P(L) = 200L + 10L^2 - L^3
P'(L) = 200 + 20L - 3L^2
A(L) = 200 + 10L - L^2
A' = 10 - 2L
Max avg P is where L=5 That is L0
A(5) = 200+50-25 = 225
P'(5) = 200 + 100 - 75 = 225 = A(5)
A'(L0) = 0 means (P(L)/L)' = 0 at L0
(P'*L - P)/L^2 = 0
P'*L - P = 0
P' = P/L = A
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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