Chapter 2 Permanent income theory 2.1 Introduction

Chapter 2

Permanent income theory 2.1 Introduction

Modern consumption theory begins with the analysis of Keynes of the consumption behavior and his most well-known characteristics are that the marginal propensity to consume (MPC) is a decreasing function of income, like the average propensity to consume (APC). This means that redistributing income from high to low income households is likely to raise aggregate consumption since low-income households have a higher MPC. But in the 1946 Kuznets showed that long time series consumption data (for USA) economy are characterized by a constant aggregate APC, inconsistent with Keynesian consumption theory. In response to this puzzle M. Friedman proposed the permanent income hypothesis (PIH), defined as the annuity value of lifetime income and wealth. Milton Friedman’s PIH comes from the basic intuition that individuals aim to smooth consumption and not let it fluctuate with short-run fluctuations in income. This model can explain much empirical facts, for example, why income is more volatile than consumption and why the long-run marginal propensity to consume out of income is higher than the short-run one. Friedman has suggested that individuals base their consumption on a longer-term expectation of income (long horizon). The basic hypothesis states that individuals consume a fraction of this permanent income in each period and thus the average propensity to consume should equal the marginal propensity to consume out of this permanent income component; the propensity could vary with a number of factors, including the interest rate and taste shifter variables. Friedman has taken on the task of testing his hypothesis against a number of empirical facts from time series data and budget studies. The theoretical construct is ex ante, while

the empirical data are ex post, so one needs a correspondence between the theoretical constructs and the observed data. The main direct method to do this, is to construct estimates of consumption (cp), permanent income (yp), transitory consumption (ct), transitory income (yt). Measured income is the sum of permanent and transitory income (yt) and measured consumption is the sum of permanent and transitory consumption (ct). We can state this as follows:

c=cp+ct

y=yp+yt

so permanent income is determined by the equation:

cp=k(r,z)yp

In the first equation c represents a consumer unit’s expenditure for some time period and it is designed as the sum of permanent component cp

and transitory component ct. Some of the factors producing transitory

components of consumption are specific to particular consumer units, like specially favorable opportunity to purchase; these effects tend to average out. Others affect consumer units in the same way, for example cold or bountiful harvest; these instead produce positive or negative mean transitory components for groups of consumers. It is an attempt to interpret the permanent components as corresponding to average lifetime values and the transitory components as the difference between such lifetime averages and the measured values in a specific time period. But it would be a mistake to accept this interpretation. So the distinction between permanent and transitory aims to interpret actual behavior. We treat consumer units as if they regarded their income and their consumption as the sum of two such components and as if the relation between the permanent components is the one suggested by the analysis. The line between permanent and transitory components is best left to be determined by the data themselves. In the same way, let y represent a consumer unit’s measured income for some time period (a year), and let us treat it as the sum of permanent component ,yp,and a

transitory component, yt, so that we obtain the second equality. The

permanent component is to be interpreted as reflecting the effect of those factors that the units regards as determining its capital value or wealth: for example the nonhuman wealth it owns, the personal attributes of the earners in the unit, like their abilities, personality, and training. Also the attributes of the economic activity of the earners, such as the type of occupation followed or the location of the economic activity. The transitory component is to be interpreted as reflecting all other factors that are likely to be treated by the unit affected as “accidental” occurrences even if they may be the predictable effect of specifiable forces, as cyclical fluctuation in the economic activity.

Transitory component, in statistical data, includes also random errors of measurement that in general is not possible to separate from the transitory component. In fact some of the factors that give rise to transitory components of income are specific to particular consumer units, for example, illness, wrong guess about when to buy or sell, and so on. For any group of consumer unit, the resulting transitory components tend to average out so that if they alone accounted for the discrepancies between permanent and measured income, the average measured income of the group would equal the average permanent component, and the average transitory component would be zero. In fact, not all factors giving rise to transitory component need be of this kind: indeed some may be largely common to the members of the group, like unusually good or bad weather. Or, another example, a sudden shift in the demand for some product, if the group is made of consumer units whose earners are employed in producing this product.

If these factors are favorable for any period, the mean transitory component is positive and if they are unfavorable, it is negative. Likewise a systematic bias in measurement may produce a nonzero mean transitory component in recorded data. As for the third equality, it defines the relation between permanent income and permanent consumption; also, it specifies

that the ratio between them is independent of the size of permanent income but does depend on other variables. In particular: k(r,z) is the average (or marginal) propensity to consume out of permanent income which depends on the rate of interest, or sets of rate of interest at which the consumer unit can borrow or lend, and on taste and preference shifter variables z. Recall that the transitory components may reflect fluctuations, or measurement errors: hence, the key point of PIH is that the consumption plan does not depend on the transitory components and to provide empirical content to this hypothesis, Friedman added the assumptions that the transitory components are uncorrelated to each other and uncorrelated to the permanent component. That is

𝜌_{𝑦𝑡,𝑝} = 𝜌_{𝑐𝑡,𝑝} = 𝜌_𝑦𝑡𝜌_𝑐𝑡= 0

Where ρ stands for the autcorrelation coefficient between the variables. The assumptions that the first two correlations are zero seems very bland and highly possible; indeed they have little substantive content and can almost be regarded as simply completing or translating the definitions of transitory and permanent components. The fusion of errors of measurement with transitory components helps further to the plausibility that these correlation are zero. For a group of individuals it is plausible to presume that the absolute size of the transitory component varies with the size of the permanent component. a given random event produces the same percentage rather than the same absolute increase or decrease in the incomes of units with different permanent components. This can make more convenient an alternative definition of transitory component said before, and it is not inconsistent with zero correlation. This latter, in fact, implies only that average transitory component is the same for all values of the permanent component. The plausibility of taking this definition of transitory components to

imply a zero correlation for a group of consumer units depends somehow on the criteria determining membership in the group. The easiest example is a classification of units by the size of their measured income: for every group, the correlation between permanent and transitory components is automatically negative, since with a common measured income the permanent component can be relatively high only if the transitory component is relatively low, and conversely.

The assumption that the third correlation is zero, in the latter equality, is a very strong assumption because is this assumption that brings important substantive content into the hypothesis; the ultimate test of its acceptability is if the its accordance with the observed data. As for savings, is this terms simply a residual? this notion implies that consumption is determined by rather long-term variables, so that any transitory changes in income lead mainly to additions or reductions of previously accumulated assets rather than to changes in consumption. But from another point of view, the assumption seems to be implausible: for example, would a person who receives an unexpected windfall, he will use at least one part for consumption expenditures? It will depend greatly on how is defined “consumption”. A first improvised answer considers the standard definition of consumption in terms of purchases, including durable good, rather than in terms of value of services.It is worth making other two considerations for the plausibility of the assumption that the transitory components of income and consumer are uncorrelated. First of all the identification of windfall with transitory income is not precise: for example suppose that inheritances are included in a particular concept of measured income. Assume a consumer whose receipts remain unchanged over a succession of time periods but in the final period, he receives an inheritance: if the inheritance was expected to occur soon or later, it would already have been included in the permanent income. The transitory component of income is only the excess of inheritance over this element of permanent income; it seems that there is no reason why the receipt

of the inheritance should make consumption different, in the final period, except through inability to borrow in advance against the inheritance. But this implies that the receipt of the inheritance changes the ratio of wealth to income. So it is already taken into account in the hypothesis, and therefore there is no essential difference if the inheritance is unexpected. The effect of the inheritance is then to increase the permanent income of the unit, justifying, then, a higher consumption in the final period.

The second consideration is that as there exist in which one person would expect a transitory increase in income to produce a transitory increase in consumption, there are also instances in which one would expect the contrary. For example when a transitory increase in income reduces opportunities for consumption like when it is obtained by working longer hours or going to a underdeveloped country.

Still, Friedman, goes on to show that the slope coefficient of a regression of observed consumption on observed income would lead to an underestimate of the marginal propensity to consume and to a positive estimated intercept. Moreover, he shows that the rate of decrease of the marginal propensity to consume is equal to the ratio of the variance of the permanent income to total income in the data (cross section or time series). This is now known as attenuation bias due to classical measurement error.

Why is it the case that the marginal propensity to consume in cross sectional data is lower than the average propensity and how can this be reconciled with the fact that the consumption function shifts upwards over time? His interpretation was that the results from a single cross section suffer from attenuation bias and the estimated consumption function using observed data shifts over time because permanent income goes up. So for a given level of observed income, permanent income is higher in later years than in earlier ones. For this reason if one joins the average observed pairs of consumption-income points across time one recovers a function that implies that the

marginal propensity is equal to the average one, in the data that Friedman used. The idea is that average income reflects (average) permanent income, because by the law of large numbers the transitory components average out. The pattern is repeated in a number of other data sets. For example an interpretation of why blacks save more than whites with the same observed income is that the former have lower permanent income than whites. When we condition on blacks being in the same observed income class, the transitory component of income (which does not affect consumption) is likely to be larger among blacks than whites. Similar arguments can be made when we confront the self-employed to the salaried workers or farm to non-farm households, the first in each pair having larger transitory components to their income. By comparing averages over time or across different groups of individuals, Friedman is implicitly using instrumental variables, now a well recognized technique for dealing with classical measurement error in linear models.2 Take the comparisons over time. The basic premise is that because of overall growth, later time periods are associated with higher permanent income. Moreover, if neither the mean of the transitory component of income nor preferences change over time we can use time as an instrument; the procedure used is what has recently been used to describe the generalized Wald estimator. Wald (1940) suggested to compare group averages to overcome measurement error, although the way he approached was not quite right. This is because he defined groups by class intervals of the variable that was supposed to be measured with error. The group averages will not except from of measurement error in this case. So, Friedman’s approach, was actually correct and bypassed Wald’s error be-cause he constructed groups based on other variables, such as time or ethnicity or occupation.

2.2 Statement of the permanent income

The PIH provides a flexible framework for the study of consumption and savings. The basic hypothesis needs a infinite horizon, which allows some degree of smoothing of consumption. The flexibility of the hypothesis comes at a price, because it is hard to define the theoretical reinforcement of the model and consequently to make more detailed statements about policy.

The definition of permanent income leaves open the question of how it should be measured. A moving average or distributed lag of past incomes are both options that seems to be the best; however the way one measures permanent income should be part of the model and has to do with the way individuals form expectations as well as with the stochastic process underlying income. Now we can describing the theoretical foundation of the hypothesis and the subsequent analysis. PIH includes the notion of shocks to income, which may reflects real but either transient or even permanent fluctuations to income; the theoretical framework should allows for i.e. the individual to be able to borrow and lend at a constant interest rate; we have, so, that the optimal consumption plan at time zero maximizes the sum of expected utility subject to a life-time budget constraint

Max c

, t=1,…T [ E

∑

𝛽

𝑈(c

)]

subject to

A0 + ∑𝑇_𝑡=1𝑅𝑡(𝑦_𝑡 + 𝑐_𝑡) = 0

Where R is the discount factor , i.e. 𝑅 = 1

1+𝑟 and A0 is initial non-human wealth;

under uncertainty the problem is solved in each period to accommodate news on income. The main implication of this model is that the marginal utility of consumption in each period is equal to the expected marginal utility of wealth multiplied by a factor depending on the interest rate and the rate of time

preference:

U’(ct) =

𝑅 𝛽 Et λt

Under certain circumstances this gives us a version of the PIH, where permanent income is defined as total lifetime wealth; for example if the rate of return is equal to the discount rate, R = 𝛽, with no uncertainty, consumption will be constant if preferences do not change over time. we can write consumption as a constant proportion to total wealth, which gives a version of the PIH when transitory shocks are all measurement error. But we can go even deeper by using quadratic utility function and assuming expected utility maximization; in this way, we get that consumption is expected to remain consistent

ct = Et ct+1

which has formed the basis of a test of the PIH by Hall (1978). Combined with the budget constraint gives the result that consumption is proportional to expected wealth

c

=

𝑟

[(1+𝑟)−(1+𝑟)−𝑇_]

E

W

where EtW =At + Et ∑𝑇_𝑠=𝑡𝑅𝑠ys represents the expected life-cycle wealth. This shows how the average (marginal) propensity to consume relates to the interest rate. Now we can see how consumption reacts to change in current income (one of the key motivations for the PIH). Sargent (1978), Flavin (1981) and Campbell (1987) elaborated this version of PHI implying that consumption changes are equal to the annuity value of all revised changes in future incomes. So we can write

∆𝑐

=

∑

( E

+ E

) y

( Et + Et-1) yt+s represents revisions in expectations on the income flow. In the simplest case where the income process contains a deterministic component plus a transitory component, we get that consumption does not react much to current income fluctuations. The reaction, in fact is zero for anticipated changes and the reaction to unanticipated transitory shocks equals the annuity value of the shock. This reflects the thinking of Friedman and it is an intuition that finds expression in modern economic analysis. However, there are doubts about if this is a good description of the income process. In fact, if the shocks to income are permanent, then all future levels of income will be revised upwards or downwards by the same amount that leads to a reaction of consumption, that is equal to the change in current income. This is exactly consistent with the PIH, since Friedman explicitly argued that changes over time should reflect (unexpected) changes in permanent income.

2.3 Precautionary savings

In this section we’re going to study the role of a precautionary saving motive in shaping consumers’ behavior; this theory predicts that risk depresses consumption, and it increases the accumulation of wealth as a type of self insurance. Precautionary saving motive exists and affects virtually every household. But we also find that this motive does not give rise to high amounts of wealth at the aggregate level. Desired precautionary wealth represents approximately 8 percent of total net worth and 20 percent of total financial wealth in the economy and various examination suggest that not just income risk should be taken into account, but also other risks; moreover another key element is the great heterogeneity in precautionary accumulation across households of objectively similar types.

Theoretical intertemporal models of consumption/saving with income uncertainty predict that precautionary wealth can explain a large share of total wealth accumulation. For example Skinner (1988) calculates that about half of household wealth can be explained by precautionary savings due to

income risk, like Caballero (1991), Gourinchas and Parker (2002), and Cagetti (2003) that report similar results. But, the empirical evidence based on micro data yields decidedly mixed results. We identify three sets of papers. In the first, Skinner (1988) finds no evidence that households in riskier occupations save more. Similarly, Dynan (1993) argues the empirical estimates of the coefficient of relative prudence are too small to generate precautionary saving. The second set of papers, including Guiso, Jappelli and Terlizzese (1992), Lusardi (1997, 1998), and Arrondel (2002), uses subjective measures of income risk and finds modest values for precautionary wealth–2 to 8 percent of total household wealth. The final set of papers, including Dardanoni (1991), Hubbard, Skinner and Zeldes (1995), Carroll and Samwick (1997, 1998), Kazarosian (1997), and Engen and Gruber (2001), finds that precautionary savings can explain a sizable share of wealth. These papers are distinguished by their thoughtful approach and careful execution; But this disturbingly large range of estimates implied suggests that there may be important latent conceptual and empirical factors that confound the analysis of the precautionary saving motive. Most of the micro-empirical work on precautionary saving has focused on the estimation of the following equation:

g(Wh) = f (riskh, 𝑌_ℎ𝑃, Xh)

where Wh indicates wealth of household h; riskh is a measure of the risk faced by household h; 𝑌_ℎ𝑃is their permanent income; Xh is a set of controls for wealth including age, demographics, and other household characteristics; and g and f indicate the functions to measure wealth and the relationship between wealth and the right-hand-side variables respectively.

Browning and Lusardi (1996) noted that the most straightforward measure, directly controlled net worth, turns out to be inappropriate except in the extreme case of certainty equivalence; because of the differing risk and liquidity characteristics of the underlying portfolio elements, they cannot, in

general, be aggregated in this model. Some authors like Engen and Gruber (2001), and Alan (2004) have simply considered a measure of liquid wealth when estimating precautionary wealth. But this approach may be overly restrictive in its implicit assumption of no substitution across assets; in fact the large majority of US households hold other assets in their portfolios very often financial assets are a relatively small part of total wealth. In general, the largest asset is housing equity, and instruments such as home equity lines of credit served to make that wealth more liquid.

Another large component of wealth, particularly for middle-aged and older workers, is designated retirement accounts. Another complication is business equity, which forms a large part of the portfolios of many wealthy households; this wealth is hard to measure and may be hard to liquidate or leverage. Even the treatment of debt is not necessarily straightforward (Engen and Gruber (2001)), and for this reason most households only need to service their debt. So it may be that only the required loan payments over some period need to be netted from assets, rather than subtracting all the short and long term debt.

Much of the empirical work on precautionary saving has concentrated on one risk factor, that is income risk. For example, in the third set of papers mentioned above, researchers have modeled a household-specific stochastic process for income, estimated it using panel data, and then used the variance of earnings or non-capital income as a proxy for risk. But it may often be difficult to distinguish empirically between transitory income and measurement error, because the estimated variability may be already insured against (Caballero (1991) and Browning and Lusardi (1996)). Precautionary accumulation depends not just on risk, but also on preferences regarding risk; for example, a key factor is the rate of coefficient of risk aversion (Deaton (1991) and Carroll (1992, 1996)), but available information suggests that there may be substantial variation in this measure across households. Consider a framework, where preferences for consumption depend on

household composition; it may be, for example, reasonable to think that more consumption is shifted towards the household when it consists of more members. The quadratic preference specification, used to show the foundations of the basic version of the PIH, does not allow for precautionary savings. The individual behaves in exactly the same way as if she were given with certainty the expected lifetime wealth: this is of course very restrictive. Thus, in addition to controlling for changes in preferences as a function of household circumstances we also change the structure of the utility function to have a positive third derivative, which has been proven to be the key to allowing for precautionary savings (Sandmo, 1969; Drèze and Modigliani, 1972). Here the utility function takes the constant relative risk aversion form with a discount rate or a risk aversion coefficient which is a function of demographic variables and written as:

𝑈(𝑐_𝑡|𝑧_𝑡) = 1

𝜌 + 1𝑎(𝑧𝑡)𝑐𝑡

𝜌+1

𝜌 < 0

where a(zt) is a function of variables that can affect preferences. The choice of these variables is of course a key issue. First, one might include demographic variables, like family size and the number and ages of children, since these will shift the needs and preferences of the household over time. Zeldes (1989) bases a test for liquidity constraints on a model including variables that reflect needs and rejects the model. Precautionary accumulation indeed, is strongly affected by the presence of liquidity constraints; Zeldes (1989) bases a test for liquidity constraints on a model including variables that reflect needs and rejects the model. However, his model is based just on food consumption.

This specification is likely to reject the hypothesis of no liquidity constraints, unless preferences across commodities are additively separable and homothetic, because we will be attributing changes in consumption

growth that are due to changes in prices, to changes in real incomes. So we should include variables describing the labour supply behavior of the household members. To the extent that households can borrow, they may not need as much wealth to shield themselves against shocks: for this reason, theoretical models of precautionary saving do not necessarily predict that wealth will be (strictly) positive. For example if households face differences in borrowing opportunities, they may want to hold different levels of precautionary savings; but individual borrowing opportunities are indirectly largely unobservable in most data sets. In general, it is not possible to find an explicit solution to the consumption function when preferences are isoelastic. So the empirical approach to test for this model is to use the Euler equation that can be approximated in this way:

∆𝑙𝑛𝑐_𝑡 = 𝜃𝑟_𝑡+ 𝜃(𝐸_𝑡−1𝜎_𝑡2− 𝛿) + 𝛽′∆𝑧_𝑡 + 𝑢_𝑡

According to this equation consumption growth increases when interest rates are predicted to be high; Moreover consumption growth will be positive, implying higher savings, when the conditional variance of consumption is higher than the personal rate of discount (the trade-off between precaution and impatience).

The other source of changes in consumption growth, are changes in taste shifter variables and labour supply, summarized in z. Finally the error term ut reflects changes due to unexpected events. The estimation The estimation of the precautionary saving motive is a very complex task.

There exist many pitfalls and difficulties in assessing the empirical importance of this motive. One of the major problems is how to measure accurately the amount of reserves people use to shield themselves against risk. The commonly used measures of wealth have many problems, and much more attention should be devoted to this important issue

2.4 Distribution of permanent income

Wealth distribution figures for different countries, or different years prompt speculation about the causes of observed differences: for example disparities in wealth distribution across countries are often seen as resulting from different social traditions, general economic environment, and relative importance of inheritance life-cycle accumulation.

Let consider the United States wealth, where it is highly concentrated and very unequally distributed: the richest 1% of the households owns one third of the total wealth in the economy; the development of various data sets in the past 30 years (in particular the Survey of Consumer Finances) has allowed economists to quantify more precisely the degree of wealth concentration in the United States and what emerged from the different waves of these surveys confirmed that a large fraction of the total wealth in the economy is concentrated in the hand of the richest percentiles.

Also income is unequally distributed and this leads to wealth inequality as well, but income much less concentrated than wealth, and with economic models is difficult in generating the observed degree of wealth concentration from the observed income inequality. But the question is: what mechanisms are necessary to generate saving behavior that leads to a distribution of asset holdings consistent with the actual data? Some models consider a dynasty as a single, infinitely-lived agent while others consider more explicitly the life-cycle aspect of the saving decision.

The basic versions of these models are unable to replicate the observed wealth concentration; by the way, more recently, some works have shown that certain ingredients are necessary, and sometimes enable the model to replicate the data. Bequests are the main determinants of inequality, and careful modelling of bequests is essential to understand wealth concentration. In addition, entrepreneurs constitute a large fraction of the very rich, and models that explicitly consider the entrepreneurial saving decision succeed in

increasing wealth dispersion. The most important source of microeconomic data on wealth for the U.S. is the Survey of Consumer Finances (SFC) which, every three years collects detailed information about wealth for a cross-section of households; the SCF was explicitly designed to measure the balance sheet of house-holds and the distribution of wealth.

It has a large number of detailed questions about different assets and liabilities, which allows highly disaggregated data analysis on each component of the total net worth of the household.

More importantly, the SCF oversamples rich households by including, in addition to a national area probability sample (representing the entire population), a list sample drawn from tax records (to extract a list of high income households). Oversampling is especially important given the high degree of wealth concentration observed in the data. For this reason, the SCF is able to provide a more accurate measure of wealth inequality and of total wealth holdings.

However there is a defect: the SCF does not follow households over time, unlike the Panel Study of Income Dynamics (PSID). The PSID is a longitudinal study, which started in 1968, follows families and individuals over time and it focuses on income and demographic variables, but since 1984 it has also included (every 5 years) a supplement with questions on wealth.

The PSID includes a national sample of low-income families, but it does not oversample the rich: as a result, this data set is unable to describe appropriately the right tail of the wealth distribution. So , because of its careful sample choice, the SCF remains the main source of information about the distribution of wealth in the U.S.

2.5 Wealth concentration in the U.S.

degree of concentration. In the Table 1 we can see the households in the top 1% of the wealth distribution hold around one third of the total wealth in the economy, and those in the top 5% hold more than half. At the other extreme, more than 10% of households have little or no assets at all.

Percentile Year Group 1989 1992 1995 1998 2001 0-49.9 2.7 3.3 3.6 3.0 2.8 50-89.9 29.9 29.7 28.6 28.4 27.4 90-94.9 13.0 12.6 11.9 11.4 12.1 95-98.9 24.1 24.4 21.3 23.3 25.0 99-100 30.3 30.2 34.6 33.9 32.7

Table 1: Percent of net worth held by various groups defined in terms of percentiles of the wealth distribution (taken from Kennickell [44], p. 9)

Net Year worth 1989 1992 1995 1998 2001 < $0 7.3 7.2 7.1 8.0 6.9 $0-$1,000 8.0 6.3 5.2 5.8 5.4 $1,000-$5,000 12.7 14.4 15.0 13.1 12.8 $25,000-$100,000 23.2 25.4 26.4 22.9 22.0 $100,000-$250,000 20.2 21.6 22.1 22.6 19.2 $250,000-$500,000 11.0 9.3 9.3 12.0 13.0 $500,000-$1,000,000 5.4 4.6 5.1 6.0 7.8 ≥ $1,000,000 4.7 3.8 3.6 4.9 7.0

Table 2: Percent distribution of household net worth over wealth groups, 2001 dollars (taken from Kennickell [44], p. 9).

The data in Table 1 and 2 refer to total net worth. There are many possible measures of wealth and the most appropriate one depending on the problem under study. Net worth includes all assets held by the households (real estate, financial wealth, vehicles) net of all liabilities (mortgages and other debts); it is, so, a comprehensive measure of most

marketable wealth. This measure thus includes the value of most defined contribution plans (such as IRAs), but excludes the implied values of defined benefit plans and social security. To study other questions it may be useful to look at more restricted measures of wealth, that for example exclude less liquid assets (such as housing), and focus on financial wealth instead. There is also significant wealth inequality within various age and demo-graphic groups.

For instance, Venti, Wise and Bernheim (1988) show that wealth is highly dispersed at retirement even for people with similar lifetime incomes, and argue that this differences cannot be explained only by events such as family status, health and inheritances, nor by portfolio choice. Another studies have also highlighted the differences in wealth holdings across different groups. There is a very large inequality in wealth holdings by race.

In addition to income differences, wealth inequality can be driven by differences in the saving behavior, or in the intergenerational transfers received. In fact Individual saving cannot be measured directly, but must be computed from other data, as the first difference in wealth, or as income minus consumption. For this motive few studies document the differences in saving rates across the population even if they suggest significant differences in saving behavior across various groups. Bequests also play an important role in shaping wealth inequality. Kotlikoff and Summers (1981) were the first to argue that life-cycle savings for retirement account for a small fraction of total capital accumulation, while intergenerational transmission of wealth it is important for the vast majority of capital formation. Another studies have confirmed the importance of intergenerational transfers; for instance, Gale and Scholz (1944) find that bequests account for about 30% of total wealth accumulation, and various other types of intended

intervivos transfers for an additional 20%. It is more difficult to measure the size of intended bequests relative to that of purely accidental ones, due to uncertainty about the life-span. Hurd (1989) estimates a very low marginal utility from leaving bequests, while Altonji and Villanueva (2003) also find relatively small values for the elasticity of bequests to permanent income, even if they show that this number increases with life-time resources.

2.6. Models

In this section, we describe the main class of models used to

study wealth concentration; most of these models are general-equilibrium, quantitative models with heterogeneous agents. We distinguish these works into three categories: models with infinitely-lived dynasties, models with overlapping-generations (OLG), and models that mix both of these features. The first type of models ignore the life-cycle structure, but consider each dynasty as a single agent who lives forever. The second type explicitly introduces an age and life-cycle structure, with various degrees of intergenerational transmission of wealth and abilities. The third type relaxes the infinitely-lived dynasty assumption of the first type of models, but greatly simplifies the life-cycle structure.

2.6.1 infinitely-lived dynasty models

In these models there is a continuum of agents and all agents have identical preferences. They have the following utility function when they first enter the model economy:

E[{∑ 𝛽𝑡_𝑢(𝑐 𝑡 ∞ 𝑡=1 }] 19

u(ct) is the constant relative-risk aversion flow of utility from consumption. The labor endowment of each household is given by an labor productivity shock z that assumes a finite number of possible values and follows a first order Markov process with transition matrix (Γ(z)). There is only one asset, a, that people can use to self-insure against earnings risk. A constant returns to scale production technology transforms aggregate capital (K) and aggregate labor (L) into aggregate output (Y). During each period each household chooses how much to consume (c) and save for next period by holding risk free assets (a’). The household’s state variables are denoted by x = (a,z), where a is asset holdings carried into the period and z is the labor shock endowment. The household’s recursive problem can thus be written as follows:

V(x) = max{𝑢(𝑐) + 𝛽𝐸[𝑉(𝑎′, 𝑧′)|𝑥]}

Subject to c+a’=(1+r)a+zw c ≥ 0, a’ ≥ a

r is the interest rate net of taxes and depreciation, w is the wage, and a is a net borrowing limit. A stationary equilibrium for this economy is a set of consumption and saving rules, prices, aggregate capital and labor, and invariant distribution of households over the state variables of the system such that:

- Given prices, the decision rules solve the household’s recursive problem described

-Aggregate capital is equal to total savings of all the households of the economy and aggregate labor is equal to total labor supplied by all of the households of the economy

- Prices, that is the interest rate and the wage rate, gross of taxes, equal the marginal product of capital, net of depreciation, and the marginal product of

labor

-The constant distribution of people is the one induced by the law of motion of the system, which is determined by the exogenous earnings shocks and by the endogenous policy functions of the households. In the table below we can see the results, drawn by the work of Quadrini and Rìos-Rull (1997)

% wealth in top Gini 1% 5% 20% U.S. data .78 29 53 80 Baseline Aiyagari .38 3.2 12.2 41.0 High variability Aiyagari .41 4.0 15.6 44.6 Quadrini: entrepreneurs .74 24.9 45.8 73.2

Table 3: Dynasty models of wealth inequality

The table display significantly less wealth concentration than in the data; The motives why households save in this type of models is to create a stock of assets to self-insure against earnings fluctuations. When the stock is reached, the agents do not save any more, and the model is not capable to explain why the rich people keep saving at very high rates. The failure of the basic model to explain wealth inequality tells us that we need to look at other mechanisms, that are entrepreneurship and preference heterogeneity. A recent contribution made by Quadrini introduces entrepreneurial choice in a dynastic framework, i.e.: during each period the households decide if to be entrepreneurs or not. In his model, three key elements are decisive to generate this result.

First, the existence of capital market imperfections induces workers that have entrepreneurial ideas to accumulate more wealth to reach minimal capital requirements.

rate on borrowing is higher than the return from saving. Third, there is additional risk associated with being an entrepreneur, hence risk averse individuals will save more. Another mechanism to generate wealth inequality is heterogeneity in preferences; the decision to save depends crucially on the specific parameter values of the utility function. Specifically, a higher degree of patience (a higher discount factor β) leads people to save more. In the presence of precautionary savings, a higher coefficient of risk aversion can also induce higher savings.

2.6.2 Overlapping-generation models

We use Huggett’s (1996) formulation: agents live at most N periods, and face an age-dependent survival probability st of surviving up to age t, conditional on surviving up to age t − 1. The demographic patterns are stable, so age t agents make up a constant fraction µt of the population at every point in time. All agents have identical preferences, and have the following utility function when they first enter the model economy:

E{∑𝑁𝑡=1𝛽𝑡(∏𝑡_𝑗=1𝑠_𝑡)𝑢(𝑐𝑡)}

u(ct) is the constant relative-risk aversion flow of utility from consumption, and the expected value is calculated with respect to the household’s earnings shocks. The labor endowment of each household is given by a function e(z,t), which depends on the agent’s age t, and on an idiosyncratic labor productivity shock z, that assumes a finite number of possible values and that follows a first order Markov chain with transition matrix Γ(z). People save to insure against earnings risk, for retirement, and in case they live a long life and who die prematurely leave accidental bequests.

There is a constant returns to scale production technology that converts aggregate capital (K) and labor (L) into output (Y).

During each period each household choose how much to consume (c) and save for next period by holding risk free assets (a’). The household’s state variables are denoted by x = (a,z), where a is asset holdings carried into the period and z is the labor shock endowment. The household’s recursive

problem can be written as follows:

V(x,t)=max{𝑢(𝑐) + 𝛽𝑠_𝑡+1𝐸[𝑣(𝑎′, 𝑧′, 𝑡 + 1)|𝑥]} (c,a’)

Subject to

c+a’=(1+r)a+e(z,t)w+T+bt c ≥ 0, a’ ≥ a and a’ ≥ 0 if t = N

r is the interest rate net of taxes and depreciation, w is the wage net of taxes, T are accidental bequests that left by all of the deceased in a period, that are assumed to be redistributed by the government to all people alive, and bt are social security payments to the retirees. At every point in time this model economy can be described by a probability distribution of people over age t, assets a, and earnings shocks z. A stationary equilibrium for this economy can be defined in the same way as the one described for the infinitely-lived model, with the additional requirements that during each period total lump-sum transfers received by the households alive equal accidental bequests left by the deceased. The results is that Huggett finds that relaxing the household’s borrowing constraint increases the fraction of people bunched at zero or negative wealth, but it does not increase much the asset holdings of the rich, and hence does not help in generating a distribution of wealth closer to the observed one.

2.7 Bequest motives

De Nardi (2004), instead, introduces two types of intergenerational links in the OLG model used by Huggett: voluntary bequests and transmission of human capital. Here, parents and their children are linked by voluntary and accidental bequests and by the transmission of earnings ability. The households save to self-insure against labor earnings shocks and life-span risk, for retirement, and possibly to leave bequests to their children. There is ,so an extra term in the value function of a retired person that faces a positive probability of death:

V(a,t)= max {𝑢(𝑐) + 𝑠_𝑡𝛽𝐸_𝑡𝑉(𝑎′, 𝑡 + 1) + (1 − 𝑠_𝑡) 𝜙(𝑏(𝑎′))} (c,a’) Where φ(b) = φ1 + (1+ 𝑏 𝜑2) 1−𝜎

The utility from leaving bequests thus depends on two parameters: φ1, which represents the strength of the bequest motive, and φ2, which measures the extent to which bequests are a luxury good. The table below summarizes the results:

Transfers percentage wealth in the top percentage with Wealth wealth negative or

Ratio Gini 1% 5% 20% 40% 60% zero wealth U.S. data

.60 .78 29 53 80 93 98 5.8-15.0 No intergenerational links, equal bequests to all

.67 .67 7 27 69 90 98 17

No intergenerational links, unequal bequests to children

.38 .68 7 27 69 91 99 17 One link: parent’s bequest motive

.55 .74 14 37 76 95 100 19

Both links: parent’s bequest motive and productivity inheritance

.60 .76 18 42 79 95 100 19

The of the table refers to the U.S data and the second to a version of Huggett’s model economy where there are only accidental bequests, redistributed equally to all people alive every year. The third line still refers to an economy in which there are only accidental bequests, but these are received by the children of the deceased upon their parent’s death, and are so unequally distributed. These experiment show the fact that measure on intergenerational transfers is sensitive to the timing of transfers: if children inherit only once, when their parent dies (rather than every year as in line three), this measure generates a fraction of wealth due to intergenerational transfers that is much lower than the one computed by Huggett. The fourth line allows for a voluntary bequest motive, and we can see that voluntary bequests can explain the emergence of large estates, which are often accumulated in more than one generation, and characterize the upper tail of the wealth distribution in the data. The fifth line allows for both voluntary bequests and transmission of ability and shows that a human-capital link, by which children partially inherit the productivity of their parents, generates a further more concentrated wealth distribution; more productive parents accumulate larger estates and leave larger bequests to their children, who, in turn, are more productive than average in the workplace.

The conclusions are that the presence of a bequest motive also generates lifetime saving profiles more consistent with the data: saving for precautionary purposes and saving for retirement are the primary factors for wealth accumulation at the lower tail of the distribution, while saving to leave bequests significantly affects the shape of the upper tail. Also, with this parameterization of the voluntary bequest motive, and consistently with the data, the rich elderly do not decumulate their assets as fast as predicted by a standard a OLG model. De Nardi finds that φ2 is a large number, so bequests are a luxury good, and that the extent to which they are a luxury good is key in generating more concentration in the hands of the richest and producing a more realistic lifetime savings profiles. With this parameterization, and

consistently with the data, the bequest motive to save is much stronger for the richest households, who, even when very old, keep some assets to leave to their children.

2.8 Mixture of life cycle and dynastic behavior

The third class of models mixes features of both life-cycle models and infinitely-lived dynasties, simplifying some aspects of either model to make them more computationally tractable.

For example Laitner (2001) assumes that all households save for life-cycle purposes, but only some of them care about their own descendants. There are perfect annuity markets, so all bequests are voluntary, and no earning risk over the life cycle, hence no precautionary savings. Laitner’s model is simple to compute and provides a number of interesting insights; the

concentration in the upper tail of the wealth distribution is matched by choosing the fraction of households that behave as a dynasty and also depends on the assumptions on the distribution of wealth within the dynasty, which is indeterminated in the model.

Another work is of Nishiyama (2002) adopts an OLG model with bequests and intervivos transfers in which households in the same family line behave strategically. Like De Nardi, he concludes that the model with intergenerational transfers better explains, although not fully, the observed wealth distribution.

Castaneda, Dıaz-Gimenez and Rıos–Rull (2003) consider a model economy populated by dynastic households that have some life-cycle flavor: i.e. workers have a constant probability of retiring at each period and once they are retired they face a constant probability of dying. Each household is perfectly altruistic toward its household.

Cagetti and De Nardi build on Quadrini’s (2001) model of wealth inequality by endogenizing the firm size distribution, the interest rate at which firms borrow and lend, and the amount of borrowing as a function of

the entrepreneur’s collateral, and by modeling the life-cycle and the intergenerational linkages; they adopt Castaneda, Dıaz-Gimenez and Rıos– Rull demographic structure. The key reason why their model succeeds in generating this large amount of wealth concentration is linked to the fact that, while entrepreneurs could invest capital at a higher rate of return, the presence of borrowing constraints and collateral requirements makes the entrepreneur to save to exploit the high rate of return even when the entrepreneur becomes “rich”. This key intuition does not depend on the demographic structure assumed; Cagetti and De Nardi chose to formulate it in an economy with more realistic life-cycle features to study the effects of government policies such as estate taxation.

References

-Friedman, M. (1957), “A Theory of the consumption function” (Princeton University Press, Princeton)

-Caggetti M., and De Nardi M., (2006) “Wealth inequality: data and models” ,Working Paper 12550

-Meghir C., (2004), “A Retrospective on Friedman’s theory of permanent income”, The economic journal

- Lusardi, A. and Kennickell, A. (2005) “Disentangling the importance of the precautionary saving motive”