Theory of Income · Part III: OLG Extensions (Class Note III) · Lesson 16 of 25

Heterogeneity and Long Lives in the OLG Model

Covers Class Note III, pp. 20–24 · About 35 min · Prerequisites: Lesson 10, Lesson 11

By the end you can:

derive each type's consumption under log utility and show that the equilibrium interest rate depends on the distribution of endowments only through the mean $\bar\alpha$;
construct a feasible allocation that Pareto-dominates the equilibrium when $\bar\alpha\lt \beta$, and verify both properties;
show with numbers that a uniform PAYG system can hurt some types even though the economy is dynamically inefficient;
prove that with $N$-period lives and negative interest rates, a small transfer from younger to older ages is a Pareto improvement.

Primary source Alvarez, Class Note III, pp. 20–24. Samuelson's original consumption-loan model actually has three-period lives, and its argument is the ancestor of §4: Samuelson, P. (1958), “An Exact Consumption-Loan Model of Interest with or without the Social Contrivance of Money,” JPE 66(6): 467–482.

Notation alert

The heterogeneity slides return to the Class Note II convention: young endowment $1-\alpha_n$, old endowment $\alpha_n$. This is the reverse of the privatization slides (Lesson 14), where young had $\alpha$. Here $\alpha$ is an endowment share again, not the capital share of Lesson 15. Preferences are $(1-\beta)\log c_y + \beta\log c_o$ for every type.

1. Heterogeneity within a generation

Each generation has $N$ types $n=1,\dots,N$, each of mass $1/N$, so total population per generation is 1. Type $n$ has endowments $e^{t,n}_t = 1-\alpha_n$ when young and $e^{t,n}_{t+1} = \alpha_n$ when old III p.20. Define the average old-age endowment

$$\bar\alpha = \frac1N\sum_{n=1}^N \alpha_n .$$

The aggregate endowment per period is still $\frac1N\sum_n(1-\alpha_n) + \frac1N\sum_n\alpha_n = 1$.

Individual demands

Lifetime wealth of a type-$\alpha$ agent at interest rate $r$: $W(\alpha,r) = (1-\alpha) + \dfrac{\alpha}{1+r}$.
Log utility spends fixed shares of wealth: $c_y = (1-\beta)W$ and $c_o/(1+r) = \beta W$. So $$c_y(\alpha,r) = (1-\beta)\Big[(1-\alpha) + \frac{\alpha}{1+r}\Big], \qquad c_o(\alpha,r) = \beta\big[(1+r)(1-\alpha) + \alpha\big].$$
Ratio across types. Both consumptions are proportional to $W$, so $$\frac{c_y(\alpha,r)}{c_y(\bar\alpha,r)} = \frac{c_o(\alpha,r)}{c_o(\bar\alpha,r)} = \frac{W(\alpha,r)}{W(\bar\alpha,r)} = \frac{(1+r)(1-\alpha)+\alpha}{(1+r)(1-\bar\alpha)+\bar\alpha} = \frac{1+(1-\alpha)r}{1+(1-\bar\alpha)r} .$$

Every type chooses the same shape of lifetime consumption, the bundle of a hypothetical average type scaled up or down by relative wealth.

Aggregation and the interest rate

Saving of type $\alpha$ is

$$s(\alpha,r) = (1-\alpha) - c_y(\alpha,r) = \beta(1-\alpha) - (1-\beta)\frac{\alpha}{1+r},$$

which is the same formula as in Lesson 9 and is linear in $\alpha$. Averaging a linear function gives the function at the average: $\frac1N\sum_n s(\alpha_n,r) = s(\bar\alpha,r)$. Equilibrium requires zero aggregate saving (there is no asset in positive net supply), so III p.21

$$s(\bar\alpha,\bar r) = 0 \quad\Longleftrightarrow\quad \bar r = \frac{\bar\alpha-\beta}{\beta(1-\bar\alpha)} .$$

This is the homogeneous-economy interest rate with $\alpha=\bar\alpha$. In particular, $\bar r<0$ iff $\bar\alpha\lt \beta$.

One big difference: now there is trade

In the homogeneous economy the equilibrium was autarky (Lesson 8). Here only aggregate saving is zero. Types with $\alpha_n\lt \bar\alpha$ (poor when old) save, $s>0$, and types with $\alpha_n>\bar\alpha$ borrow from them. Young agents trade with each other in a within-generation credit market. The old still cannot trade with the young, since nobody would lend to someone about to die.

2. The equilibrium is still inefficient when $\bar\alpha\lt \beta$

The best aggregate symmetric allocation is $c_y^* = 1-\beta$, $c_o^*=\beta$, as before. The slides III p.21 propose distributing it across types in proportion to equilibrium wealth:

The rescaled allocation

Let $\lambda_n \equiv \dfrac{1+(1-\alpha_n)\bar r}{1+(1-\bar\alpha)\bar r}$. Give every young type-$n$ agent $\big(\lambda_n(1-\beta),\ \lambda_n\beta\big)$ over its life, and give every initial-old type-$n$ agent $\alpha_n + (\beta-\bar\alpha)$.

Claim

If $\bar\alpha\lt \beta$, the rescaled allocation is feasible and Pareto-dominates the competitive equilibrium.

Proof

The weights average to one. $\lambda_n$ is linear in $\alpha_n$, so $\frac1N\sum_n\lambda_n = \dfrac{1+(1-\bar\alpha)\bar r}{1+(1-\bar\alpha)\bar r} = 1$.
Feasibility at $t\ge2$. Average young consumption plus average old consumption is $(1-\beta)\cdot 1 + \beta\cdot 1 = 1$, the aggregate endowment.
Feasibility at $t=1$. The initial old consume on average $\bar\alpha + \beta - \bar\alpha = \beta$, and the young of generation 1 consume $1-\beta$ on average. The total is 1.
The CE bundle of type $n$ is $\lambda_n$ times the average type's. By the ratio formula, type $n$ consumes $\lambda_n\big(c_y(\bar\alpha,\bar r), c_o(\bar\alpha,\bar r)\big)$. At $\bar r$ the average type saves zero, so its bundle is its endowment $(1-\bar\alpha,\bar\alpha)$. Type $n$'s CE bundle is therefore $\lambda_n(1-\bar\alpha,\bar\alpha)$.
Young generations gain. With log utility, $v(\lambda c_y,\lambda c_o) = \log\lambda + v(c_y,c_o)$. So type $n$'s gain is $v(1-\beta,\beta) - v(1-\bar\alpha,\bar\alpha)$, the same for every type. It is strictly positive, because $(1-\beta,\beta)$ uniquely maximizes $v$ on $c_y+c_o=1$ and $\bar\alpha\ne\beta$.
Initial old gain. In the CE they consume $\alpha_n$ (nothing to trade). Now they get $\alpha_n + \beta-\bar\alpha > \alpha_n$ since $\bar\alpha\lt \beta$.

∎

So heterogeneity does not remove the negative-interest-rate inefficiency. But notice how carefully the improvement had to be constructed: each type's share $\lambda_n$ depends on its own $\alpha_n$.

3. A uniform PAYG system is not enough

Now suppose the government uses the simple tool of Lesson 11: every young person pays $\tau$ and every old person receives $\tau$ III p.22. After-tax endowments become $1-\alpha_n-\tau$ and $\alpha_n+\tau$, so the average old endowment rises to $\bar\alpha+\tau$ and

$$r(\tau) = \frac{\bar\alpha+\tau-\beta}{\beta(1-\bar\alpha-\tau)}, \qquad \text{increasing in } \tau .$$

With log utility, type $n$'s equilibrium utility is (up to a constant common to all types)

$$U_n(\tau) = \log W_n(\tau) + \beta\log\big(1+r(\tau)\big), \qquad W_n(\tau) = (1-\alpha_n-\tau) + \frac{\alpha_n+\tau}{1+r(\tau)} .$$

(Plug $c_y=(1-\beta)W$ and $c_o=\beta(1+r)W$ into $v$.) A higher $r$ helps lenders and hurts borrowers. The types with high $\alpha_n$ are borrowers, and the policy destroys their cheap credit.

Worked example

$\beta = 0.5$, two equal-sized types with $\alpha_1 = 0.05$ and $\alpha_2 = 0.55$, so $\bar\alpha = 0.3\lt \beta$.

$\bar r = (0.3-0.5)/(0.5\cdot 0.7) = -4/7\approx -0.571$, so $1+\bar r = 3/7$. Type 1 saves $s_1 = 0.5(0.95) - 0.5(0.05)(7/3) \approx 0.417$; type 2 borrows $s_2\approx -0.417$.
Choose $\tau = \beta-\bar\alpha = 0.2$. Then $\bar\alpha+\tau=\beta$, so $r(\tau) = 0$, $W_n = 1$ and $U_n = 0$ for both types.
Before the policy: $U_1 = \log(0.95 + 0.05\cdot\tfrac73) + 0.5\log\tfrac37 \approx 0.0645 - 0.4236 = -0.359$, and $U_2 = \log(0.45 + 0.55\cdot\tfrac73) + 0.5\log\tfrac37 \approx 0.550 - 0.424 = +0.126$.
Type 1 gains 0.359; type 2 loses 0.126. The uniform PAYG system is not a Pareto improvement, even though the rescaled allocation of §2 gives both types a gain of $v(0.5,0.5)-v(0.7,0.3)\approx 0.087$.

Interactive: welfare change by type. The blue curve is the utility change $U_n(\tau)-U_n(0)$ of a young agent with old-age endowment $\alpha_n$, under a uniform PAYG tax $\tau$. The orange line is the gain from the rescaled allocation of §2, which is the same for every type and exists only when $\bar\alpha\lt \beta$. Try: the defaults reproduce the worked example (types 0.05 and 0.55 marked). Shrink $\tau$ toward 0: the cutoff type moves toward $\bar\alpha$, but some borrowers always lose. Then set $\bar\alpha>\beta$: now $\bar r>0$, the orange line disappears, and borrowers still lose. No allocation can help everyone, because the First Welfare Theorem applies.

Lesson from the slides III p.22

Correcting aggregate dynamic inefficiency, i.e. raising $r$ toward the growth rate, is one problem. Distributing the gains so that nobody loses is a second one. A uniform tax solves the first but not the second, because it changes a relative price, $r$, at which heterogeneous agents are on opposite sides of the market. Exercise 16.5 shows that adding within-generation lump-sum transfers fixes the second problem.

4. Agents who live $N$ periods

Now return to a single type per generation, but let each generation live $N\ge2$ periods III p.23. Generation $t$ has utility $u^t = v^t(c^t_t,\dots,c^t_{t+N-1})$, and at every date one generation of each age $a=1,\dots,N$ is alive. At the equilibrium allocation $\bar c^t$, the first-order conditions are

$$v^t_a(\bar c^t) = \lambda^t p_{t+a-1}, \qquad a = 1,\dots,N,$$

where $v^t_a$ is the marginal utility of consumption at age $a$, $\lambda^t>0$ is the multiplier on the budget, and $p_s$ is the date-$s$ price.

The "young-to-old" transfer $(\tau, Y)$

Fix $Y<N$. Every agent of age $1,\dots,Y$ gives up $\tau$. Every agent of age $Y+1,\dots,N$ receives $\dfrac{Y\tau}{N-Y}$. For generation $t$, the value at birth of this change is $$P^t(\tau,Y) = \frac{1}{p_t}\Big[-\tau\sum_{a=1}^{Y} p_{t+a-1} + \frac{Y\tau}{N-Y}\sum_{a=Y+1}^N p_{t+a-1}\Big].$$

Claim III p.24

In the pure-exchange economy, if all equilibrium interest rates are negative, then for $\tau>0$ small enough the transfer $(\tau,Y)$ Pareto-dominates the equilibrium allocation.

Proof

Feasibility, date by date. At each date there is exactly one generation of each age. Total consumption changes by $-Y\tau + (N-Y)\cdot\frac{Y\tau}{N-Y} = 0$. No goods are created or destroyed.
Prices rise over time. $p_{s+1}/p_s = 1/(1+r_s) > 1$ when $r_s<0$. So every price $p_{t+a-1}$ with $a>Y$ exceeds every price with $a\le Y$.
The transfer has positive value. Rewrite $P^t(\tau,Y) = \frac{Y\tau}{p_t}\big[\overline{p}_{\text{late}} - \overline{p}_{\text{early}}\big]$, where $\overline{p}_{\text{late}}$ is the average of the $N-Y$ late prices and $\overline{p}_{\text{early}}$ the average of the $Y$ early ones. By step 2, $P^t(\tau,Y)>0$.
Marginal welfare. Differentiating utility at $\tau=0$ and using the FOCs: $$\frac{d u^t}{d\tau}\Big|_{\tau=0} = \sum_a v^t_a\,\frac{dc^t_a}{d\tau} = \lambda^t \sum_a p_{t+a-1}\frac{dc^t_a}{d\tau} = \lambda^t p_t\, P^t(1,Y) > 0 .$$ This holds for every generation born at $t\ge 1$.
Generations already alive at date 1. A generation that is age $a_0>1$ at date 1 experiences only the changes at ages $a_0,\dots,N$. It skips some of the negative terms in $P$, so its marginal value is even larger (if $a_0>Y$, it only receives). So it gains too.
Since every generation's marginal utility gain is strictly positive at $\tau=0$, there is a $\bar\tau>0$ such that all gain for $\tau\in(0,\bar\tau)$. (With infinitely many generations one needs a single $\bar\tau$ that works for all of them. That holds, for example, in a stationary equilibrium with constant $r$, where every generation faces the same problem. The slides take this step as given.) ∎

Check: with $N=2$, $Y=1$, we get $P = (p_{t+1}-p_t)/p_t = \frac{1}{1+r_t}-1$, positive iff $r_t<0$. This is exactly the PAYG argument of Lesson 11.

Why this needs pure exchange III p.24

Step 1 checked feasibility only against a fixed endowment. With intertemporal production, a reallocation of consumption changes saving and therefore the capital path, as in Lesson 15. The new consumption plan must be matched by a feasible production and capital path, which is not automatic.

Interactive: value of the transfer. $P^t(1,Y)$ as a function of a constant interest rate $r$, for each $Y=1,\dots,N-1$. With $p_{t+a-1}/p_t = (1+r)^{-(a-1)}$, every curve is positive exactly when $r<0$. Try: increase $N$. The curves get steeper, because longer lives mean more periods over which a negative return compounds.

5. Check your understanding

1. Why does the equilibrium interest rate depend on the $\alpha_n$ only through $\bar\alpha$?

Because each type's saving is linear in its α_n
Because each type faces a different interest rate
Because nobody trades with anyone in equilibrium
Because all types share the same lifetime wealth

$s(\alpha,r) = \beta(1-\alpha) - (1-\beta)\alpha/(1+r)$ is affine in $\alpha$, so average saving is $s(\bar\alpha,r)$. With non-log preferences this exact aggregation generally fails. Types do trade, and their wealths differ.

2. In the worked example, why does type 2 (α = 0.55) lose from uniform PAYG?

It pays more tax than type 1 when it is young
It receives a smaller transfer when it is old
It borrows, and the policy raises interest rates
It lends, and the policy lowers interest rates

Taxes and transfers are identical across types. Type 2 is a borrower ($s_2<0$), and the policy raises $r$ from $-4/7$ to 0, which makes borrowing much more expensive.

3. In the $N$-period argument, what guarantees that the transfer is feasible?

Prices increase over time when interest rates are negative
Each date has one generation of each age, so flows net out
The first-order conditions hold for each generation alive
Each generation's budget constraint holds with equality

At any date, $Y$ age groups each give $\tau$ and $N-Y$ age groups each receive $Y\tau/(N-Y)$, so the net is zero. Rising prices are what make the transfer valuable, not what make it feasible.

4. With $N=2$ and $Y=1$, $P^t(1,1)$ equals which expression?

$r_t/(1+r_t)$, positive exactly when r is positive
$1+r_t$, the gross return on saving from t to t+1
$r_t$, the net interest rate between t and t+1
$1/(1+r_t)-1$, positive exactly when r is negative

$P = (-p_t + p_{t+1})/p_t = p_{t+1}/p_t - 1 = 1/(1+r_t) - 1$. This is the value at birth of paying 1 when young and receiving 1 when old, i.e. two-period PAYG.

6. Exercises

These follow the format of Alvarez's exams: true/false items with short justifications, then long derivations broken into small steps. Exercises 16.3 and 16.4 are the problem-set questions that Class Note III pp.20–24 walk through. Attempt each part before opening the solution.

Exercise 16.1 True or false? Justify. Exam format

For each statement, say whether it is true or false and give a justification of one to four lines.

In a two-period OLG endowment economy where agents of the same generation differ in their endowments and preferences, the competitive equilibrium is autarky. (Past exam: 2023 Midterm 1 Q2.1, reworded)
Show answer

False. At autarky, different types generally have different marginal rates of substitution between youth and old age, so they gain from lending to each other within the generation. Only aggregate trade between generations is zero. Some student solutions to the 2025 problem set claim that "each agent consumes its endowment". That is the homogeneous case only.
In the same economy, with no money and no government, each cohort's total consumption when young equals the cohort's total endowment when young. (Past exam: 2023 Midterm 1 Q2.2, reworded)
Show answer

True. The initial old only value date-1 consumption and have nothing to offer later, so they consume their endowment. Date-1 market clearing then pins generation 1's total young consumption. Summing generation 1's budgets over types pins its total old consumption, which pins generation 2's young consumption at date 2. By induction, each cohort consumes its own aggregate endowment.
With identical preferences $(1-\beta)\log c_y+\beta\log c_o$, the equilibrium interest rate depends on the distribution of old-age endowments $\alpha_n$ only through their mean $\bar\alpha$. (Problem set 2025 P4 Q5)
Show answer

True. Each type's saving $s(\alpha,r)=\beta(1-\alpha)-(1-\beta)\alpha/(1+r)$ is affine in $\alpha$, so aggregate saving is $N\,s(\bar\alpha,r)$, and $s=0$ gives $\bar r=\frac{\bar\alpha-\beta}{\beta(1-\bar\alpha)}$. The same holds for any identical homothetic preferences, since consumption is then linear in wealth (Gorman aggregation, Lesson 24). It fails if types have different preferences.
If $\bar\alpha\lt \beta$, a uniform PAYG tax $\tau=\beta-\bar\alpha$ makes every young type better off. (Problem set 2025 P4 Q8; Slides III p.22)
Show answer

False. The policy raises the interest rate to 0. For type $n$, $W(1+\bar r)\,dU_n=-\bar r\,d\tau+s_n\,dr$ (Exercise 16.3(h)). The transfer term helps everyone, but the rate term hurts borrowers ($s_n<0$). In the worked example of §3, type $\alpha=0.55$ loses.
In an OLG model where agents live $k\ge2$ periods, there is one commodity per period and the aggregate endowment is finite, if interest rates are strictly negative in all periods, the competitive equilibrium is not efficient. (Past exam: Fall 2025 Midterm 1 I.3)
Show answer

True. With $r_t<0$, $p_{t+1}/p_t=1/(1+r_t)>1$, so the value of the aggregate endowment is infinite and the First Welfare Theorem's proof breaks. The $(\tau,Y)$ transfer of §4 is an explicit Pareto improvement. Moving consumption from young ages to old ages has positive value $P^t>0$ for every generation, because later goods are more expensive.
In the heterogeneous economy with $\bar\alpha\lt \beta$, there is a feasible allocation that makes every type of every generation, including every initial-old type, strictly better off. (Problem set 2025 P4 Q7)
Show answer

True. This is the rescaled allocation of §2. Every young type gets $\lambda_n(1-\beta,\beta)$ instead of $\lambda_n(1-\bar\alpha,\bar\alpha)$, a strict gain with log utility. The initial old get $\alpha_n+\beta-\bar\alpha>\alpha_n$. The weights $\lambda_n$ average to one, which makes it feasible.
With $N$-period lives and intertemporal production, negative interest rates in every period no longer imply that the equilibrium is inefficient, because reallocating consumption changes the capital path. (Problem set 2025 P5 Q5)
Show answer

False. Keep the equilibrium production and capital plan unchanged. The $(\tau,Y)$ transfer leaves total consumption at each date unchanged, so it is still feasible. Its welfare effect is still $\lambda^tp_tP^t(1,Y)>0$ by the households' FOCs. The slides' caveat (III p.24) is that the argument "does not automatically go through": one must check that some feasible production plan supports the new consumption path. Holding the old plan fixed does that. What changes with production is implementation by a tax: PAYG alters saving and hence capital (Lesson 15).

Exercise 16.2 Heterogeneous agents with satiation points Past exam: 2023 Midterm 1 Q2

Dates are $t=1,2,\dots$, and each generation lives two periods. Agents are indexed by birth date $t=0,1,2,\dots$ and by type $i=1,\dots,I$ within the cohort, with one agent of each type. Agent $(t,i)$ with $t\ge1$ has utility

$$u^{t,i}=-\pi_y\frac{(\theta^i_y-x^{t,i}_t)^{1+\sigma}}{1+\sigma}-\pi_o\frac{(\theta^i_o-x^{t,i}_{t+1})^{1+\sigma}}{1+\sigma},\qquad \sigma>0,$$

on consumption below the satiation points $\theta^i=(\theta^i_y,\theta^i_o)$, which depend on type but not on cohort. The initial old have $u^{0,i}=x^{0,i}_1$. Endowments are $\alpha^i_y$ when young and $\alpha^i_o$ when old (the initial old have $\alpha^i_o$), all below the satiation points. Write $\bar\alpha_y=\sum_i\alpha^i_y$, $\bar\alpha_o=\sum_i\alpha^i_o$, $\bar\theta_y=\sum_i\theta^i_y$ and $\bar\theta_o=\sum_i\theta^i_o$.

Given (proved in Q1 of the same exam): in a finite exchange economy with these preferences, every Pareto-optimal allocation gives $x^i_l-\theta^i_l=\delta^i(\bar e_l-\bar\theta_l)$ with weights $\delta^i$ summing to one. Supporting prices satisfy $\frac{p_l}{p_k}=\frac{\pi_l(\bar\theta_l-\bar e_l)^\sigma}{\pi_k(\bar\theta_k-\bar e_k)^\sigma}$, the MRS of a representative agent with satiation point $\bar\theta$ evaluated at the aggregate endowment.

Is the equilibrium equal to autarky? Yes or no.
Is the total consumption of each cohort when young equal to $\bar\alpha_y$? Yes or no.
Is the interest rate constant over time? Yes or no.
Use the aggregation result, with the young as a representative agent, to find $\frac{1}{1+r_t}=\frac{p_{t+1}}{p_t}$ as a function of $\bar\theta_y,\bar\theta_o,\pi_y,\pi_o,\bar\alpha_y,\bar\alpha_o,\sigma$.
Suppose $\pi_o\lt \pi_y$, $\bar\alpha_y=\bar\alpha_o$ and $\bar\theta_y=\bar\theta_o$. Is the interest rate positive or negative? Is the allocation Pareto optimal?
[extension] The aggregation result characterizes Pareto-optimal allocations, but an OLG equilibrium need not be Pareto optimal. Why may we still use it in (d)?
[extension] Now suppose $\pi_o>\pi_y$, with the other assumptions of (e). Sign $r$, decide whether the equilibrium is Pareto optimal, and if not, exhibit a Pareto improvement.

Show solution

(a) No. Types differ in $\theta^i$ and $\alpha^i$, so at autarky their MRSs generally differ and they trade within the cohort.

(b) Yes. The initial old consume $\alpha^i_o$. Then the same induction as in Lesson 8, applied to cohort totals, shows that each cohort consumes $(\bar\alpha_y,\bar\alpha_o)$ in aggregate.

(c) Yes. Every cohort faces the same within-cohort problem: the same types and the same aggregate endowment. So the intertemporal price $p_{t+1}/p_t$ is the same for every $t$.

(d) Treat the cohort's two "goods" as youth and old-age consumption, with aggregate endowment $(\bar\alpha_y,\bar\alpha_o)$. The representative agent's MRS gives $$\frac{1}{1+r_t}=\frac{p_{t+1}}{p_t}=\frac{\pi_o}{\pi_y}\left(\frac{\bar\theta_o-\bar\alpha_o}{\bar\theta_y-\bar\alpha_y}\right)^{\sigma}.$$

(e) Here $1/(1+r)=\pi_o/\pi_y<1$, so $r>0$, constant over time. Then $p_t$ falls geometrically and the value of the aggregate endowment, $\sum_tp_t(\bar\alpha_y+\bar\alpha_o)$, is finite. The First Welfare Theorem applies, so the allocation is Pareto optimal.

(f) By (b), the cohort's trades net out. The equilibrium restricted to cohort $t$ is therefore a competitive equilibrium of a finite exchange economy: $I$ agents, two goods $(t,t+1)$, endowments $\alpha^i$, prices $(p_t,p_{t+1})$. Preferences are locally non-satiated below the satiation points, so the finite First Welfare Theorem makes that sub-allocation Pareto optimal within the cohort. That is all the aggregation result needs. Efficiency across generations is a separate question, answered in (e).

(g) Now $1/(1+r)=\pi_o/\pi_y>1$, so $r<0$. Prices rise over time, the value of the aggregate endowment is infinite, and the theorem is silent. The equilibrium is in fact not Pareto optimal. Take a small $\varepsilon$ from every young agent at every date and give it to the old alive at that date, equally across types. Every initial old gains. Each young agent $(t,i)$ gains to first order $\lambda^{t,i}(p_{t+1}-p_t)\varepsilon>0$, because every type's FOC uses the same prices.

Exercise 16.3 Heterogeneity within generations Problem set 2025 P4

Each generation has types $n=1,\dots,N$, one agent each, with endowments $e^{t,n}_t=1-\alpha_n$ and $e^{t,n}_{t+1}=\alpha_n$ and preferences $(1-\beta)\ln c^{t,n}_t+\beta\ln c^{t,n}_{t+1}$. The initial old of type $n$ have $e^{0,n}_1=\alpha_n$ and utility $c^{0,n}_1$. Let $\bar\alpha=\frac1N\sum_n\alpha_n$.

[5] Write the date-$t$ market-clearing condition.
[10] Show that there is no trade across generations. Hint: start with the initial old, then the initial young, then use induction.
[10] Find $c_y(\alpha,r)$, $c_o(\alpha,r)$ and the aggregate saving of the young $s(r;\{\alpha_n\},\beta)$.
[10] Show that $\frac{c_y(\alpha,r)}{c_y(\bar\alpha,r)}=\frac{c_o(\alpha,r)}{c_o(\bar\alpha,r)}=\frac{1+(1-\alpha)r}{1+(1-\bar\alpha)r}$ and that $\sum_nc_y(\alpha_n,r)=Nc_y(\bar\alpha,r)$.
[5] Show that the equilibrium interest rate depends only on $\bar\alpha$ and $\beta$.
[5] Find the best aggregate symmetric allocation $(c_y^*,c_o^*)$ subject to $c_y+c_o=1$.
[15] Suppose $\bar\alpha\lt \beta$. Show that the allocation giving young type $n$ the bundle $\lambda_n(c^*_y,c^*_o)$, with $\lambda_n=\frac{1+(1-\alpha_n)\bar r}{1+(1-\bar\alpha)\bar r}$, and initial-old type $n$ the amount $\alpha_n+(\beta-\bar\alpha)$, is feasible and Pareto-dominates the competitive equilibrium.
[15] Starting from $\bar\alpha\lt \beta$, introduce a uniform PAYG tax $\tau$. How does the equilibrium interest rate change? Show that, at $\tau=0$, type $n$'s utility changes according to $W_n(1+\bar r)\,dU_n=-\bar r\,d\tau+s_n\,dr$, where $s_n$ is its saving. Who can lose?
[10] Let $\tau=\beta-\bar\alpha$. Find the cutoff type $\hat\alpha$ above which types lose, in terms of $\bar r$ and $\beta$, and show that $\hat\alpha>\bar\alpha$.
[5] What does this imply for the design of social security?

Show solution

(a) $\sum_nc^{t,n}_t+\sum_nc^{t-1,n}_t=\sum_n(1-\alpha_n)+\sum_n\alpha_n=N$.

(b) The initial old consume $\alpha_n$. Date-1 clearing then gives $\sum_nc^{1,n}_1=\sum_n(1-\alpha_n)$. Sum generation 1's budgets over types: $p_1\sum_nc^{1,n}_1+p_2\sum_nc^{1,n}_2=p_1\sum_n(1-\alpha_n)+p_2\sum_n\alpha_n$. The young-age sums are equal, so the old-age sums are equal too. Date-2 clearing then pins generation 2's young sum. Induction finishes. Note: individual types do trade, with each other. One 2025 solution set says "each agent consumes its endowment", which is wrong here.

(c) Wealth is $W(\alpha,r)=(1-\alpha)+\frac{\alpha}{1+r}$. Log utility gives $c_y=(1-\beta)W$ and $c_o=\beta(1+r)W=\beta[(1+r)(1-\alpha)+\alpha]$. Aggregate saving is $\sum_n[(1-\alpha_n)-c_y(\alpha_n,r)]=N\big[\beta(1-\bar\alpha)-\frac{(1-\beta)\bar\alpha}{1+r}\big]$.

(d) Both consumptions are proportional to $W$, so the ratio is $\frac{W(\alpha,r)}{W(\bar\alpha,r)}$. Multiply top and bottom by $1+r$ and use $(1+r)(1-\alpha)+\alpha=1+(1-\alpha)r$. The sum identity holds because $c_y$ is affine in $\alpha$.

(e) Aggregate saving is zero iff $\beta(1-\bar\alpha)(1+\bar r)=(1-\beta)\bar\alpha$, i.e. $\bar r=\frac{\bar\alpha-\beta}{\beta(1-\bar\alpha)}$. At $\bar r$ the average type's bundle is its endowment $(1-\bar\alpha,\bar\alpha)$.

(f) $c^*_y=1-\beta$, $c^*_o=\beta$.

(g) $\bar r<0\iff\bar\alpha\lt \beta$. Feasibility: $\lambda_n$ is affine in $\alpha_n$, so $\frac1N\sum_n\lambda_n=1$. Average young consumption is $1-\beta$ and average old consumption is $\beta$. At date 1 the initial old average $\bar\alpha+\beta-\bar\alpha=\beta$. Totals equal $N$ at every date. Dominance: by (d)–(e), type $n$'s CE bundle is $\lambda_n(1-\bar\alpha,\bar\alpha)$. With log weights summing to one, $u(\lambda x)=\ln\lambda+u(x)$. So type $n$ compares $u(1-\beta,\beta)$ with $u(1-\bar\alpha,\bar\alpha)$: a strict gain, since $(1-\beta,\beta)$ uniquely maximizes $u$ on $c_y+c_o=1$. (Also $\lambda_n>0$ since $1+(1-\alpha_n)\bar r>1+\bar r>0$.) The initial old gain $\beta-\bar\alpha>0$.

(h) After-tax endowments are $(1-\alpha_n-\tau,\alpha_n+\tau)$, so $\bar\alpha$ rises to $\bar\alpha+\tau$ and $r(\tau)=\frac{\bar\alpha+\tau-\beta}{\beta(1-\bar\alpha-\tau)}$ rises, with $\frac{dr}{d\tau}\big|_0=\frac{1-\beta}{\beta(1-\bar\alpha)^2}$. Indirect utility is $U_n=\ln W_n+\beta\ln(1+r)+\text{const}$, with $W_n=(1-\alpha_n-\tau)+\frac{\alpha_n+\tau}{1+r}$. At $\tau=0$, $dW_n=-\frac{\bar r}{1+\bar r}d\tau-\frac{\alpha_n}{(1+\bar r)^2}dr$. So $$W_n(1+\bar r)\,dU_n=-\bar r\,d\tau+\Big(\beta W_n-\frac{\alpha_n}{1+\bar r}\Big)dr=-\bar r\,d\tau+s_n\,dr,$$ using $s_n=(1-\alpha_n)-(1-\beta)W_n=\beta W_n-\frac{\alpha_n}{1+\bar r}$. The first term is a gain for everyone because $\bar r<0$. The second is a loss for borrowers ($s_n<0$, high $\alpha_n$), and heavy borrowers can lose overall.

(i) With $\tau=\beta-\bar\alpha$ the new rate is 0, so $W_n=1$ and $U_n=0$ for every type. Before the policy, $U_n=\ln W(\alpha_n,\bar r)+\beta\ln(1+\bar r)$. Type $n$ loses iff $W(\alpha_n,\bar r)>(1+\bar r)^{-\beta}$. Since $W(\alpha,\bar r)=1+\alpha\big(\frac{1}{1+\bar r}-1\big)$ is increasing in $\alpha$ when $\bar r<0$, $$\hat\alpha=\frac{(1+\bar r)^{-\beta}-1}{(1+\bar r)^{-1}-1}.$$ The average type gains: before, it consumes its endowment $(1-\bar\alpha,\bar\alpha)$; after, it consumes $(1-\beta,\beta)$, the best point on the same line. So $\hat\alpha>\bar\alpha$: the lenders and some mild borrowers gain, and the heavy borrowers lose.

(j) Fixing the aggregate inefficiency (raising $r$ to 0) does not settle distribution. A uniform tax changes a relative price at which types are on opposite sides of the market. A Pareto improvement needs type-specific transfers (Exercise 16.5).

Exercise 16.4 Agents who live $N$ periods Problem set 2025 P5

Generation $t\ge1$ has utility $v^t(c^t_t,\dots,c^t_{t+N-1})$. Generations $t=-N+2,\dots,0$ are already alive at date 1 and value $c^t_1,\dots,c^t_{t+N-1}$. Endowments are positive while alive, there is one good per date, and $\bar c$ is a competitive equilibrium with all $r_t<0$.

[5] Write the market-clearing condition at date $t$.
[5] Write the first-order conditions for generation $t\ge1$, and for a generation alive at date 1.
[15] For the $(\tau,Y)$ transfer of §4, write the value $P^t(\tau,Y)$ at birth (or at date 1, for generations already alive) for the three groups $t\ge1$, $-Y+2\le t\le0$ and $-N+2\le t\le-Y+1$. Show that each is positive when $p_t$ is increasing.
[15] Show that the transferred allocation is feasible for every $\tau$, and that for small $\tau>0$ it Pareto-dominates $\bar c$. Hint: differentiate each generation's utility at $\tau=0$.
[5] With a constant interest rate $r$, let $q=1/(1+r)$. Show that for $t\ge1$, $P^t(1,Y)=\frac{Y}{N-Y}\sum_{a=Y+1}^Nq^{a-1}-\sum_{a=1}^Yq^{a-1}$. Check that $N=2$, $Y=1$ recovers two-period PAYG.
[10] Suppose there are intertemporal production possibilities, so that period consumption need not equal the period endowment. Does the conclusion change?

Show solution

(a) $\sum_{a=1}^Nc^{t-a+1}_t=\sum_{a=1}^Ne^{t-a+1}_t$: a sum over the $N$ cohorts alive at date $t$. Common error: one 2025 solution set writes $\sum_ac^t_{t+a-1}=\sum_ae^t_{t+a-1}$, which is one generation's lifetime sum, not market clearing.

(b) $v^t_a(\bar c^t)=\lambda^tp_{t+a-1}$ for $a=1,\dots,N$ when $t\ge1$. For a generation alive at date 1, the same holds for its remaining ages $a=2-t,\dots,N$.

(c) For $t\ge1$: $P^t=\frac{1}{p_t}\big[-\tau\sum_{a=1}^Yp_{t+a-1}+\frac{Y\tau}{N-Y}\sum_{a=Y+1}^Np_{t+a-1}\big]=\frac{Y\tau}{p_t}\big[\bar p_{\text{late}}-\bar p_{\text{early}}\big]$, which is positive because every late price exceeds every early price. For $-Y+2\le t\le0$: the same, but the first sum starts at $a=2-t$ and the normalization is $p_1$. Fewer than $Y$ payments are made, and $\frac{Y}{N-Y}\sum_{\text{late}}p=Y\bar p_{\text{late}}>Y\max p_{\text{early}}$, which exceeds any sum of fewer than $Y$ early prices. So $P^t>0$. For $-N+2\le t\le-Y+1$: these generations are already past age $Y$ and only receive, so $P^t=\frac{1}{p_1}\frac{Y\tau}{N-Y}\sum_{a=2-t}^Np_{t+a-1}>0$.

(d) At every date there is one cohort of each age: $Y$ of them each give $\tau$ and $N-Y$ each receive $\frac{Y\tau}{N-Y}$. The net is $-Y\tau+Y\tau=0$, so the allocation is feasible for any $\tau$ (and non-negative for $\tau$ small). Differentiating at $\tau=0$ and using (b), $\frac{du^t}{d\tau}\big|_0=\sum_av^t_a\frac{dc_a}{d\tau}=\lambda^t\sum_ap_{t+a-1}\frac{dc_a}{d\tau}=\lambda^tp_tP^t(1,Y)>0$, with $p_1$ for the generations alive at date 1. Every generation gains for small $\tau$. (With infinitely many generations one needs a common $\bar\tau$; this is immediate in a stationary equilibrium.) So $\bar c$ is not Pareto optimal. Common error: checking feasibility by summing one generation's lifetime transfers. The arithmetic happens to give zero, but the object that must net to zero is the period total.

(e) With $p_{t+a-1}/p_t=q^{a-1}$, divide (c) by $\tau$. For $N=2$, $Y=1$: $P=q-1=\frac{1}{1+r}-1$, positive iff $r<0$, as in Lesson 11.

(f) No, provided the equilibrium production plan is held fixed. The transfer leaves total consumption at each date unchanged, so the original production and capital path still supports it, and (d) goes through verbatim. The caveat on slide III p.24 is that this must be checked, not assumed. Implementing the improvement with a tax in a competitive equilibrium is a different matter: saving and capital would respond, as in Lesson 15.

Exercise 16.5 Design a Pareto-improving policy Exam-inspired

Show that the rescaled allocation of §2 can be implemented as a competitive equilibrium with (i) a uniform PAYG tax $\tau=\beta-\bar\alpha$ and (ii) lump-sum transfers $T_n$ among the young of each generation, summing to zero. Find $T_n$.

Show solution

With $\tau=\beta-\bar\alpha$, the post-tax average old endowment is $\beta$, so $r=0$. This holds for any zero-sum within-generation transfers, since saving remains affine in endowments. At $r=0$, a type-$n$ young agent's wealth is $(1-\alpha_n-\tau+T_n)+(\alpha_n+\tau)=1+T_n$. It consumes $(1-\beta)(1+T_n)$ and $\beta(1+T_n)$. Matching the rescaled allocation requires $1+T_n=\lambda_n$, i.e. $$T_n=\lambda_n-1=\frac{(\bar\alpha-\alpha_n)\,\bar r}{1+(1-\bar\alpha)\bar r}.$$ These sum to zero because $\lambda_n$ averages to one. With $\bar r<0$, the high-$\alpha_n$ types, the borrowers hurt by the rise in $r$, receive $T_n>0$. The initial old receive $\tau=\beta-\bar\alpha$, as in §2.

Exercise 16.6 The reverse transfer when rates are positive Exam-inspired

Suppose instead that all equilibrium interest rates are positive. Consider the reverse transfer, from old ages to young ages. Show that it has positive value $P^t$ for every generation born at $t\ge1$, but that it cannot be a Pareto improvement. Identify a generation that must lose.

Show solution

With $r>0$ prices fall over time, so the reverse transfer has positive value for every generation born at $t\ge1$: it receives early, when goods are expensive, and pays late, when they are cheap. But a generation aged $a_0>Y$ at date 1 only pays, receiving nothing in its remaining life, so it strictly loses. This is consistent with the First Welfare Theorem: with $r>0$, the value of the aggregate endowment is finite (Lesson 12), so the equilibrium is Pareto optimal.

Key takeaways

With log utility, type-$\alpha$ consumption is the average type's bundle scaled by $\frac{1+(1-\alpha)r}{1+(1-\bar\alpha)r}$, and saving is linear in $\alpha$. So $\bar r = \frac{\bar\alpha-\beta}{\beta(1-\bar\alpha)}$ depends only on $\bar\alpha$.
There is trade within generations: low-$\alpha$ types lend to high-$\alpha$ types.
If $\bar\alpha\lt \beta$, the wealth-rescaled best symmetric allocation, plus $\beta-\bar\alpha$ to the initial old, Pareto-dominates the CE.
Uniform PAYG raises $r$ and hurts borrowers: correcting aggregate inefficiency does not settle the distribution of gains.
With $N$-period lives and $r<0$, any small young-to-old transfer has positive value $P^t$ for every generation and is a Pareto improvement in pure exchange.

💬 Stuck or curious? Ask me (your Claude teacher) anything. Try "does exact aggregation survive with CRRA utility?" or "redo the N-period argument for a generation that's already age Y+1 at date 1."

← Lesson 15: OLG with capital Course index Glossary Lesson 17: Perpetual youth, households →

Heterogeneity and Long Lives in the OLG Model

1. Heterogeneity within a generation

Individual demands

Aggregation and the interest rate

2. The equilibrium is still inefficient when \(\bar\alpha\lt \beta\)

3. A uniform PAYG system is not enough

4. Agents who live \(N\) periods

5. Check your understanding

6. Exercises

Key takeaways