Chapter 4 - Model of Production

Chapter 4: A Model of Production
Ryan W. Herzog
Spring 2021
Ryan W. Herzog (GU) Production Spring 2021 1 / 44

1 Introduction
2 A Model of Production
3 Analyzing the Production Model
4 Understanding Differences in TFP
5 Evaluating the Production Model

Introduction
Learning Objectives
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.

Introduction
Levels and Growth Rates of Per Capita GDP

Introduction
Building a 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.
Macroeconomists document facts, build a model to understand the
facts, and examine the model to how effective it is.

Production
A Model of Production
Vast oversimplifications of the real world in a model can still allow it
to provide important insights.
Our model will consist of a single, closed economy with one
consumption good.
Labor (L) and capital (K) will be the main inputs.

Production
The Production Function
Shows how much output (Y ) can be produced given any number of
inputs
Y = F(K, L) = AK1/3
L2/3
(1)
where A is a constant measure of productivity.
Equation 1 is a specific form of a Cobb-Douglas production function:
Y = F(K, L) = AKa
Lb
For the case where a + b = 1 we say the production function exhibits
constant returns to scale, i.e. if doubling each input exactly doubles
output.

Production
Constant Returns to Scale - Example
Suppose: Y = AK1/3L2/3 with A = 1, L = 27, K = 8 then
Y = 1 ∗ 81/3
∗ 272/3
= 1 ∗ 2 ∗ 9 = 18
Now suppose A = 1, L = 54, K = 16 then
Y = 1 ∗ 161/3 ∗ 542/3 = 1 ∗ 2.5198 ∗ 14.288 = 36

Production
Returns to Scale
If the sum of the exponents on the inputs....
Sum to 1 then the function has constant returns to scale
Sum to more than 1 then the function has increasing returns to scale
Sum to less than 1 then the function has decreasing returns to scale
The argument for constant returns to scale:
A firm can build an identical factory, hire identical workers, double
production stocks, and can exactly double production.

Production
Allocating Resources - The Firm’s Problem
The firm choosing to maximize profit:
max
K,L
Π = F(K, L) − rK − wL (2)
where r is the rental rate of capital and w is the wage rate.
The rental rate and wage rate are taken as given under perfect
competition.
The solution to the problem is where MPK = r and MPL = w.

Production
Production Function
If the production function has constant returns to scale in capital and
labor, it will exhibit decreasing returns to scale in capital alone.

Production
Key Equations
The marginal product of capital (MPK) is the extra amount of
output that is produced when one unit of capital is added, holding
other inputs constant. For our Cobb-Douglas equation:
MPK =
1
3
A

L
K
2/3
=
1
3
×
Y
K
(3)
The marginal product of labor (MPL) is the extra amount of output
that is produced when one unit of labor is added, holding other inputs
constant. For our Cobb-Douglas equation:
MPL =
2
3
A

K
L
1/3
=
2
3
×
Y
L
(4)

Production
Solving the Model
The model has five endogenous variables (Y , K, L, r, w)
The model has five equations:
Production function: Y = AK1/3
L2/3
Return for hiring capital: 1
3 × Y
K
Return for hiring labor: 2
3 × Y
L
Equilibrium in labor market (supply of labor L)
Equilibrium in capital market (supply of capital K)
Exogenous variables A, K, L, we are taking the supply of capital and
labor as fixed.

Production
Labor and Capital Markets

Production
General Solution
Equilibrium levels of capital and labor are found where r = MPK and
w = MPL which then...
Y ∗
= AK
1/3
L
2/3
r∗
= 1
3 × Y ∗
K∗ = 1
3 A

L
K
2/3
w∗
= 2
3 × Y ∗
L∗ = 2
3 A

K
L
1/3
K∗
= K and L∗
= L

Production
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.

Production
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, (Y /L).
The equilibrium rental rate is proportional to output per capital,
(Y /K).

Production
United State - Empirical Evidence
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 Cobb-Douglas function.
Y ∗
= F(K, L) = AK
1/3
L
2/3
so:
w∗L∗
Y ∗
=
2
3
, and
r∗K∗
Y ∗
=
1
3
(5)

Production
National Income - All Income Paid to K or L
Results in zero profit in the economy
This verifies the assumption of perfect competition.
Also verifies that production equals spending equals income.
w∗
L∗
+ r∗
K∗
= Y ∗
(6)

Analyzing
Analyzing the Production Model
In this model we are assuming per capita is the same as per worker.
We can perform a change of variables to define output per capita (y)
and capital per person (k).
y∗
=
AK
1/3
L
2/3
L
=
AK
1/3
L
1/3
= Ak
1/3
(7)
where y∗ is output per person and k is capital per person.

Analyzing
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?

Analyzing
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 and predict
accurately
To compare across countries we start by setting A = 1 so
y∗
= Ak
1/3
(8)
= (1)k
1/3
= k
1/3
(9)

Analyzing
Key Results
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

Analyzing
The Model’s Prediction for Per Capita GDP (US =1)

Analyzing
Predicted per capita GDP

Analyzing
Across all Countries

Analyzing
Analyzing the Fit of the Model
How can we improve the fit of the model?
We assumed capital’s income share was 1/3 across all countries.
We assumed all countries had the same level of productivity (A).

Analyzing
Case Study: Why Doesn’t Capital Flow from Rich to Poor
Countries
This is referring to the Lucas Paradox, where traditionally low income
countries offer higher returns (a greater MPK).
If MPK is higher in poor countries with low K, why doesn’t capital
flow to those countries?
To start the simple production model with no difference in
productivity across countries is misguided. We must consider the
productivity parameter.

Analyzing
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

Analyzing
Measuring TFP

Analyzing
Comparing Chinese and US Production Functions

Analyzing
Measuring TFP

Analyzing
Comparing Rich and Poor Countries
Output differences between the richest and poorest countries?
Differences in capital per person explain about one-third of the
difference.
TFP explains the remaining two-thirds.
Thus, rich countries are rich because:
They have more capital per person.
More importantly, they use labor and capital more efficiently.

Analyzing
Comparing the United States with Burundi
Real GDP per capita in the US is approximately 70 times greater than
the real GDP per capita in Burundi
If both countries had the same level of TFP our model predicts real
GDP per capita in the US would only be 5 times greater (Burundi
would have 1/5 our income).
If we allow for differences in TFP (assuming the same level of capital
per person) our model predicts real GDP per capita in the US would
be 14 times greater (Burundi would have 1/14 our income).
This shows differences in TFP account for about 3/4 of the income
differential.

Analyzing
Comparing Rich and Poor Countries
Consider the five richest and five poorest countries
On average real GDP per capita was 66 times higher in the rich
countries than the average of the five lowest.
Using the example above means:
y∗
rich
y∗
poor
| {z }
70=
=
Arich
Apoor
| {z }
14×
×

krich
kpoor
1/3
| {z }
5
(10)

TFP
Factors that Explain Differences in TFP
Why are some countries more efficient at using capital and labor?
Human Capital
Technology
Institutions
Misallocation

TFP
Human Capital
Human capital is the stock of skills that individuals accumulate to
make them more productive including education and training.
Returns to education includes the value of the increase in wages from
additional schooling.
Accounting for human capital reduces the residual from a factor of 11
to a factor of 6.

TFP
Technology
Richer countries may use more modern and efficient technologies than
poor countries.
Increases productivity parameter

TFP
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

TFP
Misallocation
Resources not being put to their best use which include
Inefficiency of state-run resources
Political interference

Evaluating
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.

Evaluating
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.

Evaluating
Review Questions
What is a Cobb-Douglas production function?
What are increasing returns, constant returns, and decreasing returns,
and how are the last two relevant in this lecture?
What is the standard replication argument and how is it used?
Why are profits equal to zero under perfect competition?
Explain the equation y = Ak1/3.
How does a production function approach account for income
differences across countries?
What are the limitations of the production function approach?
What other explanations for income differences are important?

Chapter 4 - Model of Production

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