Answer to Question #250000 in Microeconomics for Gabs

Question

 Show that the quantity of labor(X1) and capital(X2) that a firm demands decreases with a factor’s own factor price (w for labor and r for capital) and increases with the output price (P) when the production function is a Cobb-Douglas of the form q=AX1^αX2^β

Answer 1

$Q=AX_1^\alpha X_2^\beta$

$\alpha$ represents the elasticity of production of labor.

$\beta$ represents the elasticity of production of capital.

A - represents the technology of the production process. The higher the value of A, the higher would be the level of output that can be produced by any combination of the inputs.

The Cobb Douglas function is a homogeneous function, the degree of homogeneity of the function being $\alpha+\beta$ .

$A(tX_1)^\alpha (tX_2)^\beta=t^{\alpha-\beta} AX_1^\alpha X_2^\beta$

$=t^{\alpha-\beta}$ .

where t is a positive real number.

From the above equation, we find out that if L and K are increased by the factor t, Q would increase by the factor $t^{\alpha-\beta}$ .

When the factor price of labor and capital increases, less of them will be demanded by the firm.

When Q is increased by the factor $t^{\alpha-\beta}$ ,the firm's demand is increased by the same factor.