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While these results are encouraging for proponents of equity investment by the Social Security system, we note two caveats. First, lower Social Security tax rates reduce welfare by 0.6% of consumption in a model in which investors are extremely impatient, with a low time discount factor of 0.8, but the government judges their welfare using a higher time discount factor of 0.96. This calculation is a crude way to capture factors that might lead to inadequate private saving and justify a mandatory retirement saving system. In this model, however, there is a substantial gain of 2.0% from Social Security equity investment with fixed tax rates. These findings suggest that the appropriate adjustment of tax rates will depend on detailed assumptions about the behavior of households, but that under a wide range of assumptions there are welfare gains to be had by investing some retirement wealth into equities.
Second, a system with partial investment of retirement wealth in the equity market has greater variability of outcomes across cohorts. Particularly negative outcomes for some cohorts might provoke pressure for political bailouts, and the anticipation of bailouts might change the consumption behavior modelled here. This is an important issue for research on Social Security reform.
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INVESTING RETIREMENT WEALTH: A LIFE-CYCLE MODEL - Welfare Analysis 4The top panel of Table 9 repeats the first three rows of Table 7 for the benchmark case. The second and third panels consider two alternative scenarios in which the equity premium is 1 percentage point lower at 3%. In the first alternative, the equity premium falls but the riskless interest rate is unchanged, so the expected equity return falls by 1 percentage point. This is a scenario envisaged by critics of Social Security investment in equities who worry that such a reform would drive up stock prices and drive down expected stock returns. In the second alternative, the equity premium falls but the riskless interest rate rises by 1 percentage point, leaving the expected equity return unchanged. This scenario is predicted by general equilibrium models in which the return to risky capital is fixed by technology, such as Cox, Ingersoll, and Ross (1985) or Abel (1999).
Within each of the alternative scenarios, the welfare gains produced by risky investment of retirement wealth are similar to but slightly smaller than those in the benchmark case. In the rows marked with double asterisks, Table 9 compares welfare in the alternative scenarios with risky investment of retirement wealth, to welfare in the benchmark case with riskless investment of retirement wealth. This is a crude way to capture the possibility that risky investment of retirement wealth might lower the equity premium. It turns out that the results are critically dependent on the way in which the equity premium falls. If it falls through lower stock returns, as in the first alternative, then welfare gains are reduced from 3.7% to 2.8%. If it falls through a higher riskless rate, as in the second alternative, then welfare gains are actually increased to 5.0%.
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The last panel of Table 7 reports results for a household headed by a self-employed college graduate. This household has risky labor income that is unusually correlated with the stock market, so its desired stock holdings are smaller than those of the benchmark household; it actually loses slightly from investment of retirement wealth in equities with a fixed tax rate. On the other hand, this household also has a particularly pronounced hump shape in labor income so it is particularly anxious to smooth consumption over the life cycle. The household gains 3.8% from a risky retirement system with a lower payroll tax rate, comparable to the results for the benchmark household.
All the results in Table 7 can be criticized on the grounds that they are derived from a model in which there is no role for a Social Security system. Since households save and invest their liquid wealth optimally, the mandatory saving and rigid asset allocation of the retirement system can only reduce their welfare. In this setting, any reform that effectively reduces the scale of Social Security will increase welfare.
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INVESTING RETIREMENT WEALTH: A LIFE-CYCLE MODEL – Effects of Heterogeneity on Portfolio Choice

INVESTING RETIREMENT WEALTH: A LIFE-CYCLE MODEL - Effects of Heterogeneity on Portfolio ChoiceIn this section we illustrate the effects of investor heterogeneity on optimal consumption and portfolio choice. First, we consider heterogeneity of preferences, calculating optimal behavior for highly risk-averse investors with 7 = 10 and impatient investors with B = 0.8. Second, we consider differences in labor income risk of the sort illustrated in section 4.1. To highlight these differences, we simulate the behavior of households whose income is particularly risky and highly correlated with asset returns: self-employed college graduates.
Table 6 shows average consumption and liquid wealth (in thousands of dollars) and the share of liquid wealth invested in stocks for different age groups. This is a more compact way for us to summarize the information presented graphically in Figures 2 through 4. The first three columns of Table 6 use the baseline parameters and consider retirement systems with aR = 0; aR = 0.5 and the same 10% tax rate as with aR = 0; and aR = 0.5 and a lower 6% tax rate that maintains the same average replacement ratio as with aR = 0.
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INVESTING RETIREMENT WEALTH: A LIFE-CYCLE MODEL – Measuring Heterogeneity in Labor Income 2

Table 3 reports the total variance of income, and its decomposition into permanent and transitory components, for each different education-industry cell. Agriculture has by far the highest variance of labor income shocks. Other industries subject to significant labor income shocks are construction and business services. The variance decomposition indicates that labor income shocks for construction workers without a high school degree are entirely temporary. At the other extreme, permanent income shocks are especially important for college graduates in public administration. As a general pattern, the relative importance of permanent shocks seems to increase with educational attainment. This was already documented for the column totals, but seems robust within individual industries.
The bottom of Table 3 splits the sample in a different way, by distinguishing selfemployed from non-self-employed households. (We included both types of households in the industry analysis, since there are too few self-employed households to allow an industry decomposition.) Income variability is dramatically larger for self-employed households. Income shocks are entirely temporary for the self-employed without a high school degree, but are disproportionately permanent for the self-employed in the two higher education groups.
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INVESTING RETIREMENT WEALTH: A LIFE-CYCLE MODEL - A Graphical Summary 2Figure 3 plots liquid wealth and liquid holdings of equities and bills over the life cycle. In each retirement system the borrowing constraint binds for young households; they would like to take more equity risk but are unable to do so. For approximately the first 20 years of life, they hold 100% of their portfolios in the form of equity. Households in midlife hold bills, but these holdings decrease again after retirement.
Figure 4 plots the portfolio share of stocks in liquid wealth. The crucial variables for portfolio composition are liquid wealth, retirement wealth and future labor income. In the model future labor income, although risky, can be thought of as implicit holdings of a riskless asset. Innovations to labor income are positively correlated with innovations to stock returns, but this correlation is not sufficiently large for future labor income to resemble more closely stocks than bills. Since early in life the implicit holdings of the riskless asset in the form of future labor income are large, the investor wishes to invest what little liquid wealth he has fully into stocks.

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The first comparison we consider is between a system with riskless retirement accumulation (ад = 0) and one in which at each age half of retirement wealth is invested in stocks (ад =.5). In the latter system, we reduce the social security tax rate from 10% to 6% so that on average the replacement ratio is the same in both systems and equal to 0.6. At retirement, the account is annuitized at the riskless interest rate, so that on average, and given survival probabilities, the system has zero balance.
To study the behavior of the variables in the model, we calculate cross-sectional averages across 10,000 households receiving different draws of income and asset returns, and plot them against age. Figure 2 plots labor income net of social security contributions, consumption, liquid wealth and retirement wealth for households with a high-school degree (the life-cycle patterns for other education groups are similar). Figure 2a illustrates the system in which retirement wealth is fully invested in the riskless asset, and Figure 2b illustrates the system in which retirement wealth is partially invested in stocks.
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Return risk also increases the dispersion of utility. The standard deviations of current utility across households with different income and return realizations are higher in the 50/50 system than in the 100/0 system. Figure 5b reports the ratio of these standard deviations in the two systems. These ex-post differences raise important practical issues for designers of retirement systems because they may create an incentive to bail out cohorts negatively affected by lower stock return realizations.
Of course, a proper welfare analysis requires the discounting of current utility over the life cycle. We defer such an analysis to section 5 below.
One limitation of the results reported so far is that they counterfactually predict 100% stock market participation among younger investors. However we can modify this prediction, with little effect on other aspects of the model, by adding a fixed cost of stock market participation. Figure 6 reports results with a $10,000 fixed cost. The fraction of households who have paid the fixed cost and the average share of assets invested in stocks are plotted for each retirement system. Early in life the two series move almost perfectly together, showing that young investors are either entirely in or entirely out of the market; later in life all investors have paid the fixed cost, and the model behaves much as it did in the absence of the cost.
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INVESTING RETIREMENT WEALTH: A LIFE-CYCLE MODEL - Other parametersAs an empirical measure for the excess return on our stylized risky asset, we use CRSP data on the New York Stock Exchange value-weighted stock return relative to the T-bill rate. For all education groups, the regression coefficients are strikingly low and insignificant. To allow for potential lags in the realization of labor income, we repeat the exercise with the excess stock return lagged one year. The relationship becomes much stronger: the regression coefficient now varies from 0.06 to 0.10, and the correlation coefficient from 0.32 to 0.52, as reported in Table 1. Interestingly, the correlation of labor income with the stock market is larger and more significant for households with higher education.
In our portfolio choice model, allowing for lags in the relationship between innovations in stock returns and permanent shocks to labor income unfortunately requires an additional state variable. We therefore assume the correlation is contemporaneous. The model requires the variances of both 1, the aggregate permanent labor income shock that is correlated with stock market risk and шц, the idiosyncratic permanent shock to labor income. The first variance is obtained immediately as the variance of Alog(Y*). Subtracting this variance from the total variance of uit gives then the variance of wit.
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