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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5

Log- D ensity

0

-2

-4

-6

-8

-10

-12

12

13

14

15

16

17

18

19

20

21

log-Wealth

6

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[17] Castaneda, A., J. Diaz-Gimenez and J.V. Rios-Rull (2003), "Accounting for the U.S. Earnings and Wealth Inequality", Journal of Political Economy, Vol. 111, 818-857. [18] Champernowne, D. (1953), "A Model of Income Distribution", Economic Journal, Vol. 63, 318-351. [19] Covas, F. (2006), "Uninsured Idiosyncratic Production Risk with Borrowing Constraints", Journal of Economic Dynamics and Control, Vol. 30, 21672190. [20] De Nardi, M. (2004), "Wealth Inequality and Intergenerational Links", Review of Economic Studies, Vol. 71, 743-768. [21] Elwood, Paul, S. M. Miller, Marc Bayard, Tara Watson, Charles Collins and Chris Hartman (1997), "Born on Third Base: The Sources of Wealth of the 1996 Forbes 400," Boston: Uni...ed for a Fair Economy. See also http://www.faireconomy.org/press/archive/Pre_1999/forbes_400_study_1997.html [22] Flavin, M. and T. Yamashita (2002), "Owner-Occupied Housing and the Composition of the Household Portfolio", American Economic Review, Vol. 92, 345-362. [23] Friedman, J. and R. Carlitz (2005), "Estate Tax Reform Could Raise MuchNeeded Revenue, Center on Budget and Policy Priorities", Washington, available at http://www.cbpp.org/3-16-05tax.htm [24] Fuster, L. (2000), "capital accumulation in an Economy with Dynasties and Uncertain Lifetimes", Review of Economic Dynamics, Vol. 3, 650-674. [25] Gabaix, X. (1999), "Zipf's Law for Cities: An Explanation", Quarterly Journal of Economics, Vol. 114, 739-767. [26] Gale, W. G. and J. K. Scholz (1994): "Intergenerational Transfers and the Accumulation of Wealth", Journal of economic perspectives, Vol. 8, 145-160. [27] Gastwirth, J. L. (1971), "A General De...nition of the Lorenz Curve", Econometrica, Vol. 39, 1037-1039. [28] Hendricks, L. (2007), "How important is discount rate heterogeneity for wealth inequality?", Journal of Economic Dynamics and Control, Vol. 31, 3042-3068. 8

[29] Huberman, B. and L. Adamic (1999), "Growth Dynamics of the World-Wide Web", Nature, Vol. 399, 131. [30] Huggett, M. (1993), "The Risk-Free Rate in Heterogeneous-Agent Incomplete Insurance Economies", Journal of Economic Dynamics and Control, Vol. 17, 953-969. [31] Huggett, M. (1996), "Wealth Distribution in Life-cycle Economies", Journal of Monetary Economics, Vol. 38, 469-494. [32] Kamien, M. and Schwartz, N. (1991), "Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management", Second Edition, Elsevier Science Publishing Co., New York. [33] Karlin, S. and H. Taylor (1981), "A Second Course in Stochastic Process", Academic Press, San Diego. [34] Klass, O.; O. Biham; M. Levy; O. Malcai and S. Solomon (2006), "The Forbes 400 and the Pareto Wealth Distribution", Economics Letters, Vol. 90, 290-295. [35] Kopczuk, W. and J. Lupton (2007), "To Leave or Not to Leave: The Distribution of Bequest Motives", Review of Economic Studies, Vol. 74, 207-235. [36] Kotliko¤, L. and L. Summers (1981), "The Role of Intergenerational Transfers in Aggregate Capital Accumulation", Journal of Political Economy, Vol. 89, 706-732. [37] Krusell, P. and A. Smith (1998), "Income and Wealth Heterogeneity in the Macroeconomy", Journal of Political Economy, Vol. 106, 867-896. [38] Levy, M. (2003), "Are Rich People Smarter?", Journal of Economic Theory, Vol. 110, 42-64. [39] Merton, R. (1971), "Optimum Consumption and Portfolio Rules in A Continuous-Time Model", Journal of Economic Theory, Vol. 3, 373-413. [40] Merton, R. (1992), "Continuous-Time Finance", Blackwell Publishers, Cambridge, MA. [41] Meyn, S. P. and R. L. Tweedie (1993), "Markov Chains and Stochastic Stability", Springer, Berlin. 9

[42] Mizuno, T., M. Katori, H. Takayasu and M. Takayasu (2002), "Statistical Laws in the Income of Japanese Companies", in H. Takayasu, Editor, empirical science of Financial Fluctuations: The Advent of Econophysics, Springer, Tokyo. [43] Moskowitz T. and A. Vissing-Jorgensen (2002), "The Returns to Entrepreneurial Investment: A Private Equity Premium Puzzle?", American Economic Review, Vol. 92, 745-778. [44] Nygard, F. and A. Sandstrom (1981), "Measuring Income Inequality", Almqvist & Wiksell International, STOCKHOLM, SWEDEN. [45] Oksendal, B. (1995), "Stochastic Di¤erential Equations", Fourth Edition, Springer, Berlin. [46] Quadrini, V. (2000), "Entrepreneurship, Saving, and social mobility", Review of Economic Dynamics, Vol. 3, 1-40. [47] Reed, W. (2001), "The Pareto, Zipf and Other Power Laws", Economics Letters, Vol. 74, 15-19. [48] Reed, W. (2006), "A Parametric Model for Income and Other size distributions and Some Extensions", International Journal of Statistics, Vol. ????? [49] Richard, S. (1975), "Optimal Consumption, Portfolio and Life Insurance Rules for An Uncertain Lived Individual in a Continuous Time Model", Journal of Financial Economics, Vol. 2, 187-203. [50] Ross, S. M. (1996), "stochastic processes", Second Edition, John Wiley & Sons, Inc., New York. [51] Schechtman, J. and V. L. Escudero (1977), "Some Results on "An Income Fluctuation Problem"", Journal of Economic Theory, Vol. 16, 151-166. [52] Vaughan, R. (1988), "Distributional Aspects of the Life Cycle Theory of Saving," in Kessler, D., and Masson, A. (eds.), Modelling the Accumulation and Distribution of Wealth, Oxford University Press, New York. [53] Wang, N. (2007), "An Equilibrium Model of Wealth Distribution," Journal of Monetary Economics, Vol. 54, 1882-1904 10

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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

Water surface topography in river channels and implications for meander development, 30 pages, 1.32 Mb

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