Question

The market demand curve is P = 90 − 2Q, and each firm’s total cost function is C = 100 + 2q2.

1. Suppose there is only one firm in the market. Find the market price, quantity, and the firm’s profit.

2.Show the equilibrium on a diagram, depicting the demand function D (with the vertical and horizontal intercepts), the marginal revenue function MR, and the marginal cost function MC. On the same diagram, mark the optimal price P, the quantity Q, and the average total cost ATC. Illustrate the firm’s profit. Hint: You don’t need to draw the AT C curve.

3.Using the demand function, find the elasticity of demand at the monopoly price and quantity.

4.Verify that the monopoly price and quantity satisfy the monopo- list’s rule of thumb for pricing.

5.What is the monopolist’s factor markup of price over marginal cost?

6.How does the monopolist’s factor markup of price over marginal cost compare to that of a perfectly competitive firm?

Answer 1

1.

Given the following information

P=90-2Q (eq 1)

C=100+2Q² (eq 2)

Therefore TR can be derived at P*Q

TR=90Q-2Q²

Marginal revenue can be derived by differentiating TR wrt Q

MR=90-4Q

Similarly marginal cost can be derived by differentiating cost function wrt Q

MC=4Q

Therefore, the equilibrium price and quantity can be derived by the condition MR=MC

Therefore, 90-4Q=4Q

90=8Q

Q=11.25

Using the value of Q in eq 1 we get the value of P,

P=90-2(11.25)

P=67.5

The profit of the firm can be seen as the difference between TR and TC

TR= 90Q-2Q²

TR= 90(11.25)-2(11.25)²

TR=759.375

TC=100+2Q²

TC = 100+2(11.25)²

TC=353.125

Therefore, profit = 759.375-353.125 = 406.25

2.

3.

Elasticity of demand = (dQ/dP)*(P/Q)

P = 90-2Q

Therefore, 1=-2(dQ/dP)

dQ/dP=-1/2

E_{d(11.25,67.5)}= -0.5(67.5/11.25)

E_{d(11.25,67.5)}= -3

4.

The rule of thumb for pricing states that the markup price should equal the inverse of the inverse of the elasticity of demand.

In our case E_d=-3, MC=45

Therefore, the price should equal = 45/(1+(1/-3))

P= 45*(3/2)

P=67.5

which is equal to the equilibrium price we derived in the 1st part therefore, it satisfies the rule of thumb.

5.

Factor markup price over marginal costs = P-MC

Therefore, Markup price = 67.5-45=22.5

6.

In a perfectly competitive firm there are no markup prices since the perfectly competetive firm is the price taker and therefore the average revenue or the demand function of the firm is equal to the marginal revenue curve leaving no place for markup prices.