Wealth Distribution in Finite Life, S Zhu

Tags: Review of Economic Studies, Wealth Distribution, Intergenerational Transfers, Capital Accumulation, Journal of Political Economy, Journal of Economic Dynamics and Control, M. Takayasu, wealth inequality, Almqvist & Wiksell International, New York University, Consumer Expenditure Survey, Household Portfolios, model economy, Idiosyncratic Risk, Income Inequality, International Journal of Statistics, Vol., A. Sandstrom, Distribution of Wealth, Household Wealth, Pareto Distribution, H. Takayasu, Journal of Financial Economics, JW
Content: Wealth Distribution in Finite Life Shenghao Zhu New York University February 23, 2009
Abstract I study the wealth distribution in a ...nite life model. Keeping the Pareto tail as in Benhabib and Bisin (2008), I try to incorporate three more realistic mechanisms to generate wealth inequality: (1) the luck within the life time, (2) the death rate from the real data, and (3) the life time income pro...le from the real data (including retirement).
1. Introduction There are several prominent reasons that cause the wealth inequality: the di¤erent life length, the di¤erent labor income due to age, the stochastic rate of return on wealth, and inheritance. In this paper I try to incorporate these factors in a model to replicate the wealth inequality that the real data display. And decompose the contribution of these factors to the wealth inequality.
2. Model
The agent lives from 0 to T . Y (t) is the labor income depending on age. Agents have portfolio selection problem between a risky asset and a riskless asset. Consumer's problem
Z T C(s)1
J(W; t) = max Et
C;!
t
1
e (s t)ds +
[(1
)W (T )]1 e (T t) 1
s:t: dW (s) = [rW (s) + ( r)!(s)W (s) C(s) + Y (s)]ds + !(s)W (s)dB(s)
1
De...ne We know that
ZT b(t) = Y (s)e r(s t)ds t db(t) = [ Y (t) + rb(t)]dt
Proposition 1. The agent's policy functions are
1 C(t) = a(t) [W (t) + b(t)]
r !(t)W (t) = 2 [W (t) + b(t)]
where 0 a(t) = @ + ( (1
( (1
)1
1 )+ 1
1
(r+
1 2
(
r)2 2
)
)1
1 )
exp([ 1
(r
+
1 2
(
) r)2 2
And
( r)2
1
d(W (t) + b(t)) = r +
2
a(t) (W (t) + b(t))dt
+ r (W (t) + b(t))dB(t)
1 A ](T t)) 1
Let X(t) = W (t) + b(t)
Thus
( r)2
1
dX(t) = r +
2
a(t) X(t)dt +
We can solve
1
a(t)
r
X(t) =
exp[(
a(0)
( r)2 + 2 2 )t +
r X(t)dB(t) r B (t)]X (0)
2
Note that
W (T ) = = =
X(T )
1
a(T )
r
exp[(
a(0)
( r)2 + 2 2 )T +
( (1
)1
1
1
r
) a(0) exp[(
+( 2
r B(T )]X(0)
r)2 2 )T +
r B(T )]X(0)
2.1. Intergenerational connection Now let T , 2T , 3T , , nT , be the born time of generation 1, 2, 3, , n, . Let X1 = X(T ); X2 = X(2T ); X3 = X(3T ); ; Xn = X(nT ); Thus
Xn+1
= X((n + 1)T )
= (1 )W ((n + 1)T ) + b(0)
1
(1 )
r
(
=
exp[(
+
a(0)
2
1
(1 )
r
(
=
exp[(
+
a(0)
2
r)2 2 )T + r)2 2 )T +
r B(T )]X(nT ) + b(0) r B(T )]Xn + b(0)
Let
1
(1 )
r
( r)2
n+1 =
a(0)
exp[(
+ 2 2 )T +
Note that n+1 is lognormally disributed. Thus Xn+1 = n+1Xn + b(0)
Thus the result of Sornette could be applied here.
r B(T )]
3. Pareto tail First I prove that the bequest distribution has a Pareto upper tail. Then by Reed (2006), I claim that the wealth distribution has an asymptotic Pareto upper tail.
3
3.1. Bequest distribution
By Sornette, the bequest follows a distribution with a Pareto upper tail, if there exists a such that E n+1 = 1 Note that n+1 is log-normally distributed. Thus
(1 )
r
E n+1 =
a(0)
exp[ (
( + 2
r)2
1
2
)T + 2
2(
r)2 2 2 T] = 1
We have
0
=
@
1 T
log
a(0) (1 )
+
( r)2 22
1 r 1A
4. Simulation reults Using the labor income estimated from Consumer Expenditure Survey (CEX), I simulate the wealth distribution of the economy. The following table shows the Gini coeў cient and quintiles of the wealth distribution in U.S. and in our model economy.1
Economy
Gini F irst Second T hird F ourth F if th
U nited States 0:78 0:39 1:74 5:72 13:43 79:49
M odel
0:63 1:78 5:13 9:42 17:47 66:2
To highlight the skewness to the right and heavy top tail, we further disaggregate the top groups, and compare the percentiles of the wealth distribution for U.S. and the benchmark model economy.
Economy
90th 95th 95th 99th 99th 100th
U nited States 12:62
23:95
29:55
M odel
13:17
20:48
16:64
1The data of the U.S. economy in the following two tables are from Castaneda, Diaz-Gimenez and Rios-Rull (2003) who calculate these tables from 1992 Survey of Consumer Finances (SCF).
4
Density
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-1
0
1
2
3
4
5
6
7
Wealth
x10 8
And the Pareto exponent is
= 1:6131
The simulated density funtion of wealth distribution is The log-log plot of the density can display the Pareto upper tail
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Log- D ensity
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-2
-4
-6
-8
-10
-12
12
13
14
15
16
17
18
19
20
21
log-Wealth
6
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5. Appendix
5.1. Proof of Proposition 1.
Proof: Hamilton-Jacobi-Bellman
C (t)1 J(W; t) = maxf C;! 1
+JW (W; t)[rW (t) + ( r)!(t)W (t)
1 + 2 JW W (W; t)
2!(t)2W (t)2
+Jt(W; t)g
C(t) + Y (t)]
We have the F.O.C.
C(t) = JW (W; t)
JW (W; t)( r) = JW W (W; t) 2!(t)W (t)
Guess
J(W; t) = a(t) (W (t) + b(t))1 1
where Thus
ZT b(t) = Y (s)e r(s t)ds t JW (W; t) = a(t)(W (t) + b(t))
JW W (W; t) = a(t)(W (t) + b(t)) 1
11
Jt(W; t)
=
a_ (t) 1
(W (t)
+
b(t))1
+ a(t)(W (t) + b(t))
We have
1 C(t) = a(t) [W (t) + b(t)]
r !(t)W (t) = 2 [W (t) + b(t)]
[ Y (t) + rb(t)]
Plugging these expressions into the HJB, we have
a(t) (W (t) + b(t))1 1
1 = a(t)
1 (W (t) + b(t))1
1
1 ( r)2
1
+a(t)(W (t) + b(t)) [rW (t) + 2
2 (W (t) + b(t)) a(t) (W (t) + b(t)) + Y (t)]
+ a_ (t) (W (t) + b(t))1 + a(t)(W (t) + b(t)) [ Y (t) + rb(t)] 1
Thus
1 a(t)
1a_ (t) + a(t) 1 [(1
)(r + 1 ( 2
r)2 2)
]+ =0
Using the boundary condition
a(T ) = (1 )1
we can have 0 a(t) = @ + ( (1 We have
( (1
)1
1 )+ 1
1
(r+
1 2
(
r)2 2
)
)1
1 )
exp([ 1
(r
+
1 2
(
) r)2 2
1 A ](T t)) 1
dW (t) = [rW (t) + ( r)!(t)W (t) C(t) + Y (t)]dt + !(t)W (t)dB(t)
( r)2
1
= [rW (t) +
2 (W (t) + b(t)) a(t) (W (t) + b(t)) + Y (t)]dt
+ r (W (t) + b(t))dB(t)
12
We know that
db(t) = [ Y (t) + rb(t)]dt
Thus
( r)2
1
d(W (t) + b(t)) = r +
2
a(t) (W (t) + b(t))dt
r + (W (t) + b(t))dB(t)
13

S Zhu

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