Question

7. For the production function q= min(K, 4L) (a) Assume that capital K is fixed at 100 units. Derive and plot Page 2 of . The total product function q(L) ii. The marginal product function MPL(L) ii. The average product function AP(L) (b) Suppose the price of labour w is $1 and the rental rate r is also $1. Derive and plot all cost functions; that is: i. Short run total cost. ii. Variable cost. iii. Fixed cost. iv. Short run average cost. v. Average variable cost. vi. Average fixed cost vii. Marginal cost Where appropriate, you may plot several functions on the same graph. Be sure clearly label your curves to avoid confusion. to

Answer 1

7. a. In the short run, the production function transforms to:

q min100, 4L)

which can be interpreted as:

q 100, If L 25, 4L If L25

i) Plotting the function:

100 80 60 40 20 40 20 60 80 C 100

ii) The marginal product is:

MP 0 if L 25, 4 if L< 25

10 -5 5 10 15 10 30 26 20 LO

iii) Average product is:

100 if L 25, 4 if L< 25 AP=

10 10 30 40 50 20

b. The firm treats four units of labor and one unit of capital as compliments. Hence, at equilibrium:

K q.L= 4

The firm's cost function is of the form:

Cost wt rK

Substituting the values of L and K derived above and r and k from the question:

5q Cost 4 qCost 4

-60 -40 20 0f 40 60 20