Question

For the production function y = KαL β , where α, β ∈ (0, 1)

(a) Derive the marginal production of K and L. Show that it is decreasing.

Answer 1

For the production function y = KαL β , where α, β ∈ (0, 1)

Marginal product of K( MP$_{K}$) is;

$\frac{\delta y}{\delta K}=\alpha K^{\alpha-1}L^{\beta}$

Marginal product of L (MP$_{L}$ ) is:

$\frac{\delta y}{\delta L}=\beta K^{\alpha}L^{\beta-1}$

To show that the marginal product of capital diminishes as the capital input increases with the labour input held constant, I calculate its derivative with respect to K:

$\frac{\delta MP_{K}}{\delta K}=$ $\alpha^{2}-\alpha K^{\alpha-2}L^{\beta}$

To show that the marginal product of labour decreases as the labour input increases with the capital input held constant, we calculate its derivative with respect to L:

$\frac{\delta MP_{L}}{\delta L}=$ $\beta^{2}-\beta K^{\alpha}L^{\beta-2}$