Income Distribution and Macroeconomics
Oded Galor and Joseph Zeira
Galor-Zeira
Inequality and Growth
1 / 57
The Galor-Zeira Model
The Galor-Zeira Model
Galor-Zeira
Inequality and Growth
2 / 57
The Galor-Zeira Model
The Galor-Zeira Model
Overlapping-Generations economy
Galor-Zeira
Inequality and Growth
2 / 57
The Galor-Zeira Model
The Galor-Zeira Model
Overlapping-Generations economy
t = 0, 1, 2, 3, ...
Galor-Zeira
Inequality and Growth
2 / 57
The Galor-Zeira Model
The Galor-Zeira Model
Overlapping-Generations economy
t = 0, 1, 2, 3, ...
One good
Galor-Zeira
Inequality and Growth
2 / 57
The Galor-Zeira Model
The Galor-Zeira Model
Overlapping-Generations economy
t = 0, 1, 2, 3, ...
One good
3 factors:
Galor-Zeira
Inequality and Growth
2 / 57
The Galor-Zeira Model
The Galor-Zeira Model
Overlapping-Generations economy
t = 0, 1, 2, 3, ...
One good
3 factors:
K
Galor-Zeira
Physical capital
Inequality and Growth
2 / 57
The Galor-Zeira Model
The Galor-Zeira Model
Overlapping-Generations economy
t = 0, 1, 2, 3, ...
One good
3 factors:
K
Physical capital
Ls
Skilled Labor
Galor-Zeira
Inequality and Growth
2 / 57
The Galor-Zeira Model
The Galor-Zeira Model
Overlapping-Generations economy
t = 0, 1, 2, 3, ...
One good
3 factors:
K
Physical capital
Ls
Skilled Labor
Lu
Unskilled Labor
Galor-Zeira
Inequality and Growth
2 / 57
The Galor-Zeira Model
Production
Galor-Zeira
Inequality and Growth
3 / 57
The Galor-Zeira Model
Production
Total output produced
Yt = Yts + Ytu
Galor-Zeira
Inequality and Growth
3 / 57
The Galor-Zeira Model
Production
Total output produced
Yt = Yts + Ytu
Production in the skilled-intensive sector:
Yts = F (Kt , Lst )
Galor-Zeira
Lst f (kt );
Inequality and Growth
kt
Kt /Lst
3 / 57
The Galor-Zeira Model
Production
Total output produced
Yt = Yts + Ytu
Production in the skilled-intensive sector:
Yts = F (Kt , Lst )
Lst f (kt );
kt
Kt /Lst
Production in the unskilled-intensive sector:
Ytu = aLut
Galor-Zeira
Inequality and Growth
3 / 57
The Galor-Zeira Model
Inverse Demand for Factors
Capital:
rt = f 0 (kt )
Galor-Zeira
Inequality and Growth
r (kt )
4 / 57
The Galor-Zeira Model
Inverse Demand for Factors
Capital:
rt = f 0 (kt )
r (kt )
Skilled labor:
wts = f (kt )
Galor-Zeira
f 0 (kt )kt
Inequality and Growth
w s (kt )
4 / 57
The Galor-Zeira Model
Inverse Demand for Factors
Capital:
rt = f 0 (kt )
r (kt )
Skilled labor:
wts = f (kt )
f 0 (kt )kt
w s (kt )
Unskilled labor:
wtu = a
Galor-Zeira
Inequality and Growth
wu
4 / 57
The Galor-Zeira Model
Factor Prices
Small open economy
Galor-Zeira
Inequality and Growth
5 / 57
The Galor-Zeira Model
Factor Prices
Small open economy
World interest = r
Galor-Zeira
Inequality and Growth
5 / 57
The Galor-Zeira Model
Factor Prices
Small open economy
World interest = r
=)
Galor-Zeira
rt = r
kt = f 0 1 (r )
wts = w s (k )
Inequality and Growth
k
ws
5 / 57
The Galor-Zeira Model
Factor Prices
Small open economy
World interest = r
=)
rt = r
kt = f 0 1 (r )
wts = w s (k )
k
ws
=)
(rt , wts , wtu ) = (r , w s , w u )
Galor-Zeira
Inequality and Growth
8t
5 / 57
The Galor-Zeira Model
Individuals
Individuals
Galor-Zeira
Inequality and Growth
6 / 57
The Galor-Zeira Model
Individuals
Individuals
Continuum of measure 1
Galor-Zeira
Inequality and Growth
6 / 57
The Galor-Zeira Model
Individuals
Individuals
Continuum of measure 1
Each Individual has 1 parent and 1 child
Galor-Zeira
Inequality and Growth
6 / 57
The Galor-Zeira Model
Individuals
Individuals
Continuum of measure 1
Each Individual has 1 parent and 1 child
Identical in:
Preferences
Innate abilities
Galor-Zeira
Inequality and Growth
6 / 57
The Galor-Zeira Model
Individuals
Individuals
Continuum of measure 1
Each Individual has 1 parent and 1 child
Identical in:
Preferences
Innate abilities
Di¤er in:
Parental income ) Inv’t in HC
Galor-Zeira
Inequality and Growth
6 / 57
The Galor-Zeira Model
Individuals
Member of Generation t: Period of Life
Galor-Zeira
Inequality and Growth
7 / 57
The Galor-Zeira Model
Individuals
Member of Generation t: Period of Life
First period of life (Period t ):
[invest in HC] or [work as unskilled]
Galor-Zeira
Inequality and Growth
7 / 57
The Galor-Zeira Model
Individuals
Member of Generation t: Period of Life
First period of life (Period t ):
[invest in HC] or [work as unskilled]
Second period of life (Period t + 1):
[work as unskilled / no inv’t in HC] or [work as skilled / inv’t in HC in
1st period]
Galor-Zeira
Inequality and Growth
7 / 57
The Galor-Zeira Model
Individuals
Member of Generation t: Endowment and Preferences
Galor-Zeira
Inequality and Growth
8 / 57
The Galor-Zeira Model
Individuals
Member of Generation t: Endowment and Preferences
Time endowment:
1 units of time in each period
Galor-Zeira
Inequality and Growth
8 / 57
The Galor-Zeira Model
Individuals
Member of Generation t: Endowment and Preferences
Time endowment:
1 units of time in each period
Capital endowment:
bt
Galor-Zeira
capital inherited in 1st period
Inequality and Growth
8 / 57
The Galor-Zeira Model
Individuals
Member of Generation t: Endowment and Preferences
Time endowment:
1 units of time in each period
Capital endowment:
capital inherited in 1st period
bt
Preferences:
u t = α ln ct +1 + (1
Galor-Zeira
α) ln bt +1
Inequality and Growth
α 2 (0, 1)
8 / 57
The Galor-Zeira Model
Individuals
Member of Generation t: Budget Constraint
Second period budget constraint:
c t + 1 + bt + 1
Galor-Zeira
ω t +1
Inequality and Growth
9 / 57
The Galor-Zeira Model
Individuals
Member of Generation t: Budget Constraint
Second period budget constraint:
c t + 1 + bt + 1
ct + 1
Galor-Zeira
ω t +1
consumption
Inequality and Growth
9 / 57
The Galor-Zeira Model
Individuals
Member of Generation t: Budget Constraint
Second period budget constraint:
c t + 1 + bt + 1
ct + 1
consumption
bt + 1
transfers to o¤spring
Galor-Zeira
ω t +1
Inequality and Growth
9 / 57
The Galor-Zeira Model
Individuals
Member of Generation t: Budget Constraint
Second period budget constraint:
c t + 1 + bt + 1
ω t +1
ct + 1
consumption
bt + 1
transfers to o¤spring
ω t +1
wealth in period t + 1
Galor-Zeira
Inequality and Growth
9 / 57
The Galor-Zeira Model
Individuals
Member of Generation t: Optimization
fct +1 , bt +1 g = arg max[α ln ct +1 + (1
s.t.
Galor-Zeira
ct + 1 + b t + 1
Inequality and Growth
α) ln bt +1 ]
ω t +1
10 / 57
The Galor-Zeira Model
Individuals
Member of Generation t: Optimization
bt + 1 = ( 1
α ) ω t +1
ct +1 = αω t +1
Galor-Zeira
Inequality and Growth
11 / 57
The Galor-Zeira Model
Individuals
Member of Generation t: Optimization
bt + 1 = ( 1
α ) ω t +1
ct +1 = αω t +1
Indirect Utility: =)
vt
Galor-Zeira
= α ln αω t +1 + (1
α) ln ω t +1
Inequality and Growth
11 / 57
The Galor-Zeira Model
Individuals
Member of Generation t: Optimization
bt + 1 = ( 1
α ) ω t +1
ct +1 = αω t +1
Indirect Utility: =)
vt
= α ln αω t +1 + (1
= [α ln α + (1
Galor-Zeira
α) ln ω t +1
α) ln(1
Inequality and Growth
α)] + ln ω t +1
11 / 57
The Galor-Zeira Model
Individuals
Member of Generation t: Optimization
bt + 1 = ( 1
α ) ω t +1
ct +1 = αω t +1
Indirect Utility: =)
vt
= α ln αω t +1 + (1
= [α ln α + (1
α) ln ω t +1
α) ln(1
α)] + ln ω t +1
=) v t is monotonic increasing in 2nd period wealth, ω t +1
Galor-Zeira
Inequality and Growth
11 / 57
The Galor-Zeira Model
Individuals
Member of Generation t: Optimization
bt + 1 = ( 1
α ) ω t +1
ct +1 = αω t +1
Indirect Utility: =)
vt
= α ln αω t +1 + (1
= [α ln α + (1
α) ln ω t +1
α) ln(1
α)] + ln ω t +1
=) v t is monotonic increasing in 2nd period wealth, ω t +1
=) maximization of ω t +1 , is the basis of occupational choices
Galor-Zeira
Inequality and Growth
11 / 57
The Galor-Zeira Model
Occupational Choice
Fundamental Assumptions
Imperfect Capital Markets:
r <i
r
interest rate for lender
i
interest rate for borrowers (for inv’t in HC)
Galor-Zeira
Inequality and Growth
(A1)
12 / 57
The Galor-Zeira Model
Occupational Choice
Fundamental Assumptions
Imperfect Capital Markets:
r <i
(A1)
r
interest rate for lender
i
interest rate for borrowers (for inv’t in HC)
Fixed cost of education (Indivisibility of inv’t in HC)
Weighted average of the
payments to teachers, administrators, and maintenance workers in the school system (i.e., weighted average of the wages
skilled and unskilled workers):
C H = θw s + (1
Galor-Zeira
θ )w u
h>0
Inequality and Growth
θ 2 [0, 1]
(A2)
12 / 57
The Galor-Zeira Model
Occupational Choice
Income: Unskilled Individuals
Galor-Zeira
Inequality and Growth
13 / 57
The Galor-Zeira Model
Occupational Choice
Income: Unskilled Individuals
ω ut+1 = (w u + bt )(1 + r ) + w u
Galor-Zeira
Inequality and Growth
13 / 57
The Galor-Zeira Model
Occupational Choice
Income: Unskilled Individuals
ω ut+1 = (w u + bt )(1 + r ) + w u
= w u ( 2 + r ) + ( 1 + r ) bt
Galor-Zeira
Inequality and Growth
13 / 57
The Galor-Zeira Model
Occupational Choice
Income: Skilled Individuals
ω st +1
Galor-Zeira
=
(
ws
(h
bt )(1 + i ) if
bt
h
w s + ( bt
h )(1 + r ) if
bt
h
Inequality and Growth
14 / 57
The Galor-Zeira Model
Occupational Choice
Income: Skilled Individuals
ω st +1
=)
ω st +1
Galor-Zeira
=
=
(
(
ws
(h
bt )(1 + i ) if
bt
h
w s + ( bt
h )(1 + r ) if
bt
h
ws
( 1 + i ) h + ( 1 + i ) bt
if
bt
h
ws
(1 + r )h + (1 + r )bt if
bt
h
Inequality and Growth
14 / 57
The Galor-Zeira Model
Occupational Choice
Assumptions
Investment in human capital is not bene…cial for individuals who must
…nance the entire cost of education via borrowing
ws
Galor-Zeira
(1 + i )h < 0
Inequality and Growth
(A3)
15 / 57
The Galor-Zeira Model
Occupational Choice
Assumptions
Investment in human capital is not bene…cial for individuals who must
…nance the entire cost of education via borrowing
ws
(1 + i )h < 0
(A3)
Investment in human capital is bene…cial for individuals who can …nance
the entire cost of education without borrowing
ws
Galor-Zeira
(1 + r )h > w u (2 + r )
Inequality and Growth
(A4)
15 / 57
The Galor-Zeira Model
Occupational Choice
Income from Being Unskilled Worker
ωtu+1 = wu (2 + r ) + (1 + r )bt
wu ( 2 + r )
bt
Galor-Zeira
Inequality and Growth
16 / 57
The Galor-Zeira Model
Occupational Choice
Income from Being Unskilled Worker
ωtu+1 = wu (2 + r ) + (1 + r )bt
ωtu+1
wu ( 2 + r )
(1 + r )
bt
Galor-Zeira
Inequality and Growth
17 / 57
The Galor-Zeira Model
Occupational Choice
Income from Being Skilled Worker: Borrowers
ws − ( 1 + i)h < 0
(A3)
ωtu+1
wu ( 2 + r )
(1 + r )
bt
w s − (1 + i )h
Galor-Zeira
Inequality and Growth
18 / 57
The Galor-Zeira Model
Occupational Choice
Income from Being Skilled Worker: Borrowers
ωts+1 = ws − (1 + i )h + (1 + i )bt if bt ≤ h
ωtu+1
wu ( 2 + r )
(1 + r )
bt
w s − (1 + i )h
Galor-Zeira
Inequality and Growth
19 / 57
The Galor-Zeira Model
Occupational Choice
Income from Being Skilled Worker: Borrowers
ωts+1 = ws − (1 + i )h + (1 + i )bt if bt ≤ h
ωtu+1
ws
wu ( 2 + r )
(1 + r )
(1 + i )
w − (1 + i )h
s
Galor-Zeira
h
Inequality and Growth
bt
20 / 57
The Galor-Zeira Model
Occupational Choice
Income from Being Skilled Worker: Borrowers
ws − ( 1 + r)h > wu( 2 + r)
(A4)
ωtu+1
ws
wu (2 + r ) + (1 + r )h
wu ( 2 + r )
(1 + r )
(1 + i )
w − (1 + i )h
s
Galor-Zeira
h
Inequality and Growth
bt
21 / 57
The Galor-Zeira Model
Occupational Choice
Income from Being Skilled Worker: Lenders
ωts+1 = ws − (1 + r )h + (1 + r )bt if bt ≥ h
(1 + r )
ws
wu ( 2 + r )
ωts+1
ωtu+1
(1 + r )
(1 + i )
w − (1 + i )h
s
Galor-Zeira
h
Inequality and Growth
bt
22 / 57
The Galor-Zeira Model
Occupational Choice
Bequest and Occupational Choice
ωts+1 = ω s (bt )
ωtu+1 = ω u (bt )
wu ( 2 + r )
w s − (1 + i )h
Galor-Zeira
f
h
Inequality and Growth
bt
23 / 57
The Galor-Zeira Model
Occupational Choice
Bequest and Occupational Choice
ωts+1 = ω s (bt )
ωtu+1 = ω u (bt )
f
Galor-Zeira
h
Inequality and Growth
bt
24 / 57
The Galor-Zeira Model
Occupational Choice
Bequest and Occupational Choice
bt
(
Galor-Zeira
<f
! xtu+1 > xts+1 (individual t becomes unskilled)
>f
! xtu+1 < xts+1 (individual t becomes skilled)
Inequality and Growth
25 / 57
The Galor-Zeira Model
Occupational Choice
Bequest and Occupational Choice
bt
(
<f
! xtu+1 > xts+1 (individual t becomes unskilled)
>f
! xtu+1 < xts+1 (individual t becomes skilled)
where
f =
Galor-Zeira
w u (2 + r )
[w s
i r
(1 + i )h ]
Inequality and Growth
>0
25 / 57
The Galor-Zeira Model
Bequest Dynamics
Bequest Dynamics
bt + 1 = ( 1
Galor-Zeira
α)xt +1
Inequality and Growth
26 / 57
The Galor-Zeira Model
Bequest Dynamics
Bequest Dynamics
bt + 1 = ( 1
bt + 1
8
(1
>
>
<
(1
=
>
>
:
(1
Galor-Zeira
α)xt +1
α)[w u (2 + r ) + (1 + r )bt ]
bt 2 [0, f ]
α)[w s
( 1 + i ) h + ( 1 + i ) bt ]
bt 2 [ f , h ]
α)[w s
(1 + r )h + (1 + r )bt ] bt 2 [h, ∞]
Inequality and Growth
26 / 57
The Galor-Zeira Model
Bequest Dynamics
Bequest Dynamics: Su¢ ciet Conditions for Multiplicity of Steady-Sate
(1
α)(1 + r ) < 1
(1
α)(1 + i ) > 1
(A5)
(1
Galor-Zeira
α )w s > h
Inequality and Growth
(A6)
27 / 57
The Galor-Zeira Model
Bequest Dynamics
Bequest Dynamics
bt +1
φ (bt )
(1 − α ) ws
(1 − α )(1 + r )
(1 − α )(1 + i )
(1 − α ) wu (2 + r )
(1 − α )(1 + r )
f
Galor-Zeira
h
Inequality and Growth
bt
28 / 57
The Galor-Zeira Model
Bequest Dynamics
Bequest Dynamics: Multiple Steady-State Equilibrium
bt +1
φ (bt )
bu
Galor-Zeira
g
Inequality and Growth
bs
bt
29 / 57
The Galor-Zeira Model
Bequest Dynamics
Bequest Dynamics: Stability of High Bequest Equilibrium
bt +1
φ (bt )
bu
Galor-Zeira
g
Inequality and Growth
bs
bt
30 / 57
The Galor-Zeira Model
Bequest Dynamics
Bequest Dynamics: Stability of Steady- State Equilibria
bt +1
φ (bt )
bu
Galor-Zeira
g
Inequality and Growth
bs
bt
31 / 57
The Galor-Zeira Model
Inequality and Skill Composition
The Distribution of the Inheritance in Period t
ξ (b0 )
bt
Galor-Zeira
Inequality and Growth
32 / 57
The Galor-Zeira Model
Inequality and Skill Composition
Income Distribution and the Long Run Decomposition of the Labor Force
ξ t ( bt )
Distribution of inheritance at time t
=)
Lt =
Galor-Zeira
Z ∞
0
ξ (bt )dbt
Inequality and Growth
1
33 / 57
The Galor-Zeira Model
Inequality and Skill Composition
The Distribution of the Inheritance in Period t
L0 ≡ 1
ξ (b0 )
bt
Galor-Zeira
Inequality and Growth
34 / 57
The Galor-Zeira Model
Inequality and Skill Composition
Income Distribution of the Long Run Decomposition of the Labor Force
limt !∞ ltu =
limt !∞ lts =
Galor-Zeira
Rg
0
R∞
g
ξ t (bt )dbt
l̄ u
ξ t (bt )dbt
l̄ s
Inequality and Growth
35 / 57
The Galor-Zeira Model
Inequality and Skill Composition
Income Distribution of the Long Run Decomposition of the Labor Force
limt !∞ ltu =
limt !∞ lts =
where
Rg
0
R∞
g
ξ t (bt )dbt
l̄ u
ξ t (bt )dbt
l̄ s
∂l̄ s /∂g < 0
Galor-Zeira
Inequality and Growth
35 / 57
The Galor-Zeira Model
Inequality and Skill Composition
Income Distribution of the Long Run Decomposition of the Labor Force
limt !∞ ltu =
limt !∞ lts =
where
Rg
0
R∞
g
ξ t (bt )dbt
l̄ u
ξ t (bt )dbt
l̄ s
∂l̄ s /∂g < 0
and
g=
Galor-Zeira
(1
α)[(1 + i )h
(1 α)(1 + i )
Inequality and Growth
ws]
>0
1
35 / 57
The Galor-Zeira Model
Inequality and Skill Composition
Income Distribution of Skill Composition
bt +1
φ (bt )
bu
l
lu
s
g
Galor-Zeira
bt
bs
Inequality and Growth
ξ (b0 )
bb
t
t
36 / 57
The Galor-Zeira Model
Inequality and Skill Composition
Income Distribution of Skill Composition
l
l
u
s
g
Galor-Zeira
Inequality and Growth
ξ (b0 )
bt
37 / 57
The Galor-Zeira Model
Skill Composition and Income Per Capita
Income Per Capita in the Long Run
Income of a skilled individual in the second period of life (wage and capital
income)
I2s = w s + (b̄ s
Galor-Zeira
Inequality and Growth
h )r
38 / 57
The Galor-Zeira Model
Skill Composition and Income Per Capita
Income Per Capita in the Long Run
Income of a skilled individual in the second period of life (wage and capital
income)
I2s = w s + (b̄ s
h )r
Income of an unskilled individual in the second period of life (wage and capital
income)
I2u = w u + (b̄ u + w u )r
Galor-Zeira
Inequality and Growth
38 / 57
The Galor-Zeira Model
Skill Composition and Income Per Capita
Income Per Capita in the Long Run
Income of a skilled individual in the second period of life (wage and capital
income)
I2s = w s + (b̄ s
h )r
Income of an unskilled individual in the second period of life (wage and capital
income)
I2u = w u + (b̄ u + w u )r
Income of an unskilled individual in the …rst period of life (only wage income)
I1u = w u
Galor-Zeira
Inequality and Growth
38 / 57
The Galor-Zeira Model
Skill Composition and Income Per Capita
Income Per Capita in the Long Run
Aggregate income in the steady-state
Ȳ = I2s l̄ s + I2u l̄ u + I1u l̄ u
Galor-Zeira
Inequality and Growth
39 / 57
The Galor-Zeira Model
Skill Composition and Income Per Capita
Income Per Capita in the Long Run
Aggregate income in the steady-state
Ȳ = I2s l̄ s + I2u l̄ u + I1u l̄ u
Aggregate income (note: l̄ s + l̄ u = 1)
Y
= [w s
rh + r b̄ s ]l̄ s + [w u (2 + r ) + r b̄ u ](1
= w u (2 + r ) + r b̄u + [(w s
Galor-Zeira
rh )
Inequality and Growth
l̄ s )
w u (2 + r ) + (b̄ s
b̄ u )]l̄ s
39 / 57
The Galor-Zeira Model
Skill Composition and Income Per Capita
Income Per Capita in the Long Run
Aggregate income in the steady-state
Ȳ = I2s l̄ s + I2u l̄ u + I1u l̄ u
Aggregate income (note: l̄ s + l̄ u = 1)
Y
= [w s
rh + r b̄ s ]l̄ s + [w u (2 + r ) + r b̄ u ](1
= w u (2 + r ) + r b̄u + [(w s
rh )
l̄ s )
w u (2 + r ) + (b̄ s
b̄ u )]l̄ s
Income per capita
ȳ = Ȳ /2
Galor-Zeira
Inequality and Growth
39 / 57
The Galor-Zeira Model
Skill Composition and Income Per Capita
Skill Composition and Income Per Capita in the Long Run
An increase in the fraction of skilled workers increases income per capita
in the steady-state
∂ȳ
= [(w s
∂l̄ s
Galor-Zeira
rh )
w u (2 + r ) + (b̄ s
Inequality and Growth
b̄ u )]/2 > 0
40 / 57
The Galor-Zeira Model
Skill Composition and Income Per Capita
Skill Composition and Income Per Capita in the Long Run
An increase in the fraction of skilled workers increases income per capita
in the steady-state
∂ȳ
= [(w s
∂l̄ s
since
rh )
ws
w u (2 + r ) + (b̄ s
b̄ u )]/2 > 0
(1 + r )h > w u (2 + r )
b̄ s > b̄ u
Galor-Zeira
Inequality and Growth
40 / 57
The Galor-Zeira Model
Skill Composition and Income Per Capita
Skill Composition and Income Per Capita in the Long Run
An increase in the fraction of skilled workers increases income per capita
in the steady-state
∂ȳ
= [(w s
∂l̄ s
since
rh )
ws
w u (2 + r ) + (b̄ s
b̄ u )]/2 > 0
(1 + r )h > w u (2 + r )
b̄ s > b̄ u
An increase in g reduces income per capita in the steady-state
∂ȳ
∂ȳ ∂l̄ s
= s
<0
∂g
∂l̄ ∂g
Galor-Zeira
Inequality and Growth
40 / 57
Inequality and Economic Development
Inequality and Development: Rich Economies
bt +1
φ (bt )
bu
bs
l
u
g
Galor-Zeira
bt
bt
bt
Inequality and Growth
41 / 57
Inequality and Economic Development
Rich Economies: Inequality is Harmful for Development
Inequality reduces human capital formation
bt +1
φ (bt )
bu
bs
l
u
g bt
Galor-Zeira
bt
Inequality and Growth
bt
42 / 57
Inequality and Economic Development
Rich Economies: Inequality is Harmful for Development
bt +1
φ (bt )
bu
bs
l
u
g bt
Galor-Zeira
bt
Inequality and Growth
bt
43 / 57
Inequality and Economic Development
Rich Economies: Inequality is Harmful for Development
l
u
g
Galor-Zeira
bt
Inequality and Growth
bt
44 / 57
Inequality and Economic Development
Inequality and Development: Poor Economies
bt +1
φ (bt )
bu
bs
l
s
bt g
Galor-Zeira
bt
Inequality and Growth
bt
45 / 57
Inequality and Economic Development
Poor Economies: Inequality may Bene…t Development
Inequality stimulates human capital formation
bt +1
φ (bt )
bu
bs
l
s
bt g
Galor-Zeira
bt
Inequality and Growth
bt
46 / 57
Inequality and Economic Development
Poor Economies: Inequality may Bene…t Development
l
bt
Galor-Zeira
s
g
Inequality and Growth
bt
47 / 57
Inequality and Economic Development
Robustness
Robustness
The qualitative results are robust to:
Education cost that is indexed to wages
Galor-Zeira
Inequality and Growth
48 / 57
Inequality and Economic Development
Robustness
Robustness
The qualitative results are robust to:
Education cost that is indexed to wages
Labor augmenting technical change
Galor-Zeira
Inequality and Growth
48 / 57
Inequality and Economic Development
Robustness
Robustness
The qualitative results are robust to:
Education cost that is indexed to wages
Labor augmenting technical change
Shocks the outcome of investment in human capital, as long as wages
are endogenous
Galor-Zeira
Inequality and Growth
48 / 57
Inequality and Economic Development
Robustness
Robustness
The qualitative results are robust to:
Education cost that is indexed to wages
Labor augmenting technical change
Shocks the outcome of investment in human capital, as long as wages
are endogenous
Concave production function of human capital (Moav (EL, 2002), GalorMoav (RES, 2004))
Galor-Zeira
Inequality and Growth
48 / 57
Inequality and Economic Development
Robustness
Robustness: Technological Progress and Endogenous Education Cost
Labor Augmenting Technological Progress: increases the productivity of
workers in both the skilled-intensive and the unskilled intensive sector.
Production in the skilled-intensive sector
Yts = F (Kt , At Lst )
Galor-Zeira
At Lst f (kt );
Inequality and Growth
kt
Kt /At Lst
49 / 57
Inequality and Economic Development
Robustness
Robustness: Technological Progress and Endogenous Education Cost
Labor Augmenting Technological Progress: increases the productivity of
workers in both the skilled-intensive and the unskilled intensive sector.
Production in the skilled-intensive sector
Yts = F (Kt , At Lst )
At Lst f (kt );
kt
Kt /At Lst
Production in the unskilled-intensive sector
Ytu = At aLut
Galor-Zeira
Inequality and Growth
49 / 57
Inequality and Economic Development
Robustness
Robustness: Technological Progress and Endogenous Education Cost
Labor Augmenting Technological Progress: increases the productivity of
workers in both the skilled-intensive and the unskilled intensive sector.
Production in the skilled-intensive sector
Yts = F (Kt , At Lst )
At Lst f (kt );
kt
Kt /At Lst
Production in the unskilled-intensive sector
Ytu = At aLut
Technological progress
At + 1 = ( 1 + λ ) At
Galor-Zeira
Inequality and Growth
λ > 0.
49 / 57
Inequality and Economic Development
Robustness
Robustness: Technological Progress and Endogenous Education Cost
Factor Prices
wts = At [f (k )
wtu = At a
f 0 (k )k ]
At w s
At w u
rt = r
Galor-Zeira
Inequality and Growth
50 / 57
Inequality and Economic Development
Robustness
Cost of Education
Weighted average of the payments to teachers, administrators, and maintenance workers in the school system
Galor-Zeira
Inequality and Growth
51 / 57
Inequality and Economic Development
Robustness
Cost of Education
Weighted average of the payments to teachers, administrators, and maintenance workers in the school system
) Weighted average of the wages skilled and unskilled workers
Galor-Zeira
Inequality and Growth
51 / 57
Inequality and Economic Development
Robustness
Cost of Education
Weighted average of the payments to teachers, administrators, and maintenance workers in the school system
) Weighted average of the wages skilled and unskilled workers
CtH = θAt w s + (1
Galor-Zeira
θ ) At w u
Inequality and Growth
At h
51 / 57
Inequality and Economic Development
Robustness
Income: Unskilled Individuals
xtu+1 = (At w u + bt )(1 + r ) + At +1 w u
Galor-Zeira
Inequality and Growth
52 / 57
Inequality and Economic Development
Robustness
Income: Unskilled Individuals
xtu+1 = (At w u + bt )(1 + r ) + At +1 w u
= At w u ( 2 + r + λ ) + ( 1 + r ) b t
Galor-Zeira
Inequality and Growth
52 / 57
Inequality and Economic Development
Robustness
Income: Skilled Individuals
xts+1
Galor-Zeira
=
(
At + 1 w s
( At h
At +1 w s + (bt
bt )(1 + i ) if
bt
At h
At h )(1 + r ) if
bt
At h
Inequality and Growth
53 / 57
Inequality and Economic Development
Robustness
Income: Skilled Individuals
xts+1
=
(
At + 1 w s
( At h
At +1 w s + (bt
bt )(1 + i ) if
bt
At h
At h )(1 + r ) if
bt
At h
=)
xts+1
=
(
Galor-Zeira
At [ w s ( 1 + λ )
( 1 + i ) h ] + ( 1 + i ) bt
if
bt
At h
At [ w s ( 1 + λ )
(1 + r )h] + (1 + r )bt if
bt
At h
Inequality and Growth
53 / 57
Inequality and Economic Development
Robustness
Threshold level of Bequest for Becoming Skilled Worker in Period t
f =
Galor-Zeira
At f w u ( 2 + r )
[w s
(1 + i )h ]
(i r )
Inequality and Growth
λ (w s
w u )g
54 / 57
Inequality and Economic Development
Robustness
Threshold level of Bequest for Becoming Skilled Worker in Period t
f =
At f w u ( 2 + r )
ft
At f w u ( 2 + r )
=
At
Galor-Zeira
[w s
[w s
(1 + i )h ]
(i r )
(1 + i )h ]
(i r )
Inequality and Growth
λ (w s
λ (w s
w u )g
w u )g
fˆ > 0
54 / 57
Inequality and Economic Development
Robustness
Threshold level of Bequest for Becoming Skilled Worker in Period t
f =
At f w u ( 2 + r )
ft
At f w u ( 2 + r )
=
At
[w s
[w s
(1 + i )h ]
(i r )
(1 + i )h ]
(i r )
λ (w s
λ (w s
w u )g
w u )g
fˆ > 0
for
w u (2 + r ) > [w s (1 + i )h ] + λ (w s w u )
Galor-Zeira
Inequality and Growth
54 / 57
Inequality and Economic Development
Robustness
Bequest Dynamics
bt + 1
8
(1
>
>
>
<
(1
=
>
>
>
:
(1
Galor-Zeira
α)fAt w u (2 + r + λ) + (1 + r )b t g
bt 2 [0, f ]
s
(1 + i )h ] + (1 + i )b t g
s
(1 + r )h] + (1 + r )bt g bt 2 [At h, ∞]
α)fAt [w (1 + λ)
α)fAt [w (1 + λ)
Inequality and Growth
b t 2 [ f , At h ]
55 / 57
Inequality and Economic Development
Robustness
Bequest Dynamics
Let b̂t +1
b̂t +1 =
b t + 1 At + 1
8
>
>
>
>
<
>
>
>
>
:
Galor-Zeira
1 α
1 +λ
fw u (2 + r + λ) + (1 + r )b̂t g
b̂t 2 [0, (fˆ )]
1 α
1 +λ
f[w s (1 + λ)
(1 + i )h] + (1 + i )b̂t g
b̂t 2 [fˆ , h ]
1 α
1 +λ
f[w s (1 + λ)
(1 + r )h] + (1 + r )b̂t g b̂t 2 [h, ∞]
Inequality and Growth
56 / 57
Inequality and Economic Development
Robustness
Bequest Dynamics
Let b̂t +1
b̂t +1 =
b t + 1 At + 1
8
>
>
>
>
<
>
>
>
>
:
1 α
1 +λ
fw u (2 + r + λ) + (1 + r )b̂t g
b̂t 2 [0, (fˆ )]
1 α
1 +λ
f[w s (1 + λ)
(1 + i )h] + (1 + i )b̂t g
b̂t 2 [fˆ , h ]
1 α
1 +λ
f[w s (1 + λ)
(1 + r )h] + (1 + r )b̂t g b̂t 2 [h, ∞]
) The dynamical system is una¤ected qualitatively by labor-augmenting
technological progress
Galor-Zeira
Inequality and Growth
56 / 57
Inequality and Economic Development
Robustness
Su¢ cient Conditions for Multiple Steady-States
(1
α)(1 + r ) < (1 + λ)
(1
α)(1 + i ) > (1 + λ)
w s (1 + λ )
(1 + i )h < 0
) The system is characterized by multiple steady-state, where the
unstable equilibrium
ĝ =
Galor-Zeira
(1
α)[(1 + i )h w s (1 + λ)]
>0
[(1 α)(i + i ) (1 + λ)]
Inequality and Growth
57 / 57