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4.2 A Model of Production
Chapter 4
A Model of
Production
By Charles I. Jones
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• Vast oversimplifications of the real world
in a model can still allow it to provide
important insights.
• Consider the following model
– Single, closed economy
– One consumption good
Dave Brown
Penn State University
4.1 Introduction
• 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
versus 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.
– Consists of equations and unknowns with real
world interpretations.
Setting Up the Model
• A certain number of inputs are used in
the production of the good
• Inputs
– Labor (L)
– Capital (K)
• Production function
– Shows how much output (Y) can be
produced given any number of inputs
• Others variables with a bar are parameters.
• Production function:
• Macroeconomists:
– Document facts.
– Build a model to understand the facts.
– Examine the model to see how effective it is.
Output
Productivity
parameter
Inputs
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• The Cobb-Douglas production function is
the particular production function that
takes the form of
Assumed to be 1/3.
Explained later.
• A production function exhibits constant
returns to scale if doubling each input
exactly doubles output.
Returns to Scale Comparison
Find the sum of
exponents on the inputs
In general for any production function.
• Show what happens to the output if we
double the inputs:
• If the output is exactly double
– constant RS
• If the output is less than double –
decreasing RS
• If the output is more than double –
increasing RS
Allocating Resources
Result
• sum to 1
• the function has constant
returns to scale
• sum to more than 1
• the function has
increasing returns to
scale
• sum to less than 1
Returns to Scale (RS)
• the function has
decreasing returns to
scale
• Standard replication argument
– A firm can build an identical factory, hire
identical workers, double production stocks,
and can exactly double production.
– Implies constant returns to scale.
Firm chooses inputs
to maximize profit
Rental rate
of capital
Wage rate
• The rental rate and wage rate are taken as
given under perfect competition.
• For simplicity, the price of the output is
normalized to one.
• The marginal product of labor (MPL)
– The additional output that is produced when
one unit of labor is added, holding all other
inputs constant.
• The marginal product of capital (MPK)
– The additional output that is produced when
one unit of capital is added, holding all other
inputs constant.
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Solving the Model:
General Equilibrium
• The solution is to use the following hiring
rules:
– Hire capital until the MPK = r
– Hire labor until MPL = w
• If the production function has constant
returns to scale in capital and labor, it will
exhibit decreasing returns to scale in
capital alone.
• The model has five endogenous
variables:
– Output (Y)
– the amount of capital (K)
– the amount of labor (L)
– the wage (w)
– the rental price of capital (r)
• The model has five equations:
– The production function
– The rule for hiring capital
– The rule for hiring labor
– Supply equals the demand for capital
– Supply equals the demand for labor
• The parameters in the model:
– The productivity parameter
– The exogenous supplies of capital and labor
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• A solution to the model
– A new set of equations that express the five
unknowns in terms of the parameters and
exogenous variables
– Called an equilibrium
• General equilibrium
– Solution to the model when more than a
single market clears
• In this model
– The solution implies firms employ all the
supplied capital and labor in the economy.
– The production function is evaluated with the
given supply of inputs.
– The wage rate is the MPL evaluated at the
equilibrium values of Y, K, and L.
– The rental rate is the MPK evaluated at the
equilibrium values of Y, K, and L.
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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.
• Output per worker = (Y/L)
• The equilibrium rental rate is
proportional to output per capital.
• Output per capital = (Y/K)
Case Study: What Is the Stock Market?
• Economic profit
– Total payments from total revenues
• Accounting profit
– Total revenues minus payments to all
inputs other than capital.
• The stock market value of a firm
– Total value of its future and current
accounting profits
– The stock market as a whole is the value of
the economy’s capital stock.
• In the United States, empirical evidence
shows:
– Two-thirds of production is paid to labor.
– One-third of production is paid to capital.
– The factor shares of the payments are equal
to the exponents on the inputs in the CobbDouglas function.
• All income is paid to capital or labor.
– Results in zero profit in the economy
– This verifies the assumption of perfect
competition.
– Also verifies that production equals spending
equals income.
4.3 Analyzing the Production
Model
• Per capita = per person
• Per worker = per member of the labor
force.
– In this model, the two are equal.
• We can perform a change of variables
to define output per capita (y) and
capital per person (k).
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• Output per person equals the productivity
parameter times capital per person raised to
the one-third power.
• Our model takes factors capital, K, and labor, L, as
inputs and returns output, Y
• The special functional form that describes output so
well, the Cobb-Douglas production function, tells us that
• In words: output per person, y, equals a productivity
constant, A, times capital per person, k, to the one-third
• y will be higher if A or k is higher
• But there are diminishing returns to scale in capital per
worker, k
Output per person
Productivity
parameter
A recap of last class:
Capital per person
Next, we will use this model to understand why
some countries are so much richer than others
•What does the model say about per capita GDP?
•Let’s use lowercases to show variables per capita. Then our model
reveals that
•Output per worker, y, is a function of productivity, A, and capital per
• What makes a country rich or poor?
• Output per person is higher if the
productivity parameter is higher or if the
amount of capital per person is higher.
– What can you infer about the value of the
productivity parameter or the amount of
capital in poor countries?
worker, k = K/L
•(But there are diminishing returns to capital per worker!)
•Our model says countries are richer if A or k is higher
Part 2
•
Comparing Models with Data
• The model is a simplification of reality,
so we must verify whether it models the
data correctly.
• The best models:
– Are insightful about how the world works
– Predict accurately
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The Empirical Fit of the Production Model
• Development accounting:
– The use of a model to explain differences
in incomes across countries.
Set productivity
parameter = 1
• Diminishing returns to capital implies that:
– Countries with low K will have a high MPK
– Countries with a lot of K will have a low MPK,
and cannot raise GDP per capita by much
through more capital accumulation
• If the productivity parameter is 1, the
model overpredicts GDP per capita.
Case Study: Why Doesn’t Capital Flow
from Rich to Poor Countries?
• If MPK is higher in poor countries with
low K, why doesn’t capital flow to those
countries?
– Short Answer: Simple production model
with no difference in productivity across
countries is misguided.
– We must also consider the productivity
parameter.
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Productivity Differences: Improving
the Fit of the Model
• The productivity parameter measures
how efficiently countries are using their
factor inputs.
• Often called total factor productivity
(TFP)
• If TFP is no longer equal to 1, we can
obtain a better fit of the model.
• However, data on TFP is not collected.
– It can be calculated because we have data on
output and capital per person.
– TFP is referred to as the “residual.”
• A lower level of TFP
– Implies that workers produce less output for
any given level of capital per person
• Output differences between the richest
and poorest countries?
– Differences in capital per person explain
about one-quarter of the difference.
– TFP explains the remaining three-quarters.
• Thus, rich countries are rich because:
– They have more capital per person.
– More importantly, they use labor and capital
more efficiently.
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4.4 Understanding TFP
Differences
• Why are some countries more efficient
at using capital and labor?
Institutions
• Even if human capital and technologies
are better in rich countries, why do they
have these advantages?
• Institutions are in place to foster human
capital and technological growth.
– Property rights
– The rule of law
– Government systems
– Contract enforcement
Human Capital
• Human capital
– Stock of skills that individuals accumulate
to make them more productive
– Education, training, etc.
• Returns to education
– Value of the increase in wages from
additional schooling
• Accounting for human capital reduces
the residual from a factor of 18 to a
factor of 9.
Technology
• Richer countries may use more modern
and efficient technologies than poor
countries.
– Increases productivity parameter
Misallocation
• Misallocation
– Resources not being put to their best use
• Examples
– Inefficiency of state-run resources
– Political interference
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Case Study: A “Big Bang” or
Gradualism? Economic Reforms in
Russia and China
• When transitioning from a planned to a
market economy, the change can be
sudden or gradual.
– A “big bang” approach is one where all old
institutions are replaced quickly by
democracy and markets.
– A “gradual” approach is one where the
transition to a market economy occurs
slowly over time.
• Russia followed a “big bang” approach,
yet GDP per capita has declined since the
transition.
• China has seen accelerated economic
growth using the “gradual” approach.
4.5 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 strong
diminishing returns because the
exponent is much less than one.
• Weaknesses of the model:
– In the absence of TFP, the production
model incorrectly predicts differences in
income.
– The model does not provide an answer
as to why countries have different TFP
levels.
Summary
• Per capita GDP varies by a factor of 50 between
the richest and poorest countries of the world.
• The key equation in our production model is the
Cobb-Douglas production function:
Output
Productivity
parameter
Inputs
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• The exponents in this production
function:
• The production model implies that
output per person in equilibrium is the
product of two key forces:
– One-third of GDP is paid out to capital.
– Two-thirds is paid to labor.
– Total factor productivity (TFP)
– Capital per person
– Exponents sum to 1, implying constant
returns to scale in capital and labor.
• The complete production model consists
of five equations and five unknowns:
-
Output Y
Capital K
Labor L
Wage rate w
Rental rate r
• The solution to this model is called an
equilibrium.
• The prices w and r are determined by
the clearing of labor and capital
markets.
• The quantities of K and L are
determined by the exogenous factor
supplies.
• Y is determined by the production
function.
• Assuming the 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.
• Economists also refer to TFP as the
residual, or a measure of our
ignorance.
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• 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.
This concludes the Lecture
Slide Set for Chapter 4
Macroeconomics
Third Edition
by
Charles I. Jones
W. W. Norton & Company
Independent Publishers Since 1923
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