Question

Smith has determined that demand for their new line of hats is given by Q = 200 - 4P and a cost equation given by C = 100 + 0.5Q‚.

a. Determine the optimal price and quantity for the firm. (Price should be calculated to nearest cent. Example: $5.43 should not be rounded to $5.00.)

b. Suppose that costs change to C = 50+ 1.0 Q‚. Determine the new optimal price and quantity. Explain why the results differ from those in part a.

Answer 1

(a) Q = 200 - 4P or P=50-Q/4

Total revenue TR=P*Q=50Q-Q^2/4

Marginal revenue MR=dTR/dQ=50-Q/2

C = 100 + 0.5Q

Marginal cost MC=dC/dQ=0.5

For Optimization: MR=MC

0.5=50-Q/2. Solving for Q we get Q=99

Thus P=50-99/4=25.25

(b) Now C = 50+ 1.0 Q

So the new MC=dC/dQ=1

So now the firm will set the marginal revenue equal to new marginal cost.

50-Q/2=1 Solving for Q we get Q=98

New price = 50-98/4=25.5

The results differ from that in part a because the marginal cost has now increased.