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Chapter 12: Problem 54

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

Short Answer

Expert verified
Answer: The optimal values of K and L that maximize the production function subject to the given constraint are K=1 and L=5.5.

Step by step solution

01

Rewrite the constraint equation as a function of L

Express L as a function of K by isolating L in the constraint equation: $$30K + 60L = 360$$ Divide both sides by 60: $$L = \frac{360 - 30K}{60}$$ This will be inserted into the production function to solve for K.
02

Substitute the L function into the production function

Write the production function P in terms of K only by substituting the L function obtained in Step 1: $$P(K)=10K^{1/3}\left(\frac{360-30K}{60}\right)^{2/3}$$
03

Find the first derivative of P(K) with respect to K

Calculate the first derivative of the production function P(K) with respect to K: $$\frac{dP(K)}{dK}=\frac{10}{9}\left(\frac{360-30K}{60}\right)^{2/3}(1-K^(-2/3})$$
04

Set the first derivative equal to zero and solve for K

To find any potential maxima or minima, set the first derivative equal to zero and solve for K: $$\frac{10}{9}\left(\frac{360-30K}{60}\right)^{2/3}(1-K^(-2/3))=0$$ Since the terms inside the parentheses are both positive, we focus on the last term: $$1-K^{-2/3}=0$$ $$K^{-2/3}=1$$ Thus, we get the optimal value for K: $$K=1$$
05

Find the optimal value for L

Plug the optimal value of K into the L function obtained in Step 1: $$L=\frac{360-30(1)}{60}$$ $$L=\frac{330}{60}=5.5$$
06

Conclusion

The values of K and L that maximize the production function subject to the given constraint are K=1 and L=5.5.

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Key Concepts

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

Production Function
In economics, a production function is a mathematical representation of the relationship between the input of factors of production (like capital 'K' and labor 'L') and the output of goods or services they produce.

Specifically, the Cobb-Douglas production function model is widely applied for its relative simplicity and its ability to translate real-world scenarios into a functional form. The formula given by \( P = CK^aL^{1-a} \) captures how changes in 'K' (capital) and 'L' (labor) proportionally affect the total production 'P'. The constants 'C' and 'a' express technology and factor allocation biases, providing us with insight into returns to scale and factor elasticity. In essence, this formula helps economists and businesses to understand how efficiently resources are being converted into outputs.
Optimization Constraints
When maximizing a production function, we often face real-world limitations such as budgetary confines. An optimization constraint ensures that resource allocation decisions are made within these practical boundaries.

In our exercise, the equation \( pK + qL = B \) represents the budget constraint, where 'p' is the cost per unit of capital, 'q' is the cost per unit of labor, and 'B' is the total budget. This constraint lays out a key economic principle: the need to make the most effective use of limited resources. It guides the process of resource allocation by establishing a trade-off between capital and labor, which must be considered when searching for the optimal combination of input factors that maximize production.
First Derivative Test
The first derivative test is a critical tool in calculus used to locate the maxima or minima of a function. Applied within the context of our production function, this test involves taking the derivative of the function with respect to one of the variables and setting it equal to zero.

This mathematical technique essentially helps us identify the points where the slope of the function changes, signaling potential peaks or troughs in the function's graph. For production optimization, finding where the derivative equals zero gives us the potential quantities of capital or labor that maximize or minimize production. In this case, we successfully used this test to determine that at a specific value of 'K', the production is at its highest point when considering the budget constraint.
Maximum Production Calculation
The maximum production calculation comes down to determining the values of input factors that result in the highest possible output. From an economic standpoint, this is the level where a firm can be the most productive under given constraints.

Following the steps outlined, we compute the optimal values of 'K' and 'L' by substituting into and deriving the production function considering the constraint. The solution to these manipulations gives us the precise input values that a firm could employ to achieve maximum production, given its budgetary limitations. As highlighted by the provided exercise, this maximization process is fundamental for efficient resource management and for maintaining competitive advantage in a market.

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Most popular questions from this chapter

The angle between two planes is the angle \(\theta\) between the normal vectors of the planes, where the directions of the normal vectors are chosen so that \(0 \leq \theta<\pi\) Find the angle between the planes \(5 x+2 y-z=0\) and \(-3 x+y+2 z=0\)

Let \(f\) be a differentiable function of one or more variables that is positive on its domain. a. Show that \(d(\ln f)=\frac{d f}{f}.\) b. Use part (a) to explain the statement that the absolute change in \(\ln f\) is approximately equal to the relative change in \(f.\) c. Let \(f(x, y)=x y,\) note that \(\ln f=\ln x+\ln y,\) and show that relative changes add; that is, \(d f / f=d x / x+d y / y.\) d. Let \(f(x, y)=x / y,\) note that \(\ln f=\ln x-\ln y,\) and show that relative changes subtract; that is \(d f / f=d x / x-d y / y.\) e. Show that in a product of \(n\) numbers, \(f=x_{1} x_{2} \cdots x_{n},\) the relative change in \(f\) is approximately equal to the sum of the relative changes in the variables.

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 \(n \rightarrow \infty ?\)

Imagine a string that is fixed at both ends (for example, a guitar string). When plucked, the string forms a standing wave. The displacement \(u\) of the string varies with position \(x\) and with time \(t .\) Suppose it is given by \(u=f(x, t)=2 \sin (\pi x) \sin (\pi t / 2),\) for \(0 \leq x \leq 1\) and \(t \geq 0\) (see figure). At a fixed point in time, the string forms a wave on [0, 1]. Alternatively, if you focus on a point on the string (fix a value of \(x\) ), that point oscillates up and down in time. a. What is the period of the motion in time? b. Find the rate of change of the displacement with respect to time at a constant position (which is the vertical velocity of a point on the string). c. At a fixed time, what point on the string is moving fastest? d. At a fixed position on the string, when is the string moving fastest? e. Find the rate of change of the displacement with respect to position at a constant time (which is the slope of the string). f. At a fixed time, where is the slope of the string greatest?

Use the definition of differentiability to prove that the following functions are differentiable at \((0,0) .\) You must produce functions \(\varepsilon_{1}\) and \(\varepsilon_{2}\) with the required properties. $$f(x, y)=x+y$$
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