Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

Consider the Cobb-Douglas production model for a manufacturing process depending on three inputs \(x, y\), and \(z\) with unit costs \(a, b\), and \(c\), respectively, given by $$ P=k x^{\alpha} y^{\beta} z^{\gamma}, \quad \alpha>0, \beta>0, \gamma>0, \alpha+\beta+\gamma=1 $$ subject to the cost constraint \(a x+b y+c z=d\). Determine \(x, y\), and \(z\) to maximize the production \(P\).

Step-by-Step Solution

Answer

The optimal values are \(x = \frac{\alpha d}{a}\), \(y = \frac{\beta d}{b}\), \(z = \frac{\gamma d}{c}\).

Step 1: Formulate the Lagrangian

To maximize the production function \( P = kx^{\alpha} y^{\beta} z^{\gamma} \) subject to the cost constraint, we set up the Lagrangian function as follows: \[ L = k x^{\alpha} y^{\beta} z^{\gamma} + \lambda (d - ax - by - cz) \] where \( \lambda \) is the Lagrange multiplier.

Step 2: Derive the first-order conditions (FOC)

Differentiate the Lagrangian with respect to \( x, y, z, \) and \( \lambda \), and set the derivatives equal to zero. These are the first-order conditions: \[ \frac{\partial L}{\partial x} = \alpha k x^{\alpha-1} y^{\beta} z^{\gamma} - \lambda a = 0 \]\[ \frac{\partial L}{\partial y} = \beta k x^{\alpha} y^{\beta-1} z^{\gamma} - \lambda b = 0 \]\[ \frac{\partial L}{\partial z} = \gamma k x^{\alpha} y^{\beta} z^{\gamma-1} - \lambda c = 0 \]\[ \frac{\partial L}{\partial \lambda} = d - ax - by - cz = 0 \]

Step 3: Solve for Lagrange multiplier \( \lambda \)

From the first-order conditions, isolate \( \lambda \) in each equation: \[ \lambda = \frac{\alpha k x^{\alpha-1} y^{\beta} z^{\gamma}}{a} = \frac{\beta k x^{\alpha} y^{\beta-1} z^{\gamma}}{b} = \frac{\gamma k x^{\alpha} y^{\beta} z^{\gamma-1}}{c} \].

Step 4: Set up proportionality equations

By equating the expressions for \( \lambda \), derive proportionality relationships:\[ \frac{\alpha}{a} = \frac{\beta}{b} \cdot \frac{x}{y} \]\[ \frac{\beta}{b} = \frac{\gamma}{c} \cdot \frac{y}{z} \]. These give the conditions \[ \frac{x}{y} = \frac{\alpha/\beta \cdot b}{a} \]\[ \frac{y}{z} = \frac{\beta/\gamma \cdot c}{b} \].

Step 5: Solve the system of equations

Using the proportionality conditions and the constraint equation \( a x + b y + c z = d \), express \( x, y, \) and \( z \) in terms of one variable and substitute into the constraint:Substitute expressions for \( x \) and \( y \) into the cost constraint to solve for \( z \), then use its value to find \( y \) and \( x \).

Step 6: Compute the values for \( x, y, \) and \( z \)

After finding one of \( x, y, z \), back-calculate the other two values using the proportional relationships and the constraint. These values are:\[ x = \frac{\alpha d}{a} \text{ where } \alpha = \frac{a}{a+b+c} \]\[ y = \frac{\beta d}{b} \text{ where } \beta = \frac{b}{a+b+c} \]\[ z = \frac{\gamma d}{c} \text{ where } \gamma = \frac{c}{a+b+c} \]

Key Concepts

Lagrangian optimization

Lagrangian optimization is a powerful mathematical strategy used to find the optimal points for functions with constraints. It involves introducing additional variables known as Lagrange multipliers, which help to incorporate the constraints directly into the optimization problem.
In the context of economic models such as the Cobb-Douglas production function, the goal is to maximize the output while considering cost constraints. The Lagrangian is formed by combining the original function and the constraint. For instance, when maximizing production given a cost constraint, you formulate a Lagrangian like this:

• The original production function (\( P = kx^{\alpha} y^{\beta} z^{\gamma} \))
• Plus the constraint term (\( \lambda(d - ax - by - cz) \)) which penalizes the solution whenever the constraint is not satisfied

This combination allows you to systematically explore the favorable inputs (labor, capital, etc.) that align with the budget constraint.
By applying the Lagrangian optimization approach, you derive conditions that ensure not only a maximum output but also adherence to budgetary limits, offering valuable insights into resource allocation.

Lagrange multiplier method

The Lagrange multiplier method is a technique used to include constraints in an optimization problem. By introducing the Lagrange multiplier \( \lambda \), it becomes possible to integrate the constraint directly into the optimization process, forming a new function (the Lagrangian).
The magic of the Lagrange multiplier lies in its ability to describe how the optimal solution changes with the constraint. In an economic production context, it helps determine the portion of the budget needed for each input to maximize production. Here's a closer look at its implementation:

• First, differentiate the Lagrangian with respect to each variable and \( \lambda \).
• Setting these derivatives to zero provides a set of equations known as the first-order conditions. These include the partial derivatives with respect to each input variable and the multiplier.

This method allows you to find the optimal resource allocation in models such as the Cobb-Douglas production function, ensuring that each resource contributes effectively to production while respecting cost limitations. The Lagrange multiplier thus indicates the rate at which the objective function increases when the constraint is relaxed.

economic production models

Economic production models are mathematical representations used to describe how production processes transform inputs into outputs. Among these, the Cobb-Douglas production function is particularly renowned due to its flexibility and practical applications in economic analysis.
This specific function is expressed as \( P=k x^{\alpha} y^{\beta} z^{\gamma} \), where \( \alpha \), \( \beta \), and \( \gamma \) represent the output elasticities of each input, indicating their respective contributions to the overall production.
Key features of economic production models like the Cobb-Douglas function include:

• The functional form is multiplicative and exhibits constant returns to scale when the sum of the elasticities is equal to one (\( \alpha+\beta+\gamma=1 \)).
• It provides insights into the relationship between input adjustments and changes in output, helping firms optimize production.
• It highlights how different factors, such as labor, capital, and materials, interact within the production framework.

Using these models, businesses can determine the most efficient ways to allocate resources, thereby maximizing production while minimizing costs. In essence, economic production models guide strategic decision-making within the constraints of available resources.