9. Some economists believe that the U.S. economy as a whole can be modeled with the following production function, called the Cobb–Douglas production function: Y = AK/³L?/3, where Y is the amount of output, K is the amount of capital, L is the amount of labor, and A is a parameter that measures the state of technology. For this production function, the marginal product of labor is MPL = (2/3) A(K/L)/3. Suppose that the price of output P is 2, A is 3, K is 1,000,000, and L is 1,000. The labor market is competitive, so labor is paid the value of its marginal product. a. Calculate the amount of output produced Y and the dollar value of output PY. b. Calculate the wage W and the real wage W/P. (Note: The wage is labor compensation measured in dol- lars, whereas the real wage is labor compensation measured in units of output.) c. Calculate the labor share (the fraction of the value of output that is paid to labor), which is (WL)/(PY). d. Calculate what happens to output Y, the wage W, the real wage W/P, and the labor share (WL)/(PY) in each of the following scenarios: i. Inflation increases P from 2 to 3. ii. Technological progress increases A from 3 to 9. iii. Capital accumulation increases K from 1,000,000 to 8,000,000.
9. Some economists believe that the U.S. economy as a whole can be modeled with the following production function, called the Cobb–Douglas production function: Y = AK/³L?/3, where Y is the amount of output, K is the amount of capital, L is the amount of labor, and A is a parameter that measures the state of technology. For this production function, the marginal product of labor is MPL = (2/3) A(K/L)/3. Suppose that the price of output P is 2, A is 3, K is 1,000,000, and L is 1,000. The labor market is competitive, so labor is paid the value of its marginal product. a. Calculate the amount of output produced Y and the dollar value of output PY. b. Calculate the wage W and the real wage W/P. (Note: The wage is labor compensation measured in dol- lars, whereas the real wage is labor compensation measured in units of output.) c. Calculate the labor share (the fraction of the value of output that is paid to labor), which is (WL)/(PY). d. Calculate what happens to output Y, the wage W, the real wage W/P, and the labor share (WL)/(PY) in each of the following scenarios: i. Inflation increases P from 2 to 3. ii. Technological progress increases A from 3 to 9. iii. Capital accumulation increases K from 1,000,000 to 8,000,000.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
Related questions
Question
Parts b,c,d,e
![1. **Economists' Model of the U.S. Economy**
Economists use the Cobb-Douglas production function to model the U.S. economy:
\[ Y = AK^{1/3}L^{2/3} \]
- **Y**: Amount of output
- **K**: Amount of capital
- **L**: Amount of labor
- **A**: Parameter measuring the state of technology
Marginal Product of Labor (MPL):
\[ MPL = (2/3)A(K/L)^{1/3} \]
2. **Scenario Assumptions:**
- Price of output \( P = 2 \)
- \( A = 3 \)
- \( K = 1,000,000 \)
- \( L = 1,000 \)
In a competitive labor market, labor is paid its marginal product value.
3. **Tasks:**
a. Calculate the output \( Y \) and dollar value of output \( PY \).
b. Calculate the wage \( W \) and real wage \( W/P \).
- Wage (W) in dollars
- Real wage in units of output
c. Calculate the labor share \((WL)/(PY)\).
d. Determine effects of:
i. Increased inflation \( P \) from 2 to 3.
ii. Advancements in technology \( A \) from 3 to 9.
iii. Increased capital \( K \) from 1,000,000 to 8,000,000.
iv. Decrease in labor \( L \) from 1,000 to 125.
e. Discuss why the labor share has been stable over time and its compatibility with the Cobb-Douglas production function.
This exercise asks students to apply mathematical concepts to an economic model and analyze various factors affecting economic output and worker compensation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6161599c-ae14-4cb3-8591-c25ebaa4a9ee%2F22538971-f967-4f9a-a9ab-bb95032ec89a%2F8jz07xy_processed.png&w=3840&q=75)
Transcribed Image Text:1. **Economists' Model of the U.S. Economy**
Economists use the Cobb-Douglas production function to model the U.S. economy:
\[ Y = AK^{1/3}L^{2/3} \]
- **Y**: Amount of output
- **K**: Amount of capital
- **L**: Amount of labor
- **A**: Parameter measuring the state of technology
Marginal Product of Labor (MPL):
\[ MPL = (2/3)A(K/L)^{1/3} \]
2. **Scenario Assumptions:**
- Price of output \( P = 2 \)
- \( A = 3 \)
- \( K = 1,000,000 \)
- \( L = 1,000 \)
In a competitive labor market, labor is paid its marginal product value.
3. **Tasks:**
a. Calculate the output \( Y \) and dollar value of output \( PY \).
b. Calculate the wage \( W \) and real wage \( W/P \).
- Wage (W) in dollars
- Real wage in units of output
c. Calculate the labor share \((WL)/(PY)\).
d. Determine effects of:
i. Increased inflation \( P \) from 2 to 3.
ii. Advancements in technology \( A \) from 3 to 9.
iii. Increased capital \( K \) from 1,000,000 to 8,000,000.
iv. Decrease in labor \( L \) from 1,000 to 125.
e. Discuss why the labor share has been stable over time and its compatibility with the Cobb-Douglas production function.
This exercise asks students to apply mathematical concepts to an economic model and analyze various factors affecting economic output and worker compensation.
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