2.1. Life-cycle consumption with rare event risk

Time is continuous and indexed by $t$ . An individual is born at $t=0$ , retires at date $t_R$ , and dies no later than $t=T$ . The probability of surviving to age $t$ is $\Psi (t)$ . The individual receives income $y(t)$ over the life cycle which consists of after-tax labor income before retirement and Social Security benefits after retirement. Consumption is denoted $c(t)$ and saving is denoted $k(t)$ . The interest rate is $r$ ; this is the rate of return at all times other than when the shock hits. The individual discounts future utility at the rate $\rho$ .

The instantaneous utility function is the CRRA function:

(1) \begin{equation} u(c)=\frac{c(t)^{1-\sigma }}{1-\sigma }. \end{equation}

The household faces a risk that they will experience a negative wealth shock at some point during their working life that makes the share $S$ of their assets disappear. The shock date $t_1$ is a random variable with probability density $\phi (t_1)$ for $t_1 \in [0,\infty ]$ . The one-time negative wealth shock reduces the household’s private savings by fraction $S$ . That is, the household only has $(1-S)$ of their assets remaining after the shock. The household knows the size of $S$ , but does not know when $S$ will occur.

The rare event financial shock that we model has an asymmetric effect on positive and negative asset holdings. That is, we allow individuals to hold negative assets (debt), and in that case there is no change in the magnitude of their debt when a financial shock hits. The shock only reduces positive asset valuations.Footnote ⁷ Therefore, notationally, we denote the state dependency of the shock

(2) \begin{equation} S(k(t_{1}))=\boldsymbol{1}\{k(t_{1})\gt 0\}S, \end{equation}

where $\boldsymbol{1}\{k(t_{1})\gt 0\}$ is an indicator function that is equal to one when assets are positive and zero when assets are zero or negative.

The household optimization problem is solved recursively, using the timing uncertainty methods from the stochastic control literature as in Cottle Hunt and Caliendo (Reference Cottle Hunt and Caliendo2021), Caliendo et al. (Reference Caliendo, Casanova, Gorry and Slavov2020), Caliendo et al. (Reference Caliendo, Gorry and Slavov2019), and Stokey (Reference Stokey2016).

2.1.1. Postshock problem

The negative wealth shock $S$ occurs at time $t_1\leq \infty$ . If the shock occurs before time $T$ , the household solves the following deterministic, fixed-end point problem from the vantage point of the shock date:

(3) \begin{equation} \max _{c(t)} \int _{t_1}^{T}e^{-\rho t}\Psi (t) u(c(t))dt, \end{equation}

subject to

(4) \begin{align} \dot{K}(t)=y(t)-c(t)+rK(t) \quad \textrm{for } t\in [t_1,T] \end{align}

(5) \begin{align} K(t_1)=k(t_1)(1-S(k(t_1))) \end{align}

(6) \begin{align} K(T)=0 \end{align}

(7) \begin{align} k(t_1) \textrm{ given, } S(k(t_1)) \textrm{ given, } t_1 \textrm{ given.} \end{align}

Here, $K(t)$ is the total financial wealth of the household after the shock takes place. $K(t_1)$ is the share of private savings $k(t_1)$ that remain immediately after the shock $S(k(t_1))$ occurs.

The solution to this problem is the consumption path $c^*_2(t|t_1,k(t_1),S(k(t_1)))$ which is the optimal path of consumption after the wealth shock has occurred. The subscript 2 indicates this is the path of post-shock consumption for $ t\in [t_1,T]$ ,

(8) \begin{equation} c^*_2(t|t_1,k(t_1),S(k(t_1))) = \frac{k(t_1)(1-S(k(t_1))) + \int _{t_1}^{T}e^{-r(v-t_1)}y(v)dv}{\int _{t_1}^{T}e^{-r(v-t_1)+(r-\rho )v/\sigma }\Psi (v)^{1/\sigma }dv}e^{(r-\rho )t/\sigma } \Psi (t)^{1/\sigma }. \end{equation}

Note changing variable notation for convenience, we replace $t$ with $z$ and $t_1$ with $t$ . Hence, consumption on $z\in [t,T]$ for shock date $t$ is

(9) \begin{equation} c^*_2(z|t,k(t),S(k(t))) = \frac{k(t)(1-S(k(t)))+ \int _{t}^{T}e^{-r(v-t)}y(v)dv}{\int _{t}^{T}e^{-r(v-t)+(r-\rho )v/\sigma }\Psi (v)^{1/\sigma }dv}e^{(r-\rho )z/\sigma } \Psi (z)^{1/\sigma }. \end{equation}

2.1.2. Preshock problem

Next we solve the dynamic stochastic problem of a household that hedges the timing risk of shock $S(k(t))$ , from the perspective of time 0.

The household’s expected utility is

(10) \begin{equation} \begin{split} \mathbb{E}(U)& =\int _{0}^{T} \phi (t) \left (\int _{0}^{t}e^{-\rho z} \Psi (z)u(c(z))dz +\int _{t}^{T}e^{-\rho z} \Psi (z)u(c^*_2(z|t,k(t),S(k(t))))dz\right ) dt\\[5pt] &\!\quad+ \int _{T}^{\infty }\phi (t)\left (\int _{0}^{T}e^{-\rho z} \Psi (z)u(c(z))dz \right )dt, \end{split} \end{equation}

which can be rewritten more compactly using the properties of iterated integrals

(11) \begin{equation} \mathbb{E}(U) = \int _0^{T} \left [ \left (\int _t^{\infty }\phi (t_1)dt_1 \right ) e^{-\rho t}\Psi (t)u(c(t)) + \phi (t) U_2(t,k(t),S(k(t))) \right ] dt \end{equation}

where

(12) \begin{equation} U_2(t,k(t),S(k(t)))=\int _t^{T}e^{-\rho z}\Psi (z)u(c^*_2(z|t,k(t),S(k(t))))dz \end{equation}

and $c^*_2(z|t,k(t),S(k(t)))$ is given by (9).

Thus, the preshock household’s stochastic optimization problem can be written neatly as a Pontryagin problem

(13) \begin{gather} \max _{c(t)_{t\in [0,T]}} \int _0^{T} \left [ \left (\int _t^{\infty }\phi (t_1)dt_1 \right ) e^{-\rho t}\Psi (t)u(c(t)) + \phi (t) U_2(t,k(t),S(k(t))) \right ] dt \end{gather}

subject to

(14) \begin{align} \dot{k}(t) = r k(t) + y(t) -c(t) \quad \textrm{for } t\in [0,T] \end{align}

(15) \begin{align} k(0)=0, k(T)=0. \end{align}

The Hamiltonian $\mathcal{H}$ with multiplier $\lambda (t)$ for this problem is

(16) \begin{equation} \begin{split} \mathcal{H}= & \left (\int _t^{\infty }\phi (t_1)dt_1 \right ) e^{-\rho t}\Psi (t)u(c(t)) \\[5pt] &+ \phi (t) U_2(t,k(t),S(k(t))) +\lambda (t) \left ( r k(t) + y(t) -c(t) \right ). \end{split} \end{equation}

The necessary conditions are

(17) \begin{align} \frac{\partial \mathcal{H}}{\partial c(t)}=\left (\int _t^{\infty }\phi (t_1)dt_1 \right ) e^{-\rho t}\Psi (t) c(t)^{-\sigma }-\lambda (t)=0 \end{align}

(18) \begin{align} \dot{\lambda }(t) = -\frac{\partial \mathcal{H}}{\partial k(t)} = -\phi (t) \frac{\partial U_2(t,k(t),S)}{\partial k(t)} - \lambda (t)r. \end{align}

Note, substituting $c^*_2(z|t,k(t),S(k(t)))$ into $U_2(t,k(t),S(k(t)))$ the partial derivative

(19) \begin{equation} \frac{\partial U_2(t,k(t),S(k(t)))}{\partial k(t)}=\left ( \frac{k(t)(1-S(k(t))) + \int _{t}^{T}e^{-r(v-t)}y(v)dv}{\int _{t}^{T}e^{-r(v-t)+(r-\rho )v/\sigma }\Psi (v)^{1/\sigma }dv}\right )^{-\sigma }(1-S(k(t)))e^{-rt}. \end{equation}

Insert (19) and (17) into (18)

(20) \begin{equation} \begin{split} \dot{\lambda }(t)=&-\phi (t) \left ( \frac{k(t)(1-S(k(t))) + \int _{t}^{T}e^{-r(v-t)}y(v)dv}{\int _{t}^{T}e^{-r(v-t)+(r-\rho )v/\sigma }\Psi (v)^{1/\sigma }dv}\right )^{-\sigma }(1-S(k(t)))e^{-rt} \\[5pt] &-r\left (\int _t^{\infty }\phi (t_1)dt_1 \right ) e^{-\rho t} \Psi (t)c(t)^{-\sigma }. \end{split} \end{equation}

Differentiate (17) with respect to $t$

(21) \begin{equation} \begin{split} 0=&-\phi (t)e^{-\rho t}\Psi (t)c(t)^{-\sigma }+\left (\int _t^{\infty }\phi (t_1)dt_1 \right ) \left (\dot{\Psi }(t)e^{-\rho t}-\rho \Psi (t)e^{-\rho t} \right )c(t)^{-\sigma }\\[5pt] &-\left (\int _t^{\infty }\phi (t_1)dt_1 \right ) \sigma e^{-\rho t}\Psi (t)c(t)^{-\sigma -1}\dot{c}(t)-\dot{\lambda }(t). \end{split} \end{equation}

Combine (20) and (21) to obtain the Euler equation in closed form

(22) \begin{equation} \begin{split} \dot{c}(t) &=\left ( \frac{c(t)^{\sigma +1}}{\Psi (t)} \left ( \frac{k(t)(1-S(k(t))) + \int _{t}^{T}e^{-r(v-t)}y(v)dv}{\int _{t}^{T}e^{-r(v-t)+(r-\rho )v/\sigma }\Psi (v)^{1/\sigma }dv}\right )^{-\sigma }(1-S(k(t)))e^{(\rho -r)t}-c(t)\right )\\[5pt] & \quad\times \left [\frac{\sigma }{\phi (t)}\int _t^{\infty }\phi (t_1)dt_1\right ]^{-1} + \left (\dot{\Psi }(t)\frac{1}{\Psi (t)} +r-\rho \right )\frac{c(t)}{\sigma }. \end{split} \end{equation}

The optimal path of consumption and assets prior to the shock $(c^*_1(t),k^*_1(t))_{t\in [0,T]}$ obey the Euler equation and the constraints.

2.2. Computational method

Although our model is solved analytically for both the preshock and postshock timepaths of consumption and savings, we must employ a recursive computational method to identify the initial consumption level. We utilize our analytical solutions to identify initial consumption as follows:

Step 1. Compute $c_{2}^{\ast }(z|t,k(t),S(k(t)))$ for given shock date $t$ , given asset holdings at that date $k(t)$ , and given shock size $S(k(t))$ .

Step 2. Guess $c(0)$ .

Step 3. Given the guessed value of $c(0)$ together with $k(0)=0$ , use the Euler equation $\dot{c}(t)$ and law of motion for capital $\dot{k}(t)$ , which themselves depend on the post-shock consumption path $c_{2}^{\ast }(z|t,k(t),S(k(t)))$ , to shoot forward in time to construct the pre-shock timepaths $c(t)$ and $k(t)$ .

Step 4. Check the terminal value of pre-shock capital to see if $k(T)=0$ . If yes, then we have found our solution pre-shock consumption and saving paths $c_{1}^{\ast }(t)$ and $k_{1}^{\ast }(t)$ . If not, go back to step 2 and repeat.

2.3. Calibration

Our baseline calibration strategy is to maximize the potential safety net feature of Social Security by assuming that a modern individual faces risk of a financial shock that is proportional to the stock market crash of the Great Depression. This means that we will calibrate things like income, career length, longevity, and Social Security to modern specifications, and we will assume the individual faces financial risk like that experienced by past generations who endured the Great Depression.Footnote ⁸

We assume the individual enters the model at age 18 ( $t=0$ ), retires at age 67 ( $t_R=49$ ), and lives to a maximum age of 100 ( $T=82$ ). We calibrate the survival function and wage income following Cottle Hunt and Caliendo (Reference Cottle Hunt and Caliendo2021). We calibrate the survival function as

(23) \begin{equation} \Psi (t) = 1-(t/T)^{2.58} \end{equation}

to ensure that the ratio of workers to retirees is 3.3, which is approximately the average value in the USA during the period 2000–2010.Footnote ⁹ The life expectancy at $t=0$ implied by this survival function is age 77.1, which corresponds closely to the life expectancy from birth for the USA reported by the World Bank of 78.5 years. We parameterize $y(t)$ to represent after-tax income before retirement and Social Security benefits after retirement. We use a 5th order polynomial to fit life-cycle wages to the data in Gourinchas and Parker (Reference Gourinchas and Parker2002)

(24) \begin{equation} w(t)=\alpha (1+0.0244t-0.0001t^{2}+4(10)^{-5}t^{3}-2(10)^{-6}t^{4}+2(10)^{-8}t^{5}) \end{equation}

where $\alpha$ is chosen such that the average wage is equal to one.

We set the Social Security tax rate to $\tau =0.106$ , which corresponds to the employer and employee Old Age Survivors portion of Social Security. We assume the Social Security system has a balanced PAYGO budget, hence

\begin{equation*} b=\tau \frac {\int _{0}^{t_{R}}w(t)\Psi (t)dt}{\int _{t_{R}}^{T}\Psi (t)dt}. \end{equation*}

Thus, $y(t)$ is given by

(25) \begin{equation} y(t) = \begin{cases} (1-\tau )w(t), & t \leq t_R\\[5pt] b, & t\gt t_R. \end{cases} \end{equation}

We set the interest rate $r=0.08$ which corresponds to the average long-term real return on equities. We set $\sigma =3$ , which is a mid-point of the standard values used in the literature. Finally, we set the discount rate $\rho =0$ .

As our baseline parameterization for the rare event shock, we assume the risk of experiencing the shock is uniform and that the individual cannot escape the shock—it is sure to hit during their life cycle. Hence

(26) \begin{equation} \phi (t)=\begin{cases} 1/T, & t\leq T\\[5pt] 0, & t\gt T. \end{cases} \end{equation}

As a robustness check later in the paper, we explore a distribution that has positive mass beyond the maximum age $T$ . We calibrate the size of the shock $S$ to correspond to the Great Depression. The S&P lost 86% of its value during the Great Depression, and so we set $S=0.86$ .Footnote ¹⁰ Our baseline parameters are summarized in Table 1.

Table 1. Summary of baseline calibration of parameters

2.4. Welfare

Our goal is to assess Social Security’s role as a safety net against rare event financial shocks to wealth. Our stylized model leaves out a number of Social Security’s features like disability insurance, spousal benefits, survivor insurance, and a benefit-earning rule that redistributes wealth from the rich to the poor. While these are important features, we view the simplicity of our baseline analysis as an advantage because it allows us to focus cleanly on the safety net rationale, which is our purpose. As a result, we are not in a position to make statements about Social Security’s overall welfare effects. Instead, we are drawing conclusions about a single mechanism in particular.

We measure the welfare effect of Social Security as the percentage of lifetime consumption the individual would be willing to give up to live in a world with Social Security relative to a Laissez Faire world with no Social Security.

Notationally, let $\mathbb{E}(U_{SS}^{\ast })$ stand for expected utility in a world with Social Security and following optimal consumption rules under uncertainty. Likewise let $\mathbb{E}(U_{LF}^{\ast })$ stand for expected utility in a Laissez Faire world without Social Security and following optimal consumption rules in that environment. With CRRA utility, the fraction of lifetime consumption the individual would give up to participate in Social Security is

(27) \begin{equation} \Delta =1-\left ( \frac{\mathbb{E}(U_{LF}^{\ast })}{\mathbb{E}(U_{SS}^{\ast })}\right ) ^{\frac{1}{1-\sigma }}. \end{equation}

We calibrate the size of the shock to correspond to the Great Depression: $S=0.86$ . Even in this extreme case, we find the welfare effect of Social Security to be negative, $\Delta =-7.64\%$ . That is, an individual would be need to receive an additional 7.64% of lifetime consumption in order to compensate for participating in Social Security.

What is the intuition for this result? The basic answer is that US equities are still a very good deal, even when individuals face catastrophic, rare event risk. An 8% return compounded annually, together with the risk of a one-time shock equivalent to the Great Depression (86% instantaneous loss in wealth), still compares favorably to the implicit rate of return in a Pay-As-You-Go Social Security program. And this is true even after accounting for the life annuity feature of Social Security.

The intuition is also visible in the possible timepaths of consumption over the life cycle with and without Social Security. Figure 1 plots a few possible paths of realized life cycle consumption with and without Social Security, given different possible dates for the shock (specifically ages 38, 48, 58, 68 and 78). The solid green line plots the Laissez Faire con- sumption paths and the dashed blue lines are with Social Security. Consumption is generally higher without Social Security.

Figure 1. Realized consumption given $S=0.86$ , with and without Social Security.

This negative welfare result is important because in some ways we have stacked our analysis in favor of Social Security in our choice of modeling assumptions. We assume the financial shock is guaranteed to happen; we assume the shock will be the biggest possible shock in modern history; and, we assume the individual’s only choice for private saving is a risky asset. By assuming the individual faces an inescapable, large shock with no access to a safe asset, the individual is left completely exposed to risk and Social Security’s safety net role should be maximized. And yet we do not find a safety net role for Social Security.

However, we will show in Section 3 that our baseline assumptions about optimal hedging under full information are driving the negative welfare result. Relaxing these assumptions will create space for large welfare gains.

2.5. Decomposition

We decompose the welfare effect of Social Security into the portion that comes from the provision of longevity insurance and the portion that comes from the provision of a financial safety net. To do this, we turn off financial risk by setting the asset shock to $S=0$ and we find that $\Delta =-10.91\%$ . Consumption with and without Social Security when $S=0$ is depicted in Figure 2. Since the shock is set to zero, there is only one realized consumption path and consumption is everywhere higher without Social Security. This welfare effect is worse (more negative) than the combined welfare effect of insurance against both longevity and asset risk which was −7.64% in our baseline model. This suggests that Social Security is less harmful in the presence of rare event risk.

Figure 2. Realized consumption given $S=0$ , with and without Social Security.

As an alternative decomposition, we turnoff both the asset risk and the asset reward by setting $S=0$ and $r=0$ . We find that $\Delta =11.57\%$ in this case. This large positive welfare effect is due to the well-known annuitization feature of Social Security. An individual would rather pay Social Security taxes and receive a Social Security annuity during retirement rather than invest in a riskless asset with zero return. Consumption with and without Social Security when $S=0$ and $r=0$ is depicted in Figure 3. In this case, consumption is everywhere higher with Social Security.

Figure 3. Realized consumption given $S=0$ and $r=0$ , with and without Social Security.

2.6. Robustness checks

Some of the parameters that we use in our baseline analysis are unobservable and could have direct bearing on our welfare calculations. In this section, we briefly discuss these issues before addressing more fundamental extensions.

Our baseline value of 3 for the coefficient of relative risk aversion ( $\sigma$ ) falls within a standard range of values used in the life-cycle consumption literature, though values as low as 0.5 and as high as 5 are also used. The welfare effect of Social Security is −8.64% and −7.08% with these values, compared to our baseline value −7.64%. Hence, while different levels of risk aversion change the specific welfare effect of Social Security, our overall conclusion about its role as a safety net is unchanged.

Similarly, our use of a 0% discount rate is not uncommon, but it is also common to see values as high as 5% used in the life-cycle literature. The welfare effect of Social Security is −7.89% under this assumption. Again, no material change to our conclusions thus far.

We assumed the financial shock was distributed uniformly over the life cycle of the individual. This is a reasonable starting assumption and it helps to stack the analysis in favor of Social Security: if the individual cannot escape experiencing the shock at some point (because the upper support of the shock coincides with the maximum age of the individual), then Social Security would be more attractive as a safety net than if there was a chance the individual may never experience the negative shock. However, our uniform assumption has a subtle side effect: if the shock has not yet happened, then as the individual ages they become increasingly sure that it will happen to them. Indeed, as the individual approaches the maximum model age, the probability that the shock strikes will spike to 100% conditional on it not happening yet. This is a mathematical definition of any truncated distribution, and yet there is no reason that a rare financial shock would increase in relative likelihood as an individual ages.

As an alternative, we consider the exponential density

(28) \begin{equation} \phi (t)=\gamma e^{-\gamma t}\textrm{, for }t\in \lbrack 0,\infty ]\textrm{ and }\gamma \gt 0. \end{equation}

This density is memoryless: if the shock has not occurred by date $x$ , the likelihood of the shock occurring between $x$ and $x+\delta$ does not depend on $x$ . As an example, we calibrate $\gamma$ by assuming there is a 50% chance that a financial crash like the Great Depression does not happen within the individual’s maximum lifespan; hence, $\gamma =-\ln (0.5)/82$ . The welfare effect of Social Security is −7.42% under this assumption. Note that while the need for a safety net is intuitively weakened if there is a chance that such a net is never needed, the exponential distribution considered here also rearranges the shock probabilities over the lifespan of the individual, with the most weight during the earliest years of life. This secondary effect virtually washes out the first effect and leaves the overall welfare effect of Social Security quite similar to our baseline estimate.

Lastly, we assumed a real interest rate of $r=0.08$ which corresponds to the average annual real return on equities in the USA over the last 50–100 years. The welfare effect of Social Security is decreasing in the interest rate. That is, Social Security reduces welfare less (or increases welfare more) if the interest rate is lower. Holding the other parameters fixed at their baseline values, the welfare effect of Social Security is −2.47% for an interest rate of $r=0.06$ , zero for $r=0.054$ , 1.82% for $r=0.05$ , and 7.9% for $r=0.04$ .

3.1. Failing to hedge

Up to this point in the paper, we have assumed the individual has full information about the risks that they face and that they behave optimally in the face of this risk. In this case, we found above that Social Security is unable to confer any welfare gains (as a safety net against rare event risk), at least at the parameterizations that we have highlighted. Our baseline model highlights that an individual who saves optimally in the face of this risk does not need a Social Security program to help manage this risk.

In this section, we consider an alternative model. In actuality, for a variety of reasons, individuals may not solve a sophisticated dynamic hedging problem. For instance, individuals may lack information on the nature of rare event risk, and without any information on the distribution of the risk, they may choose to ignore it and instead base their financial plans on expected (average) returns. Similarly, in financial planning, people often say things like, “when saving over long time horizons, equities pay better than bonds on average” and then use this fact as a justification for basing financial projections on a rate of return to saving that equals the average long-run return to equities, without explicitly modeling rare event risk. Moreover, the complexity of optimal hedging may limit its attractiveness as a saving plan, with individuals instead opting for simple rules. Whatever the cause for failing to hedge, in this section we study the welfare effect of Social Security when rare event risk exists but is not incorporated into the household’s saving plan. Unlike the previous section which struggles to find a rationale for Social Security as a safety net, in this section, the welfare gains of Social Security are large.

We consider a household that plans consumption and saving to maximize lifetime utility, ignoring the possibility of rare event risk. The household solves the following deterministic problem:

(29) \begin{equation} \max _{c(t)} \int _{0}^{T}e^{-\rho t}\Psi (t) u(c(t))dt, \end{equation}

subject to

(30) \begin{align} \dot{k}(t)=y(t)-c(t)+rk(t) \quad \textrm{for } t\in [0,T] \end{align}

(31) \begin{align} k(0)=0, k(T)=0. \end{align}

The solution to this problem is the planned consumption path

(32) \begin{equation} c^{\textrm{nh}}_1(t) = \frac{ \int _{0}^{T}e^{-rv}y(v)dv}{\int _{0}^{T}e^{-rv+(r-\rho )v/\sigma }\Psi (v)^{1/\sigma }dv}e^{(r-\rho )t/\sigma } \Psi (t)^{1/\sigma }. \end{equation}

The superscript $\textrm{nh}$ on $c^{\textrm{nh}}_1(t)$ indicates this is planned consumption with no hedging. The subscript on $c^{\textrm{nh}}_1(t)$ indicates that this is the consumption path the household follows until the rare event shock occurs (if it occurs at all).

If the rare event shock occurs during the household’s lifetime (at date $t$ ), we assume the household reoptimizes and follows the postshock consumption path $c^*_2(z|t,k(t),S(k(t)))$ for $z \in [t,T]$ , as given in equation (9). The consumption path $c^*_2(z|t,k(t),S(k(t)))$ is optimal conditional on shock date $t$ , assets $k(t)$ , and shock $S(k(t))$ . The assets the household owns at the time of the shock are determined by the law of motion equation (30) and the planned consumption path $c^{\textrm{nh}}_1(t)$ .

Since this household does not hedge the rare event risk, they are surprised if the shock occurs in their lifetime. The realized life-cycle consumption path of the household given shock date $t$ is $c^{\textrm{nh}}_1(z)$ prior to the shock for $z\lt t$ , and $c^*_2(z|t,k(t),S(k(t)))$ for $t\leq z\leq T$ following the shock.

We measure the welfare effect of Social Security from the perspective of a social planner who knows that the household does not hedge against the rare event risk. The social planner knows that the household will follow consumption path $c^{\textrm{nh}}_1(z)$ prior to the shock and the consumption path $c^*_2(z|t,k(t),S(k(t)))$ after the shock. The planner calculates ex ante expected utility by computing the life-cycle utility for the realized path of consumption the household would experience for every possible shock date and weighing each path by how likely the shock is to occur at that date. Ex ante expected utility is given by

(33) \begin{equation} \mathbb{E}(U^{\textrm{nh}}) = \int _0^{T} \left [ \left (\int _t^{\infty }\phi (t_1)dt_1 \right ) e^{-\rho t}\Psi (t)u(c^{\textrm{nh}}_1(t)) + \phi (t) U_2(t,k(t),S(k(t))) \right ] dt \end{equation}

where

(34) \begin{equation} U_2(t,k(t),S(k(t)))=\int _t^{T}e^{-\rho z}\Psi (z)u(c^*_2(z|t,k(t),S(k(t))))dz \end{equation}

and $c^*_2(z|t,k(t),S(k(t)))$ is given by (9).

Notationally, let $\mathbb{E}(U_{\textrm{SS}}^{\textrm{nh}})$ stand for expected utility in a world with Social Security and no hedging. Likewise let $\mathbb{E}(U_{\textrm{LF}}^{\textrm{nh}})$ stand for expected utility in a Laissez Faire world without Social Security and no hedging. With CRRA utility, the consumption equivalent variation is

(35) \begin{equation} \Delta ^{\textrm{nh}} =1-\left ( \frac{\mathbb{E}(U_{\textrm{LF}}^{\textrm{nh}})}{\mathbb{E}(U_{\textrm{SS}}^{\textrm{nh}})}\right ) ^{\frac{1}{1-\sigma }}. \end{equation}

The consumption equivalent variation is the percentage of lifetime consumption the individual who does not hedge rare event risk would be willing to give up to live in a world with Social Security relative to a Laissez Faire world with no Social Security. Note that our welfare comparison takes into consideration the distribution of rare event risk, even though the household itself does not incorporate this information into their consumption and saving plan. As in the previous section, a positive number indicates that Social Security makes the individual better off.

Using our baseline parameters, we find $\Delta ^{\textrm{nh}}=16.39\%$ . That is, a household who does not hedge rare event risk is made better off by participating in Social Security by an amount equivalent to about 16% of lifetime consumption. This large positive welfare effect stands in stark contrast to the negative welfare effect we found for households who optimally hedge rare event risk. Social Security is able to confer large welfare gains to households who do not hedge rare event risk by crowding out the risky assets the household owns and offering a safe asset (albeit, an asset with a low rate of return.). Households who do not hedge their rare event risk are made much better off in this scenario.

The positive welfare effect of Social Security is robust across several different parameter combinations, including when the shock distribution is calibrated to be an exponential distribution as in equation (28), in which case $\Delta ^{\textrm{nh}}=6.38\%$ .

We also conduct an alternative welfare comparison to see how much better off the household who does not hedge rare event risk would be if they were able to optimally hedge. Specifically, we compute the percentage of lifetime consumption the individual who does not hedge rare event risk would be willing to give up to optimally hedge rare event risk:

(36) \begin{equation} \Gamma =1-\left ( \frac{\mathbb{E}(U_{\textrm{LF}}^{\textrm{nh}})}{\mathbb{E}(U_{\textrm{LF}}^{*})}\right ) ^{\frac{1}{1-\sigma }}. \end{equation}

Here, $\mathbb{E}(U_{\textrm{LF}}^{\textrm{nh}})$ stands for expected utility following the no-hedging consumption path $c^{\textrm{nh}}_1(z)$ prior to the shock and the consumption path $c^*_2(z|t,k(t),S(k(t)))$ following the shock in the Laissez Faire economy. Similarly, $\mathbb{E}(U_{\textrm{LF}}^{*})$ indicates expected lifetime utility with optimal hedging.

This comparison allows us to see how much a household is harmed by not hedging rare event risk. Using our baseline parameters, we find the welfare gain of optimal hedging $\Gamma =27.44\%$ . The welfare gain of optimal hedging in Laissez Faire essentially shows the size of the problem of ignoring rare event risk. Participating in Social Security is not as good as optimally hedging, but does a reasonably good job helping households hedge rare event risk (by providing a safe asset). Recall, the welfare gain of Social Security is 16.39% for households that do not hedge rare event risk. Hence, the welfare gain of Social Security is about 60% of the welfare gain of optimally hedging, for households that ignore rare event risk.

Using the exponential shock distribution as in (28), we find the welfare gain of optimal hedging $\Gamma =14.14\%$ in Laissez Faire. With this parameterization, Social Security is able to generate about 45% of the welfare gain of optimally hedging rare event risk.

3.2. Misspecification of risk

Rare event shocks are, by definition, seldom observed in reality. Therefore, it may be difficult for decision makers to know the distribution of the risk. To infer the distribution, one would need to observe many historical instances. Such inference is possible with high frequency, stationary risk but not with truly rare events.

To further connect our model to reality, in this section, we explore the possibility that the true distribution of timing risk is unknown to the individual decision maker. Instead, the individual makes decisions to hedge risk that are based upon an incorrect distribution of the risk. This exercise tells us how the welfare gains from Social Security are affected by risk misspecification, which seems especially relevant given the difficulty of drawing accurate inferences about the distribution of rare event risk.

We assume the risk of the rare event is distributed exponentially as in equation (28)

\begin{equation*} \phi (t)=\gamma e^{-\gamma t}\text {, for }t\in \lbrack 0,\infty ]\text { and }\gamma \gt 0. \end{equation*}

Suppose the true distribution has a parameter $\gamma =-\ln (0.5)/82$ which corresponds to a 50% chance of the shock occurring within the individual’s maximum lifespan. However, the individual decision-maker does not know $\gamma$ and assumes the shock is instead distributed

(37) \begin{equation} \phi (t)=\omega e^{-\omega t}\textrm{, for }t\in \lbrack 0,\infty \rbrack, \end{equation}

where $\omega$ is a multiple of $\gamma$ . We consider four cases: $\omega =4\gamma$ , $\omega =2\gamma$ , $\omega =1/2\gamma$ , and $\omega =1/4\gamma$ , which correspond to the individual believing the shock will occur within the maximum lifespan with probability 0.94, 0.75, 0.29, and 0.16, respectively.

We solve for the individual’s optimal consumption-saving rule just like in Section 2, but now we assume the individual applies the wrong distribution of risk [equation (37)] and therefore optimizes the wrong expected utility function. Hence, the individual is sophisticated in their optimization behavior but they lack knowledge of the true distribution of risk. Then, in the welfare analysis, we apply the true distribution of risk [equation (28)] in our assessment of the welfare effect of Social Security.

Our results are presented in Table 2. We find that the welfare effect of Social Security is decreasing in the perceived likelihood of the rare event shock occurring. That is, individuals who believe the shock is more likely to occur than the true distribution are harmed more by Social Security than if they knew the risk. Similarly, individuals who do not believe the shock is as likely to occur are less harmed by Social Security. This is because Social Security is more helpful (less negative welfare effect) for households who are not as prepared for a rare event to strike. And for those who overestimate the risk they face—and therefore “over hedge”—adding an additional safety net is problematic.

Table 2. Welfare effect of Social Security, given different levels of misspecification of risk

3.3. Maximin policy making

In this section, we consider a final permutation on the information structure of the problem. We imagine a policy maker who does not have reliable information on the distribution of rare event risk. They know rare event risk exists, but do not have enough information to construct a distribution for policy evaluation. Hence, as an example, we study the case in which the policy maker evaluates the success of the Social Security safety net based upon its ability to insure against the worst case scenario, which, in our model would be the realization of the timing shock at the moment of retirement when the individual’s wealth is at a peak. Such unfortunate timing would create the largest possible loss in wealth to the individual, and we assume for the moment that this is the very scenario that the policy is designed to address. Hence, the policy maker evaluates the success of Social Security based upon its performance under this specific realization of risk.Footnote ¹¹

The policy maker calculates maximin expected utility as

(38) \begin{equation} U^{\textrm{mm}}=\int _{0}^{T}e^{-\rho t}\Psi (t) u(c^{\textrm{mm}}(t))dt \end{equation}

where $c^{\textrm{mm}}$ indicates the consumption an individual would experience if the shock hits at the date of retirement:

(39) \begin{equation} c^{\textrm{mm}}= \begin{cases} c^*_1(t) & t\lt t_R\\[5pt] c^*_2(t|t_R,k(t_R),S(k(t_R))) & t\geq t_R. \end{cases} \end{equation}

The welfare effect of Social Security from the perspective of the policy maker is the fraction of lifetime consumption the individual would give up to participate in Social Security (given the individual hedged optimally, but the shock happened exactly at the date of retirement),

(40) \begin{equation} \Delta ^{\textrm{mm}} =1-\left ( \frac{(U_{\textrm{LF}}^{\textrm{mm}})}{(U_{\textrm{SS}}^{\textrm{mm}})}\right ) ^{\frac{1}{1-\sigma }}. \end{equation}

We find the welfare effect of Social Security to be −2.28% using our baseline parameters and the uniform shock distribution. The welfare effect is less negative compared to the baseline, because the policy maker is only considering the worst possible shock date, rather than forming expected utility based on all possible shock dates. Social Security still reduces welfare because the internal rate of return is low compared to the risky asset—even when the shock hits at the worst possible moment.

However, the welfare effect is 3.55% using the exponential shock distribution. Social Security is welfare enhancing in this scenario because the individual optimally hedged risk under the assumption that the shock would only occur within their lifetime with probability 0.5, whereas the policy maker evaluates Social Security assuming the shock hits at the worst possible moment with probability 1.

Recall that the welfare effect of Social Security is −7.42% when the welfare criteria is calculated using expected utility [equation (27)] and the rare event risk is distributed exponentially. However, we have just shown, that if the welfare criteria is instead a maximin criteria—comparing utility of the worst possible realization of the shock rather than taking an expectation over all possible realizations of the shock—Social Security is welfare improving by the equivalent of 3.55%. In both cases, behavior is fully rational and the individual optimally hedges rare event risk. The only difference is which possible lifetime utility paths the policy maker includes in their welfare criteria. If all possible paths are included (and weighted according to the distribution of risk), Social Security is welfare reducing; however, if only the worst path is included, Social Security is welfare improving.