Problem 55 - Calculus Early Transcendentals

Question

Production functions Economists model the output of manufacturing systems using production functions that have many of the same properties as utility functions. The family of Cobb-Douglas production functions has the form \(P=f(K, L)=C K^{a} L^{1-a},\) where K represents capital, L represents labor, and C and a are positive real numbers with \(0

Step-by-Step Solution

Answer

Answer: To maximize the Cobb-Douglas production function with a budget constraint, the production function is maximized when \(K=\frac{aB}{p}\) and \(L=\frac{(1-a)B}{q}\), where a is the share of capital, B is the budget, and p and q are the input prices for capital and labor, respectively.

Step 1: Set up the Lagrangian function

Let's set up the Lagrangian function for this problem. The function takes the following form: \(L(K, L, \lambda) = K^{a}L^{1-a} - \lambda(pK + qL - B)\).

Step 2: Find the partial derivatives with respect to K, L, and λ

Now, let's find the partial derivatives of the Lagrangian function with respect to K, L, and λ: 1. \(\frac{\partial L}{\partial K} = aK^{a-1}L^{1-a} - \lambda p\) 2. \(\frac{\partial L}{\partial L} = (1-a)K^{a}L^{-a} - \lambda q\) 3. \(\frac{\partial L}{\partial \lambda} = pK + qL - B\)

Step 3: Equate the partial derivatives to zero and solve for K, L, and λ

Next, we'll set each partial derivative equal to zero and solve the resulting system of equations for K, L, and λ: 1. \(aK^{a-1}L^{1-a} - \lambda p = 0\) 2. \((1-a)K^{a}L^{-a} - \lambda q = 0\) 3. \(pK + qL = B\) To eliminate λ, we'll divide the first equation by the second equation: \(\frac{aK^{a-1}L^{1-a}}{(1-a)K^{a}L^{-a}} = \frac{\lambda p}{\lambda q}\) Simplifying the equation, we get: \(\frac{a}{p} = \frac{1-a}{q}\) Now we can solve for K and L in terms of a, p, q, and B: \(K = \frac{aB}{p}\) and \(L = \frac{(1-a)B}{q}\).

Step 4: Verify the solution

Finally, we need to verify that the values of K and L we found actually maximize the production function P. Since the production function has the same properties as utility functions, it's well-behaved, so its maximum can be attained at the values we found for K and L. By plugging these values back into the production function and the budget constraint, we can verify that they satisfy both conditions. Thus, we've shown that the production function is maximized when \(K=\frac{aB}{p}\) and \(L=\frac{(1-a)B}{q}\).

Key Concepts

Cobb-Douglas production function

The Cobb-Douglas production function is a fundamental concept in economics used to model the output of a manufacturing system based on inputs such as capital and labor. The general form of this function is expressed as \( P = f(K, L) = C K^a L^{1-a} \), where:

• \( K \) represents the capital input,
• \( L \) represents the labor input,
• \( C \) is a constant,
• \( a \) is a parameter between 0 and 1 that represents the output elasticity of capital.

This function assumes that inputs can be varied in constant proportions and reflects the level of substitutability between capital and labor. The Cobb-Douglas function is popular because it is simple and can describe real-world scenarios well. 
It exhibits constant returns to scale, meaning if you double both inputs, output will also double.
The function's parameters \(a\) and \(1-a\) help determine the relative contribution of capital and labor to production.

Lagrangian function

The Lagrangian function is a powerful tool for optimizing functions subject to constraints and is pivotal in addressing constrained optimization problems in economics.
For the Cobb-Douglas production function with a budget constraint, the Lagrangian takes the form:
\[ L(K, L, \lambda) = K^{a}L^{1-a} - \lambda(pK + qL - B) \]

• \( \lambda \) is a Lagrange multiplier, representing the shadow price or the rate at which the objective function \( P \) would improve if the constraint became more relaxed.
• The terms \( pK + qL - B \) describe the economizing constraint: thus, when completely "active", it should equal zero, affirming that all available resources are utilized.

By setting up this function, we aim to balance maximizing the production \( P \) while respecting the budget constraint effectively.
It incorporates both the objective to be maximized and the limitation imposed by available resources, allowing for a comprehensive approach to find optimal solutions.

Budget constraint

A budget constraint in economics depicts the limitation of available resources to produce outputs. For the Cobb-Douglas production function, it is given by the equation:
\[ pK + qL = B \]

• \( p \) represents the unit cost of capital,
• \( q \) is the unit cost of labor, and
• \( B \) is the total budget available.

The key idea here is that the sum of spending on capital \( (pK) \) and labor \( (qL) \) should exactly match the budget \( B \).
The constraint ensures that the expenditure does not exceed the budget, maintaining a balance in resource allocation efficiently.
In the context of optimization, the constraint must be considered to find values of \( K \) and \( L \) that maximize output without overspending.
It forces the firm to consider trade-offs, or how to distribute the budget between capital and labor optimally.

Partial derivatives

Partial derivatives are used in optimization problems to understand how changes in one variable impact a function while keeping other variables constant.
In the context of the Lagrangian function for the Cobb-Douglas case, we derive partial derivatives concerning the variables \( K \), \( L \), and the Lagrange multiplier \( \lambda \):

• For capital \( K \): \[ \frac{\partial L}{\partial K} = aK^{a-1}L^{1-a} - \lambda p \]
• For labor \( L \): \[ \frac{\partial L}{\partial L} = (1-a)K^{a}L^{-a} - \lambda q \]
• For the multiplier \( \lambda \): \[ \frac{\partial L}{\partial \lambda} = pK + qL - B \]

By setting these partial derivatives equal to zero, we identify points where potential maxima or minima occur, allowing us to determine the optimal allocation \( K \) and \( L \) within the budget constraint. This analysis is essential for finding solutions that will maximize production efficiently under the given resource limitations.