. 1 Accounting for Patterns of Wealth Inequality Lutz Hendricks Iowa State University, CESifo, CFS March 28, 2004.
1 Introduction 2 Wealth is highly concentrated in U.S. data: The richest 1% of households hold 35% of total wealth. The Gini of wealth is about 0.8. The question: Does life-cycle theory account for the observed concentration of household wealth? Why life-cycle theory? It is the standard theoretical framework for thinking about consumption, saving, wealth. Versions of life-cycle models replicate the cross-sectional wealth distribution.
Key features that enable life-cycle theory to replicate the wealth concentration: 3 A labor endowment (earnings) process that matches observed income inequality (Castaneda et al. 2003). Bequests (Laitner 2002). Entrepreneurship (Quadrini 2000).
4 Basic intuition of life-cycle theory: Age and earnings are the key determinants of wealth. Wealth-rich households are typically near retirement and have high lifetime earnings. The point of this paper is to make these implications precise and confront them with data.
1.1 The questions addressed in this paper 5 1. How tight is the relationship between retirement wealth and lifetime earnings? 2. How large are the wealth di erences between earnings-rich and earningspoor households? 3. How much wealth inequality remains after controlling for lifetime earnings and age? Retirement wealth is household net worth at age 65. Lifetime earnings means the discounted present value of labor income.
6 The approach: Estimate the relationship between lifetime earnings and retirement wealth in the PSID. Develop a quantitative life-cycle model that accounts for the U.S. wealth distribution. Compare the relationship between lifetime earnings and retirement wealth in model and data.
1.2 Findings 7 Life-cycle models predict a very tight relationship between lifetime earnings and retirement wealth. Large wealth inequality results from earnings rich households holding much more wealth than earnings poor households. These implications remain true when models are extended to accommodate bequests and entrepreneurship.
8 In the data: The relationship between lifetime earnings and wealth is much weaker than in the model. The wealth gap between earnings rich and earnings poor households is much smaller. Wealth inequality is in large part due to heterogeneity among households with similar lifetime earnings.
9 A preview of the ndings:. C W E Mean Gini R 90=20 PSID 0.48 0.60 1.78 Model 0.91 0.47 18.02 C W E : Correlation coe cient between lifetime earnings and retirement wealth. Mean Gini: Mean of Gini coe cients of retirement wealth within lifetime earnings deciles. R 90=20 : 90/20 ratio of Median retirement wealth Mean lifetime earnings across lifetime earnings deciles. Conclusion: Life-cycle theory fails to capture an important reason why households with similar lifetime earnings choose very di erent saving rates.
10 1.2.1 Literature Several studies show that observationally similar households hold di erent amounts of wealth (e.g., Hurst et al. 1998). Some interpret this as evidence supporting preference heterogeneity (e.g., Knowles and Postlewaite 2003). Only Venti and Wise (2000) have a good measure of lifetime earnings (social security records)..
2 The Data 11 1968 to 1992 waves of the Panel Study of Income Dynamics (PSID). Wealth includes nancial wealth, real estate including the main residence, business wealth, and the value of vehicles, but not pension wealth. Retirement wealth is the average of two wealth observations between the ages of 50 and 65, discounted to age 60. Earnings: Narrow de nition: encompasses only wages and salaries received by the household head and by the spouse. Broad de nition: covers most forms of income other than interest and dividends that are reported in the PSID.
Lifetime earnings are de ned as the present discounted value of earnings between the ages of 25 and 65. 12 They are estimated using xed e ect regressions for 4 education and sex classes. Households are deleted if they have fewer than 15 earnings observations or fewer than 2 wealth observations.
3 Empirical Findings 13 Summary of ndings: 1. The correlation between lifetime earnings and retirement wealth is at most C W E 0:5. 2. The ratio of retirement wealth to lifetime earnings rises by a factor of at most 2 between the second and the 9th lifetime earnings decile: R 90=20 2. 3. The Gini coe cient of retirement wealth within lifetime earnings deciles is at least 0.5.
Note: Holding constant age and lifetime earnings reduces the Gini coe cient of wealth by only 1/3 (from 0.75 to 0.5). This poses a challenge for life-cycle theory. 14 Venti and Wise (2000) were the rst to document large wealth inequality among households with similar lifetime earnings.
3.1 Sub-sample results 15 Consider various sub-samples to identify possible sources of wealth inequality among households with similar lifetime earnings. C W E C LW E Mean Gini R 90=20 N Baseline sample 0.48 0.51 0.60 1.78 1466 Never self-employed 0.22 0.47 0.53 2.05 910 Never divorced 0.49 0.52 0.60 1.81 1293 One/two children 0.30 0.52 0.54 2.00 376 No stock holdings 0.16 0.40 0.63 0.98 1055 No inheritance 0.52 0.53 0.60 1.91 1164
4 The Model 16 Baseline model is an extension of Huggett (1996). Agents: a continuum of households of unit mass. a single representative rm. a government. All markets are competitive and the economy is in steady state.
17 4.1 Households Each household consists of overlapping generations of parents and children. Each parent has one child which is born T G periods before the parent dies. A household solves the following problem: max E TX t=1 t u(c t ) + T b V (k T +1 ; e T +1 ; q 0 ) (1) subject to the budget constraint k t+1 = (1 + r) k t + w l t c t + (a t ) (2) and the borrowing constraint k t+1 0. t: date. T : stochastic lifetime. w; r: constant factor prices. (a): age dependent lump-sum transfer. l t = h(a t ) e t q: labor endowment.
18 4.1.1 Household dynamic program: State vector: s = (k; e; a; q). V (s) = max u(c) + P s (a) X P a (a; a 0 ) X P e e; e 0 ; a 0 V s 0 c;k 0 a 0 e 0 + (1 P s [a]) X P q q; q 0 X e 0 ^V k 0 ; e 0 ; q 0 q 0 subject to the budget constraint e 0 P e1 k 0 (s) = (1 + r) k(s) + w l(s) c(s) + (s) (3)
Inheritances: If the parent saves k 0 = (s) and dies at the end of the period, the child inherits ^b() = (1 b ) at age T G. 19 Parents and children must overlap by a realistic number of years. Otherwise the model overstates the importance of inheritances. To maintain tractability, I impose informational assumptions that allow me to solve the household problem as if parents and children did not overlap: The parent does not know his child s life history. The child learns at the beginning of life how much it will inherit and can borrow against this inheritance.
Bequest motives: No consensus as to why parents leave bequests to children. Consider 3 popular motives. 20 Sel sh parents: b V = 0. Joy-of-giving: b V (k 0 ; :) = b(k 0 ) 1 =(1 ): Altruism: b V (k 0 ; e 0 ; q 0 ) = V (1; b(k 0 ); e 0 ; q 0 ).
4.2 Firms 21 Standard static pro t maximization. Rent capital K and labor L from households so as to maximize F (K; L) q K K q L L. 4.3 Government Labor income is taxed at a proportional rate: w = (1 w ) q L. Bequests are taxed at rate b. Lump-sum transfers to retired households: (a) = R if a > a R. Government budget constraint: C G + Z (s) (s) ds = w q L L + b B
4.4 Equilibrium 22 A stationary competitive equilibrium consists of aggregates (K; L; C; C G ; B), a price system (w; r; q K ; q L ), value functions (V; ^V ), policy functions (c; ), and a distribution over household types, (s), such that: The policy functions and value function solve the household problem. Firms maximize pro ts. Markets clear. The government budget is balanced. The distribution of household types, (s), is stationary.
Households = 0:958 Matches K=Y = 2:9 = 1:5 Huggett (1996) Demographics A = 12 Number of life-cycle phases a R = 3 Work phases corresponding to ages 20-65 P S Matches mortality rates of couples. Social Security Administration, Period Life Tables 1997 P (a; a 0 ) Matches mean phase length of 15 years for work life and 3 years for retirement T G = 30 Children are born T G periods before parents die Firms = 0:30 Capital income share in NIPA k = 0:063 Matches after-tax interest rate of 4 percent = 0:93 Normalized such that q L = 1 Government w = 0:40 Trostel (1993) b = 0:25 See text T r=y = 0:40 CDR 23
5 Findings: Economies without bequests 24 C W E C LW E Mean Gini R 90=20 PSID 0.48 0.51 0.60 1.78 Determ. aging 0.93 0.79 0.46 587.93 No bequest 0.91 0.66 0.47 18.02 Model economies imply: too large wealth di erences between lifetime earnings deciles; too little wealth inequality within lifetime earnings deciles..
Log Retirement wealth 25 5 0-5 -10-15 -20-25 -30-35 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 Log Lifetime earnings Model economies imply a tight relationship between lifetime earnings and retirement wealth.
Log Retirement wealth 26 16 14 12 10 8 6 4 2 0-2 12.5 13 13.5 14 14.5 15 15.5 16 16.5 17 17.5 Log Lifetime income.
Retirement wealth / lifetime earnings 27 20 18 16 14 12 10 8 6 4 2 Determ. aging No bequest Venti/Wise 0 10 20 30 40 50 60 70 80 90 100 Lifetime earnings percentile Median retirement wealth and lifetime earnings Large wealth inequality stems from a steep relationship between lifetime earnings and wealth. Intuition: Transfer income during retirement..
Wealth / Average household earnings 28 30 25 20 Determ. aging No bequest Venti/Wise 15 10 5 0 10 20 30 40 50 60 70 80 90 100 Wealth percentile Retirement wealth distribution. 5th lifetime earnings decile. Within lifetime earnings deciles: model economies imply far too little wealth inequality..
Retirement wealth Gini 29 Why are the model Gini coe cients so high? 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 Determ. aging 0.1 No bequest Venti/Wise 0 10 20 30 40 50 60 70 80 90 100 Lifetime earnings percentile Gini coe cients of retirement wealth.
5.1 Model economies with bequests 30 C W E C LW E Mean Gini R 90=20 PSID 0.48 0.51 0.60 1.78 Accid. bequest 0.91 0.68 0.45 15.18 Joy-of-giving 0.91 0.82 0.39 6.72 Strong altruism 0.90 0.68 0.49 19.25 Results are similar to the no bequest model..
Retirement wealth / lifetime earnings 31 20 18 16 14 12 10 8 6 4 2 Accid. bequest Joy-of-giving Strong altruism Venti/Wise 0 10 20 30 40 50 60 70 80 90 100 Lifetime earnings percentile Median retirement wealth and lifetime earnings..
32 5.1.1 Why do bequests not change the ndings? Inheritances are not an important source of lifetime income. The fraction of households who inherit more than 10% of lifetime earnings is 8.3% in the accidental bequest model 15.8% in the altruism model 2% in the PSID.
Inheritances are partly consumed before retirement. In the accidental bequest model, households in the 95th percentile of the inheritance distribution hold only 7.4% more wealth than households who inherited nothing. In the altruism model, the corresponding gap is 16.1%. 33 This prediction nds some support in the PSID: Regressing retirement wealth on a lifetime earnings polynomial and on the present value of lifetime inheritances yields a coe cient on inheritances of -0.01 (s.e. 0.02). Retirement wealth is not larger for households who received higher inheritances, after controlling for lifetime earnings..
Wealth Age-wealth pro les for households with median draws of all random variables:. 34 12 10 8 0th percentile 90th percentile 95th percentile 99th percentile 6 4 2 0 20 30 40 50 60 70 80 90 100 Age Age-wealth pro les and inheritances..
6 Entrepreneurship 35 Motivation: Entrepreneurs hold a large share of total wealth. Entrepreneurship could break the tight relationship between lifetime earnings and wealth. The model: the simplest extension of the baseline model based on Cagetti and De Nardi (2002)..
36 6.1 Households Two features of the baseline model are modi ed. 1. Households randomly receive self-employment opportunities. 2. Households are sel sh. Self-Employment Opportunities: In each period, the household draws entrepreneurial productivity: 2 f1; :::; N g. Produce output according to G(k E ; ) = () k E. < 1. Evolution of is governed by P (; 0 ; a). Retired households cannot start new projects, even though they may continue existing ones. New households inherit their endowments from their parents according to the same transition matrix.
37 Entrepreneurs maximize current period pro ts: (s) = max G(k E ; ) q K k E subject to borrowing constraint B(s) =! [k(s) + Key feature: B k (s) > 0. k E k + B(s) (4) Z s2 w l(s) (s) ds] (5)
38 6.1.1 Household Dynamic Program V (s) = max u(c) + P s (a) E V (s 0 ) subject to the budget constraint k 0 (s) = y(s) c(s) (6) where income consists of pro ts, capital income, earnings, and transfers: y(s) = (s) + (1 + r) k(s) + w l(s) + (a)
6.2 Competitive Equilibrium 39 Firms and government are identical to the baseline model. Denote the rm s factor demands by K C and L C. The de nition of a competitive equilibrium is the similar to that of the baseline model, except for the market clearing conditions. Denote the set of states where households are self-employed by complement by W. Let = E [ W. E and its Labor market clearing requires L = L C. Capital market clearing requires K = K E +K C ; where K E = R s2 E (s) k E (s) ds Goods market clearing requires F (K C ; L C )+Y E = C G +X+ R s2 where (s) c(s) ds Y E = R s2 E (s) G (k E (s); (s)) ds X = R s2 (s) [k0 (s) (1 ) k(s)] ds.
6.3 Calibration 40 Households = 0:925 Matches K=Y = 2:9 Entrepreneurs = [0:00; 0:59] Matches aggregate selfemployment income = 0:81 Matches fraction of wealth held by richest 5 percent of households! = 0:04 Matches K E =K = 0:40.
6.4 Findings: Entrepreneurship 41 C W E C LW E Mean Gini R 90=20 PSID 0.48 0.51 0.60 1.78 No bequest 0.91 0.66 0.47 18.02 SE No bequest 0.35 0.51 0.76 10.84 SE Accidental bequest 0.36 0.53 0.74 8.36 Entrepreneurship helps break the tight relationship between lifetime earnings and wealth.
42 Intuition: 1. Self-employment acts like large, transitory earnings shocks. The timing and duration of these shock varies across households. Since entrepreneurs have high saving rates, this has a substantial e ect on the retirement wealth distribution. 2. Using the narrow de nition of earnings (in the model and in the data) fails to capture self-employment income, which is an important component of lifetime income for some households. 3. In part, wealth inequality among households with similar lifetime earnings re ects di erences between workers and entrepreneurs..
Workers: Households who were never self-employed look very much like households in the baseline model. 43 C W E C LW E Mean Gini R 90=20 PSID 0.22 0.47 0.53 2.05 No bequest 0.91 0.66 0.47 18.02 SE No bequest 0.78 0.61 0.65 105.50 SE Accidental bequest 0.80 0.63 0.62 50.37
Wealth / Average household earnings 44 High Gini coe cients result from large fraction of workers holding no wealth 30 25 20 15 No bequest SE No bequest SE Accidental bequ PSID Venti/Wise 10 5 0 10 20 30 40 50 60 70 80 90 100 Wealth percentile Workers. 5th lifetime earnings decile.
Broad earnings concept: Measuring lifetime earnings properly (including entreprenerial income) yields ndings similar to the baseline model. 45 C W E C LW E Mean Gini R 90=20 PSID 0.54 0.65 0.58 3.31 No bequest 0.90 0.65 0.48 16.47 SE No bequest 0.84 0.61 0.63 19.19 SE Accidental bequest 0.84 0.63 0.60 12.88
7 Conclusion 46 Life-cycle models imply: a very tight relationship between lifetime earnings and wealth; very large wealth gaps between earnings rich and earnings poor households; too little wealth inequality among households with similar lifetime earnings. These are not features of the data. Bequests and entrepreneurship do not substantially change these implications.
Life-cycle theory omits an important reason why households with similar lifetime earnings choose to save di erent amounts. 47 Preference heterogeneity?.
8 Wealth Distribution 48 Pct. class 20 40 60 80 90 95 99 100 Gini Frac=0 SCF -4.2-2.9 2.7 16.2 29.8 41.8 64.1 100.0 0.86 4.1 Determ. aging 0.0 0.5 5.5 23.2 45.0 62.9 87.0 100.0 0.74 24.5 No bequest 0.0 0.7 5.3 22.1 42.9 60.6 86.0 100.0 0.75 12.6 Accid. bequest -2.3-1.6 3.1 20.4 41.7 59.9 85.8 100.0 0.79 10.1 Joy-of-giving -5.4-3.9 3.5 22.1 43.4 61.3 86.5 100.0 0.80 1.8 Strong altruism -7.0-6.7-3.1 12.6 33.6 52.6 82.1 100.0 0.91 9.9
9 Inheritance Distribution 49 Pct. class 70 80 90 95 98 100 I/Y SCF 0.0 1.8 9.4 18.9 30.8 100.0 1.3 Accid. bequest 0.6 4.7 19.7 39.6 62.1 100.0 1.1 Joy-of-giving 38.3 48.1 62.5 74.3 84.9 100.0 2.6 Strong altruism 1.6 6.9 22.7 42.0 63.8 100.0 2.4
10 Models with intergenerational persistence 50 C W E C LW E Mean Gini R 90=20 PSID 0.48 0.51 0.60 1.78 Accid. bequest 0.91 0.68 0.45 15.18 IGP 0.96 0.70 0.42 52.90 IGP. Altruism 0.93 0.63 0.48 102.44
11 Alternative labor endowments 51 C W E C LW E Mean Gini R 90=20 PSID 0.48 0.51 0.60 1.78 Accid. bequest 0.91 0.68 0.45 15.18 CDR 1 0.94 0.64 0.55 25.15 CDR 2 0.78 0.66 0.59 34.72