2.1 Risk Premium with Higher-order Risk

Let us reconsider the Arrow-Pratt thought experiment that confronts a decision maker with a choice between consuming the outcome of a risky bet, \(c\times (1+\tilde{\delta})\), versus the expected payoff of the bet minus a risk premium, \(c\times (1-\pi)\). The question is: how much is the premium, \(\pi\), that makes the decision maker indifferent between the two options:

\[ U(c\times (1-\pi))=\mathbb{E}\left [U(c\times (1+\tilde{\delta}))\right]? \]

The usual derivation proceeds by taking a first-order Taylor approximation to the left-hand side and a second-order approximation to the right-hand side. Why do we stop at second order? Because of two implicit assumptions: (i) that the skewness of the distribution of \(\tilde{\delta}\) is not too far from zero, so the third-order term may not be too important, and (ii) the distribution is sufficiently thin tailed (no excess kurtosis) that realizations of \(\tilde{\delta}\) far from zero have very low probability, so the fourth-order term may not be too important. In light of the recent empirical evidence discussed above that income shocks have nonzero skewness and excess kurtosis, let us see what happens when we take a fourth-order approximation to the right hand side:

\[ \begin{aligned} u(c)-u'(c)c\pi & \approx \mathbb{E}\left (u(c)+u'(c)c\tilde{\delta}+\frac{u''(c)}{2}c^{2}\tilde{\delta}^{2}+\frac{u'''(c)}{6}c^{3}\tilde{\delta}^{3}+\frac{u''''(c)}{24}c^{4}\tilde{\delta}^{4}\right)\\ \pi & \approx \mathbb{E}\left (\frac{1}{2}\frac{u''(c)}{u'(c)}c\tilde{\delta}^{2}+\frac{u'''(c)}{6u'(c)}c^{2}\tilde{\delta}^{3}+\frac{u''''(c)}{24u'(c)}c^{3}\tilde{\delta}^{4}\right). \end{aligned} \]

Furthermore, let \(u(c)=(c^{1-\gamma}/(1-\gamma))\) take the CRRA form and rearrange the terms to get:

\[ \pi ^{*}\approx \underset{\text{{\color{blue}variance\;aversion}}}{\underbrace{\gamma}\times \frac{\sigma _{\delta}^{2}}{2}}\;-\underset{\text{{\color{red}neg. skew\;aversion}}}{\underbrace{\frac{(\gamma +1)\gamma}{6}\times \sigma _{\delta}^{3}}}\times{\color{red}s_{\delta}}\;+\underset{\text{{\color{cyan}{{\color{orange}kurtosis\;aversion}}}}}{\underbrace{\frac{{(\gamma +2)(\gamma +1)\gamma}}{24}\times \sigma _{\delta}^{4}}}\times{\color{blue}k_{\delta}} \]

where \(s_{\delta}\) and \(k_{\delta}\) denote, respectively, the skewness and the kurtosis coefficients (3rd and 4th standardized moments). The first term on the right-hand side is traditionally called the risk premium, and \(\gamma\) is called the relative risk aversion parameter. However, this expanded expression shows that \(\gamma\) only captures the aversion to variance, whereas the risk premium also depends on the aversion to negative skewness (second term) and aversion to kurtosis (third term). These latter terms have two counteracting effects determining their importance. On the one hand, they depend on higher powers of \(\gamma\), which is typically greater than one; on the other hand, they feature higher powers of \(\sigma _{\delta}\), which is smaller than one. Therefore, the impact of these terms on the risk premium depends on the empirical values of \(\gamma\), \(\sigma _{\delta}\), \(s_{\delta},\) and \(k_{\delta}.\) Notice that the sign of the skew aversion term is negative, indicating that a negatively skewed distribution requires a higher premium, and vice versa for a positively skewed distribution.

Note: The first column shows the risk premium under the assumption that the bet has a Gaussian distribution. The second column shows it for a Non-Gaussian distribution whose parameters are taken from Guvenen et al. (2021) for a 45-year-old male at the 90th percentile of the income distribution. The standard deviation of 0.1 is about 1/5 of its empirical counterpart (0.51), which implicitly assumes that 80% of the income shock has been insured and is not passed through to consumption.

Table 1 shows an illustrative example for a relatively high \(\gamma =10\) and a bet with a standard deviation of \(\sigma _{\delta}=0.10\) under two different assumptions. In one case, the bet is assumed to be Gaussian, so it has zero skewness and no excess kurtosis. In the second case, the bet is non-Gaussian with its skewness (–2) and excess kurtosis (27), set to equal that for a 45-year-old male US worker in the 90th percentile of the US earnings distribution (from GKOS). We should note that in the US data, the standard deviation of one-year earnings growth for the just-mentioned worker is not 0.10—in fact, it is above 0.50. So, to be conservative, the assumption in this example is that 80% of the idiosyncratic income risk has been insured, and the individual’s consumption is only subject to the remaining 20% of the risk in her earnings.

As seen in Table 1, the risk premium under the non-Gaussian risk is 22.2%, more than four times the premium under Gaussian risk (4.9%).⁵ The amplification is especially strong when risk aversion is higher, as could be predicted from the appearance of higher-order polynomials in \(\gamma\) in the formula for (2). In the rest of the paper, we will show that the much higher risk premium demanded by individuals to bear non-Gaussian risk will carry over to a properly calibrated dynamic life cycle model with borrowing and saving, as well as government insurance and transfers.

2.2 A Life-Cycle Consumption-Savings Model

We consider a life cycle consumption-savings model with income uncertainty, retirement, stochastic lifetimes, and imperfect altruism. Individuals face an age-dependent probability of death every period, and the conditional survival probability from age \(t\) to \(t+1\) is denoted with \(\delta _{t}.\) Individuals can (potentially) work for the first \(T_{W}\) years of their life, retire at age \(T_{W}+1,\) and die with certainty by age \(t>T\). An individual who dies is replaced with an offspring, who inherits her parent’s positive assets after paying a flat estate tax. Parents derive utility from leaving a bequest according to a warm-glow bequest function: \[ \phi (b)=\phi _{1}\frac{\left (b+\phi _{2}\right)^{1-\gamma}}{1-\gamma} \] as in De Nardi and Yang (2016). This is a flexible functional form that allows us to model bequests either as a necessity or luxury good depending on parameterization.

Individuals have CRRA preferences over consumption and therefore supply labor inelastically. Individuals can borrow or save using a risk-free asset with gross return \(R\), where borrowing is subject to an age-dependent, worker-type-specific limit, denoted by \(\underline{A}_{t}^{k}\), described further below. The type of a worker is given by his fixed type, \(\Upsilon ^{k},\) in the earnings equation described later below. The dynamic programming problem of an individual is

\[ \begin{aligned} V_{t}^{i}\left (a_{t}^{i},z_{t}^{i};\Upsilon ^{k}\right)= & \frac{\left (c_{t}^{i}\right)^{1-\gamma}}{1-\gamma}+\beta \left [(1-\delta _{t})\mathbb{E}_{t}\left (V_{t+1}^{i}\left (a_{t+1}^{i},z_{t+1}^{i};\Upsilon ^{k}\right)\right)+\delta _{t}\frac{\phi _{1}\left (b_{t+1}^{i}+\phi _{2}\right){}^{1-\gamma}}{(1-\gamma)}\right] \\ \text{s.t.} \\ c_{t}^{i}+a_{t+1}^{i} & =a_{t}^{i}R+Y_{t}^{\text{disp,}i},\qquad \forall t,\\ Y_{t}^{\text{disp,}i} & =\lambda \max \left \{\underline{Y},\widetilde{Y}_{t}^{i}\right \} ^{1-\tau},\qquad t=1,\ldots,T_{W},\\ Y_{t}^{\text{disp,}i} & =\lambda \left (\widetilde{Y}_{R}^{k}(z_{T_{W}}^{i})\right)^{1-\tau},\qquad t=T_{W}+1,\ldots,T, \\ a_{t+1}^{i} & \geq \underline{A}_{t}^{k},\qquad \forall t,\\ & \text{and equations}(7)\text{to}(13), \end{aligned} \]

where \(\delta _{T+1}\equiv 1,\beta\) is the time discount factor, and \(\gamma\) governs risk aversion. The budget constraint is given as in equation (3) where \(c_{t}^{i}\) and \(a_{t}^{i}\) denote consumption and asset holdings, respectively, and \(Y_{t}^{\text{disp,}i}\) is disposable income, which differs from gross income, \(\widetilde{Y}_{t}^{i}\), in two ways. First, the government provides social insurance by guaranteeing a minimum level of income, \(\underline{Y}\), to all individuals—more on this in a moment. Second, the government imposes a progressive tax on after-transfer income. Following Benabou (2000) and Heathcote, Storesletten and Violante (2014), we take this tax function to have a power form, with exponent \(1-\tau.\)

Furthermore, retirees receive pension income, \(\widetilde{Y}_{R}^{k}(z_{T_{W}}^{i})\), specified to mimic the U.S. Social Security Administration’s OASDI system’s “primary insurance amount (PIA),” which is the benefit a person would receive if she begins receiving retirement benefits at the normal retirement age. The retirement pension is a function of the average lifetime earnings below Social Security maximum taxable earnings. To avoid introducing another state variable (i.e., lifetime earnings), we approximate lifetime earnings by the persistent component of a worker’s earnings in the last year of the working life, \(z_{T_{W}}^{i}\). In particular, for each worker type \(k\), we regress the simulated average earnings on worker’s \(z_{T_{W}}^{i}\) with an intercept, which we then use to approximate worker’s lifetime earnings, \(LE^{k}(z_{T_{W}}^{i})\). The following equation specifies the pension system as a function of \(LE^{k}(z_{T_{W}}^{i})\):

\[ \begin{aligned} \widetilde{Y}_{R}^{k}(z_{T_{W}}^{i}) & =\begin{cases} 0.9LE^{k}(z_{T_{W}}^{i}) & \qquad \frac{LE^{k}(z_{T_{W}}^{i})}{AE}<0.23\\ 0.2LE^{k}(z_{T_{W}}^{i})+0.32(LE^{k}(z_{T_{W}}^{i})-0.23AE) & \qquad 0.23<\frac{LE^{k}(z_{T_{W}}^{i})}{AE}<1.38\\ 0.57AE+0.15(LE^{k}(z_{T_{W}}^{i})-1.38AE) & \qquad \frac{LE^{k}(z_{T_{W}}^{i})}{AE}>1.38 \end{cases} \end{aligned} \]

for \(t=T_{W}+1,...T\), where \(AE\) is the average earnings in the economy.

2.3 Specifications of Earnings Processes

The specification of the benchmark non-Gaussian and nonlinear earnings process follows Guvenen, Karahan, Ozkan and Song (2021) (hereafter, GKOS) and has the following components: (i) an AR(1) process (\(z_{t}^{i}\)) with innovations drawn from a mixture of normals; (ii) a nonemployment shock whose incidence probability (\(p_{\nu}^{i}(t,z_{t}\))) can vary with age or \(z_{t}\) or both, and whose duration (\(\nu _{t}^{i}\)) is exponentially distributed; (iii) a heterogeneous income profiles component (HIP); and (iv) an i.i.d. normal-mixture transitory shock \((\varepsilon _{t}^{i})\):

\[ \begin{aligned} \text{Level of earnings:} & \quad \tilde{Y}_{t}^{i}=(1-\nu _{t}^{i})e^{\left (g\left (t\right)+\alpha ^{i}+\theta ^{i}t+z_{t}^{i}+\varepsilon _{t}^{i}\right)}\\ \text{Persistent component:} & \quad z_{t}^{i}=\rho z_{t-1}^{i}+\eta _{t}^{i},\\ \text{Innovations to AR(1):} & \quad \eta _{t}^{i}\sim \begin{cases} \mathcal{N}(\mu _{\eta,1},\sigma _{\eta,1}) & \text{with prob.}p_{z}\\ \mathcal{N}(\mu _{\eta,2},\sigma _{\eta,2}) & \text{with prob.}1-p_{z} \end{cases}\\ \text{Initial condition of } z_{t}^{i}\text{:} & \quad z_{0}^{i}\sim \mathcal{N}(0,\sigma _{z_{0}})\\ \text{Transitory shock:} & \quad \varepsilon _{t}^{i}\sim \begin{cases} \mathcal{N}(\mu _{\varepsilon,1},\sigma _{\varepsilon,1}) & \text{with prob.}p_{\varepsilon}\\ \mathcal{N}(\mu _{\varepsilon,2},\sigma _{\varepsilon,2}) & \text{with prob.}1-p_{\varepsilon} \end{cases}\\ \text{Nonemployment duration:} & \quad \nu _{t}^{i}\sim \begin{cases} 0 & \text{with prob.}1-p_{\nu}(t,z_{t}^{i})\\ \min \left \{1,F_{\text{exp}}\left (\varphi \right)\right \} & \text{with prob.}p_{\nu}(t,z_{t}^{i}) \end{cases}\\ \text{Prob of Nonemp. shock:} & \quad p_{\nu}^{i}(t,z_{t})=\frac{e^{\xi _{t}^{i}}}{1+e^{\xi _{t}^{i}}}\text{, where}\xi _{t}^{i}\equiv a+bt+cz_{t}^{i}+dz_{t}^{i}t. \end{aligned} \]

In equation (7), \(g(t)\) is a quadratic polynomial—where \(t=\left (age-24\right)/10\) is a normalized age—that captures the lifecycle profile of earnings common to all individuals. The random vector \(\left (\alpha ^{i},\theta ^{i}\right)\) determines ex ante heterogeneity in the level and growth rate of earnings and is drawn from a multivariate normal distribution with zero mean and covariance matrix \(\Sigma{}_{\alpha}\). As we describe in Section 3, we will also allow \(\left (\alpha ^{i},\theta ^{i}\right)\) to be correlated across generations to capture the positive intergenerational correlation in earnings observed in the US data.

The innovations, \(\eta _{t}^{i}\), to the AR(1) component are drawn from a mixture of two normals. An individual draws a shock from \(\mathcal{N}(\mu _{\eta,1},\sigma _{\eta,1})\) with probability \(p_{z}\) and otherwise from \(\mathcal{N}(\mu _{\eta,2},\sigma _{\eta,2})\). Without loss of generality, we normalize \(\eta\) to have zero mean (i.e., \(\mu _{\eta,1}p_{z}+\mu _{\eta,2}(1-p_{z})=0\)) and assume \(\mu _{\eta,1}<0\) (without loss of generality) for identification. Heterogeneity in the initial value of \(z_{t}\) is captured by \(z_{0}^{i}\sim \mathcal{N}(0,\sigma _{z_{0}})\). Transitory shocks, \(\varepsilon _{t}^{i}\), are also drawn from a mixture of two normals (eq. 11), with analogous identifying assumptions (zero mean and \(\mu _{\varepsilon,1}<0\)). Solving a dynamic programming problem with normal mixture shocks requires modest adjustments to the computational methods commonly used with Gaussian shocks.⁶

The last component of the earnings process is a nonemployment shock (eq. 12) that is intended to primarily capture movements in the extensive margin. Specifically, a worker is hit with a nonemployment shock with probability \(p_{\nu}\) whose duration \(\nu _{t}>0\) is drawn from an exponential distribution, \(F_{\text{exp}}\) with mean \(1/\varphi\), and is truncated at 1 (corresponding to full-year nonemployment with zero annual income). This shock differs from \(z_{t}\) and \(\varepsilon _{t}\) by scaling the level of annual income—not its logarithm—which allows the process to capture the sizable fraction of workers who transition into and out of long-term nonemployment every year.⁷ Furthermore, the nonemployment incidence \(p_{\nu}\) depends on age \(t\) and \(z_{t}\) through the logistic function shown in equation 13. The dependence of \(p_{\nu}\) on \(z_{t}\)—which GKOS refer to as “state dependence”—turns out to be especially important as it induces persistence in nonemployment from one year to the next (despite \(\nu _{t}\) itself being independent over time).

Although this process has many parameters, all dynamics are captured through a single state variable (\(z_{t}^{i}\)), which makes it relatively straightforward to embed it into a dynamic programming problem. We construct a two-dimensional discrete grid over continuous ex-ante type variables \(\left (\alpha ^{i},\theta ^{i}\right)\), where each grid point corresponds to a worker type \(\Upsilon ^{k}.\) Therefore, some aspects of an individual’s problem depends on his (discrete) ex ante type \(k,\) whereas others—including the dynamics of income—are drawn from continuous distributions that are also fully individual specific. This problem can be solved using standard numerical techniques; see the computational appendix.

3.1 Demographics, Preferences, and Smoothing Opportunities

Households enter the labor market at age 25, retire at 60 (\(T_{W}=36\)), and die with certainty by 85 (\(T=60\)). We set the coefficient of relative risk aversion, \(\gamma,\) to 2 but also consider a value of 5 in the robustness analysis. The net interest rate, \(R-1\) is set to 3%. For the bequest function, we follow De Nardi and Yang (2016) and calibrate the parameters, \(\phi _{1}\) and \(\phi _{2}\), to match a bequest–wealth ratio of 0.88% (Gale and Scholz (1994)) and the 90th percentile of bequest distribution normalized by income (4.53) (Hurd and Smith (2002)).

For the borrowing limit, we follow Guvenen and Smith (2014) and assume that banks use a potentially higher interest rate to discount households’ future labor income during working years (to account for income uncertainty) in calculating the borrowing limit, but simply apply the risk-free rate for discounting retirement income, according to this formula:

\[ \underline{A}_{t}^{k}\equiv \underline{Y}\left [\sum _{j=1}^{R-t}(\frac{\psi}{R})^{j}+\psi ^{R-t+1}\sum _{j=R-t+1}^{T-t}\left (\frac{1}{R}\right)^{j}\right], \]

where \(\psi \in [0,1]\) measures the tightness of the borrowing limit. When \(\psi =0\), no borrowing is allowed against future labor income; when \(\psi =1\), households can borrow up to the natural limit. An appealing feature of the formulation in (14) is that for different values of \(\psi \in (0,1)\), it can generate a pattern of a borrowing limit that increases, decreases, or is U-shaped over the life cycle.⁸

In our baseline calibration, we set \(\psi =0.58\) which is Guvenen and Smith (2014)’s benchmark estimate, but we also experiment with lower values of \(\psi\) as well as with a more standard ad hoc borrowing limit later. Given these parameter values, \(\beta\) is calibrated to generate a wealth/income ratio of 4.⁹ Although values lower than this can also be justified on empirical grounds (and have often been used in the literature), a lower ratio implies less wealth available for smoothing income shocks and hence larger welfare costs of idiosyncratic risk. Therefore, a value of 4 represents a more conservative choice for the purposes of this paper.

To sum up, among the model parameters, four of them—\(\beta\), \(\lambda,\phi _{1},\)and \(\phi _{2}\)—are calibrated inside the model, so their values are different for each income process we consider. The values are reported below in Table VI.

The income floor, \(\underline{Y}\), is a key parameter as it limits the severity of large negative shocks to income. Guner et al. (2022) provide a comprehensive evaluation of the US welfare system and provide estimates of the magnitude of welfare assistance programs for different demographic groups. Their estimates for total government non-medical transfers for a married household with no children is 6.55% of mean household income. The same figure is similar for a single male with no children, at 6.84%. Therefore, regardless of which interpretation one adopts for the appropriate unit of analysis for the lifecycle model that we analyze in this paper, we set the income floor, \(\underline{Y}\), to 6.75% of the average earnings, which seems like a reasonable, middle-ground estimate. Given the importance of this parameter for our results, below we will also report results for an income floor of $10,000, which corresponds to 22% of average earnings (which is $45,000 in 2010 dollars in our sample).¹⁰

Finally, the curvature parameter \(\tau\), which controls the progressivity of the tax system, is set to 0.185, following Heathcote et al. (2014), and \(\lambda\) is calibrated such that average after-tax, after-transfer income over the working life is 80% of average before-tax, before-transfer labor income. Last but not least, flat estate tax is calibrated as 40% on positive inheritances. Table II summarizes the parameter values set exogenously outside the model.

3.2 Idiosyncratic Earnings Processes

We consider four specifications for the earnings process, shown in Table III. The first one (column 1) is the most widely used process in the literature, which features an individual fixed effect, a persistent shock (AR(1)), and a transitory shock (\(y_{t}^{i}=\alpha ^{i}+z_{t}^{i}+\varepsilon _{t}^{i}\), using the notation above), with all Gaussian innovations. We will refer to this as the “canonical Gaussian” process. We take the parameter values for this process from Karahan and Ozkan (2013) (hereafter KO), reported in column (1) of Table III.¹¹ As we will see below, however, this process fails to match some key features of the data, such as the (very high) level of lifetime income inequality, which are important for our analysis. Therefore, we consider a second process that keeps the same specification as the canonical one but targets a broader set of data moments used to estimate the benchmark process, except for the distribution of lifetime employment rates. This process has been estimated by GKOS, from whom we borrow the parameter values from. We call this the “Gaussian+” process (column 2 of Table III).

GKOS estimate the canonical process by also matching the distribution of lifetime employment rates, but the estimates of key parameters for this process are extremely large (column (1) of GKOS’s Table IV). To match the large fraction of persistently nonemployed individuals, the estimation requires a large variance of fixed effects. However, this process does not offer a good fit to the data in most of the other dimensions (Guvenen et al. (2021)). Therefore, we report the parameters of this process and its welfare costs in Table D.2 under column Gaussian++ in Appendix C, “Additional Figures and Tables.”

The next two specifications are slightly different versions of the full specification described in Section 2.3. The main difference is that one version restricts the heterogeneity in earnings growth rates by setting \(\theta ^{i}\equiv 0\) for all \(i\) in (7), whereas the other one does not.¹² We refer to these as the Benchmark-R (-R for restricted, column 3 of Table III) and Benchmark (column 4) processes, respectively.¹³

Intergenerational Correlation of Labor Productivity. Finally, a well-documented empirical fact in the US data is that parents’ and children’s labor earnings are positively correlated, with an estimated correlation around 0.5 (see, e.g., Halvorsen et al. (2022); Haider and Solon (2006) and the references therein). We capture this correlation by modeling \(\left (\alpha ^{i},\theta ^{i}\right)\) as imperfectly transmitted from parent to children according to an AR(1) process: =_{}+, and set \(\rho _{\alpha}=0.5\). The innovations \(\left (\varepsilon _{\alpha _{i}},\varepsilon _{\theta _{i}}\right)\) are drawn from standard normal distributions and \(\Sigma{}_{\alpha}\) denotes the covariance matrix for \(\left (\alpha ^{i},\theta ^{i}\right)\), as we define in Section 2.3.

3.3 Implications for Non-Employment Risk

Before delving into the implications of each earnings process for consumption and welfare, we first examine their implications for nonemployment risk and lifetime earnings inequality, which are both intimately related to lifetime welfare.¹⁷

Starting with full-year non-employment risk, Figure 1 shows significant heterogeneity across workers in the data: the fraction of individuals who are non-employed next year increases sharply as earnings fall below the median of the recent earnings distribution (see also Ozkan et al. (2023)).¹⁸ The benchmark and benchmark-R processes capture this highly nonlinear relationship quite well, especially for the bottom 90% of the earnings distribution.¹⁹ In contrast, both versions of the Gaussian process generate almost no full-year non-employment, so the graph is completely flat, except for a little blip at the very low end of the recent earnings distribution. Because full-year non-employment spells cause substantial earnings losses today and in the future, the inability of Gaussian processes to capture this extensive margin is a crucial shortcoming of these specifications for welfare analyses of earnings risk.

Figure: Figure 1 – Full-Year Nonemployment in \(t\) by \(\bar{Y}_{t-1}\)

3.4 Implications for Lifetime Earnings Inequality

We next ask how much lifetime earnings inequality is generated by each earnings process. This is of interest for two reasons. First, lifetime earnings inequality is intimately related to consumption inequality we study in the next section, so getting a sense about the former helps us anticipate some of the upcoming results. Second, it is not common to target lifetime earnings inequality as a moment when earnings processes are estimated in the literature, and this is also true for the four processes used in this paper. So, as a validation step, it is useful to check the extent to which each earnings process generates a plausible level of lifetime inequality compared to what we see in the data.

Table IV reports various measures of dispersion for lifetime earnings. The statistics for the US data in the first column are taken from Guvenen et al. (2022a), who calculate lifetime earnings as the sum of annual earnings between ages 25 and 55 without discounting. They report the statistics for the sample of individuals who make at least $50,000 in lifetime earnings in 2012 dollars (therefore including individuals with zero earnings in some years). The canonical Gaussian process severely understates lifetime inequality for all measures considered. For example, the standard deviation of log lifetime earnings is 0.4 compared with 1.3 in the data, the 90th- to 10th-percentile ratio (P90-P10) is only 2.8 compared with 14.9 in the data. The gap is even larger at the top, with a P99-P10 ratio that is an order of magnitude smaller than in the data (4.3 vs. 43.8).

In contrast, the benchmark process generates a much more plausible distribution of lifetime earnings, with the standard deviation of log measure slightly lower than in the data and the P90-P10 ratio close to its empirical counterpart (16.1 vs. 14.9). The benchmark process also does well at the very top, generating a P99-P10 ratio of 40.9 compared with 43.8 in the data. The only notable discrepancy from the data is that the benchmark process somewhat overstates the inequality above the median (P90-P50 ratio of 3.6 vs 2.5 in the data) and understates the inequality below the median (P50-P10 ratio of 4.5 vs 6.1 in the data). Overall, however, the benchmark process is much more closely aligned with the distribution of lifetime earnings seen in the US data compared with the canonical Gaussian process, which understates lifetime inequality up to an order of magnitude.

A natural question to ask is how much each component of the benchmark process contributes to lifetime inequality. Table V provides the answer: Each column (after the first) shuts down one component of the benchmark process at a time and reports the resulting lifetime inequality. There are several takeaways. First, with the exception of transitory shocks (last column), all components contribute nontrivially to lifetime inequality. Second, initial conditions and persistent shocks have the largest impact on overall lifetime inequality (first two rows) and top-end inequality (last row). For example, shutting down the former (\((\sigma _{\alpha},\sigma _{\theta},\sigma _{z_{0}})\equiv 0\)) lowers the P90-P10 ratio from 16.1 to 7.8, and shutting down the latter (\(z_{t}\equiv 0\)) lowers it to 8.2. Similarly, shutting down these components reduces the P99-P10 ratio to 14.3 and 12.8, respectively, from 40.9 in the full process. Third, nonemployment shocks also have a significant effect on overall inequality (reducing P90-P10 ratio from 16.1 to 10.38), and most of this effect comes from lowering inequality below the median, with the P50-P10 ratio falling from 4.5 to 3.2 while inequality above the median remains less affected (falling slightly from 3.6 to 3.3). The bottom line is that all three main components of the benchmark process matter for lifetime earnings inequality. In the next section, we will revisit this question from a slightly different angle and ask how much each contributes to the welfare costs of idiosyncratic earnings risk.

3.5 Implications for Lifecycle Profiles

Figures 2 and 3 plot the average lifecycle profiles of before-tax and after-tax income, both in levels (former) and in logs (latter). Notice that the average lifecycle profiles of both before- and after-tax earnings are quite similar across earnings processes. This is because we normalize the parameters of the deterministic lifecycle profile across earnings processes so as to generate the same average earnings (levels) profile from age 25 to 55 (Figure 2). However, the mean and variance profiles of log after-tax, after-transfer earnings look quite different across earnings processes (Figure 3) because of a Jensen inequality effect. Namely, each process generates different variance profiles of earnings over the life cycle (e.g., benchmark and Gaussian+ generate a steeper increase in within-cohort inequality than the canonical Gaussian). Conditional on having the same (levels of) average earnings profiles, those that generate a higher dispersion mechanically imply a lower log earnings over the life cycle in Figure 3. So, for example, the benchmark process has the lowest log earnings profile in Figure 3 but displays the same level as the canonical Gaussian in Figure 2.

4.1 Heterogeneity in Welfare Costs of Idiosyncratic Risk

The average welfare costs reported in Table VI mask significant heterogeneity across different types of households. Next, we discuss the between-group heterogeneity in the welfare costs of idiosyncratic risk. Specifically, we investigate how different age and income groups are affected by income risk. For this purpose, we calculate the fraction of lifetime consumption each group would be willing to give up to avoid income risk and instead receive a constant stream of income equal to the group’s average earnings at all ages. Furthermore, to focus on the role of idiosyncratic risk individuals face going forward, we assume that each individual starts with a level of assets equal to the average asset holdings of their respective group. We also compare the benchmark process with the canonical Gaussian process. Table 8 summarizes our results.

Notes: This table shows the fraction of lifetime consumption households in each cell would be willing to give up to avoid income risk and instead receive a constant stream of income equal to the group’s average earnings at all ages going forward.

First, the welfare costs systematically decrease by age for all income groups in both specifications. This is not surprising given that the ratio of risky human wealth relative to financial wealth—the safe asset in our model—declines by age (Huggett and Kaplan (2016)). Furthermore, labor earnings become less risky as individuals approach retirement, after which they receive a steady pension income. The specific functional form we use for the borrowing limit (equation 14), which is estimated from data by Guvenen and Smith (2014), also allows for a more generous borrowing limit for middle-age workers.

Next, looking at heterogeneity by income-group, we first observe that for the youngest workers there is relatively little variation in welfare costs of income risk over the income distribution, and relative differences grow over the working life. For example, at age 25 the welfare costs of the benchmark process vary only very little, from 26.5% to 27.6%, across the income distribution. In contrast, among 55-year-olds, the bottom income quartile workers are willing to give up, as a fraction of their lifetime consumption, more than three times as much (6.01%) as those in the upper-middle income group (1.85%). We also find that there is relatively little variation in welfare costs of the canonical Gaussian process among the youngest workers.

Second, and interestingly, welfare costs are not monotone over the income distribution but broadly exhibit a U-shaped pattern. For example, among 45-year-olds, the welfare cost of the benchmark process declines from 18.3% for the bottom quartile workers to 7.25% for those around the median, and then it increases to 11.4% for the workers in the top 5% of the earnings distribution. This pattern can be explained by a combination of different factors. On the one hand, lower-income workers face larger idiosyncratic risk because of their higher nonemployment risk (see Figure 1), and they have less assets to insure themselves. On the other hand, they are also more protected by the social safety net, the magnitude of which is too small to make a difference in income fluctuations of high-income workers. As a result of these opposing forces, the highest welfare costs are associated with those who have the highest and, especially, lowest incomes. As for the canonical Gaussian, the increase in welfare cost is less pronounced in the high end of the income distribution.

5.1 Measuring the Insurability of Income Shocks

The transmission rate of earnings shocks to consumption has received significant attention in the literature, because it is used to measure the extent of partial insurance—that is, insurance above and beyond self-insurance (see, e.g., Blundell et al. (2008), Primiceri and van Rens (2009), and Kaplan and Violante (2010)). In an important paper, Blundell et al. (2008) (hereafter BPP) proposed using two simple moment conditions to estimate the insurance coefficients in response to permanent shocks (\(\eta\)) and transitory shocks (\(\varepsilon\)), respectively. The formulas for the BPP insurance coefficients are:

\[ \phi ^{\eta}=1-\frac{\text{cov}(\Delta c_{t}^{i},y_{t+1}^{\text{disp,}i}-y_{t-2}^{\text{disp,}i})}{\text{cov}(\Delta y_{t}^{\text{disp,}i},y_{t+1}^{\text{disp,}i}-y_{t-2}^{\text{disp,}i})} \]

for permanent shocks and

\[ \phi ^{\varepsilon}=1-\frac{\text{cov}(\Delta c_{t}^{i},\Delta y_{t+1}^{\text{disp,}i})}{\text{cov}(\Delta y_{t}^{\text{disp,}i},\Delta y_{t+1}^{\text{disp,}i})} \]

for transitory shocks. The insurance coefficients range from 0 to 1, where 0 means no partial insurance above self-insurance (or full transmission) and 1 means perfect consumption insurance (no transmission). Here, we consider the following experiment. Suppose we give a panel dataset of income and consumption simulated from the benchmark model to an econometrician and ask her to estimate the insurance coefficients using BPP’s moments. What would the econometrician conclude about the extent of the insurability of income shocks?

The top panel of Table 9 reports the results at different ages. When the true data-generating process is the benchmark model, the BPP procedure estimates that 62% of permanent shocks are insured and the remaining 38% is transmitted to consumption at age 40. The corresponding insurance coefficient for the Gaussian process is much lower—almost half—at 32% (so 68% transmitted). Taken at face value, these findings would indicate that persistent income shocks in the benchmark model are more insurable relative to those in the Gaussian model, which seems surprising given that both the long tails and the negative skewness of the benchmark process would be expected to be harder (at least, not easier) to insure such shocks. To understand this result, note that the standard BPP moment conditions are derived under the assumption that the earnings process is given by the persistent-plus-transitory process such as the canonical Gaussian model, but unlike the benchmark process. The discrepancy could happen if the benchmark process features less persistent shocks (which is partly true, at least judging crudely based on the value of \(\rho\)—0.98 vs. 0.959—and the presence of the more transitory non-employment shock process) or if the benchmark lifecycle model features more insurance opportunities relative to the Gaussian model (which is not true—they both have self-insurance plus an income floor only).

To avoid these problems, a more direct measure of the partial insurance coefficient with respect to persistent shocks can be obtained from simulated data by directly looking at the covariation between consumption growth and the innovation, \(\eta _{t}\), to the persistent component, \(z_{t}.\) Using this covariation, we can also go one step further and measure what we call the “true” partial insurance coefficient with respect to positive and negative innovations separately using this equation:

\[ \chi _{\text{insur.}}^{\eta}=1-\frac{\text{cov}(\Delta c_{t}^{i},\eta _{t}\mid \eta _{t}>0)}{\text{var}(\eta _{t}\mid \eta _{t}>0)}. \]

The bottom panel of Table 9 reports the true coefficients for both processes, which reveal two results. First, under the Gaussian process, the insurance against positive and negative persistent shocks are similar, whereas under the benchmark process, negative shocks are better insured than positive shocks. Second, the insurability of positive persistent shocks are very similar between the two processes, whereas negative shocks are slightly more insurable under the benchmark process. This asymmetry is because the typical negative shock in the benchmark process is drawn from a thicker tail (due to both the negative skewness and excess kurtosis) and therefore reduces the earnings to a level at which the guaranteed minimum income threshold kicks in to smooth those shocks.

To complement this analysis, we consider another way to quantify the degree of partial insurance commonly used in the literature— by studying how within-cohort consumption inequality evolves over the life cycle. The idea is that if shocks are easily insurable (because either they are not persistent or the economy features rich smoothing opportunities) then consumption inequality should not rise much with age. To investigate this, Figure 4 plots the cross-sectional variance of log consumption in the benchmark and Gaussian models. In the former, consumption inequality rises by about 45 log points from age 25 to age 60, which is about four times as large as the rise in the Gaussian model (about 12 log points). Therefore, this second way of looking at the degree of partial insurance leads to the opposite conclusion: shocks in the benchmark model are less insurable, which causes consumption inequality to rise more than in the Gaussian model. This latter evidence is also more consistent with the higher welfare costs of idiosyncratic shocks that we found for the benchmark model relative to the Gaussian model above. That said, this result depends somewhat on the availability of public insurance. In the presence of a more generous income floor (i.e., \(\underline{Y}=\$10,000\)), consumption inequality rises by around 30 log points in the benchmark model, which is still much higher than in the canonical Gaussian model.

Recall that the benchmark process generates a steeper increase in within-cohort, after-tax, after-transfer income inequality than the canonical Gaussian specification even with a \(\$10,000\) income floor (see Figure 3). The total effect of earnings shocks on consumption responses also depends on the size of shocks, which depends on the shape of the distribution it is drawn from. The benchmark process with its long tails and left skewness can produce shocks that generate a larger income inequality and stronger total consumption response.

Figure: Figure 4 – Within-Cohort Variance of Log Consumption Over the Life Cycle

Taken together, we conclude from these two pieces of evidence that once we move beyond Gaussian shocks and linear models, extra care is needed to properly measure the extent of insurability of shocks. Nonlinear dynamics and higher-order moments generate interesting new patterns. For example, why does the transmission parameters indicate a low rate of transmission in the benchmark model? An important reason is that the partial insurance coefficient measures the average response of consumption growth to income shocks. But it is plausible to expect that the consumption response varies by the sign of the shock, by its size (Beraja and Zorzi, 2024), and by many of its other properties established in the previous sections. So, the average response coefficient could provide an incomplete picture of the transmission of income shocks to consumption.

A natural question that this discussion raises concerns the properties of the entire distribution of consumption growth rates implied by a model, including its higher-order moments. We turn to this point next.

5.2 The Marginal Propensity to Consume (MPC)

The marginal propensity to consume (MPC) out of earnings (or wealth) is a key ingredient in designing effective fiscal and monetary policies (e.g., Kaplan et al. (2018)). The substantial concentration of wealth and earnings in the United States and in many modern economies implies that the effectiveness of such policies can be significantly improved if they are targeted to certain segments of the population if MPCs vary in a meaningful way across the population.²⁵ In this section, we investigate the implications of non-Gaussian earnings risk for the level and distribution of MPCs in the population.

7.1 Numerical Solution of the Model

In order to be able to fully capture the features of the rich earnings dynamics we estimated in this paper, we chose not to discretize the income process when solving the consumption model. Thus it is worth discussing our numerical solution methodology to solve an otherwise standard Bewley model. We employ value function iteration to solve for the policy function for the consumption decision and then use the policy function to simulate consumption-savings decisions of individuals. We now discuss the details below.

8.1 Data and Sample

The PSID is a longitudinal study of a representative sample of U.S. households, including income and an extensive list of consumption categories from 2005 to 2021 at a biennial frequency.²⁸ For those waves, a new set of questions targeted to understanding spending dynamics allow us to track consumption on almost all expenditures measured in the cross-sectional Consumer Expenditure Survey (CEX) Andreski et al. (2014). We focus on a sample of the original SRC households that participate in the labor market and are between 25 and 65 years old. We further focus on those households in which the head of household is a male and his labor earnings are at least $1500 in the given year. This sample is comparable to that in Blundell et al. (2016). We end up with 29,644 observations, spanning 16 years at biennial frequency. We define household consumption as the sum of spending in food (at home, away, and delivery), gasoline, health, transport, utilities, clothing, and recreation. Table B.1 summarizes the characteristics of the sample and the subcomponents of consumption.

8.2 Cross-Sectional Distribution of Consumption Growth: Densities