Question

Suppose a soap-manufacturing production process is described by the following equation:

Y = a + b log K + с log L

Where,

Y= Output (number of soaps produced) K=Capital

L=Labor

a, b and c are constants

Suppose 0<a<1, 0< b<1 and 0<c<1

1. Find the Marginal Product of Labor (MPL) and Marginal Product of Capital (MPK) in the production of soap
2. Is MPL diminishing, increasing or constant as L increases?
3. Is MPK diminishing, increasing or constant as K increases? 

Answer 1

1. The marginal product of labor (MP_L) is calculated as follows:

$MP_L =\frac{ ∂Y}{∂L}\\=\frac{∂(a + b\space log\space K + c\space logL)}{∂L}\\=\frac{c}{L}$

The marginal product of labor (MP_K) is calculated as follows:

$MP_K =\frac{ ∂Y}{∂K}\\=\frac{∂(a + b\space log\space K + c\space logL)}{∂L}\\=\frac{b}{K}$

2. The marginal product of labor is:

$MP_L =\frac{ c}{L}$

The labor is in denominator. If L increases then MP_L decreases. Thus, it is diminishing.

3. The marginal product of capital is:

$MP_k =\frac{ b}{K}$

The capital is in denominator. If K increases then MP_K decreases. Thus, it is diminishing.