Question

The Widget Co. can produce widgets according to the formula: q = 5K^3/4 L^1/4 where q is the output of widgets, and K, L are the quantities of capital and labor used.

a. Are there constant, increasing or decreasing returns to scale in widget production? Explain.
b. Are there, constant, increasing or decreasing marginal products of factors? Explain
c. In the short run, the amount of capital used by company A. is fixed. Derive the short-run cost function. (Note that the short-run cost function will show C as a function of Q, K and the factor prices w and r.)
d. Derive the long-run cost function.

Answer 1

A) Constant Return to Scale

B) Decreasing Marginal Products

C) Short Run Cost Function = w*(Q/5)4 *(1/K​​​​​`3) + rK`

D) (2Q/5)*(r​​​​​3/4​​​​​​)*(w​​​​​1/4)

Step-by-step explanation:

A) Q(tL,tK) = 5*(tK)3/4*(tL)1/4

= t*Q(L,K)

Hence it exhibits constant return to scale

B) Here MPK = dQ/dK

= (3/4)* 5*L​​​​​​​​​​​1/4*K​​​​​​-1/4

So dMPK/dK = (-3/16)*5*L​​​​​​1/4*K​​​​​​-5/4

Hence dMPK/dK < 0

thus exhibits decreasing Marginal Products

(Similarly for Labor also)

C) let K is fixed at K`

So Q = 5*K`3/4*L​​​​​1/4

So L = (Q/5)4*(1/K​​​​​​`3)

So Short Run Cost Function = w*L + r*K`

C = w*(Q/5)4 *(1/K​​​​​`3) + rK`

D) in long run, MRTS = MPL / MPK = w/r

So K/L = w/r

Thus rK = wL

So from production function

Q = 5*(wL/r)3/4*L​​​​​​1/4

= 5*(w/r)3/4*L

L* = (Q/5)*(r/w)3/4

similarly K* = (Q/5)*(w/r)1/4

so Long Run Cost Function = wL* + rK*

= (2Q/5)*(r​​​​​3/4​​​​​​)*(w​​​​​1/4)