Question

Suppose the total benefit derived from a continuous decisions, Q, is B(Q) = 20Q − 2Q and the 2 total cost from deciding Q is C(Q) = 4Q + 2Q . The marginal benefit (MB) and marginal cost (MC) 2 is the first order derivative of these functions. MB(Q) = 20 − 4Q and MC(Q) = 4 + 4Q .

(1) (4 points, 2 each) What is the total benefit when Q=2? Q=10?

(2) (4 points, 2 each) What is the marginal benefit when Q=2? Q=10?

(3) (4 points) What level of Q maximizes total benefit? (hint: recall our first order condition for maximization. The first derivative is given as the MB function).

(4) (4 points, 2 each) What is total cost when Q=2? Q=10?

(5) (4 points, 2 each) What is marginal cost when Q=2? Q=10?

(6) (4 points) What level of Q minimizes total cost?

(7) (6 points) What level of Q maximizes net benefit -- that is Total Benefit - Total Cost?

Answer 1

1. total benefit, B(Q) = 20Q – 2Q²

at Q=2, B(Q)= 20(2)-2(2)² = 40-8= 32.

at Q=10, B(Q)= 0.

2. MB(Q) = 20 – 4Q

at Q=2, MB(Q)= 20-8= 12

at Q=10, MB(Q)= -20. (it is a marginal loss).

3. max. B(Q)

dB(Q)/dQ= 20-4Q.

first order condition: dB(Q)/dQ=0

= Q=5.

4. total cost, C(Q) = 4 + 2Q²

at Q=2, C(Q)= 12

at Q=10,

C(Q)= 204.

5. MC(Q) = 4Q

at Q=2, MC(Q)= 8

at Q=10, MC(Q)= 40.

6. min. C(Q)

dC(Q)/dQ= 4Q

First order condition: dC(Q)/dQ=0

4Q=0,

Q=0.

7. net benefits are maximized where MB(Q)=MC(Q)

this implies,

20 – 4Q= 4Q

8Q= 20,

Q=2.5