Do the following functions exhibit increasing, constant, or decreasing returns to scale? What happens to the marginal product of each individual factor as that factor is increased and the other factor held constant?

a. q = 3L + 2K

b. q = (2L + 2K)^1/2

c. q = 3LK²

d. q = L^1/2K^1/2

e. q = 4L^1/2 + 4K

The production function is,

q = 3L + 2K

Now multiply both capital land labor by 3, i..e. K’ = 3K and L’ = 3L. So, the output becomes:

q' = 9L + 6K

= 3 x (3L + 2K)

= 3q

Here, it is observed that as each input is increased by a factor of 3, the output level also increases by a factor of 3. Hence, the production function exhibits constant returns to scale (CRS).

In the original production function, the marginal product of capital (when labor is held fixed) is 2. The second-order derivative of MPK is zero. Thus as the capital increases, MPK does not change.

The marginal product of labor is 3. The second-order derivative of MPL is zero. Thus as the labor increases, MPL does not change.

The production function is,

q = (2L + 2K)^1/2

Now both capital and labor increase by a factor of 3, i.e. K’ = 3K and L’ = 3L. So,

q' = (6L + 6K)^1/2

$$\begin{gathered} =\sqrt{6}\times {\left(L+K\right)}^{1/2} \\ =\sqrt{6}q \end{gathered}$$

Here, it is observed that as each input is increased by a factor of 3, the output level also increases by less than the factor of 3. Hence, the production function exhibits diminishing returns to scale (DRS).

The marginal product of capital (when labor is held fixed) is

$\frac{\partial q}{\partial K}=\frac{2}{2{(2L+2K)}^{1/2}}$

Since K is in the denominator, as K increases, the MPK decreases.

The marginal product of labor is similarly

$\frac{\partial q}{\partial L}=\frac{2}{2{(2L+2K)}^{1/2}}$

Since L is in the denominator, as L increases, the MPL decreases.

Thus, both MPK and MPL decrease as the employment of capital and labor rise, respectively.

The production function is,

q =3 LK²

Now both capital and labor increase by a factor of λ, i..e. K’ = λK and L’ = λL. So,

$$\begin{gathered} q\text{'}=3\left(\lambda L\right){\left(\lambda K\right)}^2 \\ ={\lambda }^3·\left(3L{K}^2\right) \\ ={\lambda }^3·q \end{gathered}$$

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

The marginal product of capital is 6LK, so the MPK increases when K increases.

The marginal product of labor is 3K², so the MPL is constant.

The production function is,

q =L^1/2 K^1/2

Now both capital and labor increase by a factor of λ, i.e. K’ = λK and L’ = λL. So,

$q\text{'}={\left(\lambda L\right)}^{1/2}{\left(\lambda K\right)}^{1/2}=\lambda \left(L^{1/2}{K}^{1/2}\right)$

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).

The marginal product of capital is

$\frac{L^{1/2}}{2K^{1/2}}$

The MPK decreases as K increases.

The marginal product of labor is

$\frac{K^{1/2}}{2L^{1/2}}$

The MPL decreases as L increases.

Thus, both MPK and MPL decrease as the employment of capital and labor rise, respectively.

The production function is,

q =4L^1/2 + 4K

Now both capital and labor increase by a factor of λ, i.e. K’ = λK and L’ = λL. So,

$$\begin{gathered} q\text{'}=4{\left(\lambda L\right)}^{1/2}+4\left(\lambda K\right) \\ <\lambda (4L^{1/2}+4K) \end{gathered}$$

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

The marginal product of capital is 4, which is constant.

The marginal product of labor is $\frac{2}{L^{1/2}}$,

so the MPL decreases as L increases.

Thus, MPK remains constant while MPL decreases as the employment of capital and labor rise, respectively.