Economists use production functions to describe how output of a system varies with another variable such as labour or capital. For example, the production function P(L) = 200L + 10L^2 - L^3 goves the output of a system as a function of the number of labourers. The average product A(L) is the average output per labourer when L labourers are working: A(L) = P(L)/L. The marginal product M P (L) is the approximate change in output when one additional labourer is added to L labourers; that is, M P(L) = P'(L) ~ P(L + 1) - P(L).
(a) For the production function P(L) = 200L + 10L^2 - L^3, find the L-value corresponding to maximum average production, and call this value L0. Verify that P'(L0) = A(L0).
(b) Now let P(L) be any general production function (not just the one in part (a)), and suppose that the peak of the average production curve occurs at L = L0, so that A'(L0) = 0. Show that we must have M P(L0) = P'(L0) = A(L0).
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