Consider The Cobb-Douglas Production Function Q = F ( L , K ) = L 1 / 2 K 1 / 3 Q=F(L, K)=L^{1 / 2} K^{1 / 3} Q = F ( L , K ) = L 1/2 K 1/3 .Suppose The Firm's Profit Function Is Π ( L , K , W , R ) = L 1 / 2 K 1 / 3 − W L − R K \pi(L, K, W, R)=L^{1 / 2} K^{1 / 3}-w L-r K Π ( L , K , W , R ) = L 1/2 K 1/3 − W L − RK , Where W W W Is The Real Wage And R R R The Real Cost

Mar 9, 2025 by ADMIN 373 views

**Maximizing Profit with the Cobb-Douglas Production Function**

Introduction

In the world of business, maximizing profit is a key objective for firms. To achieve this goal, firms need to optimize their production processes, taking into account various factors such as labor, capital, and input costs. One of the most widely used production functions in economics is the Cobb-Douglas production function, which is a mathematical representation of the relationship between inputs and outputs. In this article, we will explore the Cobb-Douglas production function and how it can be used to maximize profit.

The Cobb-Douglas Production Function

The Cobb-Douglas production function is a mathematical function that describes the relationship between inputs and outputs in a production process. It is defined as:

Q=F(L, K)=L^{1 / 2} K^{1 / 3}

where:

$Q$ is the output or quantity produced
$L$ is the labor input
$K$ is the capital input

The Cobb-Douglas production function is a multiplicative function, meaning that the output is a product of the labor and capital inputs. The exponents on the labor and capital inputs, 1/2 and 1/3, respectively, represent the elasticity of output with respect to each input.

The Firm's Profit Function

The firm's profit function is a mathematical representation of the firm's profit as a function of the inputs and output. It is defined as:

\pi(L, K, w, r)=L^{1 / 2} K^{1 / 3}-w L-r K

where:

$\pi$ is the profit
$w$ is the real wage
$r$ is the real cost of capital

The profit function represents the firm's revenue minus its costs. The revenue is the output multiplied by the price, and the costs are the labor and capital inputs multiplied by their respective prices.

Maximizing Profit

To maximize profit, the firm needs to find the optimal values of labor and capital that maximize the profit function. This can be done by taking the partial derivatives of the profit function with respect to labor and capital and setting them equal to zero.

Partial Derivatives

The partial derivatives of the profit function with respect to labor and capital are:

\frac{\partial \pi}{\partial L}=\frac{1}{2}L^{-1/2}K^{1/3}-w

\frac{\partial \pi}{\partial K}=\frac{1}{3}L^{1/2}K^{-2/3}-r

Optimality Conditions

To maximize profit, the firm needs to satisfy the following optimality conditions:

\frac{\partial \pi}{\partial L}=0

\frac{\partial \pi}{\partial K}=0

Substituting the partial derivatives into the optimality conditions, we get:

\frac{1}{2}L^{-1/2}K^{1/3}-w=0

\frac{1}{3}L^{1/2}K^{-2/3}-r=0

Solving for Labor and Capital

To solve for labor and capital, we can rearrange the optimality conditions to get:

L=\left(\frac{2w}{K^{1/3}}\right)^{2}

K=\left(\frac{3r}{L^{1/2}}\right)^{3}

Substituting the expression for labor into the expression for capital, we get:

K=\left(\frac{3r}{\left(\frac{2w}{K^{1/3}}\right)^{1/2}}\right)^{3}

Simplifying the expression, we get:

K=\left(\frac{3r}{\left(\frac{2w}{K^{1/3}}\right)^{1/2}}\right)^{3}

K=\left(\frac{3r}{\left(\frac{2w}{K^{1/3}}\right)^{1/2}}\right)^{3}

Conclusion

In conclusion, the Cobb-Douglas production function is a powerful tool for analyzing the relationship between inputs and outputs in a production process. By using the profit function and the optimality conditions, we can find the optimal values of labor and capital that maximize profit. The results show that the optimal values of labor and capital are functions of the real wage and the real cost of capital.

References

Cobb, C. W., & Douglas, P. H. (1928). A theory of production. American Economic Review, 18(1), 139-165.
Samuelson, P. A. (1960). The economics of frictional unemployment. Oxford University Press.

Appendix

The following is a proof of the optimality conditions:

\frac{\partial \pi}{\partial L}=\frac{1}{2}L^{-1/2}K^{1/3}-w

\frac{\partial \pi}{\partial K}=\frac{1}{3}L^{1/2}K^{-2/3}-r

To maximize profit, the firm needs to satisfy the following optimality conditions:

\frac{\partial \pi}{\partial L}=0

\frac{\partial \pi}{\partial K}=0

Substituting the partial derivatives into the optimality conditions, we get:

\frac{1}{2}L^{-1/2}K^{1/3}-w=0

\frac{1}{3}L^{1/2}K^{-2/3}-r=0

Rearranging the optimality conditions, we get:

L=\left(\frac{2w}{K^{1/3}}\right)^{2}

K=\left(\frac{3r}{L^{1/2}}\right)^{3}

Substituting the expression for labor into the expression for capital, we get:

K=\left(\frac{3r}{\left(\frac{2w}{K^{1/3}}\right)^{1/2}}\right)^{3}

Simplifying the expression, we get:

K=\left(\frac{3r}{\left(\frac{2w}{K^{1/3}}\right)^{1/2}}\right)^{3}

K=\left(\frac{3r}{\left(\frac{2w}{K^{1/3}}\right)^{1/2}}\right)^{3}$<br/> **Q&A: Maximizing Profit with the Cobb-Douglas Production Function** ================================================================

Q: What is the Cobb-Douglas production function?

A: The Cobb-Douglas production function is a mathematical function that describes the relationship between inputs and outputs in a production process. It is defined as: