Chapter 13, Problem 2P

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:

$f\left(L,C\right)=c_0{L}^{c_1}{C}^{c_2},$

where c₀, c₁, and c₂ are constants. The variable L represents the units of input of labor, and the variable C represents the units of input of capital.

1. a. In this example, assume c₀ = 5, c₁ = 0.25, and c₂ = 0.75. Assume each unit of labor costs $25 and each unit of capital costs $75. With $75,000 available in the budget, develop an optimization model to determine how the budgeted amount should be allocated between capital and labor in order to maximize output.
2. b. Find the optimal solution to the model you formulated in part (a). (Hint: When using Excel Solver, use the Multistart option with bounds 0 ≤ L ≤ 3,000 and 0 ≤ C ≤ 1,000.)