Question

Consider the following production function: Q = 55⋅K^2/3L^4/3

A. Determine whether this production function has increasing, decreasing, or constant returns to scale. Explain and show your work.

B. Does this production function obey the law of diminishing marginal product in labor? Explain and show your work.

Answer 1

Answer

A. Increasing returns to scale.

The production function: Q = 55. K^2/3 . L^4/3 ......(1)

Let the inputs capital(K), and labor(L) both are increased by 'v'. With the increase in capital, and labor, the production(Q) is changed to Q^*.

$\therefore$ Q^* = 55 . (vK)^2/3 . (vL)^4/3

Or, Q^* = 55 . v^2/3 . K^2/3 . v^4/3 . L^4/3

Or, Q^* = 55 . v^{2/3 + 4/3} . K^2/3 . L^4/3

Or, Q^* = 55 . v^{2/3 + 4/3} . K^2/3 . L^4/3

Or, Q^* = 55 .v^6/3 . K^2/3 . L^4/3

Or, Q^* = 55 .v² . K^2/3 . L^4/3

Or, Q^* = v² . (55. K^2/3 . L^4/3)

Or, Q^* = v² .Q [ from equation(1) , 55. K^2/3 . L^4/3 = Q]

So, we see that when inputs are increased by 'v' , the output is increased by 'v²'. Now, as the proportionate increase in output is more than the proportionate increase in inputs, the production function exhibits here increasing returns to scale.

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B. Q = 55. K^2/3 . L^4/3

The marginal product of labor(MP_L) = change in output for a change in one additional unit of labor, keeping other factors constant. If the production function obeys the law of diminishing marginal product in labor, then the change in MP_L for the change in one more unit of labor will be less than zero(0) , or negative.

Q = 55. K^2/3 . L^4/3

Differentiating the above production function with respect to labor, keeping capital constant, we get,

dQ / dL = 55 . 4/3 . L^{(4/3) - 1} . K^2/3

Or, dQ / dL = MP_L = 220 / 3 . L^{(4-3) /3} . K^2/3

Or, MP_L = 220 / 3 . L^1/3 . K^2/3

Differentiating the above equation with respect to 'L', and let us see if the production function obeys the law of diminishing marginal product in labor;

d(MP_L) / dL = (1/3) . (220/3) . L^{(1/3) -1} . K^2/3

Or, d(MP_L) / dL = (220 / 9) . L^-2/3 . K^2/3$>$0

Here we see the the 'd(MP_L) / dL^' is greater than '0'. So, the production function does not obey the law of diminishing marginal product in labor here.

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