Let's take a step-by-step approach to solving this problem:
1. Understanding the Problem:
- We have a budget of [tex]$3533.
- The prices of inputs \( K \) and \( L \) are $[/tex]25.578 and $17.77 respectively.
- Our goal is to maximize the output given by the production function [tex]\( Q = 125 \cdot K^{0.3756} \cdot L^{0.1872} \)[/tex].
2. Setting Up the Equations:
- The budget constraint can be expressed as:
[tex]\[
25.578 \cdot K + 17.77 \cdot L = 3533
\][/tex]
3. Production Function Parameters:
- The production function has the parameters [tex]\( \alpha = 0.3756 \)[/tex] and [tex]\( \beta = 0.1872 \)[/tex].
4. Optimization Principle:
- To maximize the output, we need to equalize the marginal product per dollar spent on both [tex]\( K \)[/tex] and [tex]\( L \)[/tex]. This is done using the conditions derived from the marginal products and the budget constraint.
5. Optimal Values:
- After working through the necessary economic derivations and solving the constraint equation and the production optimization problem, the optimal values for [tex]\( K \)[/tex] and [tex]\( L \)[/tex] are found to be:
[tex]\[
K \approx 92.183
\][/tex]
[tex]\[
L \approx 66.131
\][/tex]
6. Resulting Output:
- Substituting these values back into the production function gives us the maximum output [tex]\( Q \)[/tex]:
[tex]\[
Q \approx 1498.347
\][/tex]
Thus, the combination of inputs [tex]\( K \)[/tex] and [tex]\( L \)[/tex] that maximizes output given the production function and the budget constraint is approximately:
[tex]\[
K = 92.183
\][/tex]
[tex]\[
L = 66.131
\][/tex]
with the resulting maximum output:
[tex]\[
Q \approx 1498.347
\][/tex]
