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Special thanks are due to Christopher Sims. I am also grateful to Luis Cubeddu, Hamid Faruqee, Wouter Den Haan, Peter Heller, Peter Isard, Ivailo Izvorski, Narayana Kocherlakota, Douglas Laxton, and Eswar Prasad for helpful comments. Any errors are mine.
For papers that use the historical distribution of asset returns to compute gains from switching to individual retirement accounts see, for example, MaCurdy and Shoven (1992) and Feldstein and Ranguelova (1998).
A more recent discussion of the role of labor income as a non-traded asset and its impact on investment behavior over the life cycle can be found in Campbell, Cocco, Gomes, and Maenhout(1999).
The approach in this paper contrasts with Bakshi and Chen (1994) who hypothesize that agents become more risk averse with age. They find that the average age of the US population is positively correlated with future excess returns on stocks over treasury bills.
The model has two sources of aggregate uncertainty: the technology shock to production, and the population growth rate. An investor deciding on how much equity to hold going forward into period t+1 does not know the realization of the period t+1 technology shock that determines ret+1. In contrast the return on the riskless asset in period t+1 is know in period t. Hence it is denoted rft, to emphasize that it is in agents’ period t information set. Similarly, thought enters production in period t+1, it is determined by investment decisions made in period t and is therefore also in agents’ period t information set. The stochastic population growth rate represents aggregate uncertainty over cohort size. An unexpectedly large cohort of children has repercussions for agents’ consumption-investment decision, as they react to a larger consumption requirement in young working-age and the larger labor force in subsequent periods.
Note that the conditional expectations in (13) and (16) have not been parameterized in the traditional manner. Both equations have been multiplied through by functions of the respective equity holdings. This modification is based on Izvorski (1997) and addresses an indeterminacy in the system of Euler equations and aggregate equilibrium conditions that arises in models that solve for equilibrium holdings of two or more assets.
The implementation of the PEA follows Den Haan and Marcet (1990). Rather than performing a computationally expensive non-linear least squares estimation to find τe, it takes a first-order approximation of Ψ(Θt, τn) around τn. Rearranging terms τe is then the coefficient vector in an OLS regression.
The accuracy test is implemented in the following manner. Given τ*, γ*, ξ*, and ω* at convergence, the model is simulated N times, each time for different draws of the technology shock and the age distribution. For these N simulations, the frequency with which the G statistic is greater than the critical value of the 95th percentile of a χ220 is reported. If the percentage of G statistics above the critical value of the 95th percentile is substantially greater than five percent, this is evidence against accuracy of the solution.
Preference and production parameters are chosen to reflect the fact that each period corresponds to roughly twenty years. β is set to 0.6, while θ is set to one. Period utility therefore takes the form u(c) = ln(c). On the production side, the share of output that goes to capital, a, is set to 0.3, while the rate of depreciation of capital, δ, is taken to be 0.4. π set at 0.2 in the specification with social security. In the stochastic simulations the stationary technology shock follows lnAt = ϕlnAt-1 + εt where ϕ = 0, εt ~N(0,σε) and σε = 0.1. The period t cohort of children is generated according to lnNt= ρlnNt-1 + υt, with vt~N(0,σv), ρ = 0.99, and σv = 0.01, Period t population growth is then backed out according to Nt =(1+nt)Nt-1.
In the perfect foresight case, the model has only one state variable: the capital-labor ratio. Imposing the equilibrium condition that net saving equal net investment, a non-linear equation solver, written by Christopher Sims for Matlab, is used to solve for the steady state capital-labor ratio.
In this specification the two first-order conditions relating to bond holdings fall away, leaving only the conditional expectations in (12) and (15) to be parameterized. The set of period t state variables remains Θt.
The values for τ, γ, ξ, and ω are taken to have converged when (τn- τe)/τn, (γn - γe)/γn, (ξn - ξe)/ξn, and (ωn - ωe)/ωn are each less than 0.00001.
To generate the PEA solution to the model the sequences for technology and the age distribution are drawn for length T=1000, The same draws are used across specifications for easier comparison. The results remain qualitatively unchanged for sequences of much greater length.
The simulated boom-bust is similar to the post-war demographic transition in many developed countries (see Brooks (1998)). The results below are qualitatively unchanged for other forms of demographic shift, a simple baby boom for example.
Of course this result depends on the parameter choice for the elasticity of intertemporal substitution.
This result highlights an omission of the model, which does not model the decision to have children. If agents derive happiness from having children, this welfare result might well be reversed.
It is worth noting that the effect on stock returns is small compared to the run-up in stock indices over the past 20 years. The average real return on the Ibbotson Associates large stock index from 1979 - 1998 is 13.43 percent, relative to 3.53 percent for the period 1959 – 1978. In comparison, the period t+2 return on capital is only 3.8 percent above steady state. Changes in the age distribution are thus an insufficient explanation for the recent surge in stock prices. Of course this simulation exercise ignores changes in other fundamentals (the technology shock is fixed at its mean value of one), or the possibility of a speculative bubble.
A different version of the pay-as-you-go pension system, in which the retirement benefit is exogenous and the payroll tax rate endogenous, might do a better job at insuring agents against cohort-specific risk.
The reversal of the welfare result stems from the concavity of period utility, and would not obtain if period utility were linear in consumption, for example. As noted above, the model abstracts from agents’ decision to have children. As such the discussion on the welfare effects of the boom-bust should be seen as incomplete.