Question

A competitive firm has a production function described as follows. “Weekly output is the square root of the minimum of the number of units of capital and the number of units of labor employed per week.” Suppose that in the short run this firm must use 16 units of capital but can vary its amount of labor freely. a. Write down a formula that describes the marginal product of labor in the short run as a function of the amount of labor used. (Be careful at the boundaries.) b. If the wage is w = $1 and the price of output is p = $4, how much labor will the firm demand in the short run? c. What if w = $1 and p = $10? d. Write down an equation for the firm’s short-run demand for labor as a function of w and p.

The answers are:
a. MP=1/(2L^1/2) if L<16, MP=0 if L>16.
b. 4.
c. 16.
d. L=(p/2w)<sup>2

Please show work</sup>

Answer 1

A) Production function is perfect Complements type

Q = √(Min(L,K))

thus, Q^{​​​​​​2} = Min(L,K)

now K is fixed at 16,

So, Q = Min(L,16)^.5

Now if L< 16, then Q = √L

So MPL = dQ/dL = (1/2)*(1/√L) = 1/(2√L)

If L > 16 , Q = √16 =4, so MPL = 0

so , if L< 16, MPL = 1/(2√L)

if L>16 , MPL = 0

.

b) now if L<16,

Then at eqm, VMPL = w

P*MPL = w

4/2√L = 1

L*= 4

.

C) now when P=10

So, 10/2√L = 1

L' = 25

But since Maximum possible L = 16

So L" = 16

.

D) now demand for L

VMPL = w

P/2√L = w

L* = (P/2w)²