Leibniz

3.1.2 Diminishing marginal productivity

Alexei’s production function has the property of diminishing marginal productivity. We can see it graphically: the graph gets flatter as the hours of study per day increase. What does this mean for the mathematical properties of the production function?

If the production function is $y=f(h)$, the marginal product of labour is $\frac{dy}{dh}=f’(h)$, so the marginal product of labour diminishes as $h$ increases if:

$$\frac{d}{dh}\left( \frac{dy}{dh} \right) = \frac{d^{2}y}{dh^{2}} < 0$$

or equivalently:

$$f’’(h) \lt 0$$

That is, the second derivative of the production function is negative.

An example

Consider again the production function:

$$y = Ah^\alpha$$

where $A$ and $\alpha$ are constants such that $A \gt 0$ and $0 \lt \alpha \lt 1$. We showed in Leibniz 3.1.1 that, for this production function

$$\text{MPL} = \frac{dy}{dh} = \alpha Ah^{\alpha -1}$$

which is positive as long as the hours of study $h$ are positive. What we want to show next is that as hours $h$ increase, this marginal product becomes smaller and smaller.

One way to see it is to focus on the expression

$$\alpha Ah^{\alpha -1}$$

$h$ is raised to the power $\alpha -1$, which is negative, because $\alpha \lt 1$. Recall from the property of negative exponents that as $h$ increases, $h^{\alpha -1}$ decreases, and since $A$ and $\alpha$ are positive, so does $\alpha Ah^{\alpha -1}$, the marginal product of labour.

Alternatively we can show that the marginal product is decreasing by differentiating it:

$$\frac{d^2 y}{dh^2} = (\alpha -1)\alpha Ah^{\alpha -2}= \alpha(\alpha -1)\frac{Ah^\alpha}{h^2} =\alpha(\alpha -1)\frac{y}{h^2}$$

When $h$ is positive, we know that $y$ is positive too. Then when $0 \lt \alpha \lt 1$, $\alpha(\alpha -1) \lt 0$, and so:

$$\frac{d^2 y}{dh^2} =\alpha(\alpha -1)\frac{y}{h^2} \lt 0$$

This is what we wanted to show: the second derivative of the production function is negative, so the marginal product falls as $h$ increases. In other words, the marginal product of labour is diminishing.

In Leibniz 3.1.1 we showed that when $\alpha \lt 1$, the marginal product is less than the average product. This property is closely related to the concept of diminishing marginal product: if the marginal product of a production function is diminishing for all values of the input, then it is also true that the marginal product is less than the average product (MPL < APL).

Figure 2 shows the graph of the production function $y = Ah^\alpha$ for the case where $\alpha =0.4$ and $A=30$, together with the graph of the marginal product of labour. For each value of $h$, the upper graph shows the value of $y$, and the lower graph shows the slope of the production function, $dy/dh$. You can see that the marginal product of labour decreases with $h$.

Read more: Sections 6.4 and 8.4 of Pemberton and Rau (2016).