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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.
In the previous section we studied a representative household with a high school education but no college degree. Results are similar for representative households in the other two education groups. However households may differ along other dimensions. For example, labor income processes may differ for households who work in different industries, and for self-employed households. Also some households may be more impatient or risk-averse than others as found by Barsky, Juster, Kimball, and Shapiro (1997). These differences across households may have important effects on optimal investment strategies. In this section we consider this issue.


Figure 5b – Utility Dispersion (Ratio of Standard Deviations)

Figure 6a

Figure 6a – Liquid Portfolio Share invested in Stocks and Participation Rate (Retirement Wealth Fully Invested in the Safe Asset & Fixed Cost of Entering in the Stock Market)

Figure 6b

Figure 6b – Liquid Portfolio Share invested in Stocks (Retirement Wealth Invested 50/50 in Risky/Safe Asset & Fixed Cost of Entering the Stock Market )