The questions are here in below

Homework 4

Managerial Economics

Due Thursday, October 19, 2017

Instructions: Show all of your work!!!!

1) Aggregation Practice – in each of the following situations, solve for either market supply or marked demand

a. Derive market demand for a market with three consumers, having the respective individual demand functions of: 𝑞1(𝑃) = 10 − 𝑃; 𝑞2(𝑃) = 20 − 2𝑃; and 𝑞3(𝑃) = 50 − 5𝑃.

b. Derive market supply for a market with 15 producers, each of which has an individual supply function given by 𝑞(𝑃) = 2 + 1

5 𝑃

c. Derive market supply for a market with two producers having the following for individual

inverse supply functions: 𝑝1(𝑞) = 4 − 2𝑞 and 𝑝2(𝑞) = 4 − 1 2 𝑞.

2) Writing Profits Practice – in each of the following situations, write an expression for profits. Then take a derivative with respect to Q and solve for the profit-maximizing Q.

a. The case where market price is equal to $30 and the cost function is 𝑐(𝑄) = 5 + 3𝑄2.

b. The case where market price is equal to $130 and the cost function is 𝑐(𝑄) = 5 + 10𝑄 +

3𝑄2. c. The case where market price is left general, equal to the constant, 𝑃, and the cost function

is 𝑐(𝑄) = 𝑄 + (2 3 )𝑄2. Note that here when you find the optimal quantity, it will be a

function of 𝑃.

3) Perfect Competition: Choosing Optimal Output

Consider potato farming in the US, a highly competitive market. Assume the market is perfectly competitive, and that the market demand for potatoes is given by 𝑄𝐷 = 120 − 10𝑃 and market supply is given by 𝑄𝑆 = 84 + 2𝑃.

a) Find the competitive equilibrium price and quantity of potatoes in this market.

b) Assume that one particular farmer, Joe, knows that his cost function is given by:

𝑐(𝑞) = 1 + 𝑞 + 0.1𝑞2 Find Joe’s profit-maximizing level of output, and calculate the profits he makes.

c) What if the price doubles? Now how much would Joe want to produce to maximize profits? What are his profits now?

4) Perfect Competition – the White Company is a member of the lamp industry, which is perfectly

competitive. The price of a lamp is $50. The firm’s total cost function is given by:

𝑐(𝑞) = 1,000 + 20𝑞 + 5𝑞2

a. What level of output maximizes profits for this firm?

b. What is the firm’s economic profit at this level of output?

c. Should the firm produce or shut down in the short run? Explain why.

d. If the other firms in the lamp industry have the same cost function as the White Company, is the industry in long-run equilibrium? Why or why not?

5) Perfect Competition – consider a perfectly competitive market that has 4 firms in it (assume it is perfectly competitive despite their being only 4 firms). Two of the firms use technology A and two of the firms use technology B. The respective costs of producing using technology A and B are given by the cost functions:

𝑐𝐴(𝑞𝐴) = 10 + 2𝑞𝐴 + 3𝑞𝐴2

𝑐𝐵(𝑞𝐵) = 3 + 3𝑞𝐵 + 2𝑞𝐵2

Demand is given by 𝑄𝐷(𝑃) = 20 − 1 5 𝑃.

a) Determine short run equilibria, i.e. optimal outputs for each type of firm, equilibrium quantity for the market as a whole, and the equilibrium price, and then calculate equilibrium profits for each type of firm.

b) What will we expect to happen in the long run? Will other firms enter? Will price stay the same? How will this impact the two types of firms differently? Will both technologies be used in the long run?