Question

Question 2: Working with Marginal Benefits and Costs Suppose the total benefit derived from a continuous decisions, Q, is B() 200-202 and the total cost from deciding Q is C(Q)-4Q +20. The marginal benefit (MB) and marginal cost (MC) is the first order derivative of these functions. MB)2040 and MC() 4 +40 (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? glish (United States) IT Focus 4

Answer 1

1) When Q = 2, TB = 20Q - 2Q² = 20(2) - 2(2)² = 40 - 8 = 32

    When Q = 10, TB = 20Q - 2Q² = 20(10) - 2(10)² = 200 - 200 = 0

2) When Q = 2, MB = 20 - 4Q = 20 - 4(2) = 20 - 8 = 12

    When Q = 10, MB = 20 - 4Q = 20 - 4(10) = 20 - 40 = -20

3) TB is maximized at the point where, MB = 0

20 - 4Q = 0

4Q = 20

Q = 20 / 4 = 5

Thus, TB is maximized when Q = 5

4) When Q = 2, TC = 4Q + 2Q² = 4(2) + 2(2)² = 8 + 8 = 10

   When Q = 10, TC = 4Q + 2Q² = 4(10) + 2(10)² = 40 + 200 = 240

5) When Q = 2, MC = 4 + 4Q = 4 + 4(2) = 4 + 8 = 12

     When Q = 10, MC = 4 + 4Q = 4 + 4(10) = 4 + 40 = 44

6) TC is maximized when MC = 0

4 + 4Q = 0

4Q = -4

Q = -1 (it is a negative number)

The second order derivative is 4 which is positive. So, we have to find the local minimum quantity which tends to infinity.

7) Net benefit = TB - TC = 20Q - 2Q² - (4Q + 2Q²) = 20Q - 2Q² - 4Q - 2Q² = 16Q - 4Q²

Net benefit is maximized when the first order derivative of NB function is equal to zero

16 - 8Q = 0

8Q = 16

Q = 16 / 8 = 2

Thus, net benefit is maximized at Q = 2