1. In this chapter, we learn:
how to set up and solve a macroeconomic model.
how a production function can help us understand differences in per capita GDP across countries.
the relative importance of capital per person and total factor productivity in accounting for these differences.
the relevance of “returns to scale” and “diminishing marginal products.”
how to look at economic data through the lens of a macroeconomic model.
A model is a mathematical representation of a hypothetical world that we use to study economic phenomena.
It consists of equations and unknowns with real world interpretations.
Macroeconomists document facts, build a model to understand the facts, and examine the model to see how effective it is at explaining the facts.
Vast oversimplifications of the real world in a model can still allow it to provide important insights.

2. Modern Growth around the World
After World War II, growth in Germany and Japan accelerated.
Convergence is the idea that poorer countries will grow faster to catch up to the level of income in richer countries.
Brazil had accelerated growth until 1980 and then stagnated, while China and India have had the reverse pattern.
When comparing levels of income on a ratio scale, recall that data points that are half below another country are actually much lower because the numbers on the vertical axis double.
Convergence
To US US
Leapfrogged UK

3. Most countries have sustained about 2 percent growth.
Some countries have exhibited a negative growth rate over a forty year period.
Other countries have sustained nearly 6 percent growth over the same period.
Most countries have sustained about 2 percent growth.
Small differences in growth rates result in large differences in standards of living.
Not all
catch up
Some
decline

4. Growth Rules For Cobb-Douglas Production Function Yt = AtKt1/3Lt2/3
Applying rules of growth rates will show that
g(Yt) = g(At) + (1/3)*g(Kt) + (2/3)*g(Lt)
We can estimate total factor productivity growth, g(At), as a residual
g(At) = g(Yt) - (1/3)*g(Kt) - (2/3)*g(Lt)

5. A is a productivity parameter.
Setting Up the Model
Consider a single, closed economy, with only one consumption good.
A certain number of laborers, L, make the consumption good.
A certain number of machines, K, are used to produce the good.
A production function tells how much output can be produced given any number of inputs, laborers, and machines.
Variables with a bar are parameters.
A is a productivity parameter.
A higher value of A means firms can produce more with the same inputs.
The Cobb-Douglas production function is the particular production function that takes the form Y = A K1/3 L2/3
We assume a = 1/3.
A production function exhibits constant returns to scale if doubling each input exactly doubles output.
If the exponents on the inputs sum to 1, the function has constant returns to scale.
If the exponents on the inputs sum to more than 1, the function has increasing returns to scale. If the exponents on the inputs sum to less than 1, the function has decreasing returns to scale.

6. MPK = dY/dK = 1/3 AK-2/3L2/3 = 1/3 AK1/3L2/3/K = 1/3 Y/K
Allocating Resources
Firms choose the amount of capital and labor to use in production by maximizing profits [= Revenue minus Costs]: Profit = P Y – r K – w L
The rental rate of capital, r, and the wage rate, w, are taken as given under perfect competition.
The price of the output, P, is normalized to one [P = 1].
The solution to the firm’s maximization problem is to hire capital until the marginal product of capital exactly equals the rental rate [MPK = r] and to hire labor until the marginal product of labor exactly equals the wage rate [MPL = w].
The marginal product of a factor (capital or labor) is the extra amount of output that is produced when one unit of the factor is added, holding all other inputs constant.
MPK = dY/dK = 1/3 AK-2/3L2/3 = 1/3 AK1/3L2/3/K = 1/3 Y/K
MPL = dY/dL = 2/3 AK1/3L-1/3 = 2/3 AK1/3L2/3/L = 2/3 Y/L
If the production function has constant returns to scale in capital and labor, it exhibits decreasing returns to capital alone and decreasing returns to labor alone.
In a Cobb-Douglas production function, the marginal product of an input is equal to the product of the factor’s exponent times the average amount that each unit of the factor produces.

7. Y = A K1/3 L2/3
MPK = dY/dK = 1/3 A K-2/3 L2/3 = 1/3 A K1/3 L2/3/K = 1/3Y/K
The more capital input with a given amount of labor, the less the MPK
MPK = slope of
production function =
dY/dK

8. Solving the Model: General Equilibrium
Y = A K1/3 L2/3
The five equations are as follows:
The production function, Y = A K1/3 L2/3
The rule for hiring capital: MPK = r
The rule for hiring labor: MPL = w
Supply equals the demand for labor.
Supply equals the demand for capital.
Note: the full employment of resources is taken for granted in the long-run
This is a long-run model
The parameters in the model are the productivity parameter, and the exogenous supplies of capital and labor.
The model has five equations and five endogenous variables:
Output, Y
the amount of capital, K
the amount of labor, L
the wage, w
the rental price of capital, r
A solution to the model is called an equilibrium.
A general equilibrium is the solution to the model when more than a single market clears.

9. Model Setup
Model Solution

10. Interpreting the solution:
If an economy is endowed with more machines or people, it will produce more.
The equilibrium wage is proportional to output per worker and the equilibrium rental rate is proportional to output per capital: w = MPL = 2/3 Y/L and r = MPK = 1/3 Y/K
If labor is becomes more abundant, MPL declines and the wage rate falls
Two-thirds of production is paid to labor and one-third of production is paid to capital.
w*L* = 2/3 (Y*/L*) L* = 2/3 Y* r*K* = 1/3 (Y*/K*) K* = 1/3 Y*
These zero-profit factor shares have been fairly constant throughout US history.

11. Per capita means per person while per worker means per member of the labor force.
In this model, the two are equal.
Define output per capita (y = Y/L) and capital per person (k = K/L).
Doubling capital per person less than doubles output per person because the exponent on the input is less than one  diminishing returns to capital alone.
What makes a country rich or poor?
Output per person is higher if productivity, A, is higher or if the amount of capital per person, K/L = k, is higher.

12. The Empirical Fit of the Production Model
Development accounting is the use of a model to explain differences in incomes across countries.
If we set the productivity parameter to 1, then output per person equals capital per person raised to the one-third power.

13. Diminishing returns to capital implies that countries with a low amount of capital have a high MPK but countries with a lot of capital cannot raise GDP per capita by much through the accumulation of more capital.
If the productivity parameter is 1, the model over-predicts GDP per capita.
Magnitudes are predicted incorrectly and several rich countries are even richer than they should be.

14. Just about all countries would be richer than they actually are if their productivity parameters, A, were equal to1, the value for the U.S.
Their 2000 per capita GDP’s would fall on the 45o line.
45o

15. Productivity Differences: Improving the Fit of the Model
The productivity parameter, A, measures how efficiently countries are using their factor inputs and is often called total factor productivity (TFP).
If TFP is not forced to equal 1, the US value, we can obtain a better fit of the model.
Data on TFP is not collected. Nonetheless, TFP can be estimated because we have data on output and capital per person.
Because TFP is calculated assuming that the model holds, TFP is referred to as the “residual”…it reflects the model’s error when a country’s A is assumed equal to 1.
A lower level of TFP implies that for any given level of capital per person, workers produce less output than a country with higher TFP.

16. Differences in capital per person explain about one-third of the difference in output per person between the richest and poorest countries, while TFP explains the remaining two-thirds.
Thus, rich countries are rich because they have more capital per person, but more importantly, they use labor and capital more efficiently.
Explaining differences in TFP
Human capital … education
Human capital reduces the residual from a factor of 10 to a factor of 5.
Technology
Institutions
Property rights
Rule of law
Contract enforcement

17. Evaluating the Production Model
Per capita GDP is higher if capital per person is higher and if factors are used more efficiently.
Constant returns to scale imply that output per person can be written as a function of capital per person.
Capital per person is subject to diminishing returns and the diminishing returns are very strong because the exponent is much less than one.
In the absence of TFP differences, the production model incorrectly predicts differences in income.
Additionally, the model does not provide an answer as to why countries have different TFP levels.

18. Case Study: A “Big Bang” or Gradualism
Case Study: A “Big Bang” or Gradualism? Economic Reforms in Russia and China
When transitioning from a planned to a market economy, a “big bang” approach is one where all old institutions are replaced quickly by democracy and markets.
Russia followed a “big bang” approach, yet GDP per capita has declined since the transition.
A “gradual” approach is one where the transition to a market economy occurs slowly over time, perhaps even absent political democratization.
China has seen accelerated economic growth using this approach.

19. CHAPTER 4 A Model of Production

20. Summary
Per capita GDP varies by something like a factor of 50 between the richest and poorest countries of the world. If we really understand why this is the case, we ought to be able to build a toy world in which this enormous difference can be observed.
The key equation in our production model is the Cobb-Douglas production function:
Output Y depends on the productivity parameter , the capital stock K and labor L.

21. The exponents in this production function indicate that about one-third of GDP is paid out to capital and two-thirds is paid to labor. The fact that these exponents sum to 1 implies that the production function exhibits constant returns to scale in capital and labor.
The complete production model consists of five equations and five unknowns: the quantities Y, K, and L, and the prices w (wage) and r (the rental price of capital).

22. The solution to this model is called an equilibrium
The solution to this model is called an equilibrium. The prices are determined by the clearing of labor and capital markets, the quantities of capital and labor are pinned down by the exogenous factor supplies, and output is determined by the production function.
The production model implies that output per person in equilibrium is the product of two key forces, total factor productivity (TFP) and capital per person raised to the power 1/3:

23. Assuming the productivity parameter , or TFP, is the same across countries, the model predicts that income differences should be substantially smaller than we observe. Capital per person actually varies enormously across countries, but the sharp diminishing returns to capital per person in the production model overwhelm these differences.

24. Making the production model fit the data requires large differences in TFP across countries. Empirically, these differences “explain” about two-thirds of the differences in income, while differences in capital per person explain about one-third. This “explanation” really just assigns values to that make the model hold; for this reason, economists also refer to TFP as the residual, or a measure of our ignorance.

25. Understanding why TFP differs so much across countries is an important question at the frontier of current economic research. Differences in human capital (such as education) are one reason, as are differences in technologies. These differences in turn can be partly explained by a lack of institutions and property rights in poorer countries.