ANNEX: An Overlapping Generations Model of Public Pensions
This annex presents an overlapping generations model of the effects of a public pension system. The overlapping generations framework is an obvious candidate for such an analysis since it takes explicit account of the interactions across age groups, making it particularly easy to consider the underlying structure of inter-generational transfers. In order to make the analysis more tractable two important simplifying assumptions are made. The model has no production technology, hence there is no consideration of the effect of a pension system on the supply side of the economy. 1/ The model is also limited to a small open economy facing a fixed real interest rate; hence, changes in factor returns caused by demographic shifts are ignored.
Consider an overlapping generations model in which each generation is made up of nt identical individuals who live for two periods, t and t+1. In period t, their working years, they earn an endowment et. They retire in period t+1 and earn no endowment. They consume c1t in period t and c2t+1 in order to maximize a utility function U(c1t, c2t+1) which is the same across generations. They are able to borrow and lend in asset markets using a fixed real interest rate 1+R.
The government provides a compulsory public pension scheme in which each individual pays αtet of their initial endowment to the government in period t and receives benefit βtet+1 back in period t+1. Hence, benefits at rate βt of endowment in t+1 are paid to generation t in period t+1. The government has access to the same asset markets as private individuals. The pension scheme must be solvent in the long run. This means that if its assets are defined by:
then lim At ≥ 0 as t → ∞.
The maximization problem for the individual is:
Since the real interest rate is fixed, utility depends only on the present discounted value of the endowment available to each individual, as shown in the indirect utility function:
Since utility only depends upon the discounted value of endowments, the pension scheme will improve or reduce utility depending on whether the benefit rate βt is greater than or less than (l+R)atet/et+1, the contribution rate adjusted for the real interest rate and growth in endowment.
Two possible funding schemes for the public pension system are considered. The first is an actuarially fair system in which all individuals receive the accumulated value of their contributions. In this case the relationship between the contribution rate αt and the benefit rate βt
Substituting this into equation (3) gives V(et) regardless of the values chosen for αt and βt. Hence, an actuarially fair system has no impact on the equilibrium. The reason for this result is that individuals have access to the same asset markets as the government. Accordingly, any public pension system which simply reallocates endowment over time is redundant, since Individuals can carry out the same transaction in private markets.
The analysis becomes more complex when pay-as-you-go funding schemes are considered. For a pay-as-you-go system the relationship between the contribution rate αt and the benefit rate βt is:
which can be rewritten as:
In this case, the equilibrium is a dynamic one in which the benefits for one generation depend upon the contribution rate for the next generation and the relative size of the generations. As a result, the introduction of a pay-as-you-go system always benefits the first generation, since they receive benefits without paying any money into the system. (Analogously, if the system is terminated, there is always a generation which loses.)
Turning to the situation of a mature system, in which participants both pay into the system and receive benefits from it, consider a steady state equilibrium in which population grows at rate (1+p), endowment grows at rate (1+g), and both contribution and benefit rates are kept fixed across generations (αt=αt+l,βt=βt+l). In this case the benefit rate βt is equal to α(l+p), the contribution rate adjusted for the rate of growth of population. A pension system raises endowment (and hence utility) if the benefit rate is greater than that implied by a fully funded system (α(l+R)/(l+g)), hence a pay-as-you-go system with a fixed benefit and contribution rate raises welfare if the rate of growth of the aggregate endowment ((1+p)(1+g)) is greater than the real interest rate (1+R). The intuition is that, when the present value of output is rising, it is possible to raise the welfare of all generations by systematically transferring current resources from the young to the old. 1/
Next, consider the effect of a temporary disturbance to the endowment (a productivity shock). Specifically, assume that there is a temporary fall in endowment et below its steady state level. Since contributions for any given generation depend upon their own endowment while benefits depend upon the endowment of the next generation, this will affect the absolute level of benefits received. (This explains why the levels of endowment in periods t and t+1 both enter equation (3).) Compared to autarky, the overall effect is to raise the welfare of the generation which has the lower endowment, at the cost of lowering welfare to the previous generation. 2/ In short, a pay-as-you-go public pension system can "smooth" the effects of a shock to the endowment across generations. Since the utility function is concave, this implies that a pay-as-you-go system raises expected aggregate welfare across generations. 3/
Now consider the impact of a larger-than-expected generation, nt. Since the public pension system is being operated in a pay-as-you-go manner it is impossible to keep both contribution rates and benefits rates unchanged over time, as can be seen from equation (5'). The effect of the demographic shift depends upon whether it is the contribution rate or benefit rate which is assumed to remain unchanged over time. Consider the case when benefit rates are kept fixed over time and contribution rates vary. From equation (5') this implies that contribution rates are lower than average for the large generation and higher than average for the following one. Hence, the large generation experiences a gain in income (relative to steady state) while the next generation experiences a loss in income. By contrast, if it is assumed that contribution rates remain unchanged and it is benefit rates which vary, it is the generation before the large one that has a welfare gain as benefit levels are boosted by the large number of workers who contribute, while the large generation has a welfare loss due the large number of retirees compared to contributors. Since endowment per capita has not deviated from its steady state value, these reallocations of income produce a reduction in expected aggregate welfare.
Finally, the effect of an increase in life expectancy can be analyzed. 1/ Clearly, the underlying model needs to be modified to accommodate this assumption. A simple adjustment is to assume that while individuals of generation t live for the whole of the period t, they only live for a proportion rt of period t+1. Rising life expectancy can then be captured by a rise in the value rt. It is assumed that the indirect utility function V(.) is unaffected by this change in life span.
In this case, the pay-as-you-go funding rule implies that the contribution rate, βt is equal to αt+l(nt+l/nt)/rt Since life expectancy (rt) is rising over time, both the benefit level and the contribution level cannot remain fixed over time even in the steady state. If it is assumed that it is benefit levels that remain fixed over time, then the condition for the pension system to raise welfare over time is that (1+g)(l+p)rt/rt-l is greater than (1+R). This is very similar to the condition derived earlier, except for the addition of the term in rt/rt-l. Since rt is rising over time, increasing life expectancy improves the welfare effect of a pay-as-you-go pension system relative to steady state values. 2/ This reflects the fact that with increasing life expectancy current workers tend to contribute less to fund current pensions than they get back after retirement because of their more extended lives.
To summarize, a fully funded pension system has no impact on the equilibrium. For a pay-as-you-go pension system, contribution rates in steady state equilibrium are below their fully funded values if, and only if, the growth of total endowment is greater than the real interest rate. Increasing life expectancy also tends to lower contribution rates compared to their fully funded values. In terms of movements from steady state equilibrium, a pay-as-you-go system improves expected aggregate welfare in response to a temporary shock to the endowment, but lowers it in response to a temporary demographic disturbance.
Bayoumi, Tamim, and Paul Masson, “Fiscal Flows in the United States and Canada: Lessons for Monetary Union in Europe” (forthcoming, European Economic Review).
Noord, Paul Van den, and Richard Herd, “Pension Liabilities in the Seven Major Economies,” OECD Economics Department, Working Paper No. 142 (1993).
I would like to thank Jorge Marquez-Ruarte and Christopher Towe for useful comments on an earlier version of this paper.
Another transfer program targeted at the elderly is the Guaranteed Income Supplement (GIS), which ensures that people aged 65 and over receive at least the necessary benefits to provide for a minimum standard of living. Since this program largely substitutes for insurance benefits provided to those aged less than 65 by provincial governments, it is probably best seen as a part of the social safety net rather than as a program directed specifically at the elderly, and is not analyzed in this paper.
The information provided in this chapter is derived mainly from the most recent actuarial reports for the three programs, which were issued at the end of 1991 (CPP), 1992 (QPP), and 1988 (OAS).
As discussed further below, current reserves are about three times annual expenditures.
Canada Pensions Plan, Fourteenth Actuarial Report as at December 1991, page 92.
Those with less than the minimum number of years of residence can claim partial benefits.
Derived from the relevant actuarial reports.
Total earnings are larger than earnings eligible for CPP or QPP payroll deductions. Hence, for the same expenditure level this ratio is lower than that reported for the CPP or QPP.
The QPP has not published expenditure projections after 2050. Hence, the projection to 2100 refers only to the CPP.
Including a long-term rate of inflation of prices and wages of 3.5 percent and 4.5 percent, respectively, and an interest rate of 6 percent.
Calculations of this type for all of the major industrial economies are reported in Noord and Herd (1993).
The United States has recently raised social security taxes in order to build up reserves in anticipation of the retirement of the baby boom generation; however it is still very far from being actuarially sound.
In Canada the Old Age Security program came into operation in 1951, the Canada and Quebec Pension Plans in 1966.
This is not to say that their associated payroll taxes play no role in stabilization, which they do. For example, Bayoumi and Masson (forthcoming) provide evidence that social security taxes help to stabilize incomes over the cycle within the United States and Canada.
Auerbach and Kotlikoff (1989). Blanchard and Fisher (1989, page 113) discuss the situations in which such a change raises or lowers welfare.
However, pay-as-you-go systems may also have detrimental effects on capital formation (Auerbach and Kotlikoff, 1987).
By contrast, normal business cycles are unlikely to provide effects of sufficiently long duration to have inter-generational consequences.
Life expectancy was held constant in these initial simulations since, for a given level of benefits, increases in life expectancy imply a steady rise in contribution rates, an effect which should be estimated separately.
If benefit levels were reduced, the stable contribution level would be correspondingly smaller.
These contribution rates are above the estimated actuarial rate of 9.6 percent reported earlier. Hence, in present value terms individuals will be paying more into the pension system than they receive in long-run equilibrium. This reflects the fact that, in the main case scenario, the present value of aggregate output is falling over time.
For males the life expectancy at age 65 is assumed to rise from 14.9 years in 1986 to 19.3 years in 2100, and for females the rise over the same period is from 19.1 years to 24.5 years.
This is an example of the general proposition that the welfare costs of taxes are minimized when tax rates are stable (Barro, 1979).
This is an important restriction because one of the effects of a pay-as-you-go system is to lower the capital stock (or raise foreign debt), thereby reducing aggregate income. Moreover, the impact of tax rates on labor market participation is ignored. Blanchard and Fisher (1989, page 113) report that the welfare effect of a pay-as-you-go system depends upon whether the real interest rate is greater or less than the rate of growth of population. If the real interest rate is below the rate of growth of population then welfare rises for all generations, but if it is above the increase in population then welfare falls for all generations except the first.
This implies some borrowing from the rest of the world. In addition, in a more complex model such a pay-as-you-go system may well have a negative impact on capital formation.
The effect Is symmetric, hence, in the case of a temporary rise in the endowment, a pay-as-you-go pension scheme will lower the welfare of the generation with the high endowment, and raise the welfare of the immediately previous generation.
More precisely, in the face of stochastic disturbances to the endowment, a pay-as-you-go system will raise the aggregate value of expected welfare across all generations.
Similar results can be derived for a reduction in the retirement age.
However, if contribution rates are left unchanged and it is benefit rates which are cut, an aging population has no impact on welfare.