Leibniz

3.1.3 Concave and convex functions

The concept of diminishing marginal product corresponds to the mathematical property of concavity. By looking at the mathematical idea of concave and convex functions, we can gain some further insights into the economic properties of production functions.

concave function
A function of two variables for which the line segment between any two points on the function lies entirely below the curve representing the function (the function is convex when the line segment lies above the function).

We saw in Leibniz 3.1.2 that in the case of the production function , with and , the marginal product of labour is diminishing. This means that when we move to the right along the graph of the production function, the slope of the curve decreases. A function with this property is said to be concave.

An implication of concavity (and its algebraic definition) is that ‘the function of the average is greater than the average of the function’. To illustrate what this statement means, suppose that for a function , we take any two values and . Then:

The left-hand side is the function of the average of the two values, and the right-hand side is the average of the function of the two values. (To see why the inequality holds, try drawing a concave production function, choosing two values on the horizontal axis, and finding the points on the diagram that correspond to the two averages.)

We can give this property a very neat economic interpretation. To understand what it means, consider the following example.

Suppose that Sakina has a production function like the one above, with and ; that is:

She is new to spinning cloth and is considering two different ways of organizing her time. Because she does not yet know anyone, she thinks that she should first spend some time talking to everyone around her. So, her average daily hours of work at the end of one season would be . Having established her position on the social scene, she would return to work with full fervour in the next month and work hours a day, every day. From her production function, we find that her cloth production under this regime would be for the first month, and for the second month. Her average production would thus be .

Alternatively, she could work on consolidating her social life and work time more steadily, working hours a day every day in both months. Notice that under this strategy, she gives up the same total hours of free time as under the previous approach—the total inputs are the same. What output can she then expect? She will get in each semester, which gives her on average.

Comparing these two possible strategies, Sakina realizes that in her case, slow and steady indeed wins the race, because her total output is higher when hours are constant rather than fluctuating. This is the economic implication of concavity.

By contrast, had we assumed that , we would have found that total output is higher when hours fluctuate: in this case, Sakina produces more when she works more intensely for a shorter period. When , the slope of the graph of the production function increases as hours increase, so the marginal product of labour is increasing rather than diminishing, as in Figure 3 in which . We would then describe the production function as convex rather than concave. A special case is the function : you can check by differentiating that for this production function, the graph of the marginal product of labour is an upward-sloping straight line.

Figure 3 The production function y = 1.5h1.6 and the corresponding marginal product.

Read more: Section 8.4 of Malcolm Pemberton and Nicholas Rau. 2015. Mathematics for economists: An introductory textbook, 4th ed. Manchester: Manchester University Press.