Leibniz 3.1.2

Diminishing marginal productivity

For an introduction to the Leibniz series, please see ‘Introducing the Leibnizes’.

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).