Chapter 12: The Continuous-Time Overlapping Generations Model

Demographic Transitions and the Blanchard–Yaari Framework

“The overlapping generations structure is not a technical complication but an economic necessity: it is the only way to model the redistribution between generations that is central to social security, public debt, and demographic policy.”

Cross-reference: Principles Ch. 25 (OLG demographics, Diamond model, HANK, aging); Ch. 22 (social security, Ricardian equivalence across generations) [P:Ch.25, P:Ch.22]

12.1 Why Continuous-Time OLG? The Limits of the Ramsey Model¶

The Ramsey–Cass–Koopmans model assumes an infinitely-lived representative household. This is a powerful analytical device but obscures the intergenerational dimension that is central to many of the most important macroeconomic policy debates: social security reform, public debt sustainability, the consequences of population aging, and the distributional effects of fiscal policy across cohorts.

The Diamond (1965) overlapping-generations model (Chapter 16) addresses this by explicitly modeling two-period-lived generations. But the two-period structure is too coarse for quantitative work: it cannot generate realistic life-cycle saving profiles, smooth demographic dynamics, or tractable responses to policy changes.

The Blanchard (1985) – Yaari (1965) perpetual youth model strikes a compromise: it operates in continuous time, allows for finite lifetimes, preserves tractable aggregation, and generates a clean departure from Ricardian equivalence. Households face a constant Poisson death rate $p > 0$ — they can die at any moment with probability $p\,dt$ over a small interval $dt$. This generates a population that is always young (relative to actuarial tables) but has finite expected lifetimes: $E[\text{lifespan}] = 1/p$.

12.2 Household Optimization with Uncertain Lifetime¶

12.2.1 The Modified Discount Rate¶

A household born at time $s$ maximizes expected lifetime utility:

where the expectation is taken over the random date of death $T_D$. Since death follows a Poisson process with rate $p$, the probability of surviving from $s$ to $t$ is $e^{-p(t-s)}$. Therefore:

The effective discount rate is $\rho + p$: the sum of pure time preference $\rho$ and the mortality rate $p$. Households discount the future both because of impatience and because they may not live to enjoy it.

12.2.2 Actuarially Fair Annuities¶

With competitive insurance markets, households can purchase actuarially fair annuities that pay out while alive and transfer accumulated wealth to the insurance company upon death. The return on an annuity is $r_t + p$: the market interest rate $r_t$ plus the mortality premium $p$ (reflecting that the insurer collects the wealth of those who die and distributes it to survivors). The budget constraint for a household of cohort $s$ at time $t$:

where $a_{s,t}$ is financial wealth (claims on the annuity) and $w_t$ is the wage.

12.2.3 Optimal Consumption: The Euler Equation¶

With CRRA utility $u(c) = c^{1-\sigma}/(1-\sigma)$, the Hamiltonian for the cohort-$s$ problem gives the Euler equation:

The mortality rate $p$ cancels from the Euler equation — the effective discount rate $\rho+p$ and the annuity return $r+p$ both shift by $p$, so the consumption growth rate is the same as in the Ramsey model. The departure from Ramsey appears in the level of consumption, not its growth rate.

12.2.4 Consumption Function¶

Solving the Euler equation forward subject to the intertemporal budget constraint (no-Ponzi condition):

where $a_{s,t}^{total}$ is total wealth (financial wealth plus human wealth). For simplicity with $\sigma = 1$ (log utility), the consumption function takes the elegant form:

where $h_t = \int_t^\infty e^{-\int_t^\tau(r_u+p)du}w_\tau\,d\tau$ is human wealth — the present discounted value of future wage income, discounted at the annuity-adjusted rate $(r+p)$.

Definition 12.1 (Human Wealth in the Blanchard–Yaari Model). Human wealth satisfies the differential equation:

This says: human wealth grows at rate $(r+p)$ (the annuity return) minus current wages received (which convert to financial wealth).

12.3 Aggregation¶

The key insight of the perpetual youth model is that aggregation across cohorts is tractable. At any time $t$, the age distribution of living households follows an exponential distribution with rate $p$ — there are $pe^{-p(t-s)}$ survivors of cohort $s$ per unit population (normalized so the total population is 1 without population growth, or $L_t = e^{nt}$ with growth).

The aggregate consumption function. Integrating $c_{s,t} = (\rho+p)(a_{s,t}+h_t)$ over all living cohorts (using $\sigma=1$):

Since $\int_{-\infty}^t p e^{-p(t-s)}a_{s,t}ds = A_t$ (total financial wealth per capita) and $\int_{-\infty}^t p e^{-p(t-s)}h_t\,ds = h_t$ (human wealth is common across cohorts):

where $H_t = h_t$ is aggregate human wealth (equal to individual human wealth since all living cohorts have the same $h_t$).

Definition 12.2 (Aggregate Dynamics in Blanchard–Yaari). Aggregate consumption $C_t$ and human wealth $H_t$ satisfy:

The term $-p(\rho+p)A_t$ is the key departure from Ramsey: newly born households start with zero financial wealth ($a_{s=t,t} = 0$), so aggregate consumption growth is lower than individual consumption growth by the amount $p(\rho+p)A_t$ — newly born consumers drag down the aggregate.

Derivation of the aggregate Euler equation: Differentiate $C_t = (\rho+p)(A_t + H_t)$:

Using $\dot{A}_t = (r_t+n)A_t + w_t - C_t$ (capital accumulation) and $\dot{H}_t = (r_t+p)H_t - w_t$:

Substituting $C_t = (\rho+p)(A_t+H_t)$:

After simplification (assuming $n=0$ for clarity):

Proposition 12.1 (Departure from Ramsey). The aggregate Euler equation in the Blanchard–Yaari model is $\dot{C}_t = (r_t-\rho)C_t - p(\rho+p)A_t$. This equals the Ramsey aggregate Euler equation $(r_t-\rho)C_t$ only when $p = 0$ (no mortality risk) or $A_t = 0$ (no financial wealth). The term $-p(\rho+p)A_t$ captures the wealth dilution from demographic turnover: new households enter without wealth, dragging down aggregate consumption growth.

12.4 General Equilibrium¶

In a closed economy with capital, the production side is identical to the Ramsey model:

Market clearing: $K_t = A_t L_t$ (capital equals aggregate financial wealth) and $Y_t = C_t + \dot{K}_t + \delta K_t$.

In per-capita terms (setting $n = 0$ and $g = 0$ for analytical clarity), the equilibrium is characterized by the two-dimensional system:

12.4.1 Steady State¶

Setting $\dot{k} = 0$ and $\dot{c} = 0$:

From $\dot{c} = 0$: $f'(k^*) = \delta + \rho + p(\rho+p)k^*/c^*$.

From $\dot{k} = 0$: $c^* = f(k^*) - \delta k^*$.

Substituting: $f'(k^*) = \delta + \rho + p(\rho+p)k^*/(f(k^*)-\delta k^*)$.

This is a nonlinear equation in $k^*$ that must be solved numerically. However, we can immediately see the qualitative difference from Ramsey: the RCK steady state required $f'(k^*) = \delta + \rho$ (no mortality term), while the Blanchard–Yaari steady state requires a higher net marginal product $f'(k^*) > \delta + \rho$ (when $p > 0$ and $k^*, c^* > 0$). Since $f'' < 0$, this means $k^*_{BY} < k^*_{RCK}$: the Blanchard–Yaari economy has a lower capital stock than the Ramsey economy.

Intuition: Households with finite lifetimes save less than infinitely-lived households (they don’t need to accumulate wealth to support themselves in infinite retirement). Lower saving → lower capital → lower steady-state income. This is the macroeconomic manifestation of the finite planning horizon.

12.4.2 Stability Analysis¶

The Jacobian of the Blanchard–Yaari two-dimensional system at the steady state:

The determinant:

provided $f'' < 0$ and the mortality correction is not too large. The saddle-point property is preserved (one positive, one negative eigenvalue) for standard calibrations.

12.5 Social Security and Ricardian Non-Equivalence¶

The Blanchard–Yaari model restores Ricardian non-equivalence: a deficit-financed transfer (reducing taxes today, increasing taxes in the future) raises current aggregate consumption because the future tax burden falls partly on households not yet born — households who are not in the current economy’s budget constraint.

Proposition 12.2 (Ricardian Non-Equivalence in Blanchard–Yaari). In the Blanchard–Yaari model, a lump-sum tax cut $\Delta T$ financed by government debt that will be repaid at rate $(r+p)\Delta B$ (where $\Delta B$ is new debt) raises current aggregate consumption by:

Proof sketch. The tax cut increases household wealth $A_t$ by $\Delta B$ (the present value of future tax increases is lower than $\Delta B$ because some of the future burden falls on not-yet-born households). Using the consumption function $C = (\rho+p)(A+H)$, the effect on consumption is $(\rho+p)\Delta A_{effective}$ where $\Delta A_{effective} = [p/(r+p)]\Delta B$ — the fraction of the debt not to be borne by current households. $\square$

The departure from Ricardian equivalence is proportional to $p/(r+p)$: as $p \to 0$ (infinite lives), this goes to zero (full Ricardian equivalence); as $p \to\infty$ (very short lives), it goes to $1/(1 + r/p) \to 1$ — the full tax cut is spent because households expect to be dead before repayment.

This result has an important implication for social security analysis: a PAYG pension system that taxes the young and pays the old redistributes from (not-yet-born) future young to current old. In the Blanchard–Yaari model, this redistribution has real effects — it raises current consumption and reduces capital accumulation. The welfare analysis requires computing whether the consumption gain to current old outweighs the capital-stock loss for all future generations.

12.6 Population Aging: The Demographic Dividend and Its Reversal¶

Definition 12.3 (Population Aging in Continuous Time). In the Blanchard–Yaari model, population aging is modeled as a reduction in the mortality rate $p$ (people live longer) or the birth rate $n$ (fewer new households entering). Both reduce the population growth rate $n - p$.

Effects on the steady state:

1. Reduction in $p$: Lower mortality rate → longer planning horizons → households save more for retirement → higher capital accumulation → higher $k^*$ and $y^*$. But also: the Blanchard–Yaari departure from Ricardian equivalence shrinks ($p/(r+p)$ falls) — the economy becomes more Ricardian.

2. Reduction in $n$: Fewer new workers → the capital-labor ratio rises mechanically (same capital, fewer workers) → higher $k$ per worker → higher $y$ per worker but lower aggregate output (fewer workers).

The demographic dividend refers to the period during which the working-age population is large relative to dependents (both young and old) — generating high labor supply, high saving, and rapid growth. East Asian economies exploited their demographic dividend from the 1960s to 1990s. The dividend reverses when the old-age dependency ratio rises and saving falls.

12.7 Worked Example: Steady-State Comparison with Ramsey¶

Calibration: $\alpha = 0.35$, $\delta = 0.05$, $\rho = 0.04$, $p = 0.04$ (expected life $= 1/p = 25$ years in the adult phase), $n = 0$, $g = 0$.

Ramsey steady state: $f'(k^*_{RCK}) = \delta + \rho = 0.09$. With $f'(k) = \alpha k^{\alpha-1}$:

Blanchard–Yaari steady state: Solve $f'(k^*_{BY}) = \delta + \rho + p(\rho+p)k^*_{BY}/c^*_{BY}$ with $c^*_{BY} = f(k^*_{BY}) - \delta k^*_{BY}$ numerically.

Substituting: $\alpha(k^*_{BY})^{\alpha-1} - \delta - \rho = p(\rho+p)k^*_{BY}/[(k^*_{BY})^\alpha - \delta k^*_{BY}]$.

Numerical solution (Newton–Raphson): $k^*_{BY} \approx 6.84$, $c^*_{BY} \approx 1.71$.

Comparison:

The finite-lifetime assumption (mortality rate $p = 0.04$) reduces the steady-state capital stock by 20% and output by 9% relative to the Ramsey benchmark. This is a quantitatively significant difference — the finite planning horizon matters economically, not just theoretically.

APL¶

Python¶

12.8 Programming Exercises¶

Exercise 12.1 (APL — Mortality Sensitivity)¶

Using the Newton–Raphson dfn from Section 12.7, compute the Blanchard–Yaari steady-state capital $k^*_{BY}$ for $p \in \{0.02, 0.04, 0.06, 0.08, 0.10, 0.15\}$. Produce a table showing $k^*_{BY}$, $y^*_{BY}$, $c^*_{BY}$, and the percentage deviations from the Ramsey benchmark for each $p$.

Exercise 12.2 (Python — Transition Dynamics)¶

Simulate the Blanchard–Yaari economy from $k_0 = k^*_{BY}/2$ using RK4. Plot the transition path $k(t)$ and $c(t)$ together with the nullclines in the phase plane. Compare the speed of convergence to the Ramsey model: which converges faster to its respective steady state?

Exercise 12.3 (Julia — Ricardian Non-Equivalence)¶

# Compute the degree of Ricardian non-equivalence as a function of p and r
r_vals = [0.02, 0.04, 0.06, 0.08]
p_vals = [0.01, 0.02, 0.04, 0.06, 0.10]
println("Fraction of tax cut consumed: p/(r+p)")
println("(1.0 = full Keynesian; 0.0 = full Ricardian)\n")
println(lpad("p\\r", 6), join([lpad("r=$(r)", 8) for r in r_vals]))
for pv in p_vals
    row = lpad("p=$(pv)",6) * join([lpad(round(pv/(r+pv),digits=3), 8) for r in r_vals])
    println(row)
end

Exercise 12.4 — Aging Demographic Transition ($\star$)¶

Model a permanent reduction in the mortality rate from $p_{old} = 0.04$ to $p_{new} = 0.02$ (people live longer). (a) Compute the new steady state $k^*_{new} > k^*_{old}$ (longer lives → more saving → higher capital). (b) Simulate the transition path from $k^*_{old}$ to $k^*_{new}$ using RK4. (c) Compute the welfare change along the transition: do current households benefit or lose from the demographic shift? (d) Compare to the effect of a change in the birth rate from $n_{old} = 0.02$ to $n_{new} = 0.01$ — same population aging outcome but different mechanism.

Exercise 12.5 — PAYG Social Security ($\star\star$)¶

Introduce a PAYG social security system that taxes workers $\tau w$ and pays retirees a lump-sum $b = \tau w L_{workers}/L_{retired}$. In the Blanchard–Yaari model: (a) derive the modified aggregate Euler equation with the social security tax-transfer mechanism; (b) show that the steady-state capital stock falls (PAYG reduces private saving); (c) derive the optimal PAYG transfer $\tau^*$ that maximizes a utilitarian social welfare function across generations.

12.9 Chapter Summary¶

Key results:

• The Blanchard–Yaari perpetual youth model introduces finite lifetimes via a Poisson death rate $p > 0$, modifying the effective discount rate to $\rho + p$ and the annuity return to $r + p$.

• Individual consumption follows $c_{s,t} = (\rho+p)(a_{s,t}+h_t)$ — a fraction $\rho+p$ of total wealth (financial plus human).

• Aggregation yields $C_t = (\rho+p)(A_t + H_t)$ and the aggregate Euler equation $\dot{C}_t = (r_t-\rho)C_t - p(\rho+p)A_t$.

• The departure from Ramsey is the term $-p(\rho+p)A_t$: newly born households enter without wealth, dragging down aggregate consumption growth relative to individual growth.

• The Blanchard–Yaari steady state has $k^*_{BY} < k^*_{RCK}$ — finite lifetimes reduce capital accumulation (by 20% in the calibrated example).

• Ricardian non-equivalence is restored: a deficit-financed tax cut raises aggregate consumption by $(\rho+p)\frac{p}{r+p}\Delta B$ — the fraction falling on yet-to-be-born households.

• In APL: the Newton–Raphson steady state is computed using ⍣ converged; transition dynamics via {by_ode_step ⍵}⍣T.

Connections forward: Chapter 16 develops the Diamond discrete-time OLG model — the discrete counterpart — with explicit two-period household optimization, dynamic inefficiency, and social security welfare analysis. Chapter 25 of Principles [P:Ch.25] connects the OLG structure to the HANK model with heterogeneous households.

Next: Chapter 13 — Adjustment Cost Models: Tobin’s q and Investment Dynamics