Problem 35 - Calculus with Concepts in Calculus

Question

Let \(x\) represent capital and \(y\) labor in the manufacture of \(f(x, y)\) units of a given product. Assume that capital costs \(a\) dollars per unit and labor costs \(b\) dollars per unit and that there are \(c\) dollars available, so that \(a x+b y=c\). a. Using Lagrange multipliers, show that production is maximum at the point \(\left(x_{0}, y_{0}\right)\) such that $$ \frac{f_{x}\left(x_{0}, y_{0}\right)}{a}=\frac{f_{y}\left(x_{0} y_{0}\right)}{b}=\lambda $$ where \(\lambda\) is the Lagrange multiplier, called the equimarginal productivity of the production function \(f\). b. Let \(f\) be the Cobb-Douglas production function given by $$ f(x, y)=x^{\alpha} y^{\beta} \quad \text { for } x>0 \text { and } y>0 $$ where \(\alpha\) and \(\beta\) are positive constants less than 1 . Using (a), show that at maximum production \(f\left(x_{0}, y_{0}\right)\) we have $$ \frac{y_{0}}{x_{0}}=\frac{\beta a}{\alpha b} $$ which is independent of the money available.

Step-by-Step Solution

Answer

The production is maximized when \( \frac{y_0}{x_0} = \frac{\beta a}{\alpha b} \), independent of \( c \).

Step 1: Set up the Lagrangian

To maximize the production function \( f(x, y) \) subject to the constraint \( ax + by = c \), we define the Lagrangian: \( L(x, y, \lambda) = f(x, y) + \lambda(c - ax - by) \).

Step 2: Calculate Partial Derivatives

Find the partial derivatives of the Lagrangian with respect to \( x, y, \) and \( \lambda \): \( \frac{\partial L}{\partial x} = f_x(x, y) - \lambda a = 0 \), \( \frac{\partial L}{\partial y} = f_y(x, y) - \lambda b = 0 \), and \( \frac{\partial L}{\partial \lambda} = c - ax - by = 0 \).

Step 3: Solve for the Ratio of Marginal Products (Part a)

From \( \frac{\partial L}{\partial x} = 0 \), we have \( f_x(x, y) = \lambda a \). From \( \frac{\partial L}{\partial y} = 0 \), we have \( f_y(x, y) = \lambda b \). Thus, \( \frac{f_x(x_0, y_0)}{a} = \frac{f_y(x_0, y_0)}{b} = \lambda \) shows the condition for maximum production using Lagrange multipliers.

Step 4: Substitute Cobb-Douglas Function (Part b)

Given \( f(x, y) = x^\alpha y^\beta \), compute the partial derivatives \( f_x = \alpha x^{\alpha-1} y^\beta \) and \( f_y = \beta x^\alpha y^{\beta-1} \).

Step 5: Use Lagrangian Conditions on Cobb-Douglas

Substitute the partial derivatives into the Lagrangian conditions: \( \alpha x^{\alpha-1} y^\beta = \lambda a \) and \( \beta x^\alpha y^{\beta-1} = \lambda b \). From these, we get \( \frac{\alpha x^{\alpha-1} y^\beta}{a} = \frac{\beta x^\alpha y^{\beta-1}}{b} \).

Step 6: Simplify to Find Ratio of \( y_0/x_0 \)

Divide one equation by the other from Step 5: \( \frac{\alpha y}{a} = \frac{\beta x}{b} \). Solving for \( \frac{y}{x} \), we find \( \frac{y_0}{x_0} = \frac{\beta a}{\alpha b} \). This shows that \( \frac{y_0}{x_0} \) depends only on the cost coefficients \( a \) and \( b \) and the production parameters \( \alpha \) and \( \beta \), not on total funds \( c \).

Key Concepts

Cobb-Douglas production function

The Cobb-Douglas production function is a key concept in economics, particularly in understanding how different quantities of input resources like capital and labor influence the quantity of output in production. This function is commonly expressed as \(f(x, y) = x^{\alpha} y^{\beta}\), where \(x\) and \(y\) represent inputs such as capital and labor, respectively. The exponents \(\alpha\) and \(\beta\) are constants that reflect the output elasticities of capital and labor. This means they show how responsive the output is to a change in the inputs.
Several properties make the Cobb-Douglas function appealing:

• Constant Returns to Scale: If \( \alpha + \beta = 1 \), it indicates constant returns to scale. Doubling the inputs results in a doubling of output.
• Output Elasticities: The value of \(\alpha\) is the percentage increase in production for a 1% increase in capital, while \(\beta\) measures the same for labor.
• Isoquants: These are curves that represent combinations of inputs that result in the same level of output, which are typically curved due to the diminishing marginal rate of technical substitution.

Understanding the Cobb-Douglas function helps analyze how to allocate resources efficiently to maximize output.

equimarginal productivity

Equimarginal productivity is a fundamental principle in economics that relates to how resources should be distributed among different uses to achieve optimal output. In the context of the original problem, it is used to determine the point of maximum production by using the concept of Lagrange multipliers.
By setting up the Lagrangian \(L(x, y, \lambda) = f(x, y) + \lambda(c - ax - by)\), where \(c\) is the total amount of resources available, and \(ax + by = c\) represents the budget constraint, we can find the optimal allocation of capital \(x\) and labor \(y\) that maximizes production. The condition \( \frac{f_x(x_0, y_0)}{a} = \frac{f_y(x_0, y_0)}{b} = \lambda \) represents equimarginal productivity.
This means that the marginal rate of substitution between capital and labor should equal the ratio of their prices for resources to be utilized most effectively. The intuition is that if one input provides more marginal benefit per cost unit than the other, resources should be shifted towards it until equilibrium is reached. This helps in making decisions about resource allocation to maximize output under cost constraints.

constraint optimization in economics

Constraint optimization is a powerful mathematical approach used in economics to determine the best way to allocate resources, subject to a set of limitations or constraints. In the exercise, the goal is to maximize the production function subject to budget constraints using the method of Lagrange multipliers.
The Lagrangian \( L(x, y, \lambda) \) is constructed to incorporate both the objective function (production function \( f(x, y) \)) and the constraint derived from the available budget \( ax + by = c \). By solving the simultaneous equations \( \frac{\partial L}{\partial x} = 0 \), \( \frac{\partial L}{\partial y} = 0 \), and \( \frac{\partial L}{\partial \lambda} = 0 \), the optimal levels of \(x\) (capital) and \(y\) (labor) can be found.
This type of optimization helps you find the point where the production function is balanced with the available budget, ensuring production is maximized given the cost constraints. The result, \( \frac{y_0}{x_0} = \frac{\beta a}{\alpha b} \) in this setup, indicates that the optimal mix of inputs is determined by their respective cost and contribution to production, not the total budget itself. Understanding constraint optimization allows economists and businesses to make informed decisions about allocating limited resources efficiently.