The future of Social Security is in question. Even Federal Reserve chairman Ben Bernanke warns of the rapidly approaching Social Security “fiscal crisis.”. Individuals at the beginning or middle of their prime working years are unsure of how large (or small) their Social Security benefits will be when they retire.
An NBER working paper by Gomes, Kotlikoff and Viceira (WP #12859) aims to calculate the cost of this uncertainty due to government indecision regarding Social Security benefits. Below I describe the model the authors derive, briefly explain the empirical section of the paper and state their conclusions.
Model
Individuals have constant relative risk aversion (CRRA) utility functions of the form:
- U=C1-γ(1-γ)-1 (1)
The agents learn at time L whether the social security benefits they will receive in retirement will be large, B, or small, b. AL are the assets accumulated at time L. T represents the last year of one’s life and R represents the year of retirement. Optimal consumption is given by:
- AL+B(T-R)=CB(T-L), with a large retirement benefit; and (2a)
- AL+b(T-R)=Cb(T-L), with a small retirement benefit. (2b)
- AL+=A0-CL (3)
Expected Utility and first order conditions are given by:
- EU= (1-γ)-1{ C1-γL + (T-L)[pCB1-γ + (1-p)Cb1-γ] } (4)
- FOC: C-γ= + pCB-γ + (1-p)Cb-γ (5)
One can solve equations (2a) and (2b) for CB and Cb respectively, plug equations (2a), (2b) and (3) into (4), and take the derivative with respect to C to arrive at equation (5). The authors aim to estimate how the agents’ expected utility changes when the date (L) when the government reveals the whether the benefit level is high (B) or low (b) changes. More formally, the authors find that ¶EU/¶L < 0 when the individual is risk averse (i.e.: γ>1). In words, expected utility decreases when the date the individual is informed of the social security benefits is moved further into the future.
The authors model income as a concave, quadratic function of age which has an error term which is MA(1) with probability π and ln(0.1)»-2.3 with probability (1-π). Thus, income is hump-shaped throughout one’s lifetime with some persistence in the idiosyncratic error term. Also, there is some probability of a negative income shock, which can be thought of as the probability an agent loses their job. The individual invests their savings optimally at each age, dividing their assets between stocks and a risk-free asset.
The authors then calibrate their model using parameter estimates from other sources. The authors conclude that the excess burden of government indecision can be as high as 0.6 percent of the agent’s economic resources. Individuals who are a) more risk averse, b) have more income uncertainty, c) face a larger cut in benefits or d) have higher marginal tax rates are the subpopulations most adversely affected by the government indecision.
While I am usually skeptical of life-cycle model papers which predict the future using calibration and do not feel the author’s estimates are very precise, it is important to realize that increased uncertainty due to delays in social security reform can lead to suboptimal asset allocation and consumption decisions by individuals planning for retirement.