Question

A company that produces software has the following production function ?=?3?3, where  is labour and  is capital. The price of labour is $6 per hour and the price of capital is $3 per unit.

a.     Does this production function exhibit constant, decreasing or increasing returns to scale? Explain in words or give an example with different scales of inputs.

b.    What is the optimal combination of inputs that this company would use if it wants to produce 2,400 units of software?

Answer 1

?=?3?3

a.

augment this production by factor

Q(L, K) = (L)3*(K)3 = ^6Q(K, L) -----equation 1

Let a typical production function be:

Y = F(k, L)
We augment the Y to a factor "t"
f(tL, tK) = t* f(L, K)
If = 1 , the f(L, K) exhibits Constant Returns to Scale (CRS)
If > 1 , the f(L, K) exhibits Increasing Returns to Scale (IRS)
If < 1 , the f(L, K) exhibits Decreasing Returns to Scale (DRS)

Now, from equation 1

Q(L, K) = ^6Q(K, L)

Since, = 6 > 1. So, this production function exhibits Increasing returns to Scale. It implies that if we double both labor and capital inputs, then the increment in the value of output will be more than double.

b.

MRTS = (3L^2K^3) / (3K^2L^3)

= (K/L)

at equilibrium, MRTS = w/r

(K/L) = 6/3 = 2

K = 2L

Put K in the production function,

Q = L^3*(2L)^3

Q = 8*L^6

at Q = 2400

2400 / 8 = L^6

L = 2.5

K = 5

Optimal combination = (2.5, 5)