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We let t denote adult age. The investor is adult for a maximum of T periods, of which he works the first К. For simplicity К is assumed to be exogenous and deterministic. We allow for uncertain life-span in the manner of Hubbard, Skinner and Zeldes (1994). Let r denote the probability that the investor is alive at date t + 1, conditional on being alive at date t. Then, investor i’s preferences are described by the time-separable power utility function:
Formula 1
where Си is the level of date t consumption, 7 > 0 is the coefficient of relative risk aversion, and B < 1 is the discount factor. We assume that the individual derives no utility from leaving a bequest.
Investor i’s age t labor income, Yt, is exogenously given by:
Formula 2
where f (t, Zit) is a deterministic function of age and other individual characteristics Zu, Su is an idiosyncratic temporary shock distributed as N(0,j), and vu is given by
Formula 3
where uu is distributed as N(0,j) and is uncorrelated with Su. Thus log income is the sum of a deterministic component that can be calibrated to capture the hump shape of earnings over the life cycle, and two random components, one permanent and one transitory. We assume that the temporary shock Su is uncorrelated across
Formula 4
This decomposition implies that the random component of aggregate labor income follows a random walk, an assumption made by Fama and Schwert (1977) and Jagannathan and Wang (1996). While macroeconomists such as Campbell (1996), Campbell and Mankiw (1989), and Pischke (1995) have found empirical evidence for short-term persistence in aggregate quarterly labor income growth, the simplification to a random walk should have little effect on optimal consumption and portfolio choice over the life cycle.
households, but we decompose the permanent shock u into an aggregate component 1 and an idiosyncratic component шц, uncorrelated across households:
We assume that there are two assets in which the agent can invest: a riskless asset with gross real return —, which we call Treasury bills, and a risky asset with gross real return —, which we call stocks. The excess return on the risky asset, —t+i — —, is given by:
Formula 5
where 1, the period t + 1 innovation to excess returns, is assumed to be i.i.d. over time and distributed as N(0,j). We allow innovations to excess returns to be correlated with innovations to the aggregate component of permanent labor income, and we write the correlation coefficient as p£V. We also allow for fixed costs of equity market participation: to have access to the stock market the investor must pay a one-time monetary fixed cost equal to F.