A Welfare Analysis of Labor Market Inequality: Inspecting the Mechanism∗ Jonathan Heathcote, Kjetil Storesletten, Giovanni L. Violante

Preliminary and incomplete: please, do not quote without permission.

Abstract This paper develops a series of simple heterogeneous-agents economies in order to study the welfare implications of labor market inequality. We perform two types of welfare calculations. First we compute the welfare cost of a rise in labor market risk, given a particular asset market structure (complete markets, incomplete markets, autarky). Second, we compute the expected gain from completing markets (starting from autarky or incomplete markets) for a given level of wage inequality. We are able to derive intuitive closed-form expressions for these welfare effects, expressions which depend only on preference parameters and the size of (the change in) wage dispersion. We then exploit the positive implications of our incomplete-markets economy for the cross-sectional joint distribution over earnings, hours and consumption to estimate the preference parameters. Plugging these estimates into our welfare expressions allows us to quantify the welfare consequences of the rise in wage inequality observed in the United States over the past three decades as 2.5% of lifetime consumption. Moreover, we calculate that completing the asset markets, given the current level of inequality, would lead to a welfare improvement equivalent to 40% of lifetime consumption.

∗

We would like to thank, without implicating, Randy Wright. Several ideas in this paper originated from a conversation we had with him. We also thank participants to the NYU Macro Reading Group, and to the Richmond Fed Conference on Labor Markets for comments.

1

1

Introduction

There has been a large increase in wage inequality in the United States over the past thirty years. The primary goal of this paper is to improve our understanding of the welfare implications of this phenomenon. We view this as an important exercise, since crosssectional wage inequality and individual wage fluctuations over the life-cycle are large. For example, in the United States the cross-sectional variance of the growth rate of wages for male workers is over 200 times bigger than the time variance of the growth rate of average wages for the same group.1 Thus the welfare implications of idiosyncratic labor market risk are likely to be much larger than the welfare costs of aggregate business-cycle risk. In this paper we develop and estimate a series of simple models which serve as transparent laboratories for isolating the welfare implications of rising wage inequality and for understanding the source of these welfare effects. Agents in our models are heterogeneous with respect to their labor productivity and thus their market wage. Welfare costs will depend crucially on the set of instruments available to insure against this risk. We therefore analyze a range of alternative assumptions regarding the set of financial assets that may be traded, in order to understand the role of access to explicit insurance against labor market risk. In addition, we incorporate a labor supply choice, since the ability to adjust hours is potentially an important margin of self-insurance against shocks to wages. We consider three alternative asset market structures. In the first - complete markets – all wage inequality is insurable. At the other extreme – autarky – we rule out all asset trade between agents. Finally, as an intermediate case, we put some structure on the idiosyncratic wage generating process and assume that it has two components: a permanent (fixed-effect) piece, and a transitory piece. We then assume that transitory wage inequality is perfectly insurable, while permanent inequality is wholly uninsurable. We label this the incompletemarkets economy, since this economy closely approximates an economy in which agents trade only a non-contingent bond, as in Heathcote, Storesletten and Violante (2003). We assume that agents are infinitely-lived, that preferences are separable between consumption and hours worked, and that wage shocks are drawn from log-normal distributions. 1

This number is calculated from the PSID, 1967-1996. See section 4 for details on the sample selection. In particular, the variance of the mean wage growth over the period is 0.008 and the cross-sectional variance of individual wage growth, averaged over the period, is .16.

1

These assumptions in combination with the permanent-transitory wage decomposition allow us to derive analytic expressions for expected lifetime utility. We use these expressions to compute two sets of welfare numbers. First, we compare welfare costs of rising wage inequality. In this case, we hold the asset market structure constant and increase the variance of one or both components of the wage-generating process. Second, we compute the welfare costs of market incompleteness, which we think of as the difference between welfare in the autarky or incomplete markets economy on the one hand, and the complete markets economy on the other. In this case, we hold the wage-generating process constant across economies, and think about welfare comparisons from the point of view of an agent being dropped into different economies from behind a veil of ignorance. Our first set of findings is that the welfare cost of rising wage inequality is extremely sensitive to preference parameters, and to the assumed asset market structure. For example, when asset markets are assumed to be complete, an individual would prefer to be dropped at random into an economy with high wage inequality rather than one with low inequality. The reason is simply that the planner is able to exploit differences in productivity across agents to increase average labor productivity. By contrast, when no financial assets are traded, the welfare cost of rising wage inequality can be very large, especially if agents are unwilling to substitute consumption and hours inter-temporally. To compute plausible welfare estimates we therefore need to take a stand on which combination of preference parameters and market structure is most appropriate for the United States. In the complete markets and autarky economies, only the overall increase in wage inequality matters – the extent to which this reflects widening permanent wage differences as opposed to greater volatility is irrelevant. In the incomplete markets economy, by contrast, the extent to which rising wage inequality is insurable is critical for the model’s positive and welfare predictions, so deciding how to allocate increased wage inequality between bigger transitory and permanent shocks is an additional important task for doing welfare analysis. To pick preference parameters for the incomplete markets model, and to decide what fraction of the observed increase in wage inequality was directly insurable, we exploit the positive predictions of the model, adopting a strategy in the spirit of Blundell and Preston (1998). Under our assumptions, we can obtain analytical expressions for individual con2

sumption and labor supply choices in the model, and for cross-sectional first and second moments of wages, hours, earnings and consumption. We adapt data on U.S. males in the PSID from Heathcote, Storesletten and Violante (2003) to compute a large set of empirical cross-sectional moments for 1970 and for 1995. These moments are mean hours, the variances of log wages, log hours and log earnings, and the covariance between hours and wages. To this set we add estimates from Krueger and Perri (2002) for the variance of log consumption and the covariance between hours and consumption. We extend the model economy to incorporate preference heterogeneity in the form of permanent cross-sectional differences in the relative taste for consumption versus leisure. We then use a simple Minimum-Distance estimation procedure to pick values for the curvature parameters for consumption and hours in preferences, for the variance of preference shocks, and for the fraction of total wage inequality that is insurable in each steady state. This fraction is the only parameter that is assumed to potentially vary across the two reference dates. We find that at the estimated parameter values, the model comes extremely close to replicating all the cross-sectional moments described above in both 1970 and in 1995. In particular, it accounts for the facts that there has been a large increase in the cross-sectional variance of log wages, but a much smaller increase in the variance of log consumption, and little change in the variance of log hours. The point estimates for preference parameters are well within the range of previous estimates – the risk aversion coefficient for consumption is between 1.5 and 2, while the Frisch elasticity for labor supply is around 0.3. Preference shocks to the relative taste for leisure accounts for almost three quarters of the crosssectional variance of log hours, but only one quarter of the cross-sectional variance of log consumption. We estimate that transitory (insurable) shocks account for two thirds of the total rise in U.S. wage inequality between the two reference years. Finally we use our estimated preference parameters combined with the estimated wage dynamics to revisit the welfare calculations. We find that the welfare cost of the observed rise in labor market risk in the incomplete-market economy is 2.5% of lifetime consumption, and that households would be willing to give up almost 40% of their lifetime consumption to be able to fully insure both preference shocks and permanent wage shocks. These numbers are 1, 000 times larger than commonly estimated welfare losses of business cycles fluctua3

tions. The rest of the paper is organized as follows. Section 2 describes the model economy and discusses the characterization of the equilibrium allocations through the appropriate Planner’s problem. Section 3 contains the qualitative welfare analysis. Section 4 performs the positive analysis that allows us to identify the key parameters of the model. Section 5 uses the estimated parameters as an input in order to quantify the welfare implications studied in Section 3. Finally, in Section 6, we make our concluding remarks.

2

The Economy

Demographics and preferences: The economy is populated by a continuum of infinitelylived agents. Each agent has the same time-separable utility function U (c, h) over streams ∞ of consumption c = {ct }∞ t=1 and hours worked h = {ht }t=1 ,

U (c, h) =

∞ X

β t u (ct , ht ) ,

t=0

where β ∈ (0, 1) is the agents’ discount factor.2 Within a period, preferences over consumption and hours worked are given by h1+σ c1−γ −ψ , u (c, h) = 1−γ 1+σ

(1)

where 1/γ is the elasticity of intertemporal substitution for consumption, and 1/σ is the Frisch elasticity of labor supply.3 The parameter ψ measures the utility weight on leisure, relative to consumption. Similar functional forms have frequently been estimated in the micro literature (see Browning, Hansen and Heckman 1999 for a survey) Individual labor productivity: We denote by wit the stochastic endowment of labor productivity for individual i at time t. Endowments are independently and identically distributed across the agents in the economy, and also vary stochastically over time. In order to obtain sharp welfare results, it is convenient to make distributional assumptions on wit . We assume that the mean population wage (in levels) is equal to one, i.e. E (wit ) = 1. The normalization to unity is without loss of generality, but the important implication of 2

Later we will introduce heterogeneity in the population regarding the relative taste for consumption versus leisure. 3 The Frisch elasticity of labor supply measures the elasticity of hours worked to changes in wages, keeping the marginal utility of consumption constant.

4

assuming a constant mean wage is that mean productivity will be invariant with respect to dispersion when we study the comparative statics with respect to the variance of wages, v. In parts of our positive and welfare analysis, it will be essential to put more structure on the labor productivity shocks. Thus in the incomplete markets economy we will assume that individual wages have two orthogonal components: a fixed (uninsurable) component αi and, a transitory (insurable) component εit , i.e. log wit = αi + εit . It is useful to specify a particular distribution for the shocks. We assume that ³ v ´ log wit ∼ N − , v , 2

(2)

(3)

where v is the variance of the log-normal productivity shock. To be consistent with (3), we also assume that the unconditional distributions of εit and αi are Normal, with ³ v ´ ε εit ∼ N − , vε , ´ ³ v2 α α i ∼ N − , vα , 2 where v = vε + vα is the total variance. Note that the mean value for each component of the log wage is chosen appropriately to ensure that the mean level wage is equal to one. Note that we do not specify any time series properties for εit . However, since this is the potentially insurable component of wage risk in the incomplete markets model, it is natural to think of this source of wage fluctuations as capturing shocks that are not too persistent. The α component is modelled as a time-invariant fixed effect. In principle, one could visualize this component as also capturing permanent (random walk) shocks. However, this makes the equilibrium characterization slightly more cumbersome, so for expositional simplicity we have opted for the representation in (2).4 Production technology, goods and labor markets: Production takes place through a constant-returns-to-scale aggregate production function with labor as the only input: Ã I ! X Yt = A wit hit . i=1 4

One complication that arises in the context of a model with infinitely-lived agents is that the crosssectional wage distribution is not stationary if individual wages follow a random walk.

5

In what follows, without loss of generality, we set the scaling parameter A to one. The labor market and the goods market are competitive. Insurance markets– Every household starts out with zero wealth. We consider three different insurance market structures: autarky, complete markets, and “incomplete” markets. – Autarky (AUT): No financial instruments are available, and households simply consume their labor income every period. All individual wage fluctuations are uninsurable, so we do not need the additional structure on the labor productivity process described in (2). – Complete markets (CM): Households are free to trade contracts contingent on every possible realization of the individual productivity shock at every date. All individual wage fluctuations are insurable, so as in autarky there is no need to distinguish between permanent and transitory shocks. In terms of that decomposition, agents are allowed to trade state contingent claims at time zero, before the realization of the permanent component. An arbitrarily loose constraint on borrowing rules out Ponzi-schemes. – Incomplete markets (IM): This is our intermediate benchmark. Here, households have access to perfect insurance against the transitory ε-shocks and no insurance against the permanent α-shocks. To understand the market structure in this case, consider the following “metaphor”: agents first draw a value for αi from the distribution. Next, according to their draw, they move to an “island” where they join all other agents with the same draw of the permanent component. On each island, they face a perfect insurance market against every possible future fluctuations in their productivity level, but trade across different islands is not possible.

2.1

Discussion

While autarky and complete markets are two typical benchmarks in welfare analysis, our incomplete markets economy is somewhat less standard. This model is essentially a convex combination of the autarky and complete markets economies. As the fraction of the total 6

variance (v) of labor market risk accounted for by the transitory shock (ε) goes to zero (one), IM converges to AUT (CM). There are several ways of interpreting our IM benchmark. First, one can literally argue that insurance is available (through markets or the government) for certain types of labor market risks, but not for all risks. For example, government-provided unemployment insurance helps smoothing consumption during short non-employment spells, and certain labor contracts partially shield worker’s salaries from productivity fluctuations. However, in many countries there is little market or government insurance available against long periods of non-employment, or against large drops in labor productivity due for example to skill obsolescence. Similarly, one cannot buy insurance in advance against the event of being a “low-ability” worker. Second, one can loosely interpret our economy as an approximation to a BewleyHuggett-Aiyagari economy in which agents only trade a state-noncontingent bond. The existing quantitative literature based on these environments shows that agents can smooth consumption very effectively through borrowing and precautionary saving, as long as shocks are not too persistent.5 In terms of our model, the ε-shock would correspond to this not-toopersistent (therefore insurable) component of labor market risk, whereas the α-component, being a fixed effect, would be uninsurable by definition in this environment. Third, there could be a distinction between the information set of the econometrician and that of the agent. Hence, not all productivity fluctuations map into risk and uncertainty for the agent. Important labor market events like promotions, demotions, and dismissals can be pre-announced or foreseen. The insurable ε-shocks would represent anticipated changes in wages according to this interpretation of the incomplete markets economy.

2.2

Social optimum and competitive equilibrium

Solving for equilibrium allocations in autarky is straightforward. To simplify the characterization of the equilibrium allocations under the complete markets and incomplete markets structure described above, we make heavy use of the Welfare Theorems and the Planner’s problem. 5

See, among others, Huggett (1994), Rios-Rull, Storesletten-Telmer and Yaron (2003). In previous work (Heathcote, Storesletten and Violante, 2004) we document that in an economy with only a riskless bond, a large rise in the variance of iid shocks has virtually no impact on consumption inequality.

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For the complete markets case, we will solve the natural Planner’s problem, and then follow Negishi (1960) to show that the transfer needed to implement the equal-weights social optimum in our economy is zero in expected terms, implying that the competitive equilibrium selected is indeed the one in which all agents have zero initial wealth, as assumed. For the incomplete-markets case, we solve for the problem of a Planner that is free to choose allocations for a given group of households with the same realization of α, but unable to move resources across the various groups. In other words, exploiting the island analogy sketched above, the Planner is unconstrained within the island, but faces a “no trade across island” constraint. A similar argument based on the Negishi approach holds in this case. We are now ready to state: Proposition 1: There exists an “appropriately specified” equal-weights social planner’s problem that implements the competitive equilibrium allocations of our economy, both under complete-markets and under incomplete-markets. Proof: See Appendix. The equivalence established in Proposition 1 enables us, in what follows, to talk interchangeably about equilibrium allocations and planners’ problem allocations. It is useful to reproduce here these allocations and discuss them briefly. In complete markets, µ ¶ −1 v 1+σ CM γ+σ c = ψ exp , 2 σ (γ + σ) ³w ´ it , hCM (wit ) = ψ −1/σ (¯ c)−γ/σ exp σ

(4)

hence there is full-risk sharing and consumption is constant across states and over time. Note that average consumption is increasing in the variance of the shock v, a point we develop in Lemma 1 below. Hours worked are increasing in productivity, in particular there is no distinction between permanent and transitory shocks in the labor supply decision. The Frisch elasticity measures precisely the marginal effect of either productivity shock on hours. With incomplete markets, the allocations are µ ¶ −1 1+σ ³ vε ´ IM σ+γ cα = ψ exp α+ , σ+γ 2σ ¶ µ ¶ µ 1 γ (1 + σ) 1−γ − σ+γ IM α exp − 2 vε exp (ε) . hα (ε) = ψ exp σ+γ 2σ (σ + γ) 8

(5)

Individual consumption is independent of the realization of the transitory shock, since that can be fully insured, but is rescaled by the individual permanent effect α. Average consumption across the group of households with a given fixed effect is increasing in the variance of the insurable shock. Hours worked are increasing in the transitory shock ε, since these shocks have no income effect, only a substitution effect; permanent shocks have an income effect, and whether hours increase or decrease with α depends on whether γ is below or above one. Note that the variance of the transitory shock reduces average hours worked, through its effect on the marginal utility of consumption. To complete the discussion, we report the autarkic allocations −1

1+σ

−1

1−γ σ+γ

cAU T (wit ) = ψ σ+γ witσ+γ ,

(6)

hAU T (wit ) = ψ σ+γ wit . Consumption equal earnings every period. Individual hours worked are chosen optimally and whether they increase or decrease with wit depends on the relative strength of income vs. substitution effect, regulated by the value of γ. Note that, as in the CM economy, the allocations depend only on wit , not on the two shocks separately. We conclude this section by stating a result that follows, almost directly, from Proposition 1 and it will be useful to interpret some of our findings in the welfare analysis of Section 3: Lemma 1: Under complete markets, the indirect utility function of the planner is quasiconvex in the productivities wit . Proof: See Appendix. We are now ready to entertain our qualitative welfare analysis.

3

Qualitative Welfare Analysis

Throughout our welfare analysis, we compare and rank allocations using the utilitarian social welfare function (expressed in terms of average lifetime utility) W (c1 , c2 , ..., cI ; h1 , h2 , ..., hI ) = (1 − β)

∞ X t=0

9

βt

I X i=1

φi u (cit , hit ) ,

(7)

where φi is the “social weight” on individual i, with the normalization

PI i=1

φi = 1. Specif-

ically, we will focus on the equal-weights case φi = 1/I corresponding to equilibrium allocations where all agents start with zero initial wealth, as assumed in Section 2. This equal-weights welfare function has the additional property that allows to measure welfare “under the veil of ignorance”. We propose two distinct ways to assess the welfare costs associated to labor market uncertainty. First, one can ask, for a given insurance market structure, what are the welfare costs of a rise in labor market risk. Second, one can compute, for a given level of risk, what are the gains from completing the markets. We always express our welfare calculations as an “equivalent variation”, in terms of lifetime consumption of the benchmark economy. In the last part of this section, we extend our welfare analysis to economies where households are ex-ante heterogeneous in preferences.

3.1

The Welfare Implications of Rising Labor Market Risk

We begin by fixing the market structure of the economy and measuring the welfare implications from increasing the dispersion of productivities, i.e. the degree of labor income risk, by an amount ∆v, from v ∗ to vˆ = v ∗ + ∆v. In the case of incomplete markets, we will distinguish between a change in the transitory component ∆vε and a change in the permanent component ∆vα . We focus on the equivalent variation, expressed as a fraction χ of consumption (i.e., the percentage increase in consumption in the initial economy with v ∗ required to achieve the same utility as in the economy with vˆ). Formally, it is immediate to see that in a stationary equilibrium of the economy with market structure m, the welfare gain χm solves 1X 1X ³ ˆ´ u ((1 + χm ) c∗i , h∗i ) |v∗ = u cˆi , hi |vˆ, I i=1 I i=1 I

I

m = CM, IM, AU T

(8)

where (c∗it , h∗it ) denote the optimal choices of consumption and hours worked for individual ³ ´ ˆ it denote the optimal pair of choices i at time t, in the economy with risk v ∗ , and cˆit , h for individual i at time t, in the economy with risk vˆ. Strictly speaking, the solution for χm allows one to compare the welfare level across two economies with different level of inequality. One can interpret it as a welfare gain from 10

rising inequality only when households hold no wealth and the transition across the two economies is instantaneous. We return on this point in Section 2.2. With this caveat in mind, we are ready to state: Proposition 2: The (approximated) welfare gain of a change in labor market risk equal to ∆v, where ∆v = ∆vε + ∆vα , is: (i) under complete markets, χCM '

1 ∆v , σ 2

(ii) under autarky, χAU T ' −

(γ − 1) + γ (1 + σ) ∆v , σ+γ 2

(iii) under the incomplete markets economy where transitory shocks (εit ) are insurable and permanent shocks (αi ) are uninsurable, χIM '

1 ∆vε (γ − 1) + γ (1 + σ) ∆vα − . σ 2 σ+γ 2

Proof: See Appendix. The Proof of Proposition 2 shows that it is possible to obtain the exact closed-form solution for χ in all three cases. However, these expressions are cumbersome and not particularly transparent. Through a set of log-approximations of the class ln (1 + x) ' x and ex ' 1+x, one can arrive at the much simpler and useful solutions stated in Proposition 2. Note also that the approximated expressions, as well as the exact ones, do not depend on ψ, but only on the triplet of parameters (γ, σ, ∆v). However, the linearity of χm in ∆v is a feature of the approximation.6 Arguably, the most surprising result in Proposition 2 is that under complete markets, increasing productivity dispersion strictly increases welfare, as long as labor supply elasticity is positive (σ finite). The source of this result comes from the endogeneity of labor supply: an unconstrained planner can achieve better allocative efficiency with larger dispersion, without any loss in terms of consumption smoothing, by commanding longer hours to the high-productivity workers and higher leisure to the low-productivity ones. The proof of 6 In Figure 3, we show that when (γ, σ) lie in the region of the parameter space commonly used in macroeconomics, and for values of ∆v in the range of the empirical estimates, the approximation is remarkably good.

11

Lemma 1 connects this result to the standard result from classical consumer theory that the indirect utility function of a static consumer is quasi-convex in prices, hence a meanpreserving spread of the price distribution raises its welfare. Panel (A) in Figure 1 plots χCM as a function of σ for ∆v normalized to one: note for example, when the disutility of labor effort is quadratic (σ = 1), an increase in wage dispersion translates into a rise in welfare half its size. The complete-markets result sheds some light on the fact that, even in autarky, the change in welfare following a rise in v, is not necessarily negative, see panel (B) in Figure 1. When γ = 0 (the risk-neutrality case), it is easy to see that χAU T = χCM > 0. For low levels of risk-aversion, γ ∈ (0, 1/2), there can still be a welfare gain as long as σ < 1/γ − 2 i.e., in the presence of enough willingness to substitute labor supply intertemporally. The trade-off implicit in this result is between the loss in consumption smoothing and the rise in output allowed by endogenous labor supply: when γ < 1, the substitution effect is larger than the income effect in preferences and agents work harder in good times and enjoy more leisure in bad times. When γ > 1/2, or if σ is sufficiently high, there is always a welfare loss from raising productivity dispersion in autarky. Note that when σ → ∞, the welfare cost of rising productivity fluctuations in autarky is

∆v , 2 the expression computed by Lucas (1987) for the welfare costs of aggregate consumption χAU T ' −γ

fluctuations in an economy with inelastic labor supply. Clearly, in autarky when labor supply is rigid, wage fluctuations translate one-to-one into consumption fluctuations. Realistic insurance market structures lie, arguably, strictly between complete markets and autarky. Hence, one can think of χCM and χAU T as, respectively, an upper bound and a lower bound on the welfare consequences of a rise in wage inequality. Our intermediate incomplete-markets case has precisely the purpose of reproducing an economy closer to actual ones. In the IM economy, the welfare gain can be re-expressed as χIM = χCM

∆vε ∆vα + χAU T , ∆v ∆v

i.e. exactly as a weighted average between the welfare loss in autarky and the welfare gain in complete markets, with weights equal to the share of insurable and uninsurable 12

shocks in the economy. In Section 4, we attempt an estimation of these two shares from cross-sectional U.S. data.

3.2

The Welfare Gains from Completing the Markets

Exploiting the previous welfare analysis, we show that it is possible to derive some simple analytical expressions for the welfare costs of market incompleteness (or, the welfare gains from completing the markets) when labor supply is endogenous. Our objective is to compare the same measure of household welfare defined in (7) across economies with different market structures, given the same stochastic process for idiosyncratic labor market risk. To compute the welfare gain associated with completing the insurance markets, for a given level of risk v = vα +vε , we focus on the “equivalent variation”, expressed as a fraction ξ of consumption, i.e. the percentage increase in consumption in the incomplete-markets (or autarkic) economy required to achieve the same welfare as in the economy with complete insurance markets. In particular, define the welfare gain as the value of ξ m that solves I

I

1X 1X u ((1 + ξ m ) ci , hi ) |m(v) = u (ci , hi ) |CM (v) , I i=1 I i=1

m = AU T, IM.

(9)

where we denoted, respectively, by IM and by CM the incomplete-markets and completemarkets economy. We are now ready to state Proposition 3: The (approximated) welfare gain from completing the markets in an economy with labor market risk equal to v = vα + vε is: (i) under autarky,

¸ 1 (γ − 1) + γ (1 + σ) v + , ' σ σ+γ 2 ·

ξ

AU T

(ii) under the incomplete markets economy where transitory shocks (εit ) are insurable and permanent shocks (αi ) are uninsurable, · ¸ 1 (γ − 1) + γ (1 + σ) vα IM + ξ ' σ σ+γ 2 Proof: See Appendix. The first part of the Proposition contains an intuitive expression for the welfare gains of moving from autarky to complete markets. The second part of the Proposition simply 13

states that the welfare gain of completing the markets in an economy where only some shocks are perfectly insurable is equal to that in autarky, once we rescale appropriately the variance of the uninsurable shocks. The parametric expression in ξ AU T multiplying the variance has two separate components. The first term (1/σ) is associated to the allocative efficiency gain of full risk-sharing in presence of elastic labor supply: more productive households work relatively harder and less productive households enjoy more leisure. This efficient “specialization” improves with the value of the Frisch elasticity, allowing to achieve higher aggregate welfare. The second term is a measure of the insurance value of moving towards complete markets and being able to smooth consumption across states and over time more effectively. Figure 2 plots ξ AU T for different values of σ and γ in their admissible range (0, ∞) and for v normalized to 1. Notice first that the welfare gain of completing the markets under autarky is always weakly positive and strictly increasing in γ, the degree of riskaversion. A few benchmarks are of interest. First, for γ = 0 (risk-neutrality), the welfare gain is exactly zero, since consumption fluctuations are not costly for individuals. Second, in absence of flexible labor supply, σ → ∞, the welfare gain is ξ AU T ' γ v2 , which is exactly Lucas (1978) formula for the welfare costs of business cycles. This is intuitive, since in both cases the calculation quantifies the gain from eliminating uninsurable consumption fluctuations: in autarky, without flexible labor supply, labor productivity fluctuations with variance v translate one to one into consumption fluctuations. Third, when γ = 1 and σ = 1, ξ AU T ' v, i.e. the welfare gain of completing the markets equals exactly the variance of log-productivity. Surprisingly, in general the welfare gain from completing markets is not monotone in the Frisch labor supply elasticity 1/σ. For large values of the labor supply elasticity (σ < 1), ξ AU T is decreasing in σ. For σ ≥ 1, whether or not the welfare gain ξ AU T is decreasing in σ (and increasing in the Frisch elasticity 1/σ) depends on whether γ ≤ 2σ/ (σ − 1). Hence, for γ ≤ 2, ξ AU T is always declining in σ.7 For large values of risk-aversion γ (above 2), there is always a region where agents are 7

The condition under which the welfare costs of completing the markets is increasing in the Frisch 1/σ elasticity has an intuitive interpretation. From the proof of Lemma 1, recall that hCM = ψ −1/σ c−γ/σ wit i −1

1−γ

T and hAU = ψ σ+γ wiσ+γ , thus when σ ≥ 1 the variance of log-hours worked is larger in complete markets i relative to autarky if and only if γ ≤ 2σ/ (σ − 1), the same condition we derived above.

14

relatively unwilling to adjust hours inter-temporally for which ξ AU T becomes smaller as the labor supply elasticity rises. One should always expect to be better off when more flexible in choosing hours. The question is whether this additional flexibility is more useful in a complete markets environment or in autarky. It is intuitive that for high degree of aversion to consumption fluctuations, an increase in the willingness to substitute hours intertemporally might improve more the autarkic allocation relative to the complete-markets allocation. In particular, when γ > 1, in autarky the income effect dominates the substitution effect and low-productivity households work harder, thus endogenous labor supply provides a sort of insurance.

3.3

Welfare Analysis with Preference Heterogeneity

There is a long tradition in labor economics in modelling unobserved heterogeneity. Arguably, one of the main sources of unobserved heterogeneity is preference for leisure, i.e. households differing in their weight on leisure relative to consumption in their intra-period utility function. This will be an important issue in our positive analysis, since one should account for the fact that the observed joint distribution of hours, earnings, and consumption might not be only the result of different realizations of labor productivity shock hitting ex-ante equal agents, as unobserved preference heterogeneity would induce a nontrivial dispersion even in the absence of any shock. Hence, it is important to study how our welfare analysis is affected by the presence of preference heterogeneity. We generalize individuals’ preferences described in (1) to

1+σ c1−γ −1 i ϕi hi − ψe , (10) u (ci , hi ) = 1−γ 1+σ where ϕi is an individual-specific leisure weight. Consistently with our previous analysis,

we assume that

´ ³ v ϕ ϕi ∼ N − , v ϕ , 2 8 and that (ϕ, αi , εit ) are jointly independent. 8

Alternatively, one might argue that ϕ and α should be negatively correlated. Indeed, if human capital accumulation were endogenous and some of the cost were in terms of the consumption-good, agents with low ϕ would be more prone to accumulate since they would be willing to work more hours and hence face a higher return on their investment. It would be easy to examine the implications of a model with, for example, perfectly negative correlation between α and ϕ.

15

It is straightforward to extend the characterization of the equilibrium allocations for complete-markets and incomplete-markets to this case. We think of complete markets as a situation where trade occurs before the realization of both the permanent productivity shock α and the permanent taste shock ϕ. The economy with incomplete markets is an economy where no trade is allowed before these two events, whereas there is a full set of insurance markets to insure the transitory shock εit . Generalizing the proof of Proposition 1 to the case of preference heterogeneity is simple, thus we omit the proof. It suffices to say that the socially efficient allocations that solve the equal-weights planner problem correspond to our competitive equilibrium in this case as well. It is interesting to analyze how the allocations with preference heterogeneity compare with the allocations in (4), (5), and (6) . In complete markets, µ ¶ −1 1 + σ v + vϕ CM γ+σ c = ψ exp , σ (γ + σ) 2 µ ¶ wit − ϕi −γ/σ −1/σ CM h (wit , ϕi ) = ψ (¯ c) exp . σ

(11)

Hence, interestingly the preference shock acts exactly as a (negative) productivity shock. Under incomplete markets and autarky it is easy to show that for households with preference shock ϕi , allocations are exactly as in (5), and (6) multiplied by the factor µ ¶ ϕi exp − . σ+γ Thus, intuitively both consumption and hours worked are lowered by ϕi . We are now ready to state Proposition 4: In the presence of preference heterogeneity: (i) the welfare gain χm vϕ of a change in labor market risk is independent of the degree of heterogeneity in the relative IM m taste for leisure vϕ , i.e. χm vϕ = χ ; (ii) the (approximated) welfare gain ξ vϕ of completing

the markets starting from an incomplete-markets economy with labor market risk v and variance of taste shock vϕ is equal to ¸ · γ vϕ 1 (γ − 1) + γ (1 + σ) vα IM + + ξ vϕ ' σ σ+γ 2 σ (σ + γ) 2 γ vϕ = ξ IM + σ (σ + γ) 2 16

Proof: See Appendix. The first result of this Proposition is that the expressions for the welfare gain of a rise in wage dispersion χm derived in Proposition 2 are robust to the introduction of preference heterogeneity. However, the welfare gain of completing the markets depend on the variance of the taste shock: the larger the variance, the higher the gain. The intuition for this result comes from the complete markets allocation for hours worked in (11) . Under full risk-sharing, the planner can fully exploit the differences in taste to achieve higher aggregate output.

4

Positive Analysis

The analysis in the previous sections has highlighted that the welfare implications of labor market risk depend crucially on a set of key parameters: the preference parameters (ψ, γ, σ); the parameters (vα , vε ) of the wage process determining what fraction of labor market risk is insurable; and, possibly, vϕ in the presence of taste shocks. Only with reliable estimates of these parameter we can translate our qualitative welfare study into a quantitative analysis. The objective of this section is to exploit the positive implications of our model economy to estimate the parameters of interest. The procedure we follow can be synthesized in four simple steps. First, we exploit cross-sectional U.S. data to document certain cross-sectional moments of the joint distribution of hours worked, consumption and earnings. Second, through the equilibrium allocations of the incomplete-markets economy characterized in (5) we derive the corresponding “theoretical” cross-sectional moments as a function of the parameters. Note that using the moments implied by the IM economy is not restrictive, since as vα = 0, this economy converges to the complete-markets economy, and as vε = 0, the incomplete-markets economy converges to autarky. Recall that (vα , vε ) are free parameters that will be estimated. Third, we show that a subset of the moments available in the data is needed to identify our structural parameters. Fourth, we build an estimator of the “minimum distance” class, and obtain estimates of our parameter values.

17

4.1

Data Analysis

Table 1 below reproduces the cross-sectional moments we will use for the empirical analysis. Since the allocations we computed for our economies are derived under the steady-state assumption, we focus only on two data points: 1970 and 1995. As documented extensively in the empirical literature (see Katz and Autor, 1999 for a survey), the bulk of the rise in inequality in the United States over the postwar period occurred in the 1980s. Hence, 1970 and 1995 seem far enough to justify, approximately, our stationarity assumption. In Heathcote, Storesletten and Violante (2004), we used PSID data to document certain features of the joint distribution of annual hours worked, hourly wages and annual earnings for a sample of U.S. white male workers.9 The first five columns in Table 1 reproduce our findings.

Table 1: Empirical Cross-sectional Moments

1970

mean (h/time)

var log(w)

var log(h)

corr(log w,log h)

var log(wh)

var log(c)

corr(log c,log h)

0.40

0.27

0.07

-0.05

0.31

0.18

0.25

1995 0.39 0.39 0.07 0.08 0.49 0.21 - 0.25 Columns (1)-(5), PSID; Heathcote, Storesletten, and Violante (2004) Columns (6)-(7), CEX.; Krueger and Perri (2003); Attanasio, Battistin, and Ichimura (2004)

0.18

First, notice the large rise in hourly wage dispersion (column 2), and the even larger surge in annual earnings dispersion (column 5). Interestingly, the variability of hours worked is stable over the period (column 3), so the differential increase in the variance of earnings must be accounted for by an increase in the covariance between wages and hours. Table 1 shows indeed that the cross-sectional correlation between hours and wages (column 4) grew substantially in the 25 years between the two data points. There is an ongoing debate about the measurement of the increase in consumption inequality in the United States from CEX data. Krueger and Perri (2003) find only a minor increase in the variance of log consumption over the period, roughly 0.3 points, compared to a increase in the variance of log earnings six times as large (column 6, first number for 1995). Recently Attanasio, Battistin and Ichimura (2004) challenged this result 9

See Heathcote, Storesletten and Violante (2004) for the details on sample selection.

18

and concluded that the variance of log consumption expanded by 0.7 points (column 6, second number for 1995), more than double the Krueger-Perri estimate. Given that there is no consensus yet as of what is the better estimate, we will study the sensitivity of our estimates with respect to this uncertain dimension of the data. In any case, the conclusion is that the sharp rise in earnings inequality translated only partially into consumption inequality. We report, from Krueger and Perri, another interesting fact that will turn out to be a key piece of information for our empirical analysis: the correlation between consumption and hours worked is positive and declined significantly from 1970 to 1995 (column 7). Finally, we add to these moments the average hours worked in the population (column 1), necessary to identify the relative weight on leisure ψ.

4.2

Theoretical Moments and Identification

Under our log-Normality assumption for the shocks, it is possible to derive closed-form solutions for all the theoretical moments of Table 1. In a steady state with variances of the two wage components equal to (vα , vε ) , the cross-sectional variance of hours worked are, respectively

¶2 1−γ 1 vα + 2 vε . (12) var (log hi ) = σ+γ σ The variance of log hours is increasing in both the permanent and the transitory component. µ

The latter is mediated by the Frisch elasticity. The extent to which the permanent shock translates into larger variance of labor supply depends on how far is γ from one, the value for which income and substitution effects cancel out exactly. The variance of earnings can be written as µ ¶2 µ ¶2 1−γ 1 var (log wi hi ) = 1 + vα + 1 + vε , (13) σ+γ σ which shows an important point for our empirical analysis. The variance of earnings need not necessarily increase more than the variance of wages, following a rise in the variance of wages. When γ > 1, if the rise is mostly permanent, earnings dispersion can increase only modestly. This result is due to the covariance component. The covariance between hours and wages is given by cov (log wi , log hi ) = 19

1−γ 1 vα + vε , σ+γ σ

(14)

thus transitory shocks make hours and wages covary positively, whereas the effect of permanent shocks depend on the relative strength of income and substitution effect, mediated by γ. The corresponding correlation is corr (log wi , log hi ) =

(1 − γ) vα + σ+γ vε σ q √ vα + vε vϕ + (1 − γ)2 vα +

(σ+γ)2 vε σ2

.

(15)

From this expression, it is immediate to see that for vα = 0 (all risk is insurable, as in CM), the correlation is exactly one, whereas for vε = 0 (all risk is uninsurable, as in autarky), the correlation is exactly (minus) one if γ is below (above) one, and it is zero if γ = 1. The variance of consumption µ var (log ci ) =

1+σ σ+γ

¶2 vα ,

(16)

is instead only a function of the dispersion in the permanent component, as the transitory component is insurable by assumption. Finally, the correlation between consumption and hours worked is given by (1 − γ)

corr (log ci , log hi ) = q

2

(1 − γ) +

(σ+γ)2 vε σ 2 vα

(17)

Note that the sign of this expression depends on the value of γ, exactly as the correlation between wages and hours. In our baseline empirical exercise our aim is to match the moments of Table 1 with our simple incomplete-market model where we keep fixed the preference parameters (ψ, γ, σ) over time, and we only allow the variances of the uninsurable and insurable wage components to change across the two years, in order to explain the facts. Denote the four variances in the initial and final steady state as (vαt , vεt )t=0,1 where t = 0, 1 denotes the two steady-states before and after the rise in U.S. inequality. The next Proposition characterizes the minimum set of moments needed for exact identification of our 7 parameters: Proposition 5: Suppose the following vector of data is available: © ª var (log wi )t , vart (log wi hi ) , vart (log ci ) t=0,1 , 20

then the parameters (γ, σ, vα0 , vε0 , vα1 , vε1 ) are exactly identified. Adding E (log wi )0 allows to identify ϕ. Proof: See Appendix. The result in Proposition 5 depends crucially on the availability of the variance of consumption: intuitively, it is a key moment since is depends on permanent shocks only, whereas all the other moments involve a combination of both components. The variance of earnings can instead be replaced by the joint presence of the variance of hours and the covariance of hours and wages. Note that in general, given the availability of all the data points in Table 1, the model is over-identified, albeit not exactly by the difference between the number of moments and the parameters since the variance of wages, the variance of hours, the covariance between wages and hours and the variance of earnings are linearly dependent. Finally, notice that in our Proposition 5, we have not used any covariance moment. The reason is that, in the course of our estimation below, we will sometimes abstract from either one of these moments. To explain why, note that there is a tension between the wage-hours correlation and the consumption-hours correlation. In the data, the first is not too far from zero, while the second is positive. Equation (17) shows that this latter observation requires γ < 1. However, when γ < 1, both types of shocks make wages and hours covary positively, hence we should expect this correlation to be close to one, in contrast with the data. Including both correlations among our empirical moments lead to a ill-posed estimation problem, as the theoretical model is unable to generate at the same time a small and negative wage-hours correlation and a positive consumption-hours correlation. We illustrate this point more in detail below and we argue that preference heterogeneity can solve the trade-off.

4.3

Estimation

We implement a Minimum Distance estimator that solves the following minimization problem min [s − f (Θ)]0 Ω [s − f (Θ)] , Θ

21

(18)

where s, and f (Θ) are the (M ×1) vectors of the stacked empirical and theoretical moments, Θ is the parameter vector, and Ω is a (M × M ) weighting matrix. Our parameter vector is always defined as Θ = {ϕ, γ, σ, vα0 , vε0 , vα1 , vε1 } , whereas the set of moments we include in s, hence the dimensionality M of the problem, will vary across experiments. Clearly, M ≥ 7 always. To implement the estimator, we need a choice for Ω. The bulk of the literature follows Altonji and Segal (1996) who found that in common applications there is a substantial small sample bias in the estimates of Θ, hence using the identity matrix for Ω is a strategy superior to the use of the optimal weighting matrix characterized by Chamberlain (1984). With this choice, the solution of (18) reduces to a nonlinear least square problem. Stanb is consistent, asymptotically Normal, dard asymptotic theory implies that the estimator Θ and has asymptotic covariance matrix V = (D0 D)−1 D0 ∆D (D0 D)−1 , where the matrix £ ¤ D ≡ E [∂f (Θ) /∂Θ0 ] and the matrix ∆ ≡ E (s − f (Θ)) (s − f (Θ))0 , estimated with their empirical analogs to compute standard errors. In light of the discussion above, we start by excluding the consumption-hours correlation from the estimation.10 Hence, we set © ª s = E (log wi ) , var (log wi )t , vart (log wi hi ) , cov t (log wi , log hi ) , vart (log ci ) t=0,1 . Table 2 reports the parameter estimates, and the fit of the model across all moments in Table 1. The two sets of numbers refer to two different experiments, one with the Krueger-Perri (thereafter, KP) consumption inequality numbers (top panel), and one with the AttanasioBattistin-Ichimura (thereafter, ABI) numbers (bottom panel). The estimated value for γ is above one and the one for σ is just below one. The estimated intertemporal elasticity of substitution is higher for the ABI data than for the KP data, as one should expect, since to allow a larger rise in consumption inequality households need a higher tolerance of consumption fluctuations. Both datasets agree on the fact that the uninsurable shock is the dominant component – as over 85% of the initial variance is accounted for by permanent wage differences. However, between 1970 and 1995, the transitory component is the one that increased more, propor10

Since we include the variance of wages, earnings, and the correlation between wages and hours, we exclude the variance of hours to avoid linear dependence across these moments.

22

tionately. This explains the fact that notwithstanding the surge in earnings variability, consumption inequality rose only mildly. Table 2: IM Economy – Estimation w/o corr(c,h) mean(h)

var log(h) ψ = 0.36,

corr(w,h)

γ = 1.41,

σ = 0.99,

var log(wh) v0 α=

0.23,

var log(c) v1 α=

corr(c,h)

0.32

1970 Model

0.400 0.410

0.070 0.043

-0.050 -0.048

0.310 0.303

0.180 0.161

0.250 -0.404

1995 Model

0.390 0.380

0.070 0.079

0.080 0.078

0.490 0.497

0.210 0.220

0.180 -0.347

ψ = 0.40,

γ = 1.31,

σ = 0.85,

v0 α = 0.24,

v1 α = 0.33

1970 Model

0.400 0.410

0.070 0.041

-0.050 -0.046

0.310 0.301

0.180 0.179

0.250 -0.352

1995 Model

0.390 0.380

0.0705 0.080

0.080 0.076

0.490 0.497

0.250 0.247

0.180 -0.256

Overall the fit of the model is satisfactory, except for two dimensions. First, it predicts a twofold increase in hours volatility, whereas the data show no trend. Second, as expected the sign of the consumption-hours correlation is the opposite as in the data. Next, we perform our estimation, excluding the wage-hours correlation and including instead the consumption-hours correlation. Hence, we set © ª s = E (log wi ) , var (log wi )t , vart (log wi hi ) , cov t (log ci , log hi ) , vart (log ci ) t=0,1 . Table 3 reports the results in the same format as Table 2. As expected, the model now fits very well the consumption-hours correlation with an estimate of γ below one. Given the low risk-aversion, both a lower Frisch elasticity (higher σ) and a lower fraction of permanent variance are needed to match the data on hours and consumption variation. However, the other side of the coin is a failure to match the wage-hours correlation: as explained above, this correlation is now very close to one in the model. As a result, the model generates a much larger earnings variation than the data, almost twice as big. Interestingly, changing the moments in the objective function does not help in producing a smaller increase in hours variation.

23

Table 3: IM Economy – Estimation w/o corr(w,h) mean(h)

var log(h) ψ = 0.31,

corr(w,h)

γ = 0.67,

var log(wh)

σ = 1.74,

v0 α=

0.14,

var log(c) v1 α=

corr(c,h)

0.16

1970 Model

0.400 0.417

0.070 0.045

-0.050 0.843

0.310 0.501

0.180 0.182

0.250 0.240

1995 Model

0.390 0.373

0.070 0.079

0.080 0.877

0.490 0.776

0.210 0.207

0.180 0.193

ψ = 0.33,

γ = 0.70,

σ = 1.63,

v0 α = 0.15,

v1 α = 0.19

1970 Model

0.400 0.415

0.070 0.047

-0.050 0.819

0.310 0.502

0.180 0.192

0.250 0.232

1995 Model

0.390 0.375

0.0705 0.078

0.080 0.843

0.490 0.764

0.250 0.241

0.180 0.200

The conclusion from these two exercises is that, in its simple form, the IM model provides a fair account of most of the data, with parameter estimates within a reasonable range, but misses always on a couple of key dimensions.

4.4

Introducing Preference Heterogeneity

In this section, we argue that adding preference heterogeneity can significantly improve the results. With heterogeneity in the taste for leisure, as defined in Section 3.3, the theoretical moments change in a very intuitive way. The variances of hours, earnings and consumption, and the covariance between consumption and hours, are all augmented by the term vϕ , (σ + γ)2 whereas the covariance between wages and hours remains unchanged. It is immediately obvious then that this property of the new theoretical cross-sectional moments allows an improvement upon the baseline estimation: a positive amount of preference heterogeneity can raise the covariance between consumption and hours, as households with a low realization of ϕ work harder, earn more and consume more. In particular this covariance can become positive, even when γ > 1, which allows the model to match satisfactorily also the other moments. Table 4 reports the results of the estimation with preference heterogeneity. The fit of the model is remarkable along all dimensions. Even the predicted rise in hours variability is now minor: the reason is that preference heterogeneity creates some variation in hours worked without the need for a large Frisch elasticity. The estimated value for σ is 24

indeed now over three times as large as in the previous exercises, implying a value for the Frisch elasticity around 0.3. Instead, γ is only slightly larger than the estimate of Table 2. The estimates also suggest that the fraction of permanent -uninsurable variance is roughly 60% of the total, but it accounts for only one third of the increase. Thus, from the data we conclude that roughly 2/3 of the rise in labor market risk from 1970 to 1995 in the United States was insurable in nature. Finally, note that the model performs slightly better on the KP data than on the ABI data for consumption inequality. Table 4: IM Economy with Preference Heterogeneity mean(h)

var log(h)

ψ = 0.25,

γ = 1.87,

corr(w,h)

var log(wh) v0 α=

σ = 3.35,

0.19,

v1 α=

var log(c)

0.22,

corr(c,h)

v φ = 1.45

1970 Model

0.400 0.403

0.070 0.066

-0.050 -0.048

0.310 0.323

0.180 0.183

0.250 0.248

1995 Model

0.390 0.387

0.070 0.074

0.080 0.078

0.490 0.491

0.210 0.208

0.180 0.181

ψ = 0.27,

γ = 1.67,

v0 α = 0.20,

σ = 3.00,

v1 α = 0.25

v φ = 1.16

1970 Model

0.400 0.403

0.070 0.065

-0.050 -0.046

0.310 0.323

0.180 0.202

0.250 0.251

1995 Model

0.390 0.387

0.070 0.074

0.080 0.076

0.490 0.490

0.250 0.234

0.180 0.178

Our model allows a simple decomposition to calculate what fraction of the various moments is explained by permanent wage shocks, transitory wage shocks and unobserved heterogeneity. Table 5 shows this decomposition for 1995. Preference heterogeneity explains over 70% of the variance of hours, but only a quarter of the variance of consumption, and a just over a tenth of the variance of earnings. Table 5: Variance Decomposition (1995) var(w)

var (h)

var (wh)

var (c)

Preference heterogeneity

0.00

0.72

0.11

0.26

Permanent component

0.56

0.08

0.31

0.74

Transitory component

0.44

0.20

0.58

0.00

25

5

Quantitative Welfare Analysis

The estimated parameters allows to quantify the various welfare measures we characterized in the paper. For the parametrization, we choose the estimates of the model with preference heterogeneity obtained from the Krueger-Perri data (Table 4, first panel). Thus, γ = 1.87, σ = 3.35, vϕ = 1.45. Moreover, for the welfare analysis of a rise in wage inequality in autarky and complete-markets, we use v = 0.12 (Table 1, first column) and for the incomplete-markets economy, we use vα = 0.03, as implied again by Table 4. Finally, for the welfare analysis of completing the markets we use the 1995 values, hence v = 0.39 and vα = 0.22. We summarize our results in Table 6 below. Table 6: Welfare Results (% of lifetime cons.) Welfare gain of rise in inequality

Welfare gain of going to CM

χCM

χAU T

χIM

T ξAU ϕ

ξIM ϕ

+1.81%

-9.76%

-2.55%

+65.47%

+39.97%

In the complete-markets economy there is a sizeable increase in welfare when wage inequality increases. The number is not big since the estimated Frisch elasticity is only around 0.3. The welfare loss in autarky is instead very large, close to 10%. These two number represent an upper and a lower bound. Any economy with “incomplete” market structure subject to a rise in inequality of the same size will have a welfare cost (or gain) within this range. For our IM structure, we estimate a welfare loss of roughly 2.5% of lifetime consumption. Interestingly, this number is quite close to the ones compute by Heathcote, Storesletten and Violante (2004), and Krueger and Perri (2004), using very different approaches. Let us now turn to the welfare gains from completing the markets. A household living in autarky would be willing to give up over 65% of his lifetime consumption to be able to face a complete set of insurance market against her labor market risk. Starting from an incomplete-market economy, this number drops to roughly 40% of lifetime consumption. This is the value of being able to smooth completely the permanent wage component. Note that this welfare gain was 36% in 1970, hence there was an increase of 4% over the past 25 26

years, due to the increase in inequality. Finally, we conclude by noting that these numbers are very sizeable, in general over 1, 000 times larger than commonly estimated welfare costs of business cycles.

6

Concluding Remarks

To be written

7

References

To be written

27

8

Appendix

Proof of Proposition 1 Incomplete markets– It is useful to start from the incomplete markets case and then generalize our argument to the complete-markets economy. Consider a typical “componentplanner” who chooses efficiently consumption and hours worked for all agents with the same value of the fixed-effect α. The planner’s weights on each individual within the group are identical and equal to 1/Iα . This component-planner problem for a group of households with fixed-effect α (consisting of a large number of individuals Iα ) can be expressed as P t 1 PIα max{cit, hit }i,t ∞ t=1 β Ia i=1 u (cit , hit ) s.t. (19) P∞ t PIα (α + εit ) hit − cit ] = 0, t=1 β ¡ i=1 [exp ¢ t = 1, 2, ... εit ∼ N − v2 , v , Using standard Lagrangian techniques, we can define the Lagrangian for the problem (19) as L (c1 , c2 , ..., cI ; h1 , h2 , ..., hI ; λα , α) =

∞ X t=1

¶ ¸ Iα ·µ 1−γ 1 X cit h1+σ it β −ψ + λα (exp (α + εit ) hit − cit ) . Ia i=1 1−γ 1+σ t

It is easy to show that the first-order conditions for problem (19) imply cα = (λα Iα )−1/γ , hα (εit ) = ψ −1/σ (c∗α )−γ/σ exp

µ

α + εit σ

¶

(20)

where λα is the Lagrange multiplier on the time-zero resource constraint for group α. As explained above, stationarity of the wage process implies that the static resource constraint PIα i=1 [exp (α + εit ) hit − cit ] = 0 will hold at every t. Using this constraint, it is immediate to show that under Normality of εit , the term (λα Iα ) is given by µ ³ ´ 1+σ¶ 1 v ε λα Iα = ψ 1+σ/γ exp −γ α + , 2σ σ + γ which used into equations (20) yields efficient allocations only as a function of the primitive parameters,

µ

¶ 1+σ ³ vε ´ cα = ψ exp α+ , σ+γ 2σ ¶ µ ¶ µ ³ε´ 1 γ (1 + σ) 1−γ − σ+γ α exp − 2 vε exp . hα (ε) = ψ exp σ+γ 2σ (σ + γ) σ −1 σ+γ

28

(21)

The next step is to determine the state-contigent transfer function Tα (ε) that decentralizes this efficient allocation within island α. When faced with a labor productivity endowment equal to wit = exp (α + εit ), the individual agent will optimally choose hours worked based on its first-order condition −γ/σ

hit = ψ −1/σ exp (α + εit ) cit

.

(22)

Comparing (22) with (20) makes clear that in order to implement the efficient allocations it suffices ensuring full risk sharing in consumption, i.e., to set individual consumption equal to cα in (20) for every time t and across all agents within group α. Full risk sharing in consumption can be achieved by ensuring that every period the income effect of the individual-specific transitory wage shock εit is zero. In particular, the net transfer Tα (ε) for an individual with εit = ε must equal cα minus its period t “socially efficient” earnings wit hα (εit ) or ¡ ¢ # exp σε ¡ ¢ cα , Tα (ε) = cα − exp (α + ε) hα (ε) = 1 − E exp σε "

(23)

where the second equality involves some simple algebra. The last step of the Proof, which follows by simple inspection of (23), is that the (ex-ante) expected transfer is zero. Finally, note that in the sequential equilibrium version of this allocation, all agents on all islands end up with zero wealth at the end of each period, i.e. the portfolio of one-period Arrow securities (contingent on ε) that they buy at the beginning of each period has zero cost. Complete markets– The planner with access to a storage technology with exogenous and constant return R = β −1 per period faces the constrained maximization problem P t PI max{cit, hit }i,t ∞ t=0 β i=1 φi u (cit , hit ) s.t. P∞ t PI (24) t=1 β i=1 (wit hit − cit ) = 0, ¡ ¢ log wit ∼ N − v2 , v ,

t = 1, 2, ...

The program above can be solved with standard Lagrangian techniques. Define the La-

29

grangian as, L (c1 , c2 , ..., cI ; h1 , h2 , ..., hI ; λ) =

∞ X t=1

µ 1−γ ¶ ¸ I · X cit h1+σ it β φi −ψ + λ (wit hit − cit ) . 1−γ 1+σ i=1 (25) t

The first-order conditions of (25) with respect to cit and hit can be rearranged into ci = (φi /λ)1/γ , µ ¶1/σ λ 1/σ hit = wit . φi ψ

(26)

Imposing the “equal-weights” restriction φi = 1/I, corresponding to the decentralized economy with equal (zero) endowments across households, it is immediate to argue that the planner will never borrow/save and the time-zero resource constraint will also hold with equality period by period, at each t = 1, 2, .... The reason is that the socially optimal consumption choice is equal for all agents and constant over time; since the distribution of productivity shocks is time-invariant, by saving/borrowing the planner would make aggregate (and individual) consumption fluctuate, deviating from the optimal solution.11 Using the two optimality conditions (26) with φi = 1/I into the static resource constraint of the planner, simple algebra yields that à 1

λI = ψ 1+σ/γ

I 1X

I

−1 ! 1/γ+1/σ

1+1/σ

wit

.

i=1

Hence, under the log-Normality assumption for wit , the optimal consumption level c¯ the planner delivers to every household, and the efficient allocation of hours for a household with wit = exp (α + εit ) can be expressed as µ

¶ v 1+σ c¯ = ψ exp , 2 σ (γ + σ) µ ¶ α + εit −γ/σ ¯h (α, εit ) = ψ −1/σ (¯ . c) exp σ −1 γ+σ

11

(27)

Equivalently, we could have started by stating the equal-weights planner’s problem subject to a sequence of static resource constraints I X (wit hit − cit ) = 0, i=1

for t = 1, 2, .... With this formulation, the assumption R = β −1 would not be needed.

30

Note immediately that a mean preserving spread in the distribution of wit , corresponding to an increase in v ∗ will increase average consumption.12 Following the same logic applied for the IM case, in order to implement the efficient allocation (27) in the decentralized economy it is sufficient to insure full risk-sharing in consumption across all individuals. Full risk sharing can be achieved by ensuring that the income effect of the wage shock on labor supply is zero. In particular, the net transfer T (α, ε) at time t for an individual with fixed effect α and transitory shock εit = ε must equal c¯ minus its “socially efficient” earnings at time t given ¯ (α, εit ), or by wit h ¯ (α, ε) T (α, ε) = c¯ − exp (α + ε) h " # ¡¡ ¢ ¢ exp 1 + σ1 (α + ε) © ¡¡ ¢ ¢ª , = c¯ 1 − Eα,ε exp 1 + σ1 (α + ε)

(28)

where Eα,ε denotes the expectation with respect to both the permanent and the transitory component. What remains to be shown is that the total ex-ante transfer is zero. The ex-ante transfer can be written as Z Z ∞ X t T¯ = β Pr (αi = α) Pr (εit = ε) T (α, ε) dεdα = t=1 ∞ X

= c¯

" βt 1 −

t=1

R

# ¡¡ ¢ ¢ R Pr (αi = α) Pr (εit = ε) exp 1 + σ1 (α + ε) dεdα ¢ ¢ª © ¡¡ Eα,ε exp 1 + σ1 (α + εit )

(29)

= 0 As shown by Negishi (1960), the competitive equilibrium selected by solving this equal weights planner’s problem corresponds to the Arrow-Debreu economy with zero initial wealth endowment for all agents. It is interesting to formulate the sequential trade version of this Arrow-Debreu allocation to understand what is the wealth level carried around by the agents. Assume the following 12

In deriving the expression for average consumption we have used the fact that, under log-Normality of wit , for large I, I

1X x w I i=1 it

x x = E [wit ] = E [exp {log wit }] = E [exp {x log wit }]

= exp(−

vx vx x2 v + ) = exp( (−1 + x)). 2 2 2

We will often make use of this property below.

31

trade sequence: under the veil of ignorance (before the realization of the permanent component) agents buy insurance against the realization of α. Subsequently, agents start buying one-period Arrow securities against the realization of the transitory shock ε next period. We can therefore decompose the transfer T¯ in (28) into an initial “lump-sum” transfer £ ¤ ¯ (α, ε) c¯ − Eε exp (α + ε) h T0 (α) = 1−β necessary to insure the realization of the fixed effect thereafter, and a sequence of transfers £ ¤ ¯ (α, ε) − exp (α + ε) h ¯ (α, ε) T (α, ε) = Eε exp (α + ε) h contingent on the realization of the transitory shock at each period t. It is easy to see that the second transfer has zero value in expected terms, hence each household of type α will have constant wealth T0 (α) in every period. Note that individuals with high fixed effect will carry a debt and individuals with low fixed effect will have positive assets in the sequential-trade equilibrium. This asset distribution can be interpreted, for example, as the result of an initial educational choice: high-education workers must repay the debt associated to their higher productivity.¥

Proof of Lemma 1 When the distribution of wit is time-invariant, the equal-weights Planner problem can be written as a static constrained maximization program P max{ci, li }i I1 Ii=1 u (ci , τ − li ) s.t. P τ I = Ii=1 (wi li + ci ) , where lit denotes leisure. In restating this problem we have used the time-endowment constraint τ = hit + lit , and the fact that average productivity is constant and normalized to one. Consider now the equivalent dual problem I

1X min wi li + c¯ {c,li } I i=1 s.t. I 1X ¯ U = u (c, τ − li ) , I i=1 32

where we have constrained the choice of consumption to be constant across all households i and equal to c¯. This constraint is clearly not binding at the complete-markets solution. One can interpret this problem as a simple static problem of a consumer who must choose among (I + 1) goods, where the first I goods are leisure in state i with shadow price equal to the productivity in state i, and the last good is consumption, with price normalized to one. Preferences are strictly continuous and are increasing in consumption and the I leisure goods by assumption (as long as γ and σ are finite), hence we can apply standard results of classical consumer theory to conclude that the expenditure function is concave in the relative prices wit (see for example, Mas Colell 1995, page 59). Equivalently, the indirect utility function is quasi-convex in wages wit .¥

Proof of Proposition 2 Complete markets– The welfare level associated to the complete markets equilibrium with time-invariant log-wage variance v = v ∗ can be therefore expressed as W

CM

= (1 − β)

∞ X

I X

φi u (c∗it , h∗it )

(30)

i=1

! (c∗ )1−γ (h∗it )1+σ = −ψ I 1−γ 1+σ i=1 µ ¶ 1−γ 1 (1 − γ) (1 + σ) v ∗ γ+σ − γ+σ exp , = ψ (1 − γ) (1 + σ) σ γ+σ 2 I X 1

Ã

t=0

β

t

where the first equality follows from the equal-weights assumption and the time-invariance of the distribution of the labor productivity shocks; the second equality uses the efficient allocations (27) and the log-Normality assumption. For convenience, let us restate equation (8) characterizing the welfare gain from a change in inequality from v ∗ to vˆ = v ∗ + ∆v

á

1 + χCM 1−γ

¢1−γ

I I ¢ ∗ ∗¢ 1X ³ ˆ ´ 1 X ¡¡ CM cit , hit |v∗ = u 1+χ u cˆit , hit |vˆ ⇐⇒ I i=1 I i=1 ! µ µ ¶ ¶ 1 1 + σ 1 − γ v∗ 1 + σ 1 − γ vˆ γ+σ − exp exp = , 1+σ σ γ+σ 2 (1 − γ) (1 + σ) σ γ+σ2

33

where the last expression can be rewritten, after some simple algebra, as ·

CM

1+χ where κ ≡

(1+σ)(1−γ) σ+γ

1−γ γ+σ = + exp 1+σ 1+σ

µ

κ ∆v σ 2

1 ¶¸ 1−γ

,

(31)

is a useful constant that reappears frequently in our welfare analysis.

Equation (31) provides the exact welfare gain of a rise in wage dispersion in complete markets. Using a log-approximation of the type ln (1 + x) ' x on the left-hand side of that equation, and the approximation exp (x) ' 1 + x on the right-hand side, we obtain µ ¶¸ · (1 + σ) (1 − γ) ∆v 1 1−γ γ+σ CM 1+ χ ' ln + 1−γ 1+σ 1+σ σ+γ 2σ · ¸ 1 ∆v = ln 1 + (1 − γ) 1−γ 2σ ∆v ' , 2σ the expression for χCM in Proposition 2. Autarky– In autarky, at each time t individual i solves the static problem ¾ ½ 1−γ h1+σ cit it −ψ max cit ,hit 1−γ 1+σ s.t.

(32)

cit = wit hit −1

1−γ

σ+1

−1

with solution h∗it = ψ σ+γ witσ+γ and c∗it = ψ σ+γ witσ+γ . Once again, to measure welfare we focus on the utilitarian function defined in equation (7) to obtain W

AU T

= (1 − β)

t=0

β

t

I X

φi u (c∗it , h∗it )

(33)

i=1

¶ I (σ+1)(1−γ) γ−1 X 1 1 σ+γ − ψ wi σ+γ 1−γ 1+σ i=1 µ ¶ ∗ 1−γ 1 v − σ+γ = ψ exp −κ(1 − κ) κ 2 1 = I

µ

∞ X

where the first equality uses the time-invariance in the distribution of the shocks, the equalweights assumption, and the optimal individual choices of consumption and hours worked; the second equality uses the log-Normality of wit and the definition κ ≡ 34

(1+σ)(1−γ) . σ+γ

Thus,

equation (8) characterizing the welfare gain of a change in inequality from v ∗ to vˆ = v ∗ + ∆v can be restated, after some simple algebra, as ·

AU T

1+χ

1−γ σ+γ ∆v = + exp(−κ(1 − κ) 1+σ 1+σ 2

1 ¸ 1−γ

,

(34)

Once again, using standard log-approximations, equation (34) yields the approximated expression for χAU T in the main text. Incomplete markets– We start by computing the ex-ante expected utility of a household, conditional on α. Using the time-invariance and the log-Normality of the distribution for εit and the expressions for the optimal choices of consumption and hours worked in (20) , we can show that WαIM where κ ≡

# " µ µ ¶¶ Iα 1−γ 1 1 X (c∗α )1−γ h∗α (ε)1+σ vε∗ − σ+γ = , −ψ =ψ exp κ α + Iα i=1 1 − γ 1+σ κ 2σ

(1+σ)(1−γ) , σ+γ

exactly as in the autarky case. Averaging over the entire distribution

of fixed effects, the expected utility under the veil of ignorance for an individual “dropped randomly” in this economy is then given by W

IM

=E

£

WαIM

¤

=ψ

1−γ − σ+γ

1 exp κ

µ µ ∗ ¶¶ κ vε ∗ − (1 − κ) vα . 2 σ

(35)

Consider now the welfare effects associated with a change in inequality from v ∗ to vˆ where the variance of the fixed-effect increase from vα∗ to vˆα = vα∗ + ∆vα , and where the variance of the insurable component εit increase from vε∗ to vˆε . As usual, we focus on the equivalent variation, expressed as a fraction χIM of consumption (the percentage increase in consumption in the economy with variance v ∗ required to let the planner achieve the same welfare level as in the economy with higher productivity dispersion). Using (35) into equation (8) yields 1 µ µ ¶¶¸ 1−γ 1 ∆vε ∆vα 1−γ γ+σ . + exp κ − (1 − κ) = 1+σ 1+σ σ 2 2

·

IM

1+χ

The usual log-approximations deliver the expression for χIM in the main text. ¥

35

(36)

Proof of Proposition 3 Most of the ingredients needed here have been built already in the Proof of Proposition 2. (i) Equating expressions (33) and (30) for the welfare in autarky and complete markets, respectively, it is easy to see that the equivalent consumption variation ξ AU T measuring the welfare gain of completing the markets under autarky is the solution to the equation á ! ¢1−γ ³ ³κ v´ 1 + ξ AU T v´ 1 1 exp −κ(1 − κ) − = exp . 1−γ 1+σ 2 κ σ2 Solving explicitly for ξ AU T , we arrive at 1 · µ µ ¶ ¶¸ 1−γ (1 − γ) γ+σ 1v v AU T 1+ξ = . + exp κ + (1 − κ) (1 + σ) (1 + σ) σ2 2

(37)

The standard log-approximations allow to re-write the expression as in the statement of Proposition 3. (ii) In the incomplete market economy, ex-ante welfare “under the veil of ignorance” is given by expression (35) . Equating expressions (35) and (30) , for the welfare in complete markets and complete markets, ξ IM solves the equation ! á ¢1−γ µ µ ¶¶ µ ¶ 1 + ξ IM 1 vε κ vε + vα 1 vα 1 − exp κ − (1 − κ) = exp . 1−γ 1+σ σ 2 2 κ σ 2 Simple algebra yields that 1 · µ µ ¶ ¶¸ 1−γ (1 − γ) 1 v γ + σ v α α 1 + ξ IM = + exp κ + (1 − κ) , (1 + σ) (1 + σ) σ 2 2

(38)

and the usual log-approximations deliver the expression in Proposition 3. ¥

Proposition 4 We start by computing the expected utility for agents with fixed-effect α and taste shock ϕ in an incomplete market economy with v ∗ = vα∗ + vε∗ , (¡ ) ¢1−γ Iα, ϕ 1+σ ∗ ∗ X − 1 (ε) c h 1 α,ϕ α,ϕ IM Wα,ϕ = − ψeϕ Iα,ϕ i=1 1−γ 1+σ µ ¶ ¶ µ 1−γ 1 vε∗ 1−γ − σ+γ exp κα + κ ϕ , = ψ exp − κ 2σ σ+γ 36

where to compute the expression after the second equality we have used the equilibrium allocations computed in Section 3.3. As usual, we denote κ ≡

(1+σ)(1−γ) . σ+γ

Averaging over all

pairs (α, ϕ), the welfare of the incomplete-markets allocations “under the veil of ignorance” is given by W

IM

=E

£

IM Wα,ϕ

¤

=ψ

1−γ − σ+γ

1 exp κ

µ µ ∗ ¶¶ µ ¶ κ vε κ vϕ ∗ − (1 − κ) vα exp , 2 σ σ+γ 2

Since in this welfare formula preference heterogeneity enters only multiplicatively, it will cancel out in equation (8) determining the welfare gains from a rise in wage dispersion. Thus, vϕ does not affect this welfare calculation. This proves point (i) of the Proposition. We now turn to the characterization of the welfare gain ξ IM of completing the markets ϕ from an initial incomplete markets allocation in presence taste heterogeneity. To accomplish this, we first compute the welfare in complete markets ( ) µ ¶ I 1+σ 1−γ 1 1 X c1−γ − 1 κ vα + vε + vϕ − σ+γ CM ϕ h (ε) W = − ψe =ψ exp I i=1 1−γ 1+σ κ σ 2 We then solve for the proportional increase in consumption, ξ IM ϕ , in the incomplete markets allocation that is required to give agents the same expected utility as that under complete markets. Hence, ξ IM solves the equation, ϕ ! á ¢1−γ µ µ ∗ ¶¶ µ ¶ µ ¶ 1 + ξ IM κ vε κ vϕ κ vα + vε + vϕ 1 1 ϕ ∗ − exp − (1 − κ) vα exp = exp , 1−γ 1+σ 2 σ σ+γ 2 κ σ 2 with solution ·

1+

ξ IM ϕ

1 µ µ ¶ ¶¸ 1−γ 1−γ σ+γ 1 vα κγ vϕ = + exp κ + (1 − κ) + . 1+σ 1+σ σ 2 σ (σ + γ) 2

Through the usual approximations, we obtain the expression for ξ IM in Proposition 4.¥ ϕ

Proof of Proposition 5 We first show identification of the four variances. It is enough to consider vα1 var1 (log c) = , var0 (log c) vα0 var1 (log wh) − var1 (log c) vε1 = , var0 (log wh) − var0 (log c) vε0 37

hence these two equations, together with the wage equations for t = 0, 1 vart (w) = vαt + vεt . allow to identify (vα0 , vε0 , vα1 , vε1 ) . Next, from µ 0

0

var (log wh) − var (log c) =

1+σ σ

¶2 vε0 ,

one can identify σ and finally from µ 0

var (log c) =

1+σ σ+γ

¶2 vα0 ,

also γ is identified. Finally, either one of the two observations on average log-hours allow to identify ϕ.¥

38

(A) Welfare gain of a rise in labor market risk under complete markets (∆v normalized to 1) 5

Fraction of lifetime consumption

4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

0

1

2

3

4

5

6

7

8

9

10

Inverse of the Frisch Elasticity (σ)

(B) Welfare gain of a rise in labor market risk under autarky (∆v normalized to 1) 1 0.5

Fraction of lifetime consumption

γ=0 0 γ=1 −0.5 γ=2 −1

γ=3

−1.5 γ=5 −2 −2.5

γ=10

−3 −3.5

0

1

2

3

4

5

6

7

8

9

10

Inverse of the Frisch Elasticity (σ)

Figure 1: Welfare gains and losses from a rise in wage dispersion ∆v normalized to one. Panel (A) depicts the complete-markets economy. Panel (B) depicts autarky. The parameter γ denotes the risk aversion coefficient.

Welfare gain of completing the markets ξ (variance of uninsurable shock normalized to 1) 3.5

γ =10

Fraction of lifetime consumption

3

2.5

γ=5

2

1.5

γ=3 γ=2

1 (γ=1,σ=1)

γ=1

0.5

γ=0

0 0

1

2

3

4

5

6

7

8

9

10

Inverse of the Frisch Elasticity (σ)

Figure 2: Welfare gains from completing the markets starting from an economy where the variance of the uninsurable risk is normalized to one.

Approximation bias in χCM (∆v=.1)

% Bias

0 −5 −10 −15 3 2 1

γ

0

0

2

6

4

8

10

σ

Approximation bias in χAUT (∆v=.1)

% Bias

10 5 0 −5 3 2

γ

1 0

AUT

Approximation bias in ξ

0

2

4

6

8

10

σ

(v=.3)

0

% Bias

−20 −40 −60 −80 3 2

γ

1 0

0

2

4

6

8

10

σ

Figure 3: Bias in the approximated expressions for welfare. The bias is computed as the difference between approximated value and true value, as a fraction of the true value.