 points

1. (10 points) Consider a competitive market where the demand is given by 𝐷𝐷(𝑝𝑝) = 55 − 𝑝𝑝2. The supply consists of 20 sellers each with the same total cost curve 𝐶𝐶(𝑞𝑞) = 1 + 𝑞𝑞 + 10𝑞𝑞2. (a) (2 points) Find the supply curve of each seller. (b) (3 points) Find the competitive equilibrium price, market quantity and quantity

supplied by each seller. (c) (2 points) Based on your calculation in part (b), will entry or exit occur in the long

run? (d) (3 points) What is the long run competitive equilibrium price, quantity per seller

and number of sellers.

2. (6 points) A monopoly seller faces two markets: 𝑄𝑄1(𝑝𝑝) = 100 − 2𝑝𝑝 and 𝑄𝑄2(𝑝𝑝) = 120 − 𝑝𝑝. The cost is zero for simplicity. (a) (2 points) If the monopoly can price discriminate and charge a different price in

each market, what price will it set in each market? (b) (2 points) If the monopoly must set the same price in each market, then what price

will it set? (c) (2 points) Now suppose that the seller faces uncertainty about the demand which

will either be 𝑄𝑄1(𝑝𝑝) or 𝑄𝑄2(𝑝𝑝). It must set the price 𝑝𝑝 before it finds out which demand will occur. Suppose that 𝑄𝑄1(𝑝𝑝) is twice as likely to occur as 𝑄𝑄2(𝑝𝑝). Find the price that maximizes the expected profit.

3. (6 points) Consider a consumer with utility 𝑢𝑢(𝑥𝑥, 𝑦𝑦) = 𝑥𝑥𝑦𝑦 + 2𝑥𝑥 + 𝑦𝑦. This utility function is continuously differentiable, strongly monotonic and has strictly convex preferences. You do not need to check this. (a) (4 points) Find the demands for interior solutions and any restrictions on prices

and income needed to ensure the solutions are interior. (b) (1 point) Are there any prices and income where the consumer will consume only

𝑥𝑥? If so, what are they? How much 𝑥𝑥 will be consumed? (c) (1 point) Are there any prices and income where the consumer will consume only

𝑦𝑦? If so, what are they? How much 𝑦𝑦 will be consumed?

4. (4 points) A consumer has utility 𝑢𝑢(𝑥𝑥, 𝑦𝑦) = 10�𝑦𝑦 + 𝑥𝑥. The price of 𝑥𝑥 is fixed at $1 and income is $100. Find the demand of the consumer for 𝑦𝑦 as a function of the price of 𝑝𝑝 of 𝑦𝑦.

5. (4 points) A seller has a production function 𝑦𝑦 = 𝑓𝑓(𝑥𝑥) = 200𝑥𝑥 − 𝑥𝑥2. The price of the input 𝑥𝑥 is fixed at 𝑝𝑝𝑥𝑥 = 10. (a) (2 points) Find the cost function 𝐶𝐶(𝑦𝑦) for all output levels that the seller may

produce. (b) (2 points) Find the supply function 𝑦𝑦(𝑝𝑝).