The white company is a member of the lamp industry

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: ????1(????) = 10 − ????; ????2(????) = 20 − 2????; and ????3(????) = 50 − 5????.

b. Derive market supply for a market with 15 producers, each of which has an individual supply

function given by ????(????) = 2 + 15 ????

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

inverse supply functions: ????1(????) = 4 − 2???? and ????2(????) = 4 − 12 ????.

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 ????(????) = 5 + 3????2.

b. The case where market price is equal to $130 and the cost function is ????(????) = 5 + 10???? +3????2.

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

is ????(????) = ???? + (2

3)????2. Note that here when you find the optimal quantity, it will be a

function of ????.

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 ???????? = 120 − 10???? and market supply

is given by ???????? = 84 + 2????.

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:

????(????) = 1 + ???? + 0.1????2

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:

????(????) = 1,000 + 20???? + 5????2

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:

????????(????????) = 10 + 2???????? + 3????????

2

????????(????????) = 3 + 3???????? + 2????????

2

Demand is given by ????????(????) = 20 − 1

5 ????.

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?