Fertility, Social Mobility, and Long Run Inequality
Juan Carlos Córdoba1, Xiying Liu2, and Marla Ripoll3
Abstract
We investigate social mobility and long run inequality in the presence of endogenous
fertility choices made by altruistic individuals facing uninsurable idiosyncratic risk. The
role of fertility choices in analyzing long run inequality is important for at least two
reasons. First, there is a well-documented negative relationship between fertility and
income. For example, Jones and Tertilt (2008) estimate an income elasticity of fertility of
about -0.38 using US Census data. Second, differential fertilities among rich and poor
families lead to differences in intergenerational wealth transmission, social mobility and
long run inequality. As reported by Menchik (1979), the median child-parent wealth ratio
in one-child families is three times that of families with three or more children.
Earlier theoretical work by Barro and Becker (1988, 1989), but especially Alvarez
(1999), finds that fertility choices by altruistic parents largely reduce intergenerational
persistence and increase social mobility. The reason is that wealthier parents have more
children and transfer to each child an amount that is independent of parental wealth. This
lack of intergenerational persistence is counterfactual. A key goal of this research is to
recover empirically plausible levels of persistence with altruistic models of endogenous
fertility.
We show that a calibrated version of a Barro-Becker dynastic altruistic model of fertility
choice embedded into a Bewley framework of idiosyncratic risk is able to replicate three
key aspects of the data: (i) a negative fertility-income relationship; (ii) a negative
relationship between family size and savings rates; and (iii) a significant intergenerational
persistence or lack of social mobility. We also show that the endogenous fertility model
improves upon the exogenous fertility model in a number of other dimensions such as in
generating larger wealth dispersion. Our calibration exercise also sheds light on the
technology of raising children, the shape of altruism by parents, and on the
“intergenerational elasticity of substitution.”
The following are the elements explaining our results. First, we calibrate an
intergenerational elasticity of substitution larger than one, which implies that children’s
consumption can be easily substituted by parental consumption so that richer parents are
not particularly driven to have more children. Second, the negative-fertility income
relationship naturally arises within the model because of the time cost of raising children
together with the less-than-perfect intergenerational persistence of abilities. As a result,
low ability parents have more children because they expect their children to be of higher
average ability than their own ability. Third, in our model richer families have higher
saving and inheritance rates because they have fewer children and each child weights
1
Iowa State University. E-mail: cordoba@iastate.edu
Iowa State University. E-mail: xiyingl@iastate.edu
3
University of Pittsburgh. E-mail: ripoll@pitt.edu
2
more into the utility of the parent. Different from Alvarez (1999), we assume exponential
child discounting rather than hyperbolic which implies strong decreasing benefits to
having extra children.
Ours is the first dynamic altruistic model of fertility that is consistent with a number of
features describing wealth inequality and intergenerational persistence in the United
States. Please see attached Beamer presentation summarizing some of the results of our
ongoing research.
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
Fertility, Social Mobility, and Long Run Inequality
Juan Carlos Cordoba
Xiying Liu
Marla Ripoll
Iowa State University and University of Pittsburgh
Nov 2014
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
Motivation
Wealth is highly concentrated: the wealthiest 1% owns over
30% of the nation’s wealth.
Family size is an important factor: "median child-parent
wealth ratio in one-child families was 1.84; the median in
families with three or more children was between 0.60 and
0.69." (Menchik 1979).
Standard models of inequality ("Bewley" models) implicitly or
explicitly assume equal fertility rates among individuals.
Laitner 1992, Loury 1981, Aiyagari 1994, Krusell & Smith
1998, Castañeda & Diaz-Gimenez & Rioss-Rull (CDR 2003),
Krueger and Perri (2006), Cordoba (2008), etc.
But reproduction rates are higher among poorer individuals ...
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
Fertility vs. Income
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
Motivation: Barro-Becker Families in a Bewley World
A Bewley + Barro + Becker (BBB) economy:
Bewley = In…nite life+Idiosyncractic earning risk + Incomplete
markets = optimal savings.
BB = …nite life+dynastic altruism + cost of children =
optimal fertility + optimal inheritance.
Can BBB models explain:
fertility di¤erentials among the poor and the rich?
child-parent wealth ratios di¤erentials by family size?
long run-inequality, wealth concentration, social mobility?
consumption, labor supply, savings, inheritance ....
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
What we do
Run horse races between various calibrated BBB models
Calibration strategy follows CDR (2003): use cross-section
evidence to identify parameters and time-series evidence for
overidenti…cation.
Calibration is a powerful tool for intergenerational research
given the scarcity of longitudinal data.
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
Main Findings
Properly calibrated BBB model predict:
realistic negative fertility-income relationship
a negative fertiliy-wealth relationship at certain levels of
wealth
higher inheritance rates by smaller families
more wealth concentration and wealth dispersion than
exogenous fertility models
realistic persistence
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
Main Findings
Family planning policies:
One child policy may induce perpetual growth
Two children policy:
increases average earnings, income, consumption and wealth
but also their dispersion and persistence, particularly
persistence of wealth: reduces social mobility.
Estate taxation:
Reduces average income, consumption and wealth and their
dispersion.
Reduces fertility
Increases social mobility
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
Main Findings
EIS = Elasticity of Intertemporal Substitution
EGS = Elasticity of Intergenerational Substitution
EGS seems to be di¤erent from EIS
EGS larger than 1 and probably between 1.5 to 2. EIS seems
to be less than 1.
Bewley models with EGS<1 generate too much correlation
between earnings and income, too little persistence, and too
little wealth to earnings ratio.
BBB models with EGS<1 cannot replicate a negative
fertility-income relationship for asset poor individuals.
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
Related Literature
Papers on fertility + inequality + warm glow altruism:
no assets: Kremer and Chen (2002), de la Croix and Doepke
(2003).
assets: Moav (2005), Scholz and Seshadri (2009).
Fertility + deterministic inequality + altruism: Bosi,
Boucekkine and Seegmuller (2011).
Fertility+altruism+inequality+no assets: Mookherjee, Prina
and Ray (2012) and Cordoba & Liu (2013).
Fertility+altruism+inequality+assets: Alvarez (1999)
Negative results: lack of persistence
Optimal contracts: Hosseini, Jones, Shourideh (2012)
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.1. Theory
Bewley model with endogenous fertility
Individual’s problem
c1 σ
+ βn1
σ
c, b , n 1
V (b; ω ) = Max
0
(1 + r ) b + ω (1
λn)
α
E V (b 0 ; ω 0 )jω
c + nb 0 , b 0
0; n 2 [0, 1/λ].
b =parental "transfers, bequest, inheritance"
ω parent’s ability, ω 0 children’s ability
ω0
F (ω 0 jω ) (Markov process)
Solution: c = c (b, ω ), n = n(b, ω ), b 0 = b (b, ω ).
Alvarez (1999) studies a related "household" version of this
problem where ω 0 is the same for all children.
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.1. Theory
Lack of persistence property (Alvarez)
First order conditions:
b 0 : ! u 0 (c )
βn
n :! u 0 (c ) b 0 + ωλ
α
β (1
E Vb (b 0 ; ω 0 )jω
α) n
α
E V (b 0 ; ω 0 )jω
(1)
(2)
Dividing (2) by (1):
b 0 + ωλ
(1
α)
E [V (b 0 ; ω 0 )jω ]
E [Vb (b 0 ; ω 0 )jω ]
0
b
8=
9
0
=
0
!
No
persistence!
< b (ω ) in interior solution ! ∂b
=
∂b
0 if constraint for b binds
:
;
b (b, ω ) if upper constraint for n binds
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.1. Theory
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.1. Theory
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.1. Theory
Persistence through abilities
Lack of persistence of inheritances is a big problem.
Evidence suggest large persistence (Mulligan, Piketty and
Saez, Clark, etc)
Since bt +1 = b (ω t ) , persistence of inheritances could be
driven by persistence of abilities, ω.
Can this channel generate enough plausible persistence?
We need a calibrated version.
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.2. Calibration
Calibrating an Intergenerational Bewley model with n=1.
Suppose ln ω 0 = ρ ln ω + e, e
N (0, σ2ω )
Use Tauchen Method to discretize and create F (ω 0 jω ).
Earnings(e)= ω, Income(i)=ω + rb, Wealth(b)= b.
p (b, ω ) =mass of population with wealth b and ability ω.
Wealth-Ability Distribution:
pt +1 (b 0 , ω 0 ) =
∑
∑
ω fb:b 0 =b (b,ω )g
pt (b, ω )F (ω 0 jω )
5 Parameters: [σ, β, r , ρ, σω ] . 5 targets?
1/σ is the Elasticity of Intergenerational Substitution (EGS):
it describes the willingness to substitute consumption across
generations (not time).
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.2. Calibration
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.2. Calibration
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.2. Calibration
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.2. Calibration
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.2. Calibration
Parameter Values for Exogenous Fertility Model
Parameter
r = Interest rate
σ = curvature utility function
β = discount factor
ρ ω = persistence log ability
σω = dispersion log ability
Value
2.00
0.15
0.32
0.50
1.05
Target
An annual interest rate of 4.5 for 25 years
Gini wealth
earnings income correlation
Persistence of hourly wages (Mulligan)
Gini earnings
Data
2.00
0.82
0.84
0.50
0.64
Model 1
2.00
0.80
0.88
0.50
0.60
A high EGS (= 6.66) is required to replicate earnings-income
correlation.
β is closed to
1
1 +r .
Bewley models needs β (1 + r ) < 1 to avoid unbounded
savings.
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.2. Calibration
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.2. Calibration
Performance
Exogenous Fertility Model
Coefficient variation earnings
Coefficient variation income
Coefficient variation wealth
Correlation earnings-wealth (e,b)
correlation income-wealth (i,b)
Gini Income
Gini consumption
persistence earnings
persistence income
persistence wealth
persistence consumption
average(b)/average(e)
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Data
3.60
4.32
6.02
0.48
0.57
0.58
0.32
0.48
0.71
0.5-0.8
0.77
6.80
Model 1
1.66
1.49
2.29
0.20
0.63
0.60
0.58
0.50
0.71
0.72
0.73
0.21
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.2. Calibration
Generational and intergenerational correlations
Exogenous Fertility Model
Variables in logs
ωp
ep
ip
bp
cp
lsp
ωc
fp
ec
ic
bc
cc
lsc
fc
ωp
100
ep
100
100
ip
78
78
bp
-6
-6
44
100
cp
72
72
100
52
100
lsp
0
0
0
0
0
100
fp
0
0
0
0
0
0
0
ωc
48
48
37
-3
34
0
0
100
ec
48
48
37
-3
34
0
0
100
100
ic
57
57
75
35
75
0
0
81
81
bc
48
48
89
68
92
0
0
23
23
68
100
cc
57
57
78
40
79
0
0
77
77
100
73
100
lsc
0
0
0
0
0
100
0
0
0
0
0
0
100
fc
0
0
0
0
0
0
0
0
0
0
0
0
0
100
100
100
Cordoba, LiuCorrelations
& Ripoll computed generating a random sample of 300,000 drawsIowa
University
and University
fromState
the steady
state distribution
of of Pittsburgh
Fertility, Social
anda Long
Inequality with non-zero components was used. This reduced the sample to
the Mobility,
model. Only
subsetRun
of observations
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.2. Calibration
Parameter Values for Exogenous Fertility Model 2
σ=1.5
Parameter
r = Interest rate
σ = curvature utility function
β = discount factor
ρ ω = persistence log ability
σω = dispersion log ability
Value
2.00
1.50
0.12
0.50
1.05
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Target
An annual interest rate of 4.5 for 25 years
Gini wealth
earnings income correlation
Persistence of hourly wages (Mulligan)
Gini earnings
Data
2.00
0.82
0.84
0.50
0.64
Model 1 Model 2
2.00
2.00
0.80
0.80
0.88
0.97
0.50
0.50
0.60
0.60
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.2. Calibration
Performance
Exogenous Fertility Models
Coefficient variation earnings
Coefficient variation income
Coefficient variation wealth
Correlation earnings-wealth (e,b)
correlation income-wealth (i,b)
Gini Income
Gini consumption
persistence earnings
persistence income
persistence wealth
persistence consumption
average(w)/average(e)
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Data
3.60
4.32
6.02
0.48
0.57
0.58
0.32
0.48
0.71
0.5-0.8
0.77
6.80
Model 1 Model 2
1.66
1.66
1.49
1.57
2.29
2.59
0.20
0.25
0.63
0.46
0.60
0.60
0.58
0.59
0.50
0.50
0.71
0.62
0.72
0.60
0.73
0.65
0.21
0.08
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.2. Calibration
Findings: exogenous fertility model
Exogenous fertility model produces signi…cantly persistence of
wealth and consumption...
but low dispersion of earnings, income and wealth
Explaining inequality requires high β (β !
curvature (σ ! 0) of the utility function
1
1 +r )
and low
Otherwise precautionary savings are too large, correl (e, i ) is
too strong, persistence falls signi…cantly and mean(b)/mean(e)
is too low.
Existing calibration typically use σ = 1.5 (CDR 2003,
Restuccia & Urrutia 2004)
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.2. Calibration
Calibration Alvarez endogenous fertility model
Wealth-Ability Distribution:
pt +1 (b 0 , ω 0 ) =
1
nt
∑
ω
∑
fb:b 0 =b (b,ω )g
pt (b, ω )n(b, ω )F (ω 0 jω )
where nt = ∑ω,b pt (b, ω )n(b, ω ) is average population
growth.
8 Parameters: [σ, β, r , ρ, σω ] + [α, λ] .
Need 2 more targets.
Additional targets:
E [n] = 1 (similar to U.S. total fertility rate per-capita)
Coe¢ cient of variation n = 0.6 (as suggested by Jones and
Tertil, 2008)
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.2. Calibration
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.2. Calibration
Parameter Values for Alvarez Model 3
Parameter
r = Interest rate
σ = curvature utility function
β = discount factor
α = curvature altruism
λ = cost of a child
ρ ω = persistence log ability
σω = dispersion log ability
Value
2.00
0.72
0.25
0.57
0.40
0.50
0.85
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Target
An annual interest rate of 4.5 for 25 years
Gini wealth
earnings income correlation
average fertility
coefficient of variation fertility
Persistence of hourly wages (Mulligan)
Gini earnings
Data
2.00
0.82
0.84
1.00
0.60
0.50
0.64
Model 3
2.00
0.84
0.78
1.00
0.52
0.50
0.59
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.2. Calibration
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.2. Calibration
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.2. Calibration
Performance
Exogenous Fertility Model vs Alvarez's Model
Coefficient variation earnings
Coefficient variation income
Coefficient variation wealth
Correlation earnings-wealth (e,b)
correlation income-wealth (i,b)
Gini Income
Gini consumption
persistence earnings
persistence income
persistence wealth
persistence consumption
average(w)/average(e)
income elasticity of fertility
Data
3.60
4.32
6.02
0.48
0.57
0.58
0.32
0.48
0.71
0.5-0.8
0.77
6.80
-0.20
Model 1
1.66
1.49
2.29
0.20
0.63
0.60
0.58
0.50
0.71
0.72
0.73
0.21
0.00
Model 3
1.49
1.47
2.90
0.16
0.74
0.57
0.56
0.35
0.58
0.27
0.59
0.30
-0.21
Problematic predictions: very low correlation between earnings and
consumption, strong positive association between wealth and
fertility, and therefore negative association between labor supply
Cordoba, Liu & Ripoll
State University and University of Pittsburgh
and wealth. Negative association between Iowa
consumption
and labor
Fertility, Social Mobility, and Long Run Inequality
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.2. Calibration
Generational and intergenerational correlations
Alvarez Model
Variables in logs
ωp
ep
ip
bp
cp
lsp
fp
ωc
ec
ic
bc
cc
lsc
ωp
100
ep
73
ip
93
46
100
bp
28
-28
58
cp
83
27
98
72
100
lsp
26
85
-8
-62
-27
100
fp
-48
-89
-12
65
9
-90
100
ωc
48
36
45
13
40
13
-23
100
ec
19
15
18
5
16
6
-10
91
100
ic
63
46
58
17
52
17
-30
98
83
100
bc
97
71
92
28
82
25
-46
47
18
61
cc
71
52
66
20
59
19
-34
95
76
99
69
lsc
-34
-25
-33
-10
-29
-8
16
43
76
28
-35
17
100
fc
27
19
25
8
23
6
-12
-68
-87
-53
27
-42
-86
fc
100
100
100
100
100
Correlations computed generating a random sample of 300,000 draws from the steady state distribution of
the model. Only a subset of observations with non-zero components was used. This reduced the sample to
65129 observations.
Cordoba, Liu & Ripoll
Iowa State University and University of Pittsburgh
Some problematic predictions: very low correlation between
Fertility, Social Mobility, and Long Run Inequality
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.2. Calibration
Jones-Schoonbroodt (negative utility) version
Parameter Values for Jones-Schoonbroodt Model 4
σ = 1.5
Parameter
r = Interest rate
σ = curvature utility function
β = discount factor
α = curvature altruism
λ = cost of a child
ρ ω = persistence log ability
σω = dispersion log ability
Value
2.00
1.50
0.16
1.80
0.30
0.50
1.00
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Target
An annual interest rate of 4.5 for 25 years
Gini wealth
earnings income correlation
average fertility
coefficient variation fertility
Persistence of hourly wages (Mulligan)
Gini earnings
Data
2.00
0.82
0.84
1.00
0.60
0.50
0.64
Model 4
2.00
0.81
0.74
1.00
0.52
0.50
0.61
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.2. Calibration
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.2. Calibration
Performance
Exogenous Fertility Model vs Alvarez vs JS
Coefficient variation earnings
Coefficient variation income
Coefficient variation wealth
Correlation earnings-wealth (e,b)
correlation income-wealth (i,b)
Gini Income
Gini consumption
persistence earnings
persistence income
persistence wealth
persistence consumption
average(w)/average(e)
income elasticity of fertility
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Data
3.60
4.32
6.02
0.48
0.57
0.58
0.32
0.48
0.71
0.5-0.8
0.77
6.80
-0.20
Model 1 Model 3 Model 4
1.66
1.49
1.68
1.49
1.47
1.59
2.29
2.90
3.05
0.20
0.16
0.08
0.63
0.74
0.73
0.60
0.57
0.60
0.58
0.56
0.58
0.50
0.35
0.14
0.71
0.58
0.60
0.72
0.27
0.23
0.73
0.59
0.60
0.21
0.30
0.21
0.00
-0.21
-0.12
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
2.2. Calibration
The Negative Fertility-Ability Relationship and the EGS.
First order condition:
u 0 (1 + r ) b + ω
= (1
α) n
α
n b 0 + λω
b 0 + ωλ
E V (b 0 ; ω 0 )jω
In the i.i.d. case with b 0 = b = 0:
λ (1
nλ)
σ
ω1
σ
= (1
α) n
α
E V (b 0 ; ω 0 )
Ability a¤ects the marginal cost but not the marginal bene…t.
σ 2 (0, 1) , or EGS > 1 is necessary for the the marginal cost
to be increasing with ability.
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
3.1. Theory
Extended Model
V (b; ω ) =
u (c ) + Φ(n)E V (b 0 ; ω 0 )jω
c + nb 0 + λ(n)ω, b 0
s.t. (1 + r ) b + ω
n
Max
c , b0, n
0
2 f0, 0.5, 1, 1.5, 2, ..., nmax g
Φ(n) =degree of altruism: Φ(0) = 0, Φ0 (n) > 0, Φ00 (n) < 0.
λ(n) = time cost of n children
Three channels to recover persistence:
Non-isoelastic altruistic function: Φ(n).
Non-constant costs of raising children: λ(n).
Discrete number of children.
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
3.1. Theory
First order conditions for n and b 0 :
E [V (b 0 ; ω 0 )jω ]
b 0 + ωλ0 (n) = εn (n)
|
{z
}
E [Vb (b 0 ; ω 0 )jω ]
|
{z
}
0
MRT (b ,n )
MRS (b 0 ,n )
where ε(n) = Φ0 (n) Φ(nn ) .
In Barro-Becker both MRT =
independent of n.
∂b 0
∂n
and MRS =
∂b 0
∂n
are
Persistence is recovered if MRT increases with n or MRS
decreases with n (Alvarez 1999).
In both cases parents are less willing to use fertility as a way to
obtain welfare as n increases.
Some persistence is also recovered if fertility is discrete.
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
3.2. Calibration
Calibration Extended Model
Functional forms
Altruistic weight:
Φ (n ) =
ε(nt ) =
βχn
1 + χn
1
1 + χn
(elasticity decreases with n).
Cost of children:
λ (n ) = λ (n + η ) θ
λ η θ , 0 < θ < 1.
1 /θ
λ (0) = 0, λ (nmax ) = 1 where nmax = 1/λ + η θ
η.
λ0 (n) > 0 and λ00 (n) < 0 if θ 2 (0, 1) (realistic and
necessary).
θ 1
λ0 (0) = λθη
< ∞ if η > 0 (important!)
calibrated model with η = 0 and θ < 1 implies an even higher
cost of children than in Alvarez model.
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
3.2. Calibration
Parameter Values - Full Model
Parameter
r = Interest rate
σ = curvature utility function
β = discount factor
χ = curvature altruism
λ = parameter cost of children
η = parameter cost of children
θ = elasticity cost of children
ρ ω = persistence log ability
σω = dispersion log ability
Value
2.00
0.70
0.39
2.00
3.05
1.84
0.23
0.50
0.90
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Target
An annual interest rate of 4.5 for 25 years
Gini wealth
earnings income correlation
average fertility
average cost per child (per parent)
time cost of first child (per parent)
times cost of first two children (per parent)
Persistence of hourly wages (Mulligan)
Gini earnings
Data
2.00
0.82
0.84
1.00
0.30
0.20
0.35
0.50
0.64
CL Model
2.00
0.85
0.75
1.00
0.27
0.20
0.37
0.50
0.58
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
3.2. Calibration
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
3.2. Calibration
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
3.2. Calibration
Performance
CL Model vs Alvarez Model
Coefficient variation earnings
Coefficient variation income
Coefficient variation wealth
Correlation earnings-wealth (e,b)
correlation income-wealth (i,b)
Gini Income
Gini consumption
persistence earnings
persistence income
persistence wealth
persistence consumption
average(b)/average(e)
income elasticity of fertility
wealth elasticity of fertility
Average cost per child
Efficient number of children
Maximum number of children per couple
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Data
3.60
4.32
6.02
0.48
0.57
0.58
0.32
0.48
0.71
0.5-0.8
0.77
6.80
-0.20
CL
1.49
1.53
3.29
0.21
0.80
0.60
0.59
0.37
0.65
0.51
0.65
0.24
-0.18
0.08
0.27
0.66
7.00
Alvarez
1.44
1.40
2.43
0.28
0.73
0.57
0.56
0.44
0.58
0.29
0.59
0.26
-0.21
0.19
0.40
0.60
5.00
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
3.2. Calibration
Endogenous Discounting and the Accumulation of Wealth
First order condition with respect to assets:
u 0 (c ) =
Φ (n )
n
Φ (n )
(1 + r ) E u 0 c 0
n
is the average discount factor.
Higher fertility reduces average discounting and incentives to
save.
Krussell and Smith (1998).
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
3.2. Calibration
The Negative Fertility-Wealth Relationship
Fertility goes to 1 for everyone rather than to the maximum.
Why does fertility decrease with inheritance for certain
individuals in our model?
Because of credit constraints
In addition to the pure wealth e¤ect there is a substitution
e¤ect from inheritances
Individuals who receive low inheritances may …nd themselves
credit constrained if their ability shock is low enough
For those individuals, higher inheritance reduces the shadow
price of credit (the e¤ective interest rate)
By arbitrage, the implicit rate of return to children is tied to
the rate of return to savings
A lower rate of return reduce the incentives to save in the form
of children
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
3.2. Calibration
Performance
CL Model vs Alvarez Model
Coefficient variation earnings
Coefficient variation income
Coefficient variation wealth
Correlation earnings-wealth (e,b)
correlation income-wealth (i,b)
Gini Income
Gini consumption
persistence earnings
persistence income
persistence wealth
persistence consumption
average(b)/average(e)
income elasticity of fertility
wealth elasticity of fertility
Average cost per child
Efficient number of children
Maximum number of children per couple
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Data
3.60
4.32
6.02
0.48
0.57
0.58
0.32
0.48
0.71
0.5-0.8
0.77
6.80
-0.20
CL
1.49
1.53
3.29
0.21
0.80
0.60
0.59
0.37
0.65
0.51
0.65
0.24
-0.18
0.08
0.27
0.66
7.00
Alvarez
1.44
1.40
2.43
0.28
0.73
0.57
0.56
0.44
0.58
0.29
0.59
0.26
-0.21
0.19
0.40
0.60
5.00
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
3.2. Calibration
Generational and intergenerational correlations
CL model
Variables in logs
ωp
ep
ip
bp
cp
lsp
fp
ωc
ec
ic
bc
cc
lsc
ωp
100
ep
97
ip
61
53
100
bp
-16
-21
49
cp
49
40
99
62
lsp
70
86
26
-27
14
100
fp
-84
-88
-36
43
-21
-79
ωc
41
40
25
-7
20
28
-35
100
ec
35
34
15
-12
10
24
-29
96
100
ic
61
58
68
17
62
39
-50
83
73
100
bc
58
54
93
49
91
35
-45
24
12
65
cc
63
60
74
22
69
40
-51
77
65
99
72
lsc
13
11
-12
-18
-15
6
-10
61
80
30
-16
22
100
fc
-12
-11
17
24
20
-8
10
-79
-87
-47
21
-37
-79
fc
100
100
100
100
100
100
100
Cordoba, LiuCorrelations
& Ripoll computed generating a random sample of 300,000 drawsIowa
University
and University
fromState
the steady
state distribution
of of Pittsburgh
Fertility, Social
anda Long
Inequality with non-zero components was used. This reduced the sample to
the Mobility,
model. Only
subsetRun
of observations
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
Experiments
Experiment 1: One and two child policies
One child policy: a policy that restricts n
perpetual growth into this economy.
Given that r is exogenous,
Φ (n )
n
0.5 brings
(1 + r ) > 1 for n
For bounded wealth, Bewley models needs
Φ (n )
n
0.5.
(1 + r ) < 1
We consider instead a two children policy instead: n
1.
Experiment 2: Estate taxes
Suppose a estate tax such that individuals returns are 1 + r /2
instead of 1 + r .
Collected taxes are used to …nance exogenous government
expenses.
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
Experiments
Two Children Policy (TCP) and Estate Taxation (ET)
Coefficient variation earnings
Coefficient variation income
Coefficient variation wealth
Correlation earnings-wealth (e,b)
correlation income-wealth (i,b)
Gini earnings
Gini income
Gini wealth
Gini consumption
persistence earnings
persistence income
persistence wealth
persistence consumption
average(b)/average(e)
average fertility
income elasticity of fertility
wealth elasticity of fertility
Mean(e)
Mean(i)
Mean(b')
Mean(c)
stdev(e)
stdev(i)
stdev(b')
stdev(c)
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Benchmark
1.49
1.53
3.29
0.21
0.80
0.58
0.60
0.85
0.59
0.37
0.65
0.51
0.65
0.24
1.00
-0.18
0.08
0.99
1.32
0.24
1.33
1.48
2.02
0.68
1.99
TCP*
1.46
1.44
2.42
0.20
0.81
0.57
0.58
0.81
0.58
0.49
0.71
0.67
0.72
0.29
0.92
-0.10
0.01
1.06
1.54
0.31
1.56
1.55
2.22
0.76
2.22
ST
1.49
1.49
3.90
0.15
0.46
0.57
0.57
0.93
0.56
0.45
0.54
0.15
0.55
0.08
0.97
-0.09
0.09
1.06
1.11
0.09
1.11
1.58
1.66
0.37
1.59
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
Conclusions
Properly calibrated BBB model predict:
realistic negative fertility-income relationship
a negative fertiliy-wealth relationship at certain levels of
wealth
higher inheritance rates by smaller families
more wealth concentration and wealth dispersion than
exogenous fertility models
realistic persistence
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
Conclusions
Family planning policies:
One child policy may induce perpetual growth
Two children policy:
increases average earnings, income, consumption and wealth
but also their dispersion and persistence, particularly
persistence of wealth: reduces social mobility.
Estate taxation:
Reduces average income, consumption and wealth and their
dispersion.
Reduces fertility
Increases social mobility
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh
1. Introduction
2. Horse Race Existing Models
3. A Better Model
4. Policy Experiments
5. Conclusions
Conclusions
EGS seems to be di¤erent from EIS
EGS larger than 1 and probably between 1.5 to 2.
EIS seems to be less than 1.
Cordoba, Liu & Ripoll
Fertility, Social Mobility, and Long Run Inequality
Iowa State University and University of Pittsburgh