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Question

# Economists use production functions to describe how the output of a system varies with respect to another variable such as labor or capital. For example, the production function $P(L)=200 L+10 L^2-L^3$ gives the output of a system as a function of the number of laborers $L$. The average product $A(L)$ is the average output per laborer when $L$ laborers are working; that is $A(L)=P(L) / L$. The marginal product $M(L)$ is the approximate change in output when one additional laborer is added to $L$. laborers; that is, $M(L)=\frac{d P}{d L}$. (a). For the production function given here, compute and graph $P, A$, and $M$. (b). Suppose the peak of the average product curve occurs at $L=L_0$, so that $A^{\prime}\left(L_0\right)=0$. Show that for a general production function, $M\left(L_0\right)=A\left(L_0\right)$

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$(a)$ Plotted using Desmos Graphing Calculator. ## Recommended textbook solutions #### Vector Mechanics for Engineers: Statics

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