If a business has 'L' units of labor (e.g. workers) and K units of capital (e.g. production...

Question:

If a business has 'L' units of labor (e.g. workers) and K units of capital (e.g. production machines), then its production can be modeled by a Cobb-Douglass Production Function of:

{eq}F(L,K) = \beta L^{\alpha}K^{1-\alpha} {/eq},

where the constants {eq}\alpha, \beta {/eq} depend on the business model. However, both paying for labor and buying capital cost money. The goal is to maximize the production function while keeping your total money spent under the company's budget.

Assume a business has a Cobb-Douglas production function where {eq}\beta = 20 {/eq} and {eq}\alpha = \frac{1}{4} {/eq}. If it costs $100 per unit of labor and$500 per unit of capital, with a total budget of \$20,000, then how much money whould the comapny spend on each to maximize production?

(Hint: The first step is to figure out an equation in terms of L and K, which describes how much money you can spend.)

Lagrange method of constrained optimization:

In mathematical optimization, Lagrange multipliers is a method for finding the local minima and maxima of a function subject to the some equality constraints. The fundamental idea in this procedure is to change a constrained problem into an unconstrained one so as to be able to apply derivative tests for finding stationary points. These points are further classified into minima, maxima or saddle points based on the sufficient second-order conditions. One of the primary advantages of this technique is its wide spread use in advanced optimization problems due to the freedom of avoiding explicit parameterization of the problem in the form of constraints.

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Production function of the firm:

{eq}F(L,K) = \beta L^{\alpha}K^{1-\alpha} = 20L^{1/4}K^{3/4} {/eq}

Cost budget constraint of the firm:

{eq}P_L L...