Concave and convex functions PDF

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CONCAVE AND CONVEX FUNCTIONS In the case of the production

With 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 f(h) We take two values

Then:

The left-hand side is the function of the average of the two values, and the righthand side is the average of the function of the two values. 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. 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....