concave function A function of twovariables for which the linesegment between any two pointson the function lies entirely belowthe curve representing the function(the function is convex when theline segment lies above the func-tion).

3.1.3 CONCAVE AND CONVEX FUNCTIONSThe concept of diminishing marginal product corresponds to themathematical property of concavity. By looking at themathematical idea of concave and convex functions, we can gainsome further insights into the economic properties of productionfunctions.

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 ofthe production function, the slope of the curve decreases. A function withthis property is said to be concave.

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

The left-hand side is the function of the average of the two values, and theright-hand side is the average of the function of the two values. (To see whythe inequality holds, try drawing a concave production function, choosingtwo values on the horizontal axis, and finding the points on the diagramthat correspond to the two averages.)

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

Suppose that Alexei has a production function like the one above, withand ; that is:

He has just started at university and is considering two different ways oforganizing his time. Because he does not yet know anyone, he thinks thefirst term might be better spent socializing, so that his average daily hoursof study for the end-of-semester exam would be . Having established hisposition on the social scene, he would return to study with full fervour inthe second semester and study hours a day, every day. From his produc-tion function, we find that his grades under this regime would befor the first semester exam, and for the second semester. Hisaverage exam result would thus be .

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Alternatively, he could work on his social life and academic results moresteadily, studying hours a day every day in both semesters. Notice thatunder this strategy, he gives up the same total hours of free time as underthe previous approach—the total inputs are the same. What grades can hethen expect? He will get in each semester, which gives him

on average.Comparing these two possible strategies, Alexei realizes that in his case,

slow and steady indeed wins the race, because his total output is higherwhen hours are constant rather than fluctuating. This is the economicimplication of concavity.

By contrast, had we assumed that , we would have found that totaloutput is higher when hours fluctuate: in this case, Alexei learns more whenhe studies more intensely for a shorter period. When , the slope of thegraph of the production function increases as hours increase, so the mar-ginal product of labour is increasing rather than diminishing, as in Figure 3in which . We would then describe the production function asconvex rather than concave. A special case is the function : you cancheck by differentiating that for this production function, the graph of themarginal product of labour is an upward-sloping straight line.

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

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

3.1.3 CONCAVE AND CONVEX FUNCTIONS

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