In Example 6.4, wheat is produced according to the production function

q = 100(K^0.8L^0.2)

a. Beginning with a capital input of 4 and a labor input of 49, show that the marginal product of labor and the marginal product of capital are both decreasing.

b. Does this production function exhibit increasing, decreasing, or constant returns to scale?

The production function is,

q = 100(K^0.8 L^0.2)

If labor is L = 49 units, the marginal product of capital is,

$$\begin{gathered} MPK=\frac{\partial q}{\partial K}=100\times 0.8\times K^{0.2}\times {\left(49\right)}^{0.2} \\ =174.2325\times K^{0.2} \end{gathered}$$

MPK is further differentiated with respect to capital to conclude whether the function is increasing or decreasing.

$$\begin{gathered} \frac{{\partial }^{2}q}{\partial K^2}=100\times 0.8\times 0.2\times K^{-0.8}\times {\left(49\right)}^2 \\ =34.8465\times K^{-0.8}<0 \end{gathered}$$

Thus as capital rises, the marginal productivity of each unit diminishes. Hence MPK is decreasing.

If capital is K = 4 units, the marginal product of labor is,

$$\begin{gathered} MPL=\frac{\partial q}{\partial K}=100\times 0.2\times L^{0.8}\times {\left(4\right)}^{0.2} \\ =26.39015\times L^{0.8} \end{gathered}$$

MPL is further differentiated with respect to labor to conclude whether the function is increasing or decreasing.

$$\begin{gathered} \frac{{\partial }^{2}q}{\partial K^2}=100\times 0.8\times 0.2\times L^{-0.2}\times 4^{0.2} \\ =27.8772\times L^{-0.2}<0 \end{gathered}$$

Thus, as labor employment rises, the marginal productivity of each unit falls. Hence MPL is decreasing.

The production function is,

q = 100(K^0.8 L^0.2)

Now both capital land labor increases by λ unit, i.e. K’ = λK and L’ = λL. So,

$$\begin{gathered} q=100\times {\left(\lambda K\right)}^{0.8}\times {\left(\lambda L\right)}^{0.2} \\ ={\lambda }^{\left(0.8+0.2\right)}\times 100\times K^{0.8}\times L^{0.2} \\ =\lambda q \end{gathered}$$

Here, it is observed that as each input is increased by a factor of λ, the output level also increases by a factor of λ. Hence, the production function exhibits constant returns to scale (CRS).