Limited Participation and Wealth Distribution

Transcription

1 Limited Participation and Wealth Distribution María José Prados April 2009 Abstract This paper studies the e ect that limited participation in asset markets has on the distribution of wealth in the economy. We analyze this feature in the context of a stylized model with incomplete markets that incorporates a market for a risky asset, and generates limited participation by introducing transaction costs. The performance of the model in terms of limited participation and savings behavior is analyzed for CRRA and DRRA preferences. [ DRAFT - VERY INCOMPLETE ] Introduction Signi cant wealth inequality that cannot be replicated by simple models. This paper proposes a very stylized model of heterogeneous agents with incomplete markets that incorporates a market for a risky asset, and can generate endogenously an unequal distribution of wealth. The proposed model serves to identify the amplifying e ect that limited participation in the - nancial assets market has on an initial source of divergence between earnings of di erent agents, by assuming two particular features: there are transaction costs in the risky asset market and preferences that display decreasing relative risk aversion. The objective is to match some moments of the wealth distribution in the US, so that the model can be used to perform comparative and policy exercises. The degree of inequality in net worth and in asset holdings in the U.S. is re ected in the data in Table.

2 Table : Relative mean holdings of assets by quartiles of the wealth distribution, 2007 Percentile of net worth Net Worth Stock Stock + Pooled Investment Funds Transaction Accounts Any Financial Asset Less than 25-0.% 0.9%.% 2.9% 0.5% % 2.5% 3.7% 8.6% 3.4% % 6.9% 9.2% 5.4%.% % 2.2% 5.0% 2.% 25.7% % 77.5% 7.% 52.0% 59.3% 00.0% 00.0% 00.0% 00.0% 00.0% Source: 2007 Survey of Consumer Finances, Federal Reserve Board There are two empirical observations related to asset ownership in the U.S. that motivate the interest in including these features. First, most households don t own any nancial assets. The 2007 Survey of Consumer Finances indicates that only 7.9% of the families in the U.S. hold stocks, as can be seen in Table 2. Only 4.3% of the families in the bottom quartile of the wealth distribution hold any stocks at all, and this fraction is around 0% for the bottom half of the distribution. This is known as the limited participation puzzle - since according to a standard model agents should diversify their portfolio, and thus hold at least a small amount of stock - and it has been extensively documented in the literature (Vissing-Jorgensen, 2002). Table 2: Percentage of families holding asset, 2007 Stock (direct ownership) Pooled investment funds Transaction accounts All families Percentile of net worth Less than * Source: 2007 Survey of Consumer Finances The second empirical observation is about how the saving and investment behavior of the rich di ers from that of the mean household. Using data from the Panel Study of Income Dynamics (PSID), the Survey of Consumer 2

3 Finances, and the Consumer Expenditure Survey, Dynan, Skinner and Zeldes (2004) nd a strong positive relationship between saving rates and lifetime income and a weaker but still positive relationship between the marginal propensity to save and lifetime income. Also, Kessler and Wol (99) nd that portfolios of households with low levels of net worth tend to have mostly risk free assets. Carroll (2000) shows that portfolio composition di ers between people with di erent levels of wealth: rich people tend to hold riskier assets. Mankiw and Zeldes (99) estimate unconditional Euler equations for stockholders and non-stockholders using data from the PSID that indicate large di erences in relative risk aversion between the two groups: less risk averse agents are more willing to invest in the stock market. On the other hand, those who are income rich have a higher tendency to invest in the stock market, thus implying that risk aversion may vary with the income level. Based on these empirical facts I analyze the interaction between limited participation and decreasing relative risk aversion as channels to generate this distinct attitude towards nancial asset holdings and to contribute to wealth inequality as a consequence. In order to do so, I study the implications of external habit formation in consumption, which provides this decreasing relative risk aversion feature of preferences. Models of heterogeneous agents with incomplete markets have desirable features for the study of aggregate savings. However, the standard Bewley model of income uctuations - households are ex ante identical, but su er idiosyncratic shocks to income - with incomplete markets is not able to generate a behavior of savings that can replicate the degree of inequality in wealth and asset holdings observed in the data. This is because in a Bewley economy, due to the uninsurability of the idiosyncratic risk, agents want to save in order to use savings as a bu er to self-insure against the uctuation in their income, but after this bu er stock has been built, they have no extra incentives to save, hence the inequality in asset holdings generated by this model is limited. Therefore, in this paper I study the additional inequality generated by the participation in nancial markets, through the savings decisions and portfolio choices of households that face idiosyncratic risk, borrowing and short sale constraints, and transaction costs in the market for the risky asset. The main amplifying e ect is that if wealthier people save more, and also invest proportionally more in risky asset than the poor, then di erent agents embark upon di erent paths of wealth accumulation: the lucky ones will get a higher return for their investment than the mean household, and the re- 3

4 sulting distribution of wealth will be more unequal than the distribution of earnings. Several explanations for the observed limited participation in the stock market have been o ered in the literature, and one of the most frequent is the presence of market frictions mostly in the form of xed entry and/or transaction costs. Moreover, the standard model of portfolio choice cannot generate zero stockholding unless an entry cost is assumed. This type of costs is consistent with empirical ndings that indicate there is a signi cant structural state dependence in the stock market entry decision. Ameriks and Zeldes (2000) nd that in practice most individuals make very few changes in either the allocation of stocks of assets each period, or the ows of new contributions, suggesting the presence of either transaction costs or inertia. Borrowing and short sale constraints a ect household portfolio choice and can help explain median saving behavior and the equity premium, as it has been extensively studied in the literature 2. The presence of borrowing constraints can be imputed to market frictions as asymmetric information or moral hazard problems. The rationale behind imposing short selling constraints results from the empirical evidence and institutional setting: short selling is more expensive and riskier than establishing a long position, and it is an activity that only a specialized investor can do. In the data, short-selling is relatively rare and the amount of shares sold short is small 3. The current literature has considered how several features can help explain the observed wealth inequality. The main aspects considered have been the role of entrepreneurial activity, heterogeneous preferences, life cycle considerations, implications of social security policies, and bequests motives. Some models focus on particular features of the preferences: Krusell and Smith s (998) experiments can only match the wealth dispersion observed in the data by assuming heterogeneous preferences in a dynastic model, where the discount factor changes stochastically through time. De Nardi (999) Heaton and Lucas (997) examine a portfolio choice model with stock and bonds when there are incomplete markets. They cannot obtain a positive demand for bonds under a broad scope of assumptions about utility, the stochastic process for income and asset returns, and market frictions. All these variants of the basic model give the same prediction: investors wish to borrow and invest all of their savings in stock. Haliassos and Michaelides (2003) nd an entry cost that breaks this specialization pattern. 2 Constantinides et al (2002), and Storelestten et al (998) show this in lifecycle models. 3 For further treatment of the topic, references and evidence, see Lamont (994), Dechow et al (200). 4

5 develops a life cycle model with intergenerational transmission of genes and warm glow altruism (joy-of-giving bequests motive). As an attempt to explain why savings rates are so much higher for the rich than for the poor, Carroll (2000) incorporates wealth directly in the utility function. Castañeda, Díaz Giménez and Ríos Rull (2003) study a model that combines life-cycle with dynastic behavior. They include retirement and death risk, as well as a progressive tax structure similar to that in the US. It is standard in this literature to use estimates of the earnings process obtained from micro data, but in this case, in order to match the wealth distribution the authors have to calibrate the earnings process, and this process is calibrated so that the highest productivity is more than 00 times higher than the second one. Several papers focus on entrepreneurship, by assuming some heterogeneity in innate ability. Quadrini (997) introduces entrepreneurial choice in a dynastic model with capital market imperfections. Cagetti and De Nardi (2006) obtain a large amount of wealth concentration in a model where they assume that individuals are born with either entrepreneurial ability or worker s ability, and endogenize the rm size distribution. However, not much has been said with respect to how the households portfolio choices may a ect their opportunities to accumulate wealth and magnify those mechanisms that have already been identi ed. Moreover, a stylized model that identi es inequality generating channels is useful to analyze the e ects on wealth inequality of changes in aggregate risk, nancial conditions, policies, etc. My approach is therefore somehow complementary to the existing literature. Section 2 presents the model. The theoretical aspects of the household s solution will be developed in Section 3. The numerical solution, calibration and results for partial equilibrium are described in Section 4. General equilibrium is discussed in Section 5. Section 6 provides concluding remarks. 2 The Model The economy is populated by a continuum of dynasties that maximize their lifetime expected utility by choosing how much to consume, how much to save, and the composition of their portfolio. Labor is supplied inelastically, and agents are subject to idiosyncratic risk from the endowment they receive. It is an endowment economy, with no aggregate risk. Markets are 5

6 incomplete, and there are two types of assets: a risk free bond which is in zero net supply-, and a risky asset. Shares of the risky asset represent the right to a share of the period dividends. There is a limit to the amount a household can borrow - hence those who are poor and near the lower bound for consumption will save in order to partially insure themselves against their idiosyncratic risk - and there are short sale constraints. In equilibrium with limited participation, when wealth-poor households increase their precautionary savings, they do so by buying bonds, i.e., by lending money to those less risk averse in the economy who will contract debt in order to buy more shares of the risky asset. 2. Preferences The households in this economy have preferences that display external habit formation, or "keeping up with the Joneses" e ect. The current level of aggregate consumption a ects the utility each agent derives from his own consumption. Under this con guration, a consumer s utility depends on how his own consumption compares to the mean consumption in the economy. We assume each household cares about the utility of their descendants as much as they care for their own, hence this economy is modeled as populated by in nitely lived agents, or dynasties. Preferences in period t are then described by: E 0 JX t=0 t (c i t h) where 4 h = C t Since the marginal utility of consumption tends to in nity as the consumption level approaches h; we will refer to h as the basic level of consumption for a household. The individual supra index i will be dropped from now on whenever it does not lead to confusion. Endowments Each period, agents receive an endowment w t that has a persistent and a transitory component, and evolves according to the following process, as is standard to assume in the literature: ln w t = z t + t z t = z t + t 6

7 where t ~N(0; 2 ) t ~N(0; 2 ) Since the level of income will be a state variable in the household s problem, we will discretize the space of w t in order to make the problem computationally tractable. In order to do so, following Tauchen (986) we can approximate this autoregressive process with a discrete process that follows Markov chain. Using estimates for ; 2 and 2 from micro data, the process for w t will be then approximated by y t 2 fy ; :::; y n g with transition matrix given by y : Financial Assets Risk free bond Agents can borrow or lend by issuing or buying a bond that pays a risk free interest rate. This risk free bond will be in zero net supply in equilibrium. Agents are subject to a borrowing constraint that establishes a credit limit for every period. The borrowing constraint can be speci ed as the natural borrowing limit: agents can borrow up to the amount they could repay if they had no stocks, and were to receive the least favorable outcome of their earnings process and consume their basic level of consumption h forever. That is, ^b b i ; where ^b = P t= t (y h) = (y h) 5. This natural borrowing limit is never binding for the optimizing household. In a general equilibrium analysis it is important if the borrowing constraint is not binding since if many individuals have binding borrowing constraints - i.e. the demand for funds is limited -, the equilibrium risk free interest rate will be lower than if individuals were able to borrow up to their natural limit. Therefore, this natural borrowing limit will be used in the general equilibrium analysis. 5 Since lim U 0 (c) = ; the households would not choose to borrow an amount such c!h that this could induce a state of c < h with a possitive probability (it is assumed that there is no possibility of defaulting), hence the maximum they could borrow is the present expected value of the excess income over basic consumption, assuming they set current level of stock to zero (in the worst case scenario - D = D l forever - it is best to get rid of all the stock). 7

8 In partial equilibrium, since prices are not allowed to adjust, we can allow the individual to face a binding constraint on debt and compare it to the natural limit case. For this alternative constraint we assume that the maximum amount he can borrow is given by three times the minimum labor earnings he can have. Risky asset There is a stock market where agents can buy shares to a risky asset. The risk in this market can be understood as there being two trees whose dividends are perfectly correlated so that D i 2 fd h ; D l g for i; j = ; 2; where Pr(D i = D h ) = 0:5; and Pr(D i = D l jd j = D h ) = 6 : That is, the dividends of these trees can take one of two values, and both values are observed every period. There is no history dependence of the dividend s process 7. When an agent buys shares in the stock market, he is in fact buying shares for a lottery that determines which one of these outcomes he will receive. That is, he buys shares of one of these two trees, and only from one of them (he cannot insure against the risk, no matter how many shares he buys). At the end of the period, the agent collects the dividends and can sell his shares or keep them. To capture the behavior identi ed in empirical studies that households do not adjust their nancial assets holdings frequently, and that there is limited stock market participation, we assume there is an entry cost to participate in the stock market, and there is also a cost of reinvesting in stocks, This participation cost works as a reduced form to capture the idea that the agents must exert an e ort to nd out about investment opportunities, or pay intermediation costs. There is no cost for selling or reducing an agent s shares of the risky asset. The participation cost is assumed to have the following speci cation: (s t+ ; s t ) = 0 I (st+ >0;s t=0) + s t+ where I (st+ >s t;s t=0) is an indicator function that equals one if s t+ > 0 and s t = 0; and zero otherwise: 6 This perfect correlation is assumed so that there is no aggregate risk: when half of the economy s trees give the low output, the other half gives the high output. Therefore, the aggregate dividend is constant every period. 7 In other words, the transition matrix governing the evolution of D i is just D = 0:5 0:5 0:5 0:5 8

9 That is, there is a xed cost of entering the risky asset market. After this cost has been incurred, there is a cost that is proportional to the amount invested The Household s Decision Problem Agents maximize their lifetime expected utility by choosing consumption, bond holding, and shares holding policies, taking prices as given. Their optimization problem is the following: max fc tg;fs t+ g;fb t+ g E 0 X t=0 t (c i t h) c i t+p b tb i t++p s ts i t++ 0 I (st+ >0;s t=0)+ s t+ yt+b i i t+(p s t +di t)s i t 0 c i t 0 s i t+ ^b b i t+ 2.3 Equilibrium y t v y A stationary equilibrium for this economy is fc(s; b; y); b 0 (s; b; y); s 0 (s; b; y); p b ; p s ; g satisfying: () c(s; b; y); b 0 (s; b; y); and s 0 (s; b; y) are optimal decision rules for the household problem, given p b ; and p s (2) Markets clear in every period: (i) R X ci (s; b; y)d = Y D (ii) Y D = R X yi d y (y) + D X (iii) R X bi (s; b; y)d = 0 8 An alternative speci cation for the participation cost that is also analyzed in the numerical exercise displays a cost that is decreasing on the amount of stocks held the previous period, thus incorporating the idea that there are some economies of scale in investing in the stock market: (s t+ ; s t ) = exp( s t )I (st+>s t) 9

10 (iv) R X si (s; b; y)d = (3) is a stationary probability measure de ned on (X; B X ) such that () = R X P (x; )d for all 2 B X where X = R X (s0 (s; b; y); s)d; b 2 B = [ ^b; b] 9 ; s 2 [0; s]; D = E(d) = 0:5 D h + 0:5 D l ; y 2 H with transition matrix y ; and x = (b; s; y; d) 2 X = B [0; ] H fd h ; D l g: P (x; ) is the transition function. Huggett (993) shows that a unique stationary probability measure exists under certain circumstances. 3 Theoretical Analysis [TO BE WRITTEN] 4 Numerical Analysis The model is solved using quantitative methods. The approach used is value function iteration to nd the households value function and optimal policies, and then a xed point algorithm iterating over the distribution over the state space to nd the stationary distribution in equilibrium. In order to properly identify the importance of this channel, the relative contribution of each of the characteristics of the model should be quanti ed. I calibrate the model and compare its predictions to the data on wealth inequality. 4. Partial Equilibrium Important aspects of the model are studied in partial equilibrium, feeding in empirically plausible values for the interest rate, risk premium and volatility of stocks. 4.. Preliminary Exercise This rst stage seeks to quantify and isolate the consequences of this limited participation feature. The model is extended to consider agents with di erent 9 Hugget (993) shows that the optimal choice of b in problems of this sort can be in fact bounded from above, hence b b: 0

11 earnings processes - skilled versus unskilled - and to include aggregate risk (to be done). An interesting exercise is to compare the performance of the model for a given parameterization, as the di erent assumptions are sequentially built in. Table 3 provides the parameter values used in this stage (the ones to calibrate will be changed in Section 4..2). Parameterization The risk free rate is assumed to be 2%. Consistently with evidence from nancial data, the equity premium is set at 6%, and stock returns are assumed to have a standard deviation of 5%. There is some consensus in the literature about the estimates for parameters of the labor income process; the ones used here are taken from Storesletten et al (2004). Haliassos and Michaelides (2003) nd in a numerical exercise that an entry cost of 4% of mean annual labor income is enough to generate zero stockholding, and Alan (2006) estimates an entry cost of approximately 2% of annual labor income using data from the PSID. In the calibration exercise, this cost will be kept within these bounds. Table 3 parameter value from data risk free rate =p b 2% equity premium 6% standard deviation of stock s 5% correlation of persistent shock 0.97 volatil. of temporary shock volatil. of transitory shock to calibrate risk aversion [.5,3] intertemporal discount factor 0.97 habit formation h [0.2,0.4] entry cost 0 2% min. labor income proportional cost 2% 4..2 Sequential e ect of assumptions If the model is computed for CRRA utility (h = 0), = :5; idiosyncratic earnings risk, and no participation costs, the resulting distribution of wealth

12 is given by the Lorenz Curve in Figure, and the distribution of assets and net worth is given in percentiles in Figure 2 (a) through (c): Figure Lorenz Curve, net worth Figure 2 Distribution of wealth per net worth quarter Distribution of stocks per net worth quarter Distribution of bonds per net worth quarter % 50% 75% 90% 00% Quartlies of Net Worth Distribution 0 25% 50% 75% 90% 00% Quartlies of Net Worth Distribution % 50% 75% 90% 00% Quartlies of Net Worth Distribution This standard speci cation does not generate substantial inequality. It is particularly counterfactual that those in the bottom of the distribution borrow in order to buy stock. We will see next how the di erent features of the proposed model a ect these outcomes. Figure 3 shows the resulting distribution of the case with CRRA ( = 0); the case with DRRA ( = 0:4); with and without transaction costs. 2

13 Figure 3 Lorenz Curve for Net Worth Case with Shocks to Labor Income γ=0, χ=0 γ=0.4, χ=0 γ=0, χ=2% γ=0.4, χ=2% Under the parameterization in Table 3, the model generates the distribution in Figure 4, it has a Gini coe cient of 0.38, and 78% of the households participate in the stock market. Figure 4 Lorenz Curve, net worth deciles of wealth To quantify the amplifying mechanism of the model, we can allow for some additional income inequality without changing anything else in this setting, by introducing unskilled and skilled workers. According data from the 2005 CPS, the labor force is composed by 70% of workers with less than college degree, and a 30% of workers with college degree or more. The wage 3

14 of the later is on average 90% higher than that of the former. Therefore in the calibration I set y hs = ; y c = :9; and Pr(y = y hs = 0:7); with y sh and y c being absorbing states (that is, there is no mobility). The results in term of distribution are shown in Figures 5 and 6: Figure 5 Lorenz Curve, net worth Figure 6 Distribution of stocks per net worth quartile % 50% 75% 90% 00% Quartlies of Net Worth Distribution Inequality increases substantially, the Gini coe cient goes up to 0.43, and participation falls to 50%. The distribution of wealth and of stock are close to the data for percentiles below 75%, but they overestimate the holdings of those in the 90 percentile by 25%. [TO BE COMPLETED] 4..3 Calibration [] 4

15 4.2 General Equilibrium [] 5 Concluding Remarks [] References [] S. Alan, Entry costs and stock market participation over the life cycle, Review of Economic Dynamics [2] J. Ameriks, SP Zeldes, Why do Household Portfolio Shares Vary with Age?, manuscript [3] A. Atkeson, and M.Ogaki, 996, Wealth-varying elasticity of substitution: Evidence from panel data and aggregate data, Journal of Monetary Economics 38, [4] A. Brav, G. M. Constantinides, and C. Geczy, Asset Pricing with Heterogeneous Consumers and Limited Participation: Empirical Evidence, Journal of Political Economy, 0:4, [5] M Cagetti, M De Nardi, Entrepreneurship, Frictions, and Wealth, Journal of Political Economy [6] C. Carroll, Why do the Rich Save so Much?, in Does Atlas Shrug? The Economic Consequences of Taxing the Rich, J. Slemrod ed. [7] C. Carroll, Portfolios of the Rich, in Luigi Guiso, Michael Haliassos, Tullio Jappelli eds., Household Portfolios (MIT Press, Cambridge, MA),.p.389 [8] C. Carroll, J. Overland, D. Weil, Saving and Growth with Habit Formation, American Economic Review [9] A Castañeda, J Díaz-Giménez, JV Ríos-Rull, Accounting for the US Earnings and Wealth Inequality, Journal of Political Economy 5

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