/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 42 The output \(Q\) of an economic ... [FREE SOLUTION] | 91影视

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Chapter 12: Problem 42

The output \(Q\) of an economic system subject to two inputs, such as labor \(L\) and capital \(K\), is often modeled by the Cobb-Douglas production function \(Q(L, K)=c L^{a} K^{b},\) where \(a, b,\) and \(c\) are positive real numbers. When \(a+b=1,\) the case is called constant returns to scale. Suppose \(a=\frac{1}{3}, b=\frac{2}{3},\) and \(c=40.\) a. Graph the output function using the window \([0,20] \times[0,20] \times[0,500].\) b. If \(L\) is held constant at \(L=10,\) write the function that gives the dependence of \(Q\) on \(K.\) c. If \(K\) is held constant at \(K=15,\) write the function that gives the dependence of \(Q\) on \(L.\)

Short Answer

Expert verified
Answer: The function for the dependence of Q on K with L held constant at 10 is \(Q(10, K) = 86.176 K^{\frac{2}{3}}\).

Step by step solution

01

Graph the output function

Since it is not possible to directly create and embed a graph in this format, you can use software such as a graphing calculator, Desmos, or other online graphing tools to plot the function. Given the details in the exercise, plug in the values and graph \(Q(L,K) = 40L^{\frac{1}{3}} K^{\frac{2}{3}}\) within the specified window: \([0, 20] \times [0, 20] \times [0, 500]\). Make sure to pay attention to the axes range and labels when plotting the graph.
02

Write the function for the dependence of Q on K with L held constant at 10

Given the function \(Q(L,K) = 40L^{\frac{1}{3}} K^{\frac{2}{3}}\), hold L constant at 10, i.e., \(L=10\). Replace the value of L in the function and simplify it: $$ Q(10, K) = 40(10)^{\frac{1}{3}} K^{\frac{2}{3}} = 40(2.1544) K^{\frac{2}{3}} = 86.176 K^{\frac{2}{3}} $$ So, the function for the dependence of Q on K with L held constant at \(10\) is \(Q(10, K) = 86.176 K^{\frac{2}{3}}\).
03

Write the function for the dependence of Q on L with K held constant at 15

Given the function \(Q(L,K) = 40L^{\frac{1}{3}} K^{\frac{2}{3}}\), hold K constant at 15, i.e., \(K=15\). Replace the value of K in the function and simplify it: $$ Q(L, 15) = 40L^{\frac{1}{3}} (15)^{\frac{2}{3}} = 40L^{\frac{1}{3}} (6.3496) = 253.984 L^{\frac{1}{3}} $$ So, the function for the dependence of Q on L with K held constant at \(15\) is \(Q(L, 15) = 253.984 L^{\frac{1}{3}}\).

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Key Concepts

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

Economic System Modeling
Modeling an economic system can be a complex task, as it must capture the interplay between various inputs producing outputs. A popular mathematical representation of how inputs are transformed into output in a production process is the Cobb-Douglas production function.

This function takes the form of Q(L, K) = c L^a K^b, where Q is the total quantity of output, L signifies labor, K represents capital, and a, b, and c are parameters that determine the output elasticity with respect to labor and capital. In this model, labor and capital are raised to powers that reflect their respective contributions to the production process.

For example, if we have a = 1/3, b = 2/3, and c = 40, the function would be Q(L, K) = 40 L^{1/3} K^{2/3}. This function quantifies how different amounts of labor and capital contribute to the production output, allowing economists to predict the output level based on various input combinations.
Constant Returns to Scale
In the realm of economics, the notion of 'constant returns to scale' is vitally important. It describes a situation where increasing all inputs by a certain factor results in an increase in output by the same factor. Essentially, this means the productivity of the inputs does not change with their scale.

The Cobb-Douglas function Q(L, K) = c L^a K^b exhibits constant returns to scale when the sum of the exponents a and b is equal to one, that is, a + b = 1. In our example, with a = 1/3 and b = 2/3, adding these together indeed equals one, satisfying the constant returns to scale condition. This implies that if both labor and capital are doubled, the output will also double, thus maintaining the same efficiency even at a larger scale. Understanding constant returns to scale helps businesses and economies optimize their production processes without sacrificing efficiency when expanding their operations.
Graphing Multivariable Functions
Graphing multivariable functions, like the Cobb-Douglas production function, requires more than just the standard two-dimensional Cartesian plane. With two inputs, labor (L) and capital (K), and one output (Q), such functions are three-dimensional and can be visualized using 3D graphing tools.

For the function Q(L, K) = c L^a K^b, with specific values of a, b, and c, you would typically set up a 3D coordinate system where the x-axis represents labor, the y-axis represents capital, and the z-axis represents the output. This allows students and researchers to visualize how changing input levels affect outputs.

By plotting the function within the window [0, 20] 脳 [0, 20] 脳 [0, 500] as described in the step-by-step solution, one can observe the shape of the production surface and how it responds to variations in labor and capital. This visual representation is a powerful tool for understanding the dynamic relationship between inputs and output in an economic system.

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Most popular questions from this chapter

Given three distinct noncollinear points \(A, B,\) and \(C\) in the plane, find the point \(P\) in the plane such that the sum of the distances \(|A P|+|B P|+|C P|\) is a minimum. Here is how to proceed with three points, assuming that the triangle formed by the three points has no angle greater than \(2 \pi / 3\left(120^{\circ}\right)\). a. Assume the coordinates of the three given points are \(A\left(x_{1}, y_{1}\right)\) \(\underline{B}\left(x_{2}, y_{2}\right),\) and \(C\left(x_{3}, y_{3}\right) .\) Let \(d_{1}(x, y)\) be the distance between \(A\left(x_{1}, y_{1}\right)\) and a variable point \(P(x, y) .\) Compute the gradient of \(d_{1}\) and show that it is a unit vector pointing along the line between the two points. b. Define \(d_{2}\) and \(d_{3}\) in a similar way and show that \(\nabla d_{2}\) and \(\nabla d_{3}\) are also unit vectors in the direction of the line between the two points. c. The goal is to minimize \(f(x, y)=d_{1}+d_{2}+d_{3} .\) Show that the condition \(f_{x}=f_{y}=0\) implies that \(\nabla d_{1}+\nabla d_{2}+\nabla d_{3}=0\) d. Explain why part (c) implies that the optimal point \(P\) has the property that the three line segments \(A P, B P,\) and \(C P\) all intersect symmetrically in angles of \(2 \pi / 3\) e. What is the optimal solution if one of the angles in the triangle is greater than \(2 \pi / 3\) (just draw a picture)? f. Estimate the Steiner point for the three points (0,0),(0,1) and (2,0).

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