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Chapter 12: Problem 55

Given the production function \(P=f(K, L)=K^{a} L^{1-a}\) and the budget constraint \(p K+q L=B,\) where \(a, p, q,\) and \(B\) are given, show that \(P\) is maximized when \(K=a B / p\) and \(L=(1-a) B / q\).

Short Answer

Expert verified
Question: Show that the production function \(P = f(K, L) = K^a L^{1-a}\) is maximized when \(K = a B / p\) and \(L = (1-a) B / q\), given the budget constraint \(p K + q L = B\). Answer: The production function \(P\) is maximized when \(K = a B / p\) and \(L = (1-a) B / q\).

Step by step solution

01

Define the Lagrangian function

The Lagrangian function \(L(K, L, \lambda)\) is given by: $$ L(K, L, \lambda) = K^a L^{1-a} - \lambda(p K + q L -B) $$ where \(\lambda\) is the Lagrange multiplier.
02

Calculate partial derivatives

We will now compute the partial derivatives of the Lagrangian function with respect to \(K\), \(L\), and \(\lambda\). These are given by: $$ \frac{\partial L}{\partial K} = a K^{a-1} L^{1-a} - \lambda p $$ $$ \frac{\partial L}{\partial L} = (1-a) K^{a} L^{-a} - \lambda q $$ $$ \frac{\partial L}{\partial \lambda} = p K + q L - B $$
03

Set partial derivatives equal to zero

To find the values of \(K, L\), and \(\lambda\) that maximize the production, we will set the above partial derivatives equal to zero: $$ a K^{a-1} L^{1-a} - \lambda p = 0 $$ $$ (1-a) K^{a} L^{-a} - \lambda q = 0 $$ $$ p K + q L - B = 0 $$
04

Solve the system of equations

From the first and second equations, we have: $$ \lambda = \frac{a K^{a-1} L^{1-a}}{p} = \frac{(1-a) K^{a} L^{-a}}{q} $$ Divide these two equations to eliminate \(\lambda\): $$ \frac{a K^{a-1} L^{1-a}}{(1-a) K^{a} L^{-a}} = \frac{p}{q} $$ Simplify and rearrange to obtain: $$ \frac{K}{L} = \frac{a q}{(1-a) p} $$ Using the budget constraint equation \(p K + q L = B\), we can now solve for \(K\) and \(L\): $$ K = \frac{a B}{p} $$ $$ L = \frac{(1-a) B}{q} $$ Thus, we have shown that the production \(P\) is maximized when \(K = a B / p\) and \(L = (1-a) B / q\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Production Function Optimization
Production function optimization involves finding the best combination of inputs to produce the maximum possible output. In many economic problems, a production function, like the one given by \( P=f(K, L)=K^{a} L^{1-a} \), captures how inputs like capital (\( K \)) and labor (\( L \)) are transformed into output.To optimize this function, we use the Lagrangian technique. This method allows us to account for constraints, like budgets, while maximizing output. By introducing a Lagrange multiplier (\( \lambda \)), we can combine the production function with the budget constraint in a single equation. This approach helps us to see how changes in one variable affect the others, leading to the best possible production level within the given limits.
Partial Derivatives
Partial derivatives are a mathematical tool used to understand how a function changes when one of its variables changes, holding the others constant. In the context of the Lagrangian method, calculating partial derivatives allows us to determine how small incremental changes in capital (\( K \)), labor (\( L \)), and the Lagrange multiplier (\( \lambda \)) impact the system.The partial derivatives of the Lagrangian function \( L(K, L, \lambda) \) with respect to each of these variables are set to zero. Why? Because at these points the function is optimized:
  • \( \frac{\partial L}{\partial K} = 0 \)
  • \( \frac{\partial L}{\partial L} = 0 \)
  • \( \frac{\partial L}{\partial \lambda} = 0 \)
These conditions help us find the values of \( K \), \( L \), and \( \lambda \) that maximize the production, ensuring the best use of resources.
Budget Constraint
A budget constraint is a limitation representing how much money is available to spend on inputs like capital and labor. The given budget constraint in our problem is:\[ pK + qL = B \]Here, \( p \) and \( q \) are the prices of capital and labor, respectively, and \( B \) is the total budget available. This equation dictates how resources are allocated between \( K \) and \( L \) to ensure no overspending occurs.By incorporating this constraint into the Lagrangian function, we ensure that the optimization respects the financial limits. Solving the system of equations, including this constraint, leads us to the combination of \( K \) and \( L \) that provides the highest production without exceeding the budget.So, by respecting the budget while maximizing the production function, the task shows that both capital and labor need to be apportioned precisely as \( K = \frac{aB}{p} \) and \( L = \frac{(1-a)B}{q} \) for optimal production.

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Most popular questions from this chapter

Traveling waves (for example, water waves or electromagnetic waves) exhibit periodic motion in both time and position. In one dimension, some types of wave motion are governed by the one-dimensional wave equation $$\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}},$$ where \(u(x, t)\) is the height or displacement of the wave surface at position \(x\) and time \(t,\) and \(c\) is the constant speed of the wave. Show that the following functions are solutions of the wave equation. \(u(x, t)=A f(x+c t)+B g(x-c t),\) where \(A\) and \(B\) are constants and \(f\) and \(g\) are twice differentiable functions of one variable

Two resistors in an electrical circuit with resistance \(R_{1}\) and \(R_{2}\) wired in parallel with a constant voltage give an effective resistance of \(R,\) where \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}\). a. Find \(\frac{\partial R}{\partial R_{1}}\) and \(\frac{\partial R}{\partial R_{2}}\) by solving for \(R\) and differentiating. b. Find \(\frac{\partial R}{\partial R_{1}}\) and \(\frac{\partial R}{\partial R_{2}}\) by differentiating implicitly. c. Describe how an increase in \(R_{1}\) with \(R_{2}\) constant affects \(R\). d. Describe how a decrease in \(R_{2}\) with \(R_{1}\) constant affects \(R\).

The equation \(x^{2 n}+y^{2 n}+z^{2 n}=1,\) where \(n\) is a positive integer, describes a flattened sphere. Define the extreme points to be the points on the flattened sphere with a maximum distance from the origin. a. Find all the extreme points on the flattened sphere with \(n=2\) What is the distance between the extreme points and the origin? b. Find all the extreme points on the flattened sphere for integers \(n>2 .\) What is the distance between the extreme points and the origin? c. Give the location of the extreme points in the limit as \(n \rightarrow \infty\). What is the limiting distance between the extreme points and the origin as \(n \rightarrow \infty ?\)

The flow of heat along a thin conducting bar is governed by the one- dimensional heat equation (with analogs for thin plates in two dimensions and for solids in three dimensions) $$\frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}},$$ where \(u\) is a measure of the temperature at a location \(x\) on the bar at time t and the positive constant \(k\) is related to the conductivity of the material. Show that the following functions satisfy the heat equation with \(k=1\). $$u(x, t)=10 e^{-t} \sin x$$

Use the gradient rules of Exercise 81 to find the gradient of the following functions. $$f(x, y)=\ln \left(1+x^{2}+y^{2}\right)$$
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