Intermediate Ecnomics
lanzichao2015
model_of_production.pdf
A"Model"of"Produc-on" Econ"105A,"Intermediate"Macroeconomic"Theory!
! Reading:!Chapter!4!(Charles!Jones)!
!
Tasneem"Raihan" Department"of"Economics"
University"of"California,"Riverside"
Preliminaries
Review: The Model Approach
Facts Model Predictions
New facts to look for?
Consistent with the facts?
Production April 8, 2014
Preliminaries
The Production Model
Facts: some countries are rich, some countries are poor
Model: mathematical description of output production
I System of equations to be solved (supply and demand, e.g.)
I Same approach throughout this class
I This approach shared by both academia and industry
How well does the model explain the facts?
What might be wrong, and how can it be improved?
I Goal is constructive criticism
F How is the model unrealistic?
F How do assumptions a↵ect predictions?
Production April 8, 2014
The Model Production Function
Cobb-Douglas Production Function
Production Function
Y = F(K, L) = AK1/3L2/3
L is labor (workers)
K is capital (machines, buildings, land)
A is the productivity parameter
I Follow Jones textbook notation
I Bar over the letter denotes fixed and exogenous parameter
Production April 8, 2014
The Model Production Function
Constant Returns to Scale
Y = F(K, L) = AK1/3L2/3
Suppose we double both inputs, labor (L) and capital (K):
F(2K, 2L) = A(2K)1/3(2L)2/3
= A21/3K1/322/3L2/3
= 2AK1/3L2/3 = 2F(K, L)
Double both inputs =) Double output
I Constant returns to scale
Standard replication argument
Production April 8, 2014
The Model Production Function
Constant Returns to Scale
Y = F(K, L) = AK1/3L2/3
Suppose we double both inputs, labor (L) and capital (K):
F(2K, 2L) = A(2K)1/3(2L)2/3
= A21/3K1/322/3L2/3
= 2AK1/3L2/3 = 2F(K, L)
Double both inputs =) Double output
I Constant returns to scale
Standard replication argument
Production April 8, 2014
The Model Production Function
Constant vs. Decreasing Returns to Scale
Y = F(K, L) = AK1/3L2/3
Suppose we double only one input, labor (L):
F(K, 2L) = AK1/3(2L)2/3
= A22/3K1/3L2/3
= 22/3AK1/3L2/3 = 22/3F(K, L) < 2F(K, L)
Same result if we double only capital (K)
I F(2K, L) = 21/3F(K, L) < 2F(K, L)
Constant returns to scale in K and L together
Decreasing returns to scale in both K and L alone
Production April 8, 2014
The Model Production Function
Diminishing Marginal Products
Y = F(K, L) = AK1/3L2/3
Decreasing returns in K and L =) diminishing marginal products
Marginal product of capital (MPK):
MPK = @Y
@K =
1
3 AK
�2/3 L
2/3
= 1
3 A
✓ L
K
◆ 2/3
= 1
3
AK
1/3 L
2/3
K
= 1
3 · Y
K
MPK is decreasing in capital (K)
Production April 8, 2014
The Model Production Function
Diminishing Marginal Products
Y = F(K, L) = AK1/3L2/3
Decreasing returns in K and L =) diminishing marginal products
Marginal product of labor (MPL):
MPL = @Y
@L =
2
3 AK
1/3 L
�1/3
= 2
3 A
✓ K
L
◆ 1/3
= 2
3
AK
1/3 L
2/3
L
= 2
3 · Y
L
MPL is decreasing in labor (L)
Production April 8, 2014
The Model Production Function
Diminishing Marginal Products
Production April 8, 2014
The Model Solving the Model
The Production Model So Far
Production function: Y = AK1/3L2/3
One equation (production function)
One exogenous variable (productivity parameter A)
Three endogenous variables: Y , K, L
I Need two more equations, more exogenous variables
Production April 8, 2014
The Model Solving the Model
Completing the Model
Capital market supply equals demand: K = K
I Exogenous supply of capital K
Labor market supply equals demand: L = L
I Exogenous supply of labor L
Three equations, three unknown endogenous variables
Y = AK1/3L2/3
K = K
L = L
Production April 8, 2014
The Model Solving the Model
Equilibrium of the Production Model
Equilibrium
The equilibrium quantities of output, labor, and capital in the production
model are given by
Y
⇤ = A K 1/3
L
2/3
K
⇤ = K
L
⇤ = L
Star above letter denotes an equilibrium value
Express endogenous variables as functions of exogenous variables
Key equation is Y ⇤ = A K 1/3
L
2/3
Production April 8, 2014
The Model Factor Shares
Predictions of the Production Model
Y
⇤ = A K 1/3
L
2/3
Key prediction:
more machines or more people =) more production, more output
Any predictions about where output goes?
I Capital vs. labor
I Need to determine the equilibrium wage rate w
I Need to determine the equilibrium rental rate of capital r
Production April 8, 2014
The Model Factor Shares
Marginal Products and Wages
Perfect competition =) (
w = MPL
r = MPK
Why?
I If MPL > w, firms want to hire more labor
I MPL = 2
3
· Y L
is decreasing in L, so L " =) MPL #
I If MPK > r, firms want to hire more capital
I MPK = 1
3
· Y K
is decreasing in K, so K " =) MPK #
Production April 8, 2014
The Model Factor Shares
Equilibrium Factor Shares
MPL = 2
3 · Y
⇤
L
⇤ and MPK = 1
3 · Y
⇤
K
⇤
Total labor income equals w⇤L⇤ = L⇤ · MPL = 2 3
Y
⇤
I It follows that w ⇤ L
⇤
Y
⇤ = 2
3
Total capital income equals r⇤K⇤ = K⇤ · MPK = 1 3
Y
⇤
I It follows that r ⇤ K
⇤
Y
⇤ = 1
3
Model predicts that labor gets 2/3 of income while capital gets 1/3
I Consistent with the data?
Production April 8, 2014
The Model Factor Shares
Factor Shares over Time
Production April 8, 2014
Production Model Predictions
Output per Person
Production model:
Y
⇤ = A K 1/3
L
2/3
Key prediction:
more machines or more people =) more production, more output
What about output per person (GDP per capita)?
I Better measure of country welfare
I Need to solve model for output per person
Production April 8, 2014
Production Model Predictions
From Output to Output per Person
y = Y L
is output per person
k = K L
is capital per person
Recall that Y ⇤ = A K 1/3
L
2/3 and L⇤ = L
Output per person:
y
⇤ = Y
⇤
L
⇤ = A K
1/3 L
2/3
L
= A
✓ K
L
◆ 1/3
= A k 1/3
Production April 8, 2014
Production Model Predictions
From Output to Output per Person
y = Y L
is output per person
k = K L
is capital per person
Recall that Y ⇤ = A K 1/3
L
2/3 and L⇤ = L
Output per person:
y
⇤ = Y
⇤
L
⇤ = A K
1/3 L
2/3
L
= A
✓ K
L
◆ 1/3
= A k 1/3
Production April 8, 2014
Production Model Predictions
Two Predictions
y
⇤ = A k 1/3
1 More capital per person =) higher GDP per capita
2 Higher productivity parameter =) higher GDP per capita
Facts Model Predictions
New facts to look for?
Consistent with the facts?
Production April 8, 2014
Production Model Predictions
How Do We Test the Production Model?
Equilibrium
The equilibrium quantity of GDP per capita in the production model is
given by
y
⇤ = A k 1/3
Equal productivity across countries =) A same for all countries
A = 1 =) y⇤ = k1/3
Simple prediction about GDP per capita vs. capital per capita
I Data on GDP and capital per capita across countries
Production April 8, 2014
Production Model Predictions
How Do We Test the Production Model?
Equilibrium
The equilibrium quantity of GDP per capita in the production model is
given by
y
⇤ = A k 1/3
Equal productivity across countries =) A same for all countries
A = 1 =) y⇤ = k1/3
Simple prediction about GDP per capita vs. capital per capita
I Data on GDP and capital per capita across countries
Production April 8, 2014
Empirical Fit of the Production Model
A Look at the Data Model Prediction: y⇤ = k
1/3
Country Observed capital Observed GDP Predicted GDP
per capita per capita per capita
Untied States 1.000 1.000 1.000
Turkey 0.290 0.221 0.662
Argentina 0.247 0.292 0.628
Nigeria 0.015 0.072 0.245
Thailand 0.209 0.162 0.593
Netherlands 0.747 0.916 0.908
Japan 1.173 0.713 1.055
Production April 8, 2014
Empirical Fit of the Production Model
A Look at the Data
Production April 8, 2014
Empirical Fit of the Production Model
Consistent with the Facts?
Higher capital per person =) higher GDP per person
I Data and model match decently well
Model tends to overestimate GDP per person
Change production function so that y⇤ = k x
, with x > 1/3?
I Helps with overestimation for poor countries
I Contradicts factor shares data
I Model still overestimates GDP for rich countries
Allow A to vary across countries?
I y
⇤ = A k 1/3
Production April 8, 2014
Empirical Fit of the Production Model
Total Factor Productivity
Production function: Y = F(K, L) = AK1/3L2/3
Higher value of A =) higher GDP for any values of K and L
I Measures how e�ciently inputs K and L are used to produce output
I Captures many di↵erent real-world circumstances
I Virtually impossible to measure directly
A is often called total factor productivity (TFP)
Production April 8, 2014
Empirical Fit of the Production Model
Varying TFP
Can’t measure A directly
Can measure y⇤ and k⇤ directly
) =)
Calculate value of A that
validates model
Equilibrium GDP per capita: y⇤ = A k 1/3
Assume production model is correct
Plug in observed values of y⇤ and k, and solve for A
Production April 8, 2014
Empirical Fit of the Production Model
Examples: y⇤ = A k 1/3
Spain:
I y
⇤ = 0.733, k = 0.908
I Predicted y⇤ = k 1/3
= 0.968
I Implied value of TFP A = 0.757
Spain combines inputs about 3/4 as e�ciently as U.S.
China:
I y
⇤ = 0.183, k = 0.127
I Predicted y⇤ = k 1/3
= 0.502
I Implied value of TFP A = 0.365
China combines inputs about 1/3 as e�ciently as U.S.
Production April 8, 2014
Empirical Fit of the Production Model
Diminishing Marginal Products and TFP Di↵erences
Production April 8, 2014
Empirical Fit of the Production Model
Implied TFP Across Many Countries
Production April 8, 2014
Empirical Fit of the Production Model
Capital Stock vs. TFP
How important are TFP di↵erences in explaining rich vs. poor?
How important are capital stock di↵erences?
Consider five richest and five poorest countries:
y
⇤ rich
y
⇤ poor
= A
rich
A
poor
✓ k
rich
k
poor
◆ 1/3
66 = 11 ⇥ 6
=) TFP almost twice as important as capital stock
What exactly is TFP?
Production April 8, 2014
Discussion of Total Factor Productivity
TFP Overview
Assigning two-thirds of rich-poor income gap to TFP is unsatisfying
I Residual, captures everything else that a↵ects output
I Measure of our ignorance
I Very di�cult to measure directly
I How is TFP increased? Can certain policies help?
How quickly and abruptly can TFP change?
I Russia post-reforms
Production April 8, 2014
Discussion of Total Factor Productivity
Collapse of TFP in Russia
Production April 8, 2014
Discussion of Total Factor Productivity
Human Capital
Human capital is the stock of workers’ skills
I More skills =) workers more productive
How do workers acquire skills?
I Education: literacy, high school, university
I Experience: learning to operate equipment, new techniques
Big di↵erences in education and hence human capital across countries
Enough to explain rich-poor income gap?
I Most studies conclude only about 1/7 of gap explained by education
Production April 8, 2014
Discussion of Total Factor Productivity
Technology
Countries produce output using very di↵erent technologies
I Computer and information technologies
I Mechanical, chemical, and electrical engineering
I Inventories and supply chain management
I Modern agriculture
How di↵erent are countries’ access to production technologies?
Production April 8, 2014
Discussion of Total Factor Productivity
Institutions
Douglas North:
“Institutions are the rules of the game in a society or, more formally, are
the humanly devised constraints that shape human interaction”
Enforcement of property rights and other contracts
Rule of law, separation of powers, corruption
Contrasting neighbors
I North vs. South Korea
I East vs. West Germany
How can institutions be improved?
I Additional reading: Role of Institutions in Growth and Development
Production April 8, 2014
Discussion of Total Factor Productivity
Institutional Change
Limited understanding of what makes good institutions
Virtually no understanding of how to improve institutions
Seemingly positive changes can have negligible or even negative e↵ect
I U.S. Civil War, Cambodia after 1989
I Latin America in the 1980s, African political adjustment
Political and economic institutions are highly persistent
Success stories
I Botswana, China, Chile, Korea, Singapore, Taiwan
Production April 8, 2014
