"Population Forecasts, Fiscal Policy, and Risk"

This paper describes how stochastic population forecasts are used to inform and analyze policies related to government spending on the elderly, mainly in the context of the industrialized nations. The paper first presents methods for making probabilistic forecasts of demographic rates, mortality, fertility, and immigration, and shows how these are combined to make stochastic forecasts of population number and composition, using forecasts of the U.S. population by way of illustration. Next, the paper discusses how demographic models and economic models can be combined into an integrated projection model of transfer systems such as social security. Finally, the paper shows how these integrated models describe various dimensions of policy-relevant risk, and discusses the nature and implications of risk in evaluating policy alternatives.

Derivatives of the Stochastic Growth Rate

We consider stochastic matrix models for population driven by random environments which form a Markov chain. The top Lyapunov exponent $a$, which describes the long-term growth rate, depends smoothly on the demographic parameters (represented as matrix entries) and on the parameters that define the stochastic matrix of the driving Markov chain. The derivatives of $a$ -- the "stochastic elasticities" -- with respect to changes in the demographic parameters were derived by \cite{tuljapurkar1990pdv}. These results are here extended to a formula for the derivatives with respect to changes in the Markov chain driving the environments. We supplement these formulas with rigorous bounds on computational estimation errors, and with rigorous derivations of both the new and the old formulas.Comment: 35 page

Life, Death, and the Economy: Mortality Change in Overlapping-Generations Model

Demographers have shown that there are regularities in mortality change overtime, and have used these to forecast changes due to population aging. Such models leave out potential economic feedbacks that should be captured by dynamic models such as the general-equilibrium, overlapping-generations model first studied by Yaari and Blanchard. Previous analytical and simple numerical work by economists has focused on comparative statics and used simplistic representations of mortality, such as the assumption of a constant age-independent death rate, or some parametric approximation to a survival curve. We show that it is straight forward to analyze equilibria in such models if we work with the probability distribution of the age at death. US and other data show that this distribution can be plausibly described by a normal distribution {for this case we obtain analytical results. For the general case we have numerical results. We show that a proper accounting for the uncertainty of when one dies has significant qualitative and quantitative effects on the equilibria of such economic models. There are, in turn, significant lessons to be drawn for models of future fiscal policy.

Stochastic Rates of Return for Social Security Under Various Policy Scenarios

In this paper, we compute distributions of rates of return by cohort for the Social Security retirement system, using a combination of historical data and stochastic forecasts of productivity and mortality rates. Since our forecasts of productivity and mortality are stochastic, the rate-ofreturn estimates themselves are stochastic; that is, we compute an entire distribution of rates for each cohort. We repeat these calculations under a variety of policy scenarios designed to bring the trust fund into future solvency with roughly 50% probability. This allows us to examine the impact of various schemes on different cohorts. Policies which delay reform the longest and impact taxpayers the greatest dramatically concentrate the impact of reform on the youngest cohorts. Reforms which are more immediate and focused on retirees tend to spread the cost of reform across generations more evenly. Authors’ Acknowledgements This research was funded by a grant from the Michigan Retirement Research Center, which is in turn supported by the Social Security Administration. Some closely related research was also supported by the Center for the Economics and Demography of Aging at U.C. Berkeley, which is supported by the National Institutes of Health.

Random Scenario Forecasts Versus Stochastic Forecasts

Probabilistic population forecasts are useful because they describe uncertainty in a quantitatively useful way. One approach (that we call LT) uses historical data to estimate stochastic models (e.g., a time series model) of vital rates, and then makes forecasts. Another (we call it RS) began as a kind of randomized scenario: we consider its simplest variant, in which expert opinion is used to make probability distributions for terminal vital rates, and smooth trajectories are followed over time. We use analysis and examples to show several key differences between these methods: serial correlations in the forecast are much smaller in LT; the variance in LT models of vital rates (especially fertility)is much higher than in RS models that are based on official expert scenarios; trajectories in LT are much more irregular than in RS; probability intervals in LT tend to widen faster over forecast time. Newer versions of RS have been developed that reduce or eliminate some of these differences.

Stochastic Forecasts of the Social Security Trust Fund

We present stochastic forecasts of the Social Security trust fund by modeling key demographic and economic variables as historical time series, and using the fitted models to generate computer simulations of future fund performance. We evaluate several plans for achieving long-term solvency by raising the normal retirement age (NRA), increasing taxes, or investing some portion of the fund in the stock market. Stochastic population trajectories by age and sex are generated using the Lee-Carter and Lee- Tuljapurkar mortality and fertility models. Interest rates, wage growth and equities returns are modeled as vector autoregressive processes. With the exception of mortality, central tendencies are constrained to the Intermediate assumptions of the 2002 Trustees Report. Combining population forecasts with forecasted per-capita tax and benefit profiles by age and sex, we obtain inflows to and outflows from the fund over time, resulting in stochastic fund trajectories and distributions. Under current legislation, we estimate the chance of insolvency by 2038 to be 50%, although the expected fund balance stays positive until 2041. An immediate 2% increase in the payroll tax rate from 12.4% to 14.4% sustains a positive expected fund balance until 2078, with a 50% chance of solvency through 2064. Investing 60% of the fund in the S&P 500 by 2015 keeps the expected fund balance positive until 2060, with a 50% chance of solvency through 2042. An increase in the NRA to age 69 by 2024 keeps the expected fund balance positive until 2047, with a 50% chance of solvency through 2041. A combination of raising the payroll tax to 13.4%, increasing the NRA to 69 by 2024, and investing 25% of the fund in equities by 2015 keeps the expected fund balance positive past 2101 with a 50% chance of solvency through 2077.