The Cobb$-$Douglas production function is a classic model from economics used to model output as a function of capital and labor. It has the

form

$f(L,C)=c_0{L}^{c_1{C}^c}$

where $c_0,c_1,$ and $c_2$ are constants. The variable $L$ represents the units of input of labor and the variable $C$ represents the units of input of

capital.

$($a$)$ In this example, assume $c_0=5,c_1=0\mathrm{.}25,$ and $c_2=0\mathrm{.}75.$ Assume each unit of labor costs $$25$ and each unit of capital costs $$75.$

With $$80,000$ available in the budget, develop an optimization model for determining how the budgeted amount should be allocated

between capital and labor in order to maximize output.

Max

s$.$t$.$

$80,000$

$L,C0$

$($b$)$ Find the optimal solution to the model you formulated in part $($a$).$ What is the optimal solution value $($in dollars$)?$ Hint: Put bound

constraints on the variables based on the budget constraint. Use $L3,000$ and $C1,000$ and use the Multistart option as described

in Appendix $8\mathrm{.}1.($Round your answers to the nearest integer when necessary.$)$

$

$at(L,C)=(,)$