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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.
In both systems, the average consumer is borrowing-constrained early in life. Consumption tracks net income very closely and little savings accumulate outside the retirement account until after age 40. These limited savings early in life are driven by the precautionary savings motive; thus like Gourinchas and Parker (1996) we find that younger consumers are buffer-stock savers rather than life-cycle savers in the classic sense. Consumption rises with income early in life because of borrowing constraints, and falls later as increased mortality drives up the effective rate of time preference; thus consumption profiles are hump-shaped over life, as found in the literature on life-cycle consumption behavior.
Investment of some retirement wealth in stocks has an income effect. Because the average return on stocks is higher than the average return on bills, and because younger consumers have neither the desire nor the liquid wealth to offset a shift of retirement wealth into stocks, the shift increases average lifetime resources. Since we reduce the social security tax rate to keep the average replacement ratio constant across systems, the investment of retirement wealth in stocks frees up resources in the working years. These additional resources are consumed early in life, since at this stage households are borrowing-constrained. Of course the investment of retirement wealth in stocks has a cost: It imposes additional risk on households. In midlife households react by increasing their precautionary saving, accumulating more liquid wealth and consuming less relative to income. After retirement the additional wealth is run down, since at this stage retirement wealth is rolled into a riskless annuity. These patterns show up in the paths of consumption relative to income in Figure 2.


Figure 2a – Consumption, Income, Wealth and Annuity (Retirement Wealth Fully Invested in the Safe Asset)


Figure 2b – Consumption, Income, Wealth and Annuity (Retirement Wealth Invested 50/50 in Risky/Safe Asset)