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: $$Y=A K^{1 / 3} L^{2 / 3}$$ 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 $$M P L=(2 / 3) A(K / L)^{1 / 3}$$ 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 \(P Y\). b. Calculate the wage \(W\) and the real wage \(W / P\). (Note: The wage is labor compensation measured in dollars, 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 \((W L) /(P Y)\) d. Calculate what happens to output \(Y\), the wage \(W\), the real wage \(W / P\), and the labor share \((W L) /(P Y)\) in each of the following scenarios: n 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 iv. A plague decreases \(L\) from 1,000 to \(125 .\) e. Despite many changes in the U.S. economy over time, the labor share has been relatively stable. Is this observation consistent with the Cobb-Douglas production function? Explain.

Step by step solution

01

Calculate Output (Y)

Use the Cobb-Douglas production function: $$Y = A K^{1/3} L^{2/3}$$Given: A = 3, K = 1,000,000, and L = 1,000. Plug the values into the function:$$Y = 3 * (1,000,000)^{1/3} * (1,000)^{2/3}$$ $$Y = 3 * (100)^{1/3} * (10)^{2}$$ $$Y = 3 * 10 * 100$$ $$Y = 3000$$

02

Calculate Dollar Value of Output (PY)

Multiply Y by the price of the output P to get PY:Given: P = 2$$PY = P * Y$$$$PY = 2 * 3000$$$$PY = 6000$$

03

Calculate Marginal Product of Labor (MPL)

Use the given marginal product of labor formula:$$MPL = (2/3) A (K / L)^{1/3}$$Substitute the known values:$$MPL = (2/3) * 3 * (1,000,000 / 1,000)^{1/3}$$$$MPL = 2 * (1,000)^{1/3}$$$$MPL = 2 * 10$$$$MPL = 20$$

04

Calculate Wage (W)

The wage W is the marginal product of labor times the price of output:$$W = MPL * P$$$$W = 20 * 2$$$$W = 40$$

05

Calculate the Real Wage (W/P)

Divide the nominal wage by the price of output to get the real wage:$$W / P = W / P$$$$W / P = 40 / 2$$$$W / P = 20$$

06

Calculate the Labor Share ((WL)/(PY))

Multiply the wage W by the amount of labor L, then divide by the dollar value of output PY:$$Labor Share = (W * L) / (P * Y)$$$$Labor Share = (40 * 1,000) / 6000$$$$Labor Share = 40000 / 6000$$$$Labor Share = 6.67$$

07

Analyze the impact of an increase in P from 2 to 3

Both W and PY will increase, but the real wage W/P will decrease since P will increase. Labor share remains constant as both numerator and denominator scale with P equally.

08

Analyze the impact of an increase in A from 3 to 9

This would increase both Y and W, leading to higher MPL, and PY will also increase. The labor share remains constant as this is a feature of Cobb-Douglas functions where exponents sum to 1.

09

Analyze the impact of an increase in K from 1,000,000 to 8,000,000

This would increase both Y and W, since both depend on K. The labor share would not change as these are scaled by the same factor.

10

Analyze the impact of a decrease in L from 1,000 to 125 due to a plague

With a decrease in L, Y will decrease but at a lesser rate than L due to diminishing marginal returns to labor according to the Cobb-Douglas function. MPL and W will increase as K/L go up, while the labor share will increase due to the greater proportionate fall in PY relative to WL.

11

Discuss the stability of the labor share over time

The labor share's stability over time, despite economic changes, is consistent with the properties of the Cobb-Douglas production function as it suggests a constant return to scale. The constants (exponents) of K and L explain this invariance as they determine the share regardless of the levels of K and L.

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Key Concepts

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

Marginal Product of Labor

The marginal product of labor (MPL) is a critical concept in economics. It refers to the additional output produced as a result of adding one more unit of labor, keeping other production factors constant. In the context of the Cobb-Douglas production function, it's expressed mathematically as
\[\begin{equation} MPL = \frac{\text{d}Y}{\text{d}L} = (2 / 3) A(K / L)^{1 / 3} \frac{2}{3}\end{equation}\]
By knowing the marginal product of labor, businesses can make informed decisions about hiring additional workers. It is also pivotal for determining wages in a competitive labor market, where labor is compensated based on the value of its marginal productivity.

Real Wage

Real wage represents the purchasing power of wages, i.e., the amount of goods and services workers can buy with their wages. It contrasts with the nominal wage, which is the amount paid to labor in monetary terms without considering inflation or the cost of living. In our exercise, to calculate the real wage, we divide the nominal wage by the price of output (P), resulting in:
\[\begin{equation} W / P = 20 \end{equation}\]
This acts as an important indicator of the standard of living for employees in any economy. Workers are more concerned with the real wage than the nominal wage because it more accurately reflects their ability to afford goods and services.

Labor Share

Labor share is the proportion of output that's allocated to labor compensation. Within the Cobb-Douglas framework, the labor share equation is:
\[\begin{equation} \text{Labor Share} = (W \times L) / (P \times Y) \end{equation}\]
This measure provides insights into income distribution and the relative contributions of labor and capital in production. The stability of labor share over time can imply a well-balanced, functional economy, while significant deviations could signal shifts in the dynamics between labor and capital.

Production Function

A production function quantifies the relationship between input factors—such as capital (K) and labor (L)—and the output (Y) that results from their combination. The Cobb-Douglas production function, specifically, has proved invaluable in economic modeling. It is represented as:
\[\begin{equation} Y = A K^{1 / 3} L^{2 / 3} \end{equation}\]
where A signifies the level of technology. This formula allows us to predict changes in output with variations in inputs and is integral to understanding the mechanics of an economy.

Capital and Labor

In economics, capital and labor are the two main factors of production. Capital refers to the tools, machinery, and buildings that can enhance productivity. Labor, on the other hand, signifies the human effort that goes into the production process. The Cobb-Douglas production function showcases the interplay between capital and labor in generating output, with their contributions to production often stated in terms of ‘elasticities’—the exponents in the production function formula.

Economic Modeling

Economic modeling involves using theoretical frameworks and mathematical equations to represent complex economic processes. These models aim to explain and predict economic phenomena by establishing relationships between variables. The Cobb-Douglas production function is a prime example of an economic model that illustrates the relationship between output, capital, and labor. By adjusting its variables and parameters, economists can simulate various economic scenarios and analyze the corresponding effects on output, wages, and other key economic indicators.