A firm has two variable factors and a production function f(x 1 , x 2 ) = x^1/2 x_2^1/4 . Please find the marginal product using formulas in the lecture notes by yourself. The price of its output is 4. Factor 1 receives a wage of ω1 and factor 2 receives a wage of ω2 . Required:
Write an equation that says that the value of the marginal product of factor 1 is equal to the wage of factor 1.

The equation that says that the value of the marginal product of factor 1 is equal to the wage of factor 1 is as follows:

1/2( x_2^1/4 / x_1^1/2) = ω1

Explanation:

Given:

f(x_1 , x_2 ) = x^1/2 x_2^1/4

Since x = x_1, we have:

f(x_1 , x_2 ) = x_1^1/2 x_2^1/4 ……………………………….. (1)

To obtain the marginal product of factor 1 (MP_1), the partial derivative of equation with respect to x_1 is taken as follows:

MP_1 = df(x_1 , x_2 )/dx_1

MP_1 = (1/2)x_1^((1/2)-1) x_2^1/4

MP_1 = (1/2)x_1^-(1/2) x_2^1/4, or 1/2( x_2^1/4 / x_1^1/2)

The equation that says that the value of the marginal product of factor 1 is equal to the wage of factor 1 can be obtained as follows:

MP_1 = ω1 …………………….. (2)

Substituting MP_1 = 1/2( x_2^1/4 / x_1^1/2) in equation (2), we have:

1/2( x_2^1/4 / x_1^1/2) = ω1 …………………… (3)

Equation (3) is therefore the equation that says that the value of the marginal product of factor 1 is equal to the wage of factor 1.