Question

1. Suppose that a monopolist has a patent for widgets and the market demand curve Q(P) is:

Q = 60 – 2P,

where P is the price in dollars and Q is quantity.

a. Solve for the inverse demand P(Q) curve by solving the demand curve for P in terms of Q.

b. Using your answer from (a), express the monopolist’s total revenue in terms of Q as TR(Q) = QP(Q).

c. Calculate the monopolist’s marginal revenue MR(Q) by differentiating the total revenue you found in the previous step: MR(Q) = dTR(Q) / dQ.

d. If MR(Q) > 0, then total revenue increases by selling more units. If MR(Q) < 0, then total revenue increases by selling fewer units. Calculate the Q* such that MR(Q*) = 0 – this will be where total revenue is maximized.

e. Determine the price P* the monopolist must charge in order to sell Q* units by plugging your answer for Q* into the inverse demand curve P(Q).

f. Neatly draw a graph showing the demand curve for widgets and the marginal revenue curve, carefully showing the vertical and horizontal intercepts. Also show Q* and P*.

g) Compute TR(Q*) by multiplying Q* by P(Q*). On your graph, shade the area under the marginal revenue curve from 0 over to Q*. This are should be a triangle. Compute the area of this triangle (using the usual formula of one half times base times height).

Answer 1

a) Q = 60-2P

2P = 60-Q

P = 30-(1/2)Q

P = 30-0.5Q

Inverse demand curve = P = 30-0.5Q

b) Total revenue = TR = P * Q

TR = (30-0.5Q)Q

TR = 30Q - 0.5Q^​​​2

c) Marginal revenue = MR = d(TR)/dQ

MR = d(30Q-0.5Q²)/dQ

MR = 30-(0.5*2)Q

MR = 30-Q

d) when MR = 0

30-Q = 0

Q = 30

Maximum value of Q = 30

e) P = 30-0.5*Q

P = 30-(0.5*30)

P = 15