Macroeconomics
Macroeconomics
10th Edition
ISBN: 9781319105990
Author: Mankiw, N. Gregory.
Publisher: Worth Publishers,
Question
Book Icon
Chapter 3, Problem 4PA

(a)

To determine

Fraction of income that the capital and labor receive.

(a)

Expert Solution
Check Mark

Explanation of Solution

Given information:

Production function is F(K,L)=AKαL1α.

Value of “α” is 0.3.

Calculation:

Under Cobb-Douglas production function, explain the input-output relationship in a production process. The general form of Cobb-Douglas production function is given below:

F(K,L)=AKαL1α (1)

The marginal product labor can be derived as follows:

MPL=(Total production)(Total labor)=(AKαL1α)L=(1α)YL

Thus, the marginal product of capital is (1α)YL.

The marginal product of capital can be derived as follows:

MPK=(Total production)(Total labor)=(AKαL1α)L=(α)YK

Thus, the marginal product of capital is (α)YL.

In a competitive, the profit maximizing firm hires labor when the MPL is equal to real wage rate. When MPK is equal to real rental rate, they will hire capital. Thus, by using this information, the marginal products for Cobb-Douglas production function can be written as follows:

WP=MPLWP=(1α)YL (2)

RP=MPKRP=(α)YK (3)

Rearrange the MPL equation (Equation (2)) as follows:

(WP)×L=MPL×L=(1α)YL×L=(1α)Y

Rearrange the MPK equation (Equation (3)) as follows:

(RP)×K=MPK×K=(α)YK×K=(α)Y

From the rearranged form, the term (WP)×L is the wage bill and the term (RP)×K is the total returns to capital.

When the value of “α” is 0.3, the labor receives 70 percent of total output (income) ((1α)Y=10.3=0.7). The capital receives 30 percent of total output (income) ((α)Y=0.3).

Economics Concept Introduction

Cobb-Douglas production function: Cobb-Douglas production function is a well-known production function; it shows the relationship between inputs used and output produced in a production process.

(b)

To determine

Impact of increase in labor force on total output, rental price of capital, and real wage rate.

(b)

Expert Solution
Check Mark

Explanation of Solution

Given information:

Production function is F(K,L)=AKαL1α.

Value of “α” is 0.3.

Labor increases by 10%.

Calculation:

Separate the Cobb-Douglas production function as Y1 and Y2 to explain the initial quantity of output and final quantity of output, respectively.

Y1=AK0.3L0.7 (4)

Y2=AK0.3(1.1L)0.7 (5)

In Y2 equation, L is multiplied with 1.1 to reflect the 10% increase in the labor force.

Divide the value of initial output and final value of output to calculate the total output after 10% increase in labor forces.

Y2Y1=(AK0.3(1.1L)0.7AK0.3L0.7)=(1.1)0.7=1.069

Total output is increased by 1.069 (6.9%).

The rental price after 10% increase in labor force is calculated as follows:

(RP)=MPK=αAKα1L1α

Separate the equation as (RP)1 to show the initial value of rental price of capital, and (RP)2 to show the final rental price of capital after 10% increase in labor force.

(RP)1=(0.3)AK0.7L0.7 (6)

(RP)2=(0.3)AK0.7(1.1L)0.7 (7)

In (RP)2 equation, L is multiplied with 1.1 to reflect the 10% increase in the labor force.

Divide the value of initial rental price of capital by the final rental price of capital to calculate the rental price of capital after 10% increase in labor forces.

(RP)2(RP)1=((0.3)AK0.7(1.1L)0.70.3AK0.7L0.7)=(1.1)0.7=1.069

The rental price of capital is increased by 1.069 (6.9%).

The real wage after 10% increase in labor force is calculated as follows:

(WP)=MPL=(1α)AKαLα

Separate the equation as (WP)1 to show the initial value of real wage, and (WP)2 to show the final value of real wage after 10% increase in labor force.

(WP)1=(1 0.3)AK0.3L0.3 (8)

(WP)2=(1 0.3)AK0.3(1.1L)0.3 (9)

In (WP)2 equation, L is multiplied by 1.1 to reflect the 10% increase in the labor force.

Divide the initial value of real wage and final value of real wage to calculate the real wage rate after 10% increase in labor forces.

(WP)2(WP)1=((1 0.3)AK0.3(1.1L)0.3(1 0.3)AK0.3L0.3)=(1.1)0.3=0.972

The real wage is reduced by 0.972 (28%).

(c)

To determine

Impact of increase in capital stock on total output, rental price of capital, and real wage rate.

(c)

Expert Solution
Check Mark

Explanation of Solution

Given information:

Production function is F(K,L)=AKαL1α.

Value of “α” is 0.3.

Capital stock increases by 10%.

Calculation:

Using the same logic used in sub-part (b), the total output after 10% increase in capital stock is calculated as follows:

Y2Y1=(A(1.1K)0.3L0.7AK0.3L0.7)=(1.1)0.3=1.029

Total output is increased by 1.029 (2.9%).

The rental price of capital after 10% increase in capital stock is calculated as follows:

(RP)2(RP)1=((0.3)A(1.1K)0.7L0.70.3AK0.7L0.7)=(1.1)0.7=0.935

The rental price of capital is decreased by 0.935.

The real wage rate after 10% increase in capital stock is calculated as follows:

(WP)2(WP)1=((1 0.7)A(1.1K)0.3L0.3(1 0.7)AK0.3L0.3)=(1.1)0.3=1.029

The real wage is increased by 1.029 (2.9%).

(d)

To determine

New output, new rental price of capital, and real wage.

(d)

Expert Solution
Check Mark

Explanation of Solution

Given information:

Production function is F(K,L)=AKαL1α.

Value of “α” is 0.3.

The value of “A” increases by 10%.

Calculation:

Using the same logic used in sub-part (b) and in sub-part (c), the total output with 10% increase in the value of parameter is calculated as follows:

Y2Y1=((1.1A)K0.3L0.7AK0.3L0.7)=1.1

Total output is increased by 1.1 (10%).

The rental price of capital is calculated as follows:

(RP)2(RP)1=((0.3)(1.1A)K0.7L0.70.3AK0.7L0.7)=1.1

The rental price of capital is 1.1.

The real wage is calculated as follows:

(WP)2(WP)1=((1 0.7)(1.1A)K0.3L0.3(1 0.7)AK0.3L0.3)=1.1

The real wage is 1.1.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!
Students have asked these similar questions
1) Use the supply and demand schedules to graph the supply and demand functions. Find and show on the graph the equilibrium price and quantity, label it (A). P Q demanded P Q supplied 0 75 0 0 5 65 5 0 10 55 10 0 15 45 15 10 20 35 20 20 25 25 25 30 30 15 30 40 35 40 5 0 35 40 50 60 2) Find graphically and numerically the consumers and producers' surplus 3) The government introduced a tax of 10$, Label the price buyers pay and suppliers receive. Label the new equilibrium for buyers (B) and Sellers (S). How the surpluses have changed? Give the numerical answer and show on the graph. 4) Calculate using midpoint method the elasticity of demand curve from point (A) to (B) and elasticity of the supply curve from point (A) to (C).
Four heirs (A, B, C, and D) must divide fairly an estate consisting of three items — a house, a cabin and a boat — using the method of sealed bids. The players' bids (in dollars) are:   In the initial allocation, player D Group of answer choices gets no items and gets $62,500 from the estate. gets the house and pays the estate $122,500. gets the cabin and gets $7,500 from the estate. gets the boat and and gets $55,500 from the estate. none of these
Jack and Jill are getting a divorce. Except for the house, they own very little of value so they agree to divide the house fairly using the method of sealed bids. Jack bids 140,000 and Jill bids 160,000. After all is said and done, the final outcome is Group of answer choices Jill gets the house and pays Jack $80,000. Jill gets the house and pays Jack $75,000. Jill gets the house and pays Jack $70,000. Jill gets the house and pays Jack $65,000. none of these
Knowledge Booster
Background pattern image
Recommended textbooks for you
Text book image
MACROECONOMICS FOR TODAY
Economics
ISBN:9781337613057
Author:Tucker
Publisher:CENGAGE L
Text book image
Micro Economics For Today
Economics
ISBN:9781337613064
Author:Tucker, Irvin B.
Publisher:Cengage,
Text book image
Economics For Today
Economics
ISBN:9781337613040
Author:Tucker
Publisher:Cengage Learning
Text book image
Survey Of Economics
Economics
ISBN:9781337111522
Author:Tucker, Irvin B.
Publisher:Cengage,
Text book image
Microeconomics: Private and Public Choice (MindTa...
Economics
ISBN:9781305506893
Author:James D. Gwartney, Richard L. Stroup, Russell S. Sobel, David A. Macpherson
Publisher:Cengage Learning
Text book image
Economics: Private and Public Choice (MindTap Cou...
Economics
ISBN:9781305506725
Author:James D. Gwartney, Richard L. Stroup, Russell S. Sobel, David A. Macpherson
Publisher:Cengage Learning