Question

A company A can produce widgets according to Q=5K^3/4L^1/4 where Q is the output of widgets, and K, L are quantities of capital and labor used.

1. Are there constant, increasing or decreasing returns to scale in widget production? Explain.
2. Are there, constant, increasing or decreasing marginal products of factors? Explain
3. 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.)
4. Derive the long-run cost function.

Answer 1

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

= t*Q(L,K)

Hence it exhibits CRS

2) now 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 diminishing MP

(Similarly for Labor also)

3) let K is fixed at K`

So Q = 5*K`^3/4*L^{​​​​​1/4}

So L = (Q/5)⁴*(1/K^{​​​​​​`3})

So SRTC = w*L + r*K`

C = w*(Q/5)⁴ *(1/K<sup>​​​​​`3</sup>) + rK`

4) 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 LRTC = wL* + rK*

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