Answer to Question #251590 in Microeconomics for Jude

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=AX1αX2β

Answer 1

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

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

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

A is an implication of the technology used in the production process. When the value of A is high, the level of output that can be produced by any combination of the outputs is also high.

The Cobb Douglas function given is a homogeneous function and the degree of homogeneity is given by: $\alpha$ +$\beta$ .

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

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

t is a real positive value.

From the above equation, we get the implication that if L and K by the factor t, Q will increase by the factor $t^{\alpha-\beta}$

In increase in factor prices of labor and capital will result in less of them being demanded by the firm.

However, when Q increases by the factor $t^{\alpha-\beta}$,the firm's demand for the input factors also increases by $t^{\alpha-\beta}$ .