Question

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

(a) (7 points) Suppose there is only one firm in the market. Find the market price, quantity, and the firm’s profit.

(b) (5 points) 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 ATC curve.

(c) (5 points) Using the demand function, find the elasticity of demand at the monopoly price and quantity.

(d) (2 points) Verify that the monopoly price and quantity satisfy the monopolist’s rule of thumb for pricing.

 

Answer 1

We have the following information

Demand: P = 90 – 2Q where P is the price and Q is the quantity

Total cost (TC): C = 100 + 2q²

Since there is only one firm so Q = q

a) The equilibrium is at the point where the marginal revenue (MR) is equal to the marginal cost (MC)

Total revenue (TR) = Price × Quantity

TR = (90 – 2Q)Q

TR = 90Q – 2Q²

MR = ΔTR/ΔQ = 90 – 4Q

MC = ΔTC/ΔQ = 4Q

Equating MC and MR

4Q = 90 – 4Q

90 = 8Q

Equilibrium quantity = 11.25

P = 90 – 2Q

P = 90 – 22.5

Equilibrium price = 67.5

Profit = TR – TC

Profit = (67.5 × 11.25) – (100 + 2(11.25)²)

Profit = 759.375 – 353.125

Profit = 406.25

Average total cost (ATC) = TC/Q = (100/Q) + 2Q

For Q = 11.25

ATC = (100/11.25) + (2 × 11.25)

ATC = 8.89 + 22.5

ATC = 31.39

b)

c) Price elasticity of demand is proportionate change in the quantity demanded of a good due to a proportionate change in the price of the good. Formula for calculating price elasticity of demand is following

Price elasticity = (ΔQ/ΔP) × (P/Q)

P = 90 – 2Q

Q = 45 – 0.5P

ΔQ/ΔP = – 0.5

For P = 67.5 and Q = 11.25 the price elasticity of demand is

Price elasticity = – 0.5 × (67.5/11.25)

Price elasticity (Ed) = – 3

d) Monopolist rule of thumb: P = MC/[1 + (1/Ed)]

MC = 4Q = 45

Ed = – 3

P = 45/[1 + (1/– 3)]

P = 67.5

So, we can say that the monopoly price and quantity satisfies the Monopolist’s Rule of Thumb.