### The Cobb-Douglas Production Function

``` 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.
Modern Growth around the World
 After World War II,
growth in Germany and
Japan accelerated.
Convergence
To US
US
Leapfrogged
UK
 Convergence is the idea
that poorer countries will
grow faster to catch up to
the level of income in
richer countries.
growth until 1980 and then
stagnated, while China and
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.
 Some countries have
exhibited a negative
growth rate over a forty
year period.
Not all
catch up
Some
decline
 Other countries have
sustained nearly 6 percent
growth over the same
period.
 Most countries have
growth.
 Small differences in
growth rates result in large
differences in standards of
living.
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)
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.
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.
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
Solving the Model: General Equilibrium
Y=A
 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
1/3
K
2/3
L
 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
 A solution to the model is called an
taken for granted in the long-run
equilibrium.
 This is a long-run model
 A general equilibrium is the solution
to the model when more than a single The parameters in the model are the
market clears.
productivity parameter, and the exogenous
supplies of capital and labor.
Model Setup
Model Solution
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.
 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.
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 onethird power.
 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.
45o
•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.
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.
 Differences in capital per person
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
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.
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.
CHAPTER 4 A Model of Production
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 CobbDouglas production function:
 Output Y depends on the productivity parameter , the
capital stock K and labor L.
 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).
 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:
 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.
 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.
 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.
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