Theory of Income · Part II: Overlapping Generations · Lesson 11 of 25

Pay-As-You-Go Social Security

Covers Class Note II, pp. 19–20 · About 30 min · Prerequisite: Lesson 10

By the end you can:

model a pay-as-you-go (PAYG) social security system as a change in after-tax endowments, and solve for the new CE;
find the tax $\tau^* = \beta-\alpha$ that implements the best symmetric allocation;
answer the slide question: for which parameters, and for which taxes $\tau$, does social security produce a CE that Pareto-dominates the original one? Answer: iff $\bar r<0$, and then for every $\tau\in(0,\bar\tau]$.

Primary source Alvarez, Class Note II, pp. 19–20. Samuelson's 1958 paper makes the same point with a "social compact" and with fiat money. Exercise 11.6 shows that the two are equivalent. Diamond, "National Debt in a Neoclassical Growth Model", AER (1965), extends the argument to an economy with capital (see Lesson 15).

1. The policy

Definition: PAYG social security with tax $\tau$ II p.19

At every date, each young agent pays a lump-sum tax $\tau$ and each old agent alive receives a transfer $\tau$. One young and one old are alive at each date, so taxes equal transfers period by period: the government budget balances for every $\tau$ and the government never issues debt. After-tax endowments become $$e^i_i = (1-\alpha)-\tau,\qquad e^i_{i+1} = \alpha+\tau\quad (i\ge1),\qquad e^0_1 = \alpha+\tau .$$ We need $\tau<1-\alpha$ so that young endowments stay positive. A negative $\tau$ is a transfer from old to young.

"Pay-as-you-go" means that current workers' contributions are paid out immediately to current retirees. Nothing is invested. Contrast this with a funded system that would save contributions and pay retirees out of the accumulated returns; Lesson 14 shows how to move between the two.

2. The equilibrium with social security

The policy only relabels endowments. Every assumption of the no-trade proposition (Lesson 8) still holds with the after-tax endowments, so the unique CE is again autarky, now at the after-tax point:

$$c_y(\tau) = 1-\alpha-\tau,\qquad c_o(\tau) = \alpha+\tau .$$

The equilibrium interest rate follows from Lesson 9's formula evaluated at the new endowment:

$$1+r(\tau) = \frac{v_1}{v_2}\Big|_{(1-\alpha-\tau,\ \alpha+\tau)} = \frac{(1-\beta)(\alpha+\tau)}{\beta(1-\alpha-\tau)},$$

which is increasing in $\tau$. Social security shifts resources toward old age, so the young want to save less. To keep them content with zero savings, the interest rate must rise.

Implementing the best symmetric allocation

Choosing $\tau$ so that the after-tax endowment is $(1-\beta,\beta)$ requires II p.19

$$\boxed{\;\tau^* = \beta-\alpha\;}$$

and then $1+r(\tau^*) = \frac{(1-\beta)\beta}{\beta(1-\beta)} = 1$: the interest rate becomes zero, the biological rate of return.

3. When does social security make everyone better off?

Compare the CE with policy $\tau$ to the CE without it ($\tau=0$).

Initial old: consumption $\alpha+\tau$. They gain iff $\tau>0$.
Generations $i\ge1$: utility $W(\tau) \equiv (1-\beta)\log(1-\alpha-\tau) + \beta\log(\alpha+\tau)$.

Analysis of $W(\tau)$

Strict concavity. $W''(\tau) = -\frac{1-\beta}{(1-\alpha-\tau)^2} - \frac{\beta}{(\alpha+\tau)^2} < 0$.
Peak. $W'(\tau) = -\frac{1-\beta}{1-\alpha-\tau} + \frac{\beta}{\alpha+\tau} = 0$ at $\alpha+\tau = \beta$, i.e. at $\tau^*=\beta-\alpha$.
Slope at zero. $W'(0) = \frac{\beta}{\alpha} - \frac{1-\beta}{1-\alpha} = \frac{1-\beta}{1-\alpha}\Big[\frac{1}{1+\bar r}-1\Big]$. This is positive iff $\bar r<0$ iff $\beta>\alpha$. A small PAYG system is a "bond" paying a gross return of 1. The young value it at $1/(1+\bar r)$, which exceeds 1 exactly when $\bar r<0$.
Case $\beta>\alpha$. $W$ rises on $[0,\tau^*]$ and falls after, with $W\to-\infty$ as $\tau\uparrow1-\alpha$. By continuity there is a unique $\bar\tau\in(\tau^*,1-\alpha)$ with $W(\bar\tau)=W(0)$. Hence $W(\tau)\ge W(0)\iff\tau\in[0,\bar\tau]$.
Case $\beta\le\alpha$. $\tau^*\le0$, so $W$ is strictly decreasing on $[0,1-\alpha)$. Any $\tau>0$ hurts generations $i\ge1$, and any $\tau<0$ hurts the initial old. No $\tau\ne0$ is a Pareto improvement.

∎

Answer to the slide question II p.20

A PAYG system with tax $\tau$ yields a CE that Pareto-dominates the no-policy CE if and only if $\beta>\alpha$, equivalently the no-policy CE has a negative interest rate $\bar r<0$, and $\tau\in(0,\bar\tau]$, where $\bar\tau>\beta-\alpha$ solves $$(1-\beta)\log\frac{1-\alpha-\bar\tau}{1-\alpha} + \beta\log\frac{\alpha+\bar\tau}{\alpha} = 0 .$$ At $\tau=\bar\tau$ the young are exactly indifferent, but the initial old strictly gain, so it is still a Pareto improvement. The improvement is largest for the young at $\tau^*=\beta-\alpha$. If $\beta=\tfrac12$, then $W$ is symmetric around $\tau^*$ and $\bar\tau = 2(\beta-\alpha)$ exactly. Otherwise $\bar\tau\gtrless2(\beta-\alpha)$ as $\beta\lessgtr\tfrac12$ (Exercise 11.3).

Interactive. Left: the after-tax endowment, which is also the CE, $(1-\alpha-\tau,\alpha+\tau)$ (violet), slides along $c_y+c_o=1$ as $\tau$ changes. Orange is the no-policy CE, blue is the best symmetric point. Right: welfare changes relative to $\tau=0$: blue is $W(\tau)-W(0)$ for every generation $i\ge1$; orange is the initial old's gain $\tau$. The two are in different units (utils vs. goods), so only their signs matter for Pareto comparisons. The aqua band on the axis is the Pareto-improving set $(0,\bar\tau]$. Try: lower $\beta$ below $\alpha$ and watch the band disappear, as the blue curve now peaks at a negative $\tau$.

Intuition: PAYG is an asset with return equal to population growth

From any generation's point of view, PAYG says: give up $\tau$ when young, get $\tau$ when old. That's a savings instrument with a gross return of $1$, the ratio of contributors to recipients. Private markets offered an "autarky" return $1+\bar r$. If $1+\bar r<1$, the PAYG asset dominates and everyone wants some. The initial old receive a pure windfall. The scheme's "capital" is the promise that the next young will pay, and that promise is honoured forever. In Lesson 13 the return on PAYG becomes $(1+n)(1+g)$ and the condition becomes $1+\bar r<(1+n)(1+g)$.

4. Check your understanding

1. What is the CE allocation once PAYG social security with tax $\tau$ is introduced?

The best symmetric allocation, for any positive tax τ
The original endowment, since taxes cancel out exactly
Autarky at the after-tax endowment (1−α−τ, α+τ)
Whatever allocation sets the interest rate equal to zero

The no-trade proposition applies to any positive endowments, including after-tax ones. So consumption equals after-tax endowment. Only at $\tau=\beta-\alpha$ is this the best symmetric allocation, and only then is $r=0$.

2. Why does a small PAYG system benefit generations $i\ge1$ exactly when $\bar r<0$?

It pays a zero net return, beating the market rate r̄
It lowers taxes on the young below the transfer to the old
It raises aggregate endowment above one in every period
It eliminates the initial old, who were the binding agents

$W'(0) = \frac{1-\beta}{1-\alpha}\big[\frac1{1+\bar r}-1\big]$. The marginal value of a unit received when old, relative to a unit paid when young, is $1/(1+\bar r)$. PAYG trades them one-for-one, so it helps iff $1/(1+\bar r)>1$.

3. With $\beta>\alpha$, which taxes $\tau$ give a Pareto improvement over the no-policy CE?

Only the single tax τ = β − α, and no other value
Every tax τ between zero and one minus α, inclusive
Every tax τ between zero and 2(β − α), for any β
Every tax in (0, τ̄], where young utility returns to CE

The initial old gain for any $\tau>0$. The young weakly gain on $[0,\bar\tau]$ by concavity of $W$. $\bar\tau$ equals $2(\beta-\alpha)$ only in the special case $\beta=1/2$.

4. Social security is in place with $\tau=\tau^*$. The government abolishes it at date $T$. Who loses?

Generation T, since it never pays the tax when young
Generation T − 1, which paid when young but gets nothing
Nobody, since the government budget always balanced
The initial old, since the scheme retroactively unwinds

Generation $T-1$ paid $\tau^*$ at $T-1$ and expected $\tau^*$ at $T$. Abolition leaves it with $(1-\beta,\alpha)$, worse than both the CE and the best allocation. Generation $T$ returns to autarky, and earlier generations are unaffected. The chain must never be broken.

5. Exercises

Social security in OLG shows up on exams as true/false items ("does PAYG make everyone better off?") and in the 2025 problem set, which extends this lesson to population growth and to heterogeneous types (Exercises 11.7–11.8). Answer with general parameters.

Exercise 11.1 Exam-inspired True or false? Justify in 1–4 lines.

If the no-policy CE has $\bar r<0$, introducing a PAYG system with any tax $\tau>0$ is a Pareto improvement.
Show answer
False. The young's utility $W(\tau)$ rises up to $\tau^*=\beta-\alpha$ and then falls. It drops below its CE level once $\tau>\bar\tau$. The Pareto-improving set is $(0,\bar\tau]$, not $(0,\infty)$.
If $\bar r>0$, a small PAYG system ($\tau>0$) makes every generation born at $t\ge1$ worse off.
Show answer
True. $W'(0)=\frac{1-\beta}{1-\alpha}\big[\frac1{1+\bar r}-1\big]<0$. The system is a "bond" with gross return 1, and the young value it at $1/(1+\bar r)<1$. (Only the initial old gain.)
In the pure-endowment economy, introducing PAYG raises the equilibrium interest rate.
Show answer
True. The after-tax endowment $(1-\alpha-\tau,\alpha+\tau)$ is again autarkic, and $1+r(\tau)=\frac{(1-\beta)(\alpha+\tau)}{\beta(1-\alpha-\tau)}$ is increasing in $\tau$. With more old-age resources, the young want to save less, so the rate that makes them content not to trade rises.
Under PAYG with $\tau=\beta-\alpha$, the equilibrium interest rate is zero and the allocation is the best symmetric allocation.
Show answer
True. After-tax endowments become $(1-\beta,\beta)$, and $1+r(\tau^*)=\frac{(1-\beta)\beta}{\beta(1-\beta)}=1$. This is the "golden rule": the implicit return on PAYG (zero without growth) equals the market rate.
With heterogeneous endowments within each generation and $\bar r<0$, a small uniform PAYG tax makes every young type better off. (cf. Problem set 2025, P4 Q8)
Show answer
False. Everyone gains from the transfer itself, but the policy raises $r$, and that hurts types who borrow when young. Heavy borrowers can lose on net (Exercise 11.8).
With population growth $n$, the PAYG tax that implements the best symmetric allocation is positive if and only if $n>\bar r$. (cf. Problem set 2025, P6 Q5)
Show answer
True. The required tax equals the young's desired saving at $r=n$: $\tau^*=s(n)$. Since $s$ is increasing with $s(\bar r)=0$, $\tau^*>0\iff n>\bar r$ (Exercise 11.7).
Abolishing an existing PAYG system cannot be a Pareto improvement.
Show answer
True. The generation that is old when the system ends paid $\tau$ when young and now receives nothing. Compared with keeping the system, it strictly loses (Exercise 11.4).
Valued fiat money can implement the same allocation as the optimal PAYG system.
Show answer
True when $\bar r<0$. A stationary monetary equilibrium has return 1 on money, and real balances equal $s(0)=\beta-\alpha$. So consumption is $(1-\beta,\beta)$, exactly as under $\tau^*$ (Exercise 11.6). If $\bar r\ge0$, no valued stationary monetary equilibrium exists.

Exercise 11.2 Slides II pp.19–20 Exam-inspired When does PAYG Pareto-improve?

Endowments $(1-\alpha,\alpha)$, preferences $(1-\beta)\log c_y+\beta\log c_o$, and the PAYG policy $\tau$ of §1. Answer without looking back at §3.

[5 pts] Show that the CE with policy $\tau$ is autarky at the after-tax endowments, and find $r(\tau)$.
[10 pts] Write the utility $W(\tau)$ of a generation $t\ge1$. Show that $W'(0)>0\iff\bar r<0$, and interpret.
[10 pts] Show that $W$ is strictly concave. Find the $\tau$ that maximizes it and the interest rate there.
[10 pts] Characterize the full set of $\tau$ that produce a CE Pareto-dominating the no-policy CE. Answer the slide's two questions: for which parameters, and at what no-policy interest rate?

Show solution

(a) Taxes and transfers just relabel endowments, so the Lesson 8 proposition applies: the CE is no trade at $(1-\alpha-\tau,\alpha+\tau)$, with $1+r(\tau)=\frac{(1-\beta)(\alpha+\tau)}{\beta(1-\alpha-\tau)}$.

(b) $W(\tau)=(1-\beta)\log(1-\alpha-\tau)+\beta\log(\alpha+\tau)$, so $W'(0)=\frac{\beta}{\alpha}-\frac{1-\beta}{1-\alpha}=\frac{1-\beta}{1-\alpha}\Big[\frac{1}{1+\bar r}-1\Big]$. This is positive iff $\bar r<0$. Each unit given up when young returns one unit when old, which the young value at $1/(1+\bar r)$.

(c) $W''<0$ (a sum of logs of affine functions). The FOC $\frac{\beta}{\alpha+\tau}=\frac{1-\beta}{1-\alpha-\tau}$ gives $\tau^*=\beta-\alpha$, where $r(\tau^*)=0$.

(d) The initial old gain iff $\tau>0$. If $\beta>\alpha$, $W$ rises on $[0,\tau^*]$ and then falls to $-\infty$ as $\tau\uparrow1-\alpha$. So there is a unique $\bar\tau\in(\tau^*,1-\alpha)$ with $W(\bar\tau)=W(0)$, and the improving set is $(0,\bar\tau]$. If $\beta\le\alpha$, $W$ is decreasing on $\tau\ge0$, and no $\tau\ne0$ works. Answer: PAYG Pareto-improves iff $\beta>\alpha$, i.e. iff the no-policy CE has $\bar r<0$, and then only for $\tau\in(0,\bar\tau]$.

Exercise 11.3 Proof

Show that when $\beta=\tfrac12$, $\bar\tau = 2(\beta-\alpha) = 1-2\alpha$. Then show that, more generally, $\bar\tau>2(\beta-\alpha)$ if $\beta\lt \tfrac12$ and $\bar\tau<2(\beta-\alpha)$ if $\beta>\tfrac12$.

Hint

Write young utility as a function of old-age consumption $c$: $f(c) = (1-\beta)\log(1-c) + \beta\log c$, which peaks at $c=\beta$. With $d\equiv\beta-\alpha>0$, the CE is at $c = \beta-d$, and $\tau=2d$ puts the young at $c=\beta+d$. Study $g(d)\equiv f(\beta+d)-f(\beta-d)$.

Show solution

Reduction. $\bar\tau$ is where $f$ returns to its CE level $f(\beta-d)$ on the far side of the peak. Since $f$ is strictly decreasing to the right of $\beta$: $\bar\tau > 2d \iff f(\beta+d) > f(\beta-d) \iff g(d) > 0$, and $\bar\tau = 2d\iff g(d)=0$.
Derivative. $g(0)=0$ and $g'(d) = f'(\beta+d)+f'(\beta-d)$. Using $f'(c) = \frac{\beta}{c}-\frac{1-\beta}{1-c}$: $$g'(d) = \frac{2\beta^2}{\beta^2-d^2} - \frac{2(1-\beta)^2}{(1-\beta)^2-d^2}.$$
Sign. $h(x) = \frac{x^2}{x^2-d^2} = 1 + \frac{d^2}{x^2-d^2}$ is strictly decreasing in $x>d$. So $g'(d)>0\iff h(\beta)>h(1-\beta)\iff\beta\lt \tfrac12$. For $\beta=\tfrac12$, $g'\equiv0$.
Conclusion. Integrate from $g(0)=0$. If $\beta=\tfrac12$, $g\equiv0$ and $\bar\tau = 2d = 1-2\alpha$ (indeed $f(c)=\tfrac12\log(c(1-c))$ is symmetric about $\tfrac12$). If $\beta\lt \tfrac12$, $g(d)>0$ and $\bar\tau>2(\beta-\alpha)$. If $\beta>\tfrac12$, $g(d)<0$ and $\bar\tau<2(\beta-\alpha)$. (If $\beta+d\ge1$, $\tau=2d$ is infeasible and $\bar\tau<2d$ trivially.)

Move the sliders in the figure of §3 to see all three cases.

Exercise 11.4 Exam-inspired Transitions

Suppose $\beta>\alpha$.

The government introduces PAYG with $\tau^*$ at date $T>1$, not at date 1. Rank every generation's welfare relative to the no-policy CE.
Instead it introduces the system at date 1 and abolishes it at date $T$. Rank again.
What does this say about the political economy of PAYG systems?

Show solution

(a) Generations $0,\dots,T-2$ are unaffected. Generation $T-1$ is the "initial old" of the new system: it gets $\tau^*$ for free and strictly gains. Generations $\ge T$ pay and receive $\tau^*$ and strictly gain. This is a Pareto improvement.

(b) Generations $0,\dots,T-2$ strictly gain. Generation $T-1$ pays when young and receives nothing when old, so it consumes $(1-\beta,\alpha)$ and strictly loses. Generations $\ge T$ are unaffected. Not a Pareto improvement.

(c) Introducing PAYG creates a windfall for the generation that is old at introduction. Abolishing it imposes a loss on the generation that is old at abolition. Every generation therefore wants to keep the system going: a self-sustaining "social compact" (Samuelson).

Exercise 11.5 Exam-inspired Negative taxes in Case 3

Now take $\beta\lt \alpha$ (so $\bar r>0$).

Show that a negative $\tau$ (old-to-young transfers) raises the utility of every generation $i\ge1$ for small $|\tau|$.
Why is this nevertheless not a Pareto improvement?
Would it be one if the economy started at date $-\infty$, with no initial old?

Show solution

(a) $W'(0) = \frac{1-\beta}{1-\alpha}\big[\frac1{1+\bar r}-1\big]<0$ when $\bar r>0$, so lowering $\tau$ below 0 raises $W$.

(b) The initial old's consumption falls to $\alpha+\tau\lt \alpha$, so someone loses.

(c) With a doubly infinite time line there is no initial old. Every generation is both a payer and a receiver, so $\tau<0$ would make everyone better off when $\bar r>0$. The asymmetry between Cases 1 and 3 comes entirely from having a beginning without an end (Shell 1971).

Exercise 11.6 Exam-inspired Fiat money

Remove social security. At date 1 the initial old instead hold $M>0$ units of intrinsically useless fiat money. Each period the young may buy money from the old at price $1/P_t$ (goods per unit of money) and sell it when old.

Write generation $t$'s budget constraints and show that the gross real return on money is $P_t/P_{t+1}$.
Look for a stationary monetary equilibrium with $P_t = P$. Find real balances $M/P$ and consumption.
When does such an equilibrium exist? Compare it with the PAYG system.

Show solution

(a) Young: $c_y + m_t/P_t = 1-\alpha$. Old: $c_o = \alpha + m_t/P_{t+1}$. Real savings $m_t/P_t$ earn gross return $P_t/P_{t+1}$.

(b) With constant $P$ the return is 1, i.e. $r=0$. The young must hold all the money, so $M/P = s(0) = \beta(1-\alpha)-(1-\beta)\alpha = \beta-\alpha$. Consumption is $(1-\beta,\beta)$, and the initial old consume $\alpha + M/P = \beta$.

(c) It requires $M/P>0$, i.e. $\beta>\alpha$, exactly when $\bar r<0$. The allocation is identical to PAYG with $\tau^*$: valued fiat money is a privately sustained version of social security. This is Samuelson's "social contrivance of money".

Exercise 11.7 Problem set 2025, P6 Q3–Q6 PAYG with population growth

Keep log preferences and endowments $(1-\alpha,\alpha)$, but let cohorts grow: $N_{t+1}=(1+n)N_t$, $n>0$. Under PAYG each young pays $\tau$, and each old receives $N_t\tau/N_{t-1}=(1+n)\tau$.

[5 pts] Show that the no-policy interest rate $\bar r$ does not depend on $n$. Why?
[10 pts] Using feasibility per young person, $c_y+\frac{c_o}{1+n}=(1-\alpha)+\frac{\alpha}{1+n}$, find the best symmetric allocation $(c^*_y,c^*_o)$.
[15 pts] Find the tax $\tau^*$ that implements it. Show that $\tau^*=s(n)$, the young's desired saving at $r=n$. Conclude that $\tau^*>0\iff n>\bar r$, and sign $\partial\tau^*/\partial n$. Give the intuition.
[15 pts] The system with $\tau^*$ is in place when population growth falls permanently to $n'=0$, with the tax per young held fixed. What happens to the welfare of the generation that is old at the change, and of every later generation? Show that $\tau^*$ is now larger than the optimal tax for $n'=0$, and sign the new equilibrium interest rate.

Show solution

(a) There is still no trade (Exercise 8.7(a)), so $1/(1+\bar r)$ is each generation's MRS at its endowment, $\frac{\beta(1-\alpha)}{(1-\beta)\alpha}$. Population size never enters.

(b) The FOC is $\frac{\beta}{1-\beta}\frac{c_y}{c_o}=\frac1{1+n}$. With wealth $W=(1-\alpha)+\frac{\alpha}{1+n}$: $c^*_y=(1-\beta)W$ and $c^*_o=(1+n)\beta W=\beta[(1+n)(1-\alpha)+\alpha]$.

(c) Set $1-\alpha-\tau^*=c^*_y$: $\tau^*=\beta(1-\alpha)-\frac{(1-\beta)\alpha}{1+n}$. Old-age consumption checks: $\alpha+(1+n)\tau^*=c^*_o$. Comparing with Exercise 9.3, $\tau^*=s(n)$. Since $s$ is strictly increasing with $s(\bar r)=0$, $\tau^*>0\iff n>\bar r$. Also $\partial\tau^*/\partial n=\frac{(1-\beta)\alpha}{(1+n)^2}>0$. Intuition: $n$ is the implicit return on PAYG. Social security is worth having exactly when it beats the market rate $\bar r$, and a higher $n$ makes it a better deal. (Talk of "substitution effects" in circulated solutions misses the point. The clean channel is that a higher $n$ lowers the present value of the old-age endowment.)

(d) The old at the change paid $\tau^*$ but receive $\tau^*$ instead of $(1+n)\tau^*$, so they lose. Every later generation has the same $c_y=c^*_y$ but old-age consumption $\alpha+\tau^*\lt c^*_o$, so they lose too. Also $\tau^*-(\beta-\alpha)=\frac{\alpha(1-\beta)n}{1+n}>0$, so the fixed tax overshoots the $n'=0$ optimum $\beta-\alpha$. Since $\alpha+\tau^*>\beta$, the new no-trade rate satisfies $1+r'=\frac{(1-\beta)(\alpha+\tau^*)}{\beta(1-\alpha-\tau^*)}>1$, so $r'>0=n'$: the system is now too big. Graphically, the resource line rotates down around the endowment point (Lesson 13).

Exercise 11.8 Problem set 2025, P4 Q8 Uniform PAYG with heterogeneous types

Each cohort has types $n=1,\dots,N$ with endowments $(1-\alpha_n,\alpha_n)$, log preferences with weight $\beta$, and $\bar\alpha\equiv\frac1N\sum_n\alpha_n\lt \beta$, so $\bar r=\frac{\bar\alpha-\beta}{\beta(1-\bar\alpha)}<0$ (Lesson 16). A uniform PAYG tax $\tau$ is introduced, so type $n$'s endowments become $(1-\alpha_n-\tau,\alpha_n+\tau)$.

[10 pts] How does the equilibrium interest rate change? Hint: compute the new $\bar\alpha$.
[15 pts] Show that type $n$'s indirect utility is $U_n=\log W_n+\beta\log(1+r)+\text{const}$, where $W_n$ is lifetime wealth. Then show that, at $\tau=0$, $$\frac{dU_n}{d\tau}=\frac{-\bar r+s_n\,r'(0)}{(1+\bar r)W_n},$$ where $s_n$ is type $n$'s saving at $\bar r$.
[10 pts] Which types gain and which may lose? Is the new equilibrium necessarily Pareto superior?
[5 pts] What does this imply for the design of social security?

Show solution

(a) The equilibrium rate depends only on the average old-age endowment (Lesson 16), which becomes $\bar\alpha+\tau$. So $1+r(\tau)=\frac{(1-\beta)(\bar\alpha+\tau)}{\beta(1-\bar\alpha-\tau)}$ rises, with $r'(0)=\frac{1-\beta}{\beta(1-\bar\alpha)^2}>0$.

(b) With $W_n=(1-\alpha_n-\tau)+\frac{\alpha_n+\tau}{1+r}$, log utility gives $c_y=(1-\beta)W_n$ and $c_o=\beta(1+r)W_n$, so $U_n=\log W_n+\beta\log(1+r)+\text{const}$. Differentiate: $$\frac{dU_n}{d\tau}=\frac1{W_n}\Big[-\frac{r}{1+r}-\frac{e_{o,n}}{(1+r)^2}r'\Big]+\frac{\beta}{1+r}r'.$$ Since $\beta/(1+r)=c_{o,n}/((1+r)^2W_n)$, the $r'$ terms combine to $\frac{(c_{o,n}-e_{o,n})}{(1+r)^2W_n}r'=\frac{s_n}{(1+r)W_n}r'$, using $c_o-e_o=(1+r)s_n$. This gives the formula.

(c) The transfer term $-\bar r>0$ helps everyone. The rate term helps savers ($s_n>0$, low $\alpha_n$) and hurts borrowers ($s_n<0$, high $\alpha_n$). Type $n$ loses iff $s_n\lt \bar r/r'(0)=\frac{\bar r\,\beta(1-\bar\alpha)^2}{1-\beta}$. Since $\sum_ns_n=0$, borrowers exist whenever types differ. With enough dispersion some type loses, so the new equilibrium is not necessarily Pareto superior. Lesson 16 works through an example where it fails. Both circulated solutions state this verbally; neither derives it.

(d) Fixing aggregate dynamic inefficiency doesn't settle the distribution. A uniform system can hurt heavy borrowers, and a Pareto improvement needs type-specific transfers (Lesson 16).

Key takeaways

PAYG with tax $\tau$ relabels endowments to $(1-\alpha-\tau,\alpha+\tau)$. The new CE is autarky there, with $1+r(\tau) = \frac{(1-\beta)(\alpha+\tau)}{\beta(1-\alpha-\tau)}$.
$\tau^*=\beta-\alpha$ implements the best symmetric allocation and drives $r$ to 0.
PAYG Pareto-improves iff $\bar r<0$ ($\beta>\alpha$), and then for every $\tau\in(0,\bar\tau]$, where $W(\bar\tau)=W(0)$.
PAYG is an asset with return equal to population growth (here 0). It beats markets when $\bar r$ is below that. Valued fiat money does the same job.

💬 Stuck or curious? Ask me (your Claude teacher) anything. Try "why doesn't the government budget constraint ever bind here?" or "how does this change with population growth?" (preview of Lesson 13).

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