Answer to Question #85132 in Microeconomics for Ron

Question

If the demand curve is Q(p) = p^a (where a < 0), what is the elasticity of demand? If the marginal cost is $2, and a = -4, what is the profit-maximising price?

Consider an example of the third degree price discrimination. There are two groups of consumers. Consumers are identical within each group. In group 1, the demand function is P = 6 - 3Q, in group 2, the demand function is P=5-2Q. The marginal cost of production is 1. Find out the optimal price for each group.

Answer 1

If the demand curve is Q(p) = p^a (where a < 0), then the elasticity of demand is Ed = a*p/Q.

If the marginal cost is $2, and a = -4, then the profit-maximising price is P = MR = MC = $2.

In group 1, the demand function is P = 6 - 3Q, in group 2, the demand function is P = 5 - 2Q. The marginal cost of production is MC = 1. The optimal prices for each group are:

1) if P = 6 - 3Q, then MR = TR' = (P*Q)' = 6 - 6Q.

The optimal quantity at MR = MC is:

6 - 6Q = 1,

Q = 5/6 units, P = 6 - 3*5/6 = $3.5.

1) if P = 5 - 2Q, then MR = TR' = (P*Q)' = 5 - 4Q.

The optimal quantity at MR = MC is:

5 - 4Q = 1,

Q = 1 unit, P = 5 - 4*1 = $1.