Chapter 05 - The Production Process and Costs

Chapter 5: Answers to Questions and Problems

a. When K = 81 and L = 16, Q = (81)

0.75

(16)

0.25

= 54. Thus, AP

= Q/L = 54/16 =

3.375. When K = 81 and L = 256, Q = (81)

0.75

(256)

0.25

= (27)(4) = 108. Thus, AP

= 108/81 = 0.422.

b. The marginal product of labor is MP

= (1/4)*(81)

0.75

(L)

-3/4

= (27/4)(L)

-3/4

. When L

= 16, MP

= (27/4)(16)

-3/4

= 0.844. When L = 81, MP

= (27/4)(81)

-3/4

= 0.25.

Thus, as the number of units of labor hired increases, the marginal product of

labor decreases MP

(16) = 0.844 > 0.25 = MP

(81), holding the level of capital

fixed.

c. We must equate the value marginal product of labor to the wage and solve for L.

Here, VMP

= (P)(MP

) = ($200)(27/4)(L)

-3/4

=1350(L)

-3/4.

Setting this equal to the

wage of $50 gives 1350(L)

-3/4

= 50. Solving for L, the optimal quantity of labor is

L = 81.

2. See Table 5-1.

(1) (2) (3) (4) (5) (6) (7)

Capital Labor Output

Marginal

Product of

Capital

Average

Product of

Capital

Average

Product

of Labor

Value Marginal

Product of

Capital

VMP

0 20 0 - - - -

1 20 50 50 50 2.50 200

2 20 150 100 75 7.50 400

3 20 300 150 100 15 600

4 20 400 100 100 20 400

5 20 450 50 90 22.50 200

6 20 475 25 79.17 23.75 100

7 20 475 0 67.86 23.75 0

8 20 450 -25 56.25 22.50 -100

9 20 400 -50 44.44 20 -200

10 20 300 -100 30 15 -400

11 20 150 -150 13.64 7.50 -600

Table 5-1

a. Labor is the fixed input while capital is the variable input.

b. Fixed costs are 20($30) = $600.

c. To produce 475 units in the least-cost manner requires 6 units of capital, which

cost $25 each. Thus, variable costs are ($25)(6) = $150.

5-1

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