Overlapping Generations and Pension Systems

Models of overlapping generations (OLG) make it possible to analyze long-term issues: capital accumulation, pension systems, government debt. In contrast to the representative agent model with an infinite horizon, OLG allows for dynamically inefficient equilibria and Pareto improvements through redistribution between generations.

Basic OLG Model (Samuelson, 1958; Diamond, 1965)

Structure: Each period, a new generation is born with measure 1. The population growth rate is $n$: generation $t$ has measure $(1+n)^t$. An agent lives for 2 periods: “young” ($t$) receives wage $w_t$, makes savings $s_t$; “old” ($t+1$) consumes $(1+r_{t+1})s_t$.

Optimization of the young:

$ \max_{c_{1t},\ c_{2,t+1}} u(c_{1t},\ c_{2,t+1}) \quad\text{subject to}\quad c_{1t} + s_t = w_t,\quad c_{2,t+1} = (1+r_{t+1})s_t $

Euler condition: $u'1(c_1) = (1+r{t+1})\beta\cdot u'2(c_2)$. The savings function: $s_t = s(w_t, r{t+1})$ — depends on wage and return.

Equilibrium: Capital market: $k_{t+1} = s_t/(1+n)$ — each unit of savings by a young agent finances capital per young person of the next period. For Cobb-Douglas $f(k) = k^\alpha$: $w_t = (1-\alpha)k_t^\alpha$, $r_t = \alpha k_t^{\alpha-1} - \delta$.

Dynamic Inefficiency and Pensions

“Golden rule” for OLG: Maximize sustainable consumption $c^* = f(k) - (\delta+n)k \rightarrow f'(k_{GR}) = \delta + n$. If in equilibrium $f'(k_{OLG}) < \delta + n \rightarrow k_{OLG} > k_{GR}$ — there is an excess of capital (over-accumulation). The economy is “dynamically inefficient”: it is possible to consume more by saving less.

Condition of dynamic inefficiency: $r^* < n$ — the real interest rate is below the growth rate. There are arguments that the USA and a number of European countries in the 1960–70s were close to this condition (Abel et al., 1989).

Why this is possible: In OLG, there is no “last generation” that would inherit all surplus capital. Every young generation saves for its old age → “excess” capital accumulates. In the infinite horizon, the transversality condition rules this out.

PAYG (pay-as-you-go) pension as a solution under dynamic inefficiency: The young pay contributions $\tau \cdot w_t$, the old receive $T_{t+1} = \tau \cdot w_{t+1}\cdot(1+n)$. The “return” of PAYG $= n$ (growth rate). When $r < n$: PAYG outperforms the market → improvement for all generations!

Funded pension: contributions are invested, the return is $r$. If $r > n$ — funded system is more efficient. Under dynamic efficiency ($r > n$) PAYG acts as a “tax” on the young (forced savings with lower return).

Government Debt and Ricardian Equivalence

Ricardian equivalence (Barro, 1974): In an infinite horizon model with altruistic agents: debt is financed by future taxes → agents increase savings by the size of the debt → neutrality of debt. Lowering taxes now = increasing debt = higher future taxes → no incentive to spend more.

Breakdown in OLG: Future taxes are paid by generations not yet born at the time of debt issuance → the current generation benefits. The “shifting of the burden” onto descendants. This is the main reason why OLG is an irreplaceable tool for fiscal policy analysis.

Optimal government debt: for $r > g$ ($r$ — rate, $g = n$ — growth): debt is “explosive” (needs primary surplus). For $r < g$: debt is manageable without primary surplus (Blanchard’s argument, 2019 — relevant for the USA with $r < g$ over the past 10 years).

Numerical Example

OLG: $u = \ln(c_1) + 0.9\cdot\ln(c_2)$, $f(k) = k^{0.4}$, $n = 0.01$, $\delta = 0.05$. Stationary equilibrium: $s^* = w_t/(1+\beta^{-1}) = w/(1+1/0.9) \approx 0.474w$. $w = (1-0.4)k^{0.4} = 0.6k^{0.4}$. $k = s/(1+n)$: $k = 0.474\cdot 0.6k^{0.4}/1.01 \rightarrow k^{0.6} = 0.474\cdot 0.6/1.01 \approx 0.2813 \rightarrow k^* \approx 0.092$.

Golden rule: $f'(k_{GR}) = \delta + n = 0.06 \rightarrow 0.4\cdot k^{-0.6} = 0.06 \rightarrow k_{GR} = (0.4/0.06)^{5/3} \approx 0.17$. $k^* < k_{GR}$ → no dynamic inefficiency ( $r^* = 0.4\cdot (k^*)^{-0.6} - 0.05 \approx 0.09 > n = 0.01$ ).

Task: OLG with $u = \ln(c_1) + 0.9\cdot\ln(c_2)$, $f(k) = k^{0.4}$, $n = 0.01$, $\delta = 0.05$. (1) Find stationary $k_{OLG}$. (2) Compare with $k_{GR}$. (3) Does dynamic inefficiency exist? (4) Introduce PAYG with $\tau = 0.1$. How does $k$ change in the stationary state? Is the welfare of all generations improved?