Calculating Productivity and Production. $(32\mathrm{.}5\%)$ You are managing a fairly complicated production

process of digital readers, and you are told the following information:

Important Information:

Production Function: $,-{0\mathrm{.}05}^{**}{L}^3+{0\mathrm{.}75}^{**}{L}^2+{2\mathrm{.}5}^{**}L+L^{\frac{1}{2}**}{K}^{\frac{1}{2}}$

wage per unit of labor $(w)$ is $$20$

rent per unit of capital $(r)$ is $$35$

units of capital $(K)$ is $10$

$($a$)$ Incorporating the formulae and values above, please show the formula for the following items:

$(10\%)$

a$.$ Average product of labor

b$.$ Marginal Product of Labor

c$.$ Average Variable Cost

d$.$ Marginal Cost

$($b$)$ Using Excel, provide a graph of the marginal product and average product of labor for values of

labor between $2$ and $10.$ In this graph, please indicate when the firm is experiencing increasing

and decreasing marginal returns to labor and clearly explain in words the law of diminishing

marginal returns. $(7\mathrm{.}5\%)$

$($c$)$ Using Excel and the appropriate formulae, please provide a table and graph of the average

variable, average total, and marginal costs when your labor inputs are $2-10.$ Note, your graph

should have quantity as the X$-$axis variable. Clearly describe the graphs $($i$.$e$.,$ when and where

rising and to what extent are the different lines related$).$ Given the information provided to you

up to this point, what time$/$planning horizon must we be considering given the information

provided? $(7\mathrm{.}5\%)$