Appendix I Pension Arrangements in an Overlapping-Generations Model

Alfredo Cuevas, George Mackenzie, and Philip Gerson
Published Date:
September 1997
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This appendix uses an overlapping-generations model to look at the impact of introducing funded and unfunded social security systems, and of replacing an unfunded system with a funded one. The analysis follows closely the overlapping-generations model presented in chapter 3 of Blanchard and Fischer (1989) and illustrates the main findings of the text of this paper: that the introduction of a fully funded pension scheme should have no impact on the level of private savings unless the mandatory contribution rate is very high; that the introduction of an unfunded PAYG scheme will tend to decrease private saving; and that the replacement of an existing PAYG scheme with a fully funded scheme will not typically increase saving unless it is accompanied by a fiscal consolidation.

The Model

The economy consists of individuals and firms. Individuals live for two periods and consume c1.t in period t and c2, t + 1 in period t + 1. An individual’s lifetime utility is given by


where θ > 0, u′> 0, and u″<0. Individuals work only in the first period of their lives, supplying labor inelastically and earning a wage wt. They consume part of their labor income and save the rest, investing their savings to finance their second-period consumption.

The saving of the young in period t generates the capital stock that is used in period t + 1 in combination with the labor supplied by the young generation in that period. The number of individuals born at time t and working in that period is Nt. Population growth is given exogenously at rate n.

Firms behave competitively and produce output using the constant-returns-to-scale production function Y = F(K, N). Output per worker is given by the production function y = f(k), where k is the capitallabor ratio. Assume the production function satisfies the Inada conditions33 and that firms take the wage rate and the rental price of capital, rt, as given when maximizing profits.

The maximization problem for the individuals is given as

subject to


(In the second period, individuals consume both their savings and the interest income earned by them.) The first-order condition for a maximum is

which implies a savings function

where 0<Sw < 1 and the sign of sr is ambiguous because of offsetting income and substitution effects.34

The maximization problem for firms yields the familiar first-order conditions



Finally, goods-market equilibrium requires that the demand for and the supply of goods be equal or, equivalently, that savings equal investment. In other words, the stock of capital at t + 1 must equal the savings of the young generation at t:

These first-order conditions define the equilibrium. Blanchard and Fischer (1989) show that under certain restrictions, the equilibrium will be a stable one.35

Funded Social Security

In a fully funded system, the government (or, more generally, the pension fund) in period t collects contributions of dt from the young and invests the proceeds in the capital stock. It also pays benefits of bt = (1 + rt)dt−1 to the old, whose contribution was invested in period t - 1. Equations (1) and (4) thus become


Comparing equations (1) and (4) with equations (5) and (6), it is clear that if kt solves the former set, it also solves the latter, as long as dt < (1 + n) kt + 1. That is, as long as the required contribution under the fully funded scheme does not exceed the level of voluntary saving that would exist without the pension scheme, the introduction of a fully funded scheme does not affect the level of private saving. Individuals earn the same return on pension savings as on any other form of savings and are therefore indifferent between the allocation of s and d. They simply adjust their voluntary saving s to take into account any mandatory savings d.

Unfunded System

In an unfunded system, first-period income still falls by dt and second-period income still increases by bt but now the benefit paid in period t is equal to the contribution paid in during that same period; that is, bt = (1 + n)dt. In other words, if each worker’s contribution to the pension fund is constant over time (as it is in the steady state in this model), the return on pension contributions is only n rather than r.

Under these circumstances, equations (1) and (4) become


It is straightforward to show that both the saving rate and the per capita stock of capital are decreasing functions of the required contribution rate. First, differentiating equation (7) with respect to the contribution rate (assuming that dt = dt+1) yields

which is unambiguously negative, meaning that private saving falls when an unfunded system is introduced. Differentiation of equation (8) through the implicit function rule yields

which is also negative.36 Thus, both private saving and the stock of capital fall when an unfunded pension scheme is introduced.

Replacing an Unfunded with a Funded System

Now suppose that an unfunded system is replaced by a fully funded one. For the current generation of retirees, the government has an obligation that it finances by borrowing from the young generation at the market interest rate.

Because the value of the obligation owed to retirees is unaffected by the change, the amount the government borrows per worker, zt, must be the same as the amount it would have collected from each worker under the old PAYG system, dt. Therefore, equations (1) and (4) become


Assuming, as before, that dt = dt + 1, equations (7) and (8) are identical to equations (9) and (10) except that the portion of savings going to finance the retirement consumption of the currently old earns a return of rt + 1 rather than n. This increase in the infra-marginal rate will generate a pure income effect that will tend to discourage current-period saving. Thus, the transition from a PAYG to a fully funded system generates an immediate income effect that lowers private saving.

Assuming that no fiscal consolidation accompanies the change in pension systems, the government will need to continue refloating its debt. Thus, to repay in period t + 1 the funds it borrowed in period t, the government will need to borrow zt + 1 = zt(1 + rt + 1)/(1 + n) from each worker in period t + 1. The first-order conditions for a worker in period t + 1 therefore become


If, as is commonly assumed, r > n, then the per worker stock of debt increases in each period, leading (in partial equilibrium) to a decrease in savings for capital accumulation and a lower capital stock. From equation (11), increases in z are fully offset by decreases in s (given that individuals are indifferent between buying government debt and investing in capital) until voluntary savings are exhausted. However, this is a partial equilibrium result in that it treats the interest rate as given. Clearly, as the stock of capital falls, the rate of interest will rise and labor income will fall, both of which will influence the level of saving. To move to general equilibrium, we can differentiate equation (12) with respect to zt + 1 to show

where sr is the derivative of savings with respect to the interest rate. Given that f″ < 0, if sr is nonnegative, then the effect of an increase in government borrowing per worker is to unambiguously decrease the stock of capital in general equilibrium.

Notice also that the path of zt is unsustainable over time, because it grows at the rate of (1 + r)/(l + n). Financing the transition with debt leads to spiraling government deficits, even though the underlying primary fiscal balance is constant. Although the decline in the capital stock will lead to an increase in the interest rate and generate additional savings, the explosive path of zt will eventually exhaust output. Accordingly, some adjustment in the primary fiscal balance will be required to ensure stability in the model. It may be in this indirect sense that the switch from a PAYG to a fully funded scheme encourages saving.

That is, f′ > 0, f″ < 0, limk→0f′ = ∞ and limk→∞f′ = 0.

A higher interest rate means that each additional unit of current-period saving will allow a greater increase in second-period consumption than was previously the case. At the margin, that will tend to increase savings (the substitution effect). At the same time, the higher interest rate increases the level of second-period consumption that can be financed from the existing stock of first-period savings, which tends to decrease the level of saving (the income effect).

The main restriction is that at the equilibrium per capita stock of capital k*, 1dkt+1kt|=|-swk*f"(k*)1+nsrf"(k*)|<|.

To obtain the denominator, note that ∂s/∂k = (∂s/∂r) × (∂r/∂k), and that r = f′(k).

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