Age, Luck, and Inheritance

Transcription

1 Age, Luck, and Inheritance Jess Benhabib Shenghao Zhu New York University December 7, 2007 ess Benhabib Shenghao Zhu (New York University)Age, Luck, and Inheritance December 7, / 23

2 Motivations We set up a parsimonious model and solve it analytically to match the wealth distribution in terms of Gini coe cient Right skewness Heavy upper tail Pareto distribution We disentangle the contribution of luck, age and inheritance for the wealth inequality. Jess Benhabib Shenghao Zhu (New York University)Age, Luck, and Inheritance December 7, / 23

3 Mechanism to produce a Pareto distribution Pareto distribution f (x) x β x x A standard mechanism to generate Pareto distribution is to construct a stochastic process with negative drift and a lower re ecting barrier. Champernowne (1953). Because of luck, we ll loose the re ecting barrier and get a "Double-Pareto". Jess Benhabib Shenghao Zhu (New York University)Age, Luck, and Inheritance December 7, / 23

4 Density Mechanism to produce a Pareto distribution (cont.) A Pareto Distribution looks like this: Pareto distribution y=1.6x 2.6, x x Jess Benhabib Shenghao Zhu (New York University)Age, Luck, and Inheritance December 7, / 23

5 An equivalent mechanism to generate Pareto distribution with a re ecting barrier: Positive growth rate and death. At death new agent is reinjected at a lower wealth x. w = x e gt, T pe pt =) w w ( p g +1), w x More realistically, at death the individual wealth process can jump to some place other than the lower bound x if there is inheritance, taxes, annuities or life insurance. Benhabib and Bisin (2006). Jess Benhabib Shenghao Zhu (New York University)Age, Luck, and Inheritance December 7, / 23

6 What is new of this paper? We introduce the stochastic rate of capital return (Geometric Brownian motion) to Benhabib and Bisin (2006) and generate a "Double Pareto" distribution. We produce the wealth inequality of Gini as high as 0.64 with a parsimonious model. We disentangle the contributions of luck, age and inheritance for the wealth inequality. And we nd the two opposing roles of inheritance for the wealth inequality. The mechanism is similar to Reed (2001). But Reed (2001) does not discuss inheritance. Jess Benhabib Shenghao Zhu (New York University)Age, Luck, and Inheritance December 7, / 23

7 Building blocks Yarri (1965) RES, Blanchard (1985) JPE Merton (1971) JET Richard (1975) JFE Wold and Whittle (1957) Econometrica Huberman and Adamic (1999) Nature Jess Benhabib Shenghao Zhu (New York University)Age, Luck, and Inheritance December 7, / 23

8 Model There is a continuum of agents in the economy. Constant death rate p. When the agent dies, one child is born. Investment opportunity riskless asset risky asset where r < α. dq(t) = Q(t)rdt ds(t) = S(t)αdt + S(t)σdB(t) Joy-of-giving bequest motive Z (s, t) denotes the bequest that the agent born at time s leaves at time t if the agent dies. Life insurance purchase is P(s, t). The price of the life insurance is µ. Annuities are negative life insurance. In a fair market µ = p. Z (s, t) = W (s, t) + P(s, t) µ Jess Benhabib Shenghao Zhu (New York University)Age, Luck, and Inheritance December 7, / 23

9 A portfolio selection problem with uncertain lifespan-richard (1975) The agent s utility maximization problem is Z + max E t e (θ+p)(v t) [ C 1 γ (s, v) ((1 ζ)z (s, v))1 γ + pχ ]dv C,ω,P t 1 γ 1 γ (1) subject to dw (s, t) = [(r τ)w (s, t) + (α r)ω(s, t)w (s, t) C (s, t) P(s, t)]dt + σω(s, t)w (s, t)db(s, t)(2) where θ is the time discount rate, χ is bequest motive parameter, τ is capital income tax, ζ is estate tax and ω(s, t) is the share of wealth the agent invests in risky asset. Jess Benhabib Shenghao Zhu (New York University)Age, Luck, and Inheritance December 7, / 23

10 A portfolio selection problem (cont.) Theorem The agent s optimal policies are characterized by C (s, t) = A 1 r W (s, t), ω(s, t) = α r γσ 2, Z (s, t) = ( pχ Aµ ) 1 γ (1 ζ) 1 γ γ W (s, t) with A = ( θ+p (1 γ)(r τ+µ+ (α r )2 2γσ 2 ) γ(1+(pχ) 1γ µ γ 1 γ (1 ζ) 1 γ γ ) ) γ and dw (s, t) = gw (s, t)dt + κw (s, t)db(s, t). with g = r τ+µ θ p γ + 1+γ (α r ) 2 2γ and κ = α r γσ 2 γσ. ess Benhabib Shenghao Zhu (New York University)Age, Luck, and Inheritance December 7, / 23

11 Inheritance and government redistributive policy heterogenous bequest motives A fraction of q p people leave bequest. (χ > 0) A fraction of p q p people do not leave bequest. (χ = 0) Government subsidy If the newborn s inheritance is lower than a threshold level that is proportional to the aggregate wealth, the government gives the newborn a subsidy that brings their starting wealth to the threshold level. This is nanced by estate and capital income taxes. (For example expenditures per pupil in public elementary and secondary education, approximately $9000 per annum.) If the newborn s inheritance is higher than the threshold, the newborn does not receive a government wealth subsidy. The newborn whose parent does not have a bequest motive obtains a wealth subsidy nanced by estate and capital income taxes and starts life at the threshold level of wealth. Alternatively, wealth at birth could be the discounted lifetime labor income: then our solution approximates this case. Government expenditure ηw (t). Government budget is balanced. Jess Benhabib Shenghao Zhu (New York University)Age, Luck, and Inheritance December 7, / 23

12 The connection of wealth between two consecutive generations Let x W (s) be the threshold level of wealth below which newborns pχ(1 ζ) qualify for the government wealth subsidy. Let ρ = ( Aµ ) γ 1. If W (e, s) x ρ W (s), W (s, s) = (1 ζ)( pχ Aµ ) γ 1 (1 ζ) 1 γ pχ(1 ζ) γ W (e, s) = ( ) 1 r W (e, s) Aµ where W (e, s) is wealth level of dying father who is born at period e, and W (s, s) is wealth level of son who is born at period s. W (s, s) = ρw (e, s) If the parents have a bequest motive but their wealth level W (e, s) < x ρ W (s), or if the parents do not have a bequest motive, then the government subsidizes their children. W (s, s) = x W (s) Jess Benhabib Shenghao Zhu (New York University)Age, Luck, and Inheritance December 7, / 23

13 Wealth distribution and inequality To investigate the cross-sectional distribution of the wealth, we discount the individual wealth level by the aggregate economy growth rate g. Since aggregate wealth level is growing, we investigate the distribution of the ratio of individual wealth to aggregate wealth. Let X (s, t) be the the ratio of the individual wealth to the aggregate wealth. X (s, t) = e g t W (s, t) We normalize W (0) = 1 Note that X (s, t) is also a Geometric Brownian Motion. dx (s, t) = (g g)x (s, t)dt + κx (s, t)db(s, t) where we assume that g g 1 2 κ2 0. Jess Benhabib Shenghao Zhu (New York University)Age, Luck, and Inheritance December 7, / 23

14 PDE for the evolution of wealth distribution f (x, t) t = x 2 (κ2 x 2 f (x, t)) x ((g g)xf (x, t)) pf (x, t) + qf ( x ρ, t) 1 ρ, x > x (10) f (x, t) t = 1 2 x 2 (κ2 x 2 f (x, t)) x ((g g)xf (x, t)) pf (x, t), x < x (11) 2 The endogenous value x is determined by government s subsidy policy: (p Z x q)x ρ + q (x ρx)f (x)dx 0 = q( pχ Aµ ) γ 1 1 γ (1 ζ) γ ζ + τ η (13) Jess Benhabib Shenghao Zhu (New York University)Age, Luck, and Inheritance December 7, / 23

15 Stationary distribution Theorem The stationary distribution has the following kernel C1 x f (x) = β 1 C 2 x β 2 when x < x when x > x where β 1 is the smaller root of the characteristic function κ 2 2 β2 ( 3 2 κ2 (g g))β + κ 2 p (g g) = 0 (14) and β 2 is the larger solution of the characteristic function κ 2 2 β2 ( 3 2 κ2 (g g))β + κ 2 p (g g) + qρ β 1 = 0. (15) where β 1 < 1 and β 2 > 2. ess Benhabib Shenghao Zhu (New York University)Age, Luck, and Inheritance December 7, / 23

16 Density The calibrated economy The parameters: θ = 0.03, χ = 15, σ = 0.26, γ = 3, r = 1.8%, α = 8.8%, p = 0.016, q = 0.75p = 0.012, ζ = 0.19 and τ = Stationary distribution of wealth in the simulated economy Relative ratio of individual wealth to the aggregate wealth Jess Benhabib Shenghao Zhu (New York University)Age, Luck, and Inheritance December 7, / 23

17 Wealth distribution Distributions of wealth in the United States and in the benchmark model economies (Qintile) Economy Gini First Second Third Fourth Fifth Unitied States Model Top groups (Percentile) Economy 90th 95th 95th 99th 99th 100th Unitied States Model Jess Benhabib Shenghao Zhu (New York University)Age, Luck, and Inheritance December 7, / 23

18 Density Density Density Density Wealth distribution conditional on age 3 Age=20 2 Age= Wealth Age= Wealth Wealth Age= Wealth Jess Benhabib Shenghao Zhu (New York University)Age, Luck, and Inheritance December 7, / 23

19 Gini coefficient Gini coe cient on age 0.55 Gini coef f icient conditional on age Age Jess Benhabib Shenghao Zhu (New York University)Age, Luck, and Inheritance December 7, / 23

20 Disentangle Contribution of luck The Gini coe cient in the economy without risky asset is Comparing this Gini coe cient with that of the economy with risky asset, , we nd that the Gini coe cient would decrease by about 31% if we close the investment opportunity of agents for the risky asset. Intergenerational transmission and age e ect With inheritance the wealth process is more persistent across generations. On the other hand, if people leave bequests, the aggregate economy growth rate increases because people cut back on consumption to leave bequests. Since the economy grows faster if there is inheritance the lucky who earn high returns relative to the economy will not break out and leave others behind by as much. (Life cycle savings cannot account for US capital stock. Bequests plus inter-vivos transfer account for more than 50%. Kotliko and Summers (1981), Gale and Scholtz (1994).) Inheritance can decrease the wealth inequality in the economy. Jess Benhabib Shenghao Zhu (New York University)Age, Luck, and Inheritance December 7, / 23

21 Disentangle (cont.) Contribution of age e ect We pick τ = so that the relative growth rate g g 1 2 κ2. In this economy, the Gini coe cient is In the general case of our economy, the Gini coe cient is After we close the age e ect, the Gini coe cient decreases by about 37%. Jess Benhabib Shenghao Zhu (New York University)Age, Luck, and Inheritance December 7, / 23

22 Redistributive Policies Tax and wealth inequality The minimum of Gini coe cient is , which is obtained when ζ = 0.95 and τ = The maximum of Gini coe cient is , which is obtained by ζ = 0.83 and τ = 0. Taxes and welfare The aggregate welfare is maximized by ζ = 0.18 and τ = Jess Benhabib Shenghao Zhu (New York University)Age, Luck, and Inheritance December 7, / 23

23 Conclusions We set up a parsimonious model and solve it analytically to replicate the wealth distribution which displays Right skewness Heavy upper tail Pareto distribution The Gini coe cient is as high as This number is lower than that of the data. But we have fewer parameters (5) than the other models using simulations (35). Castaneda, Diaz-Gimenez and Rios-Rull (2003). Luck contributes about 31% to wealth inequality as measured by the Gini coe cient. Age e ect contributes about 37% to wealth inequality as measured by the Gini coe cient. Surprisingly, inheritance can decrease the wealth inequality in the economy. Jess Benhabib Shenghao Zhu (New York University)Age, Luck, and Inheritance December 7, / 23