Question

Monopoly pricing: Consider a simple linear demand function that intersects the quantity 0 point at $110 and the $0 axis at 1,200 units. The marginal cost is linear and starts at $10 at 0 quantity and reaches $110 at 500 units.

1. What is the elasticity of demand? Is it constant? Why or why not?
2. What is the competitive market clearing price?
3. What is the competitive consumer surplus?
4. What is the competitive producer profit (surplus)?
5. What is the profit-maximizing monopoly price and quantity?
6. At that price, what is the consumer surplus?
7. If we allow the producer to segment the market, so that it can pick 100 consumers that have to pay their willingness to pay (i.e., according to the demand function at quantity 100), how do consumer surplus and profit change?
8. If we assume a mind-reading producer who can price-differentiate so that each consumer pays according to their position on the demand curve, how do profit and consumer surplus change? (In other words, consumers are lined up according to the demand curve and get served at that price.)

It is probably easiest to use algebraic representations for the curves, rather than trying to do this graphically. Show your work.

Answer 1

(1)

Elasticity of demand is the ratio of percentage change in quantity demanded to the percentage change in price. For a linear demand function of the form Q = a - bP,

Elasticity = (dQ/dP) x (P/Q) = -b x (P/Q)

Since b is a constant (slope of demand line) but P and Q change along the demand curve. Therefore elasticity is not constant.

(2)

(a) Linear demand equation: P = a - bQ

When Q = 0, P = 110

110 = a - 0

a = 110

When P = 0, Q = 1,200

0 = a - 1,200b

0 = 110 - 1,200b

1,200b = 110

b = 0.0917

Demand equation: P = 110 - 0.0917Q

(b) Equation of MC function: MC = c + dQ

When Q = 0, MC = c = 10

When Q = 500, MC = 110

110 = 10 + 500d

500d = 100

d = 0.2

MC equation: MC = 10 + 0.2Q

(c) In competitive equilibrium, P = MC.

110 - 0.0917Q = 10 + 0.2Q

0.2917Q = 100

Q = 343

P = 10 + (0.2 x 343) = 10 + 68.6 = 78.6

(3)

Consumer surplus ($) = Area between demand curve and price = (1/2) x (110 - 78.6) x 343 = 171.5 x 31.4 = 5385.1

(4)

Producer surplus ($) = Area between MC curve and price = (1/2) x (78.6 - 10) x 343 = 171.5 x 68.6 = 11764.9

NOTE: As per Answering Policy, 1st 4 parts are answered.