Theory of Income · Part II: Overlapping Generations · Lesson 13 of 25

The Growing OLG Economy: When Is \(r\) "Too Low"?

Covers Class Note II, pp. 28–33 and Class Note III, pp. 25–27 · About 40 min · Prerequisites: Lesson 10, Lesson 11, Lesson 12
By the end you can:
  1. solve the OLG endowment economy with population growth \(n\) and productivity growth \(g\): savings, the (unchanged) interest rate, the best symmetric allocation;
  2. derive the inefficiency condition \(1+\bar r<(1+g)(1+n)\), and explain why it replaces "\(\bar r<0\)";
  3. design the pay-as-you-go tax \(\hat\tau\) that implements the best symmetric allocation, and analyze a permanent fall in population growth.
Primary source Alvarez, Class Note II, pp. 28–33 (the questions on pp. 29–33 are all solved below) and Class Note III, pp. 25–27. Samuelson's 1958 paper introduced the "biological rate of interest" equal to the population growth rate, which is this lesson's punchline.

1. The environment

Everything is as in Lesson 8, with two growth rates added N2 p.28:

2. Savings scale with productivity; the interest rate does not move

Savings in the growing economy N2 p.29

Facing a constant rate \(r\), generation \(t\)'s optimal saving is $$s_t(r;\alpha,\beta,g)=s(r;\alpha,\beta)\,(1+g)^t,\qquad s(r;\alpha,\beta)=\beta(1-\alpha)-\frac{(1-\beta)\alpha}{1+r},$$ where \(s(r;\alpha,\beta)\) is the no-growth saving function of Lesson 9.

Proof
  1. Generation \(t\) solves \(\max_s\ (1-\beta)\log\big[(1-\alpha)(1+g)^t-s\big]+\beta\log\big[\alpha(1+g)^t+s(1+r)\big]\).
  2. Change variables to \(s=(1+g)^t\sigma\). Each log becomes \(\log(1+g)^t+\log(\cdot)\), so the objective is $$\underbrace{\log(1+g)^t}_{\text{constant}}+(1-\beta)\log(1-\alpha-\sigma)+\beta\log(\alpha+\sigma(1+r)).$$
  3. The maximizer \(\sigma^*\) is therefore the no-growth solution \(s(r;\alpha,\beta)\), and \(s_t=(1+g)^t s(r;\alpha,\beta)\). This is just homotheticity: scaling the whole endowment profile scales the optimal plan.
∎
Proposition: still no trade, same interest rate N2 p.29

The unique CE is autarky, \(x^i=e^i\), and the equilibrium interest rate is the same as without growth: $$1+\bar r=\frac{(1-\beta)\alpha}{\beta(1-\alpha)},\qquad\text{i.e.}\quad \bar r=\frac{\alpha-\beta}{\beta(1-\alpha)}\quad\text{for any } g,n.$$

Proof
  1. No trade (same induction as Lesson 8). The initial old want only date-1 goods and own only date-1 goods, so \(x^0_1=e^0_1\). Date-1 clearing, \(N_1x^1_1+N_0x^0_1=N_1e^1_1+N_0e^0_1\), then gives \(x^1_1=e^1_1\) (identical members of a cohort make identical choices, by strict concavity). Generation 1's budget then forces \(x^1_2=e^1_2\). Date-2 clearing gives \(x^2_2=e^2_2\), and so on by induction.
  2. Interest rate. Aggregate saving of generation \(t\) must be zero: \(N_t s_t(r_t)=N_t(1+g)^t s(r_t;\alpha,\beta)=0\). Since \(N_t(1+g)^t>0\), this is \(s(r_t;\alpha,\beta)=0\), the same equation as with \(g=n=0\).
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Why growth doesn't change \(\bar r\) (slide question: "is there any difference?")

The interest rate is pinned down by each cohort's MRS at its own endowment. Productivity growth scales both entries of a cohort's endowment by \((1+g)^t\), and homothetic preferences have the same MRS along a ray. Population growth only changes how many identical agents there are, and in autarky each agent's saving is zero whatever the cohort's size. Growth will matter, a lot, for efficiency, because it changes what is feasible through transfers across cohorts.

3. Feasible symmetric allocations

Call an allocation symmetric if consumption, normalized by the cohort's productivity, is the same for every cohort: \(c^t_t=\hat c_y(1+g)^t\) and \(c^t_{t+1}=\hat c_o(1+g)^t\). (The slide writes the old at date \(t\) as \(c_{ot}=\hat c_o(1+g)^{t-1}\): they belong to cohort \(t-1\).) Date-\(t\) feasibility $$N_t\hat c_y(1+g)^t+N_{t-1}\hat c_o(1+g)^{t-1}=N_t(1-\alpha)(1+g)^t+N_{t-1}\alpha(1+g)^{t-1}$$ divided by \(N_{t-1}(1+g)^{t-1}\) becomes the same equation for every \(t\) N2 p.30:

Normalized resource constraint
$$G\,\hat c_y+\hat c_o=W,\qquad G\equiv(1+g)(1+n),\qquad W\equiv(1-\alpha)G+\alpha .$$

\(G\) is the gross growth rate of the aggregate economy: there are \(1+n\) young per old, each \(1+g\) times as productive. The autarky point \((\hat c_y,\hat c_o)=(1-\alpha,\alpha)\) is always on the line. The slope is \(-G\): each unit taken from a young person yields \(G\) units for the old, because there are \(1+n\) young per old and each young unit is worth \(1+g\) in old-cohort productivity units.

4. The best symmetric allocation

Because \(\log[\hat c(1+g)^t]=\log\hat c+t\log(1+g)\), maximizing \((1-\beta)\log c^t_t+\beta\log c^t_{t+1}\) over symmetric allocations is maximizing \((1-\beta)\log\hat c_y+\beta\log\hat c_o\) subject to \(G\hat c_y+\hat c_o=W\). That is a Cobb–Douglas problem with "prices" \((G,1)\) and "income" \(W\), so expenditure shares are \(1-\beta\) and \(\beta\) N2 p.31:

$$\hat c_y^*=(1-\beta)\frac{(1-\alpha)(1+g)(1+n)+\alpha}{(1+g)(1+n)}=\frac{(1-\beta)W}{G},\qquad \hat c_o^*=\beta\big[(1-\alpha)(1+g)(1+n)+\alpha\big]=\beta W .$$

With \(g=n=0\), \(W=1\) and we recover \((1-\beta,\beta)\) from Lesson 10.

5. When does the best symmetric allocation dominate the CE?

Inefficiency condition in a growing economy N2 p.32

The best symmetric allocation Pareto-dominates the autarky CE if and only if $$1+\bar r<(1+g)(1+n)=G .$$ If \(1+\bar r=G\) the two coincide. If \(1+\bar r>G\), the initial old strictly prefer the CE, so the two are not Pareto-ranked.

Proof
  1. Generations \(t\ge1\). The CE point \((1-\alpha,\alpha)\) lies on the line \(G\hat c_y+\hat c_o=W\). The best symmetric point is the unique maximizer of \(v\) on that line (strictly concave objective). So every \(t\ge1\) strictly prefers it unless the two points coincide.
  2. The initial old consume \(c^0_1=\hat c_o(1+g)^0=\hat c_o\), so they prefer the best symmetric allocation iff \(\hat c^*_o>\alpha\): $$\beta\big[(1-\alpha)G+\alpha\big]>\alpha\iff\beta(1-\alpha)G>\alpha(1-\beta)\iff G>\frac{\alpha(1-\beta)}{\beta(1-\alpha)}=1+\bar r .$$
  3. Combine: all generations prefer \(x^*\) (with strict preference) iff \(G>1+\bar r\). At equality, \(\hat c^*_o=\alpha\), and by uniqueness of the maximizer the two allocations are identical.
∎
Intuition: two rates of return

A young person can save in two ways. The market pays gross return \(1+\bar r\). An intergenerational transfer scheme pays \(G\): give one unit (in your cohort's productivity units) when young, and when old you receive what the next cohort gives. There are \(1+n\) of them per old person, each \(1+g\) times as productive. If the scheme pays more than the market, everyone, the initial old included, can be made better off. With small rates, the condition is roughly \(\bar r\lt n+g\): the interest rate is below the growth rate of the economy. This is the growing-economy version of "\(\bar r<0\)", and it is the same condition under which the economy is dynamically inefficient in the production OLG model (Lesson 15).

Consistency with Lesson 12. Aggregate endowment at date \(t\) is \(N_{t-1}(1+g)^{t-1}W=G^{t-1}W\), and \(p_t=(1+\bar r)^{-(t-1)}\). So $$p\cdot\sum_i\bar x^i=W\sum_{t\ge1}\Big(\frac{G}{1+\bar r}\Big)^{t-1}\lt \infty\iff 1+\bar r>G .$$ The 1WT applies exactly when the best symmetric allocation fails to dominate. The two analyses agree.

6. Pay-as-you-go social security with growth

Tax each young person at \(t\) the amount \(\tau_t=\hat\tau(1+g)^t\) and pay the revenue to the old alive at \(t\) N2 p.33.

  1. Transfer per old person. Revenue is \(N_t\hat\tau(1+g)^t\), shared among \(N_{t-1}\) old. Each receives \(\hat\tau(1+g)^t(1+n)\). Normalized by the old person's own productivity \((1+g)^{t-1}\), that is \(\hat\tau(1+g)(1+n)=\hat\tau G\).
  2. Normalized after-tax endowments. These are \((1-\alpha-\hat\tau,\ \alpha+\hat\tau G)\), still on the resource line. The no-trade argument applies verbatim to after-tax endowments, so the CE consumes exactly them.
  3. Choose \(\hat\tau\) to hit the best point. Set \(\alpha+\hat\tau G=\hat c_o^*=\beta W\): $$\boxed{\hat\tau=\frac{\beta W-\alpha}{G}=\beta(1-\alpha)-\frac{\alpha(1-\beta)}{(1+g)(1+n)}}$$ Check the young: \(1-\alpha-\hat\tau=(1-\beta)(1-\alpha)+\alpha(1-\beta)/G=(1-\beta)W/G=\hat c_y^*\). ✓ With \(g=n=0\), \(\hat\tau=\beta-\alpha\), as in Lesson 11.
  4. Sign. \(\hat\tau>0\iff\beta(1-\alpha)G>\alpha(1-\beta)\iff G>1+\bar r\). A positive PAYG system is called for exactly in the inefficient region.
  5. Interest rate after the reform. Consumption is \((\hat c^*_y,\hat c^*_o)(1+g)^t\), so \(1+r=\frac{(1-\beta)\hat c^*_o}{\beta\hat c^*_y}=G\). The reform raises the interest rate to the growth rate: Samuelson's biological rate of interest, the "golden rule".

7. Class Note III: population growth only, and a fall in \(n\)

Notation check: Note III vs Note II

Class Note III's population-growth slides set \(g=0\) and keep Note II's endowments (\(1-\alpha\) young, \(\alpha\) old). They write the resource constraint divided by \(N_t\) rather than \(N_{t-1}\) N3 p.25: $$c_y+\frac{c_o}{1+n}=(1-\alpha)+\frac{\alpha}{1+n}.$$ Multiply by \(1+n\) and you get our \(G\hat c_y+\hat c_o=W\) with \(G=1+n\). (Careful: Note III's privatization slides, Lesson 14, flip the endowments to \(\alpha\) young and \(1-\alpha\) old and call the PAYG tax \(\theta\).)

With \(g=0\), the best symmetric allocation satisfies the FOC \(\frac{\beta}{1-\beta}\frac{c_y}{c_o}=\frac{1}{1+n}\) N3 p.25, and a balanced PAYG system with tax \(\tau\) per young pays \((1+n)\tau\) per old N3 p.26: $$c_y=1-\alpha-\tau,\qquad c_o=\alpha+(1+n)\tau .$$ More young taxpayers per retiree means a higher implicit return on PAYG, namely \(1+n\). Matching to the optimum gives \(\tau=\beta(1-\alpha)-\alpha(1-\beta)/(1+n)\), the \(g=0\) case of the boxed formula.

A permanent fall in population growth N3 p.27

Suppose \(n>0\) falls permanently to \(n'=0\) at date \(T\) (so \(N_T=N_{T-1}\)), with the tax per young \(\tau\) held fixed.

Interactive: normalized consumption \((\hat c_y,\hat c_o)\). The black line is the resource constraint \(G\hat c_y+\hat c_o=W\), and the grey square is the autarky CE \(E=(1-\alpha,\alpha)\). The blue dot is the best symmetric allocation \(B\), with the indifference curve through it (blue); the dotted curve is the indifference curve through \(E\). The dashed horizontal line \(\hat c_o=\alpha\) is the initial old's CE consumption: \(B\) Pareto-dominates \(E\) iff \(B\) is above it. Fall in n: lower \(n'\) below \(n\) to draw the post-fall resource line (dashed orange) and the PAYG point with \(\hat\tau\) held fixed (orange). The defaults show a fall from \(n=0.5\) to \(n'=0\). Try: set \(n'=n\) to hide the fall, then set \(\alpha=0.5,\beta=0.3\) (so \(\bar r>0\)) and raise \(g\) and \(n\) until \(B\) crosses the dashed line. That happens exactly where \(G=1+\bar r\).

8. Check your understanding

1. How does a higher population growth rate \(n\) change the equilibrium interest rate \(\bar r\) in the endowment OLG economy?
  • It raises r̄ because more young savers want assets
  • It leaves r̄ unchanged since autarky saving stays zero
  • It lowers r̄ because more young compete to lend
  • It raises r̄ one-for-one, so that 1+r̄ equals 1+n
\(\bar r\) solves \(N_t(1+g)^t s(r)=0\), i.e. \(s(r)=0\), which doesn't involve \(n\) or \(g\). Growth changes what transfers can achieve (feasibility), not the autarky MRS.
2. The best symmetric allocation Pareto-dominates the CE exactly when…
  • the net interest rate r̄ is negative, as without growth
  • the population growth rate n exceeds productivity growth g
  • the preference weight β exceeds the old endowment α
  • the gross rate 1+r̄ is below the growth factor (1+g)(1+n)
The initial old gain iff \(\hat c_o^*=\beta W>\alpha\), which rearranges to \(G>1+\bar r\). Without growth (\(G=1\)) this is \(\bar r<0\), equivalently \(\beta>\alpha\). With growth, even \(\bar r>0\) can be "too low".
3. Under PAYG with \(\tau_t=\hat\tau(1+g)^t\), what does each old person receive, normalized by their own productivity \((1+g)^{t-1}\)?
  • τ̂(1+g)(1+n), the tax scaled by the growth factor
  • τ̂(1+n), since only population growth adds payers
  • τ̂(1+r̄), the market return on the contributed tax
  • τ̂, since the budget balances person for person
Revenue \(N_t\hat\tau(1+g)^t\) is split among \(N_{t-1}\) old: \(\hat\tau(1+g)^t(1+n)\) each. Dividing by \((1+g)^{t-1}\) gives \(\hat\tau(1+g)(1+n)\). Both growth rates raise the implicit return.
4. Population growth falls permanently from \(n>0\) to \(0\) with the PAYG tax per young held fixed. Who is hurt?
  • Only the old at the date of the change
  • Only generations born after the change
  • The old at the change and all later ones
  • Nobody, since the budget remains balanced
The old at the change paid \(\tau\) but get \(\tau\) instead of \((1+n)\tau\). Every later generation also pays \(\tau\) and gets back only \(\tau\). Young consumption is unchanged on impact, but lifetime welfare falls.

9. Exercises

These follow the format of Alvarez's exams. The midterms open with true/false statements whose points are for the justification. Longer questions come as guided derivations in short steps. Class Note III's population-growth slides are Problem 6 of the 2025 problem set, so Exercises 13.4–13.5 are that problem. Write your answer before you open the solution.

Exercise 13.1 True or false? Justify. Past materials Exam-inspired

Answer each in one to four lines. State the condition you are using.

  1. "In an OLG model with positive interest rates in all periods, the competitive equilibrium may still be inefficient." Trap: a 2025 study guide's answer key marks this False
    Show answer

    True (so the answer key is wrong). With growth, the CE is dominated by a stationary reallocation iff \(1+\bar r<(1+g)(1+n)\), so a positive but low rate can come with inefficiency. Even without growth, rates \(r_t>0\) that converge to 0 fast enough can make \(\sum_tp_t\) diverge (item 8). The answer would be "False" only for a stationary no-growth economy with \(r\) bounded away from 0.

  2. "In the growing OLG endowment economy, faster population growth \(n\) raises the equilibrium interest rate." Class Note II p.29 / Problem set 2025, P6
    Show answer

    False. The equilibrium has no trade, so \(1+\bar r\) is the MRS at the endowment point, \(\frac{(1-\beta)\alpha}{\beta(1-\alpha)}\). It does not depend on \(n\) or \(g\). Population growth multiplies zero saving by \(N_t\), and productivity growth scales the endowment along a ray, where the MRS of homothetic preferences is constant.

  3. "In the growing economy, the best symmetric allocation Pareto-dominates the CE if and only if \(1+\bar r<(1+g)(1+n)\)." Class Note II p.32 / 2025 review notes
    Show answer

    True. Every generation \(t\ge1\) weakly prefers the best symmetric allocation, which is chosen to maximize its utility on a constraint that contains the CE point, and strictly prefers it unless the two coincide. The initial old prefer it iff \(\hat c^*_o>\alpha\), and \(\hat c^*_o=\beta[(1-\alpha)G+\alpha]>\alpha\iff G>\frac{\alpha(1-\beta)}{\beta(1-\alpha)}=1+\bar r\) (Exercise 13.2).

  4. "An economy with a positive real interest rate cannot have a Pareto-improving PAYG system." Exam-inspired
    Show answer

    False once the economy grows. The comparison is \(1+\bar r\) against \(G=(1+g)(1+n)\), PAYG's implicit gross return, not \(\bar r\) against 0. For example, \(\bar r>0\) with \(G>1+\bar r\) has an inefficient CE, and the PAYG system with \(\hat\tau>0\) of Exercise 13.2 improves on it.

  5. "If the risk-free interest rate is below the growth rate of the economy, a PAYG social security system can make every generation better off." Problem set 3 (2022), P4
    Show answer

    False. What matters is whether the value of aggregate consumption is finite, i.e. the return on a claim to aggregate consumption relative to growth. Under uncertainty that return includes a risk premium. Exercise 13.6 builds an OLG economy where \(R^b<e^g\), yet the value of aggregate consumption is finite, so the CE is PO and no PAYG scheme can help everyone.

  6. "After the PAYG reform that implements the best symmetric allocation, the equilibrium gross interest rate equals \((1+g)(1+n)\)." Class Note II p.33
    Show answer

    True. At the new allocation, \(1+r=\frac{(1-\beta)\hat c^*_o}{\beta\hat c^*_y}=G\). This is Samuelson's "biological" interest rate, the golden rule: the market return now equals PAYG's implicit return.

  7. "A permanent fall in population growth, with the tax per young held fixed, hurts the current old but leaves the welfare of future generations unchanged." Problem set 2025, P6 Q6
    Show answer

    False. Future generations pay the same tax when young but receive \(\tau\) instead of \((1+n)\tau\) when old, so they are also worse off. Fewer young per old lowers PAYG's implicit return from \(n\) to 0 (Exercise 13.5).

  8. "With no population or productivity growth, an OLG endowment economy can have a constant, strictly positive interest rate and still be dynamically inefficient, because \(r<n\)." Trap: a warning box in 2025 final-review notes
    Show answer

    False. With \(n=0\), "\(r<n\)" means \(r<0\), which contradicts \(r>0\). A constant \(\bar r>0\) gives \(\sum_tp_t=(1+\bar r)/\bar r<\infty\), so the CE is PO. The general point in those notes (positive rates do not rule out inefficiency) is right only with growth, or with rates that converge to zero (e.g. \(r_t=1/t\) gives \(p_t=1/t\) and \(\sum_tp_t=\infty\)).

  9. "Faster productivity growth makes a given PAYG system (fixed \(\hat\tau\)) more attractive to every generation." Exam-inspired
    Show answer

    True in this model. Holding \(\hat\tau\) fixed, each generation's normalized tax is \(\hat\tau\), while its normalized benefit \(\hat\tau(1+g)(1+n)\) rises with \(g\). The initial old also receive more. (The welfare-maximizing \(\hat\tau\) also changes with \(g\).)

Exercise 13.2 Slides N2 pp.29–33 Exam-inspired (the growing economy, start to finish)

Class Note II's growing economy: \(N_{t+1}=(1+n)N_t\), endowments \(e^t_t=(1+g)^t(1-\alpha)\) and \(e^t_{t+1}=(1+g)^t\alpha\), preferences \((1-\beta)\log c^t_t+\beta\log c^t_{t+1}\). Let \(G\equiv(1+g)(1+n)\) and \(W\equiv(1-\alpha)G+\alpha\).

  1. (10 pts) Show that \(s_t(\bar r;\alpha,\beta,g)=s(\bar r;\alpha,\beta)(1+g)^t\), where \(s(\cdot)\) is the no-growth savings function. Hint: substitute \(s=(1+g)^t\tilde s\).
  2. (5 pts) Argue that the only CE has no trade. Repeat the no-growth induction; three lines.
  3. (10 pts) Find \(\bar r\) as a function of \(\alpha,\beta,g,n\). Is there any difference from the case \(g=n=0\)? Why or why not?
  4. (10 pts) Write the feasibility constraint for symmetric allocations \(c^t_t=\hat c_y(1+g)^t\), \(c^{t-1}_t=\hat c_o(1+g)^{t-1}\) in normalized form, and derive the best symmetric \((\hat c^*_y,\hat c^*_o)\).
  5. (10 pts) For which parameters does the best symmetric allocation Pareto-dominate the CE? Express your answer as a comparison of \(1+\bar r\) with \(G\). Make sure the initial old prefer it too.
  6. (10 pts) Implement it with PAYG: tax \(\tau_t=\hat\tau(1+g)^t\) on each young, with all revenue paid to the current old. Show that each old person receives \(\hat\tau G\) in normalized units, and derive \(\hat\tau\).
  7. (5 pts) Show that after the reform \(1+r=G\).
Show solution
  1. (a) The objective is \((1-\beta)\log((1+g)^t(1-\alpha)-s)+\beta\log((1+g)^t\alpha+s(1+\bar r))\). With \(s=(1+g)^t\tilde s\), it becomes \(\log(1+g)^t\) plus the no-growth objective in \(\tilde s\). So the optimal \(\tilde s\) is \(s(\bar r;\alpha,\beta)\).
  2. (b) The initial old eat \(e^0_1\) whatever the prices. Market clearing at \(t=1\) then forces the young to eat \(e^1_1\), so they save zero and eat \(e^1_2\) when old. Market clearing at \(t=2\) forces the next young to eat \(e^2_2\), and so on by induction.
  3. (c) No trade means \(N_ts_t=0\), i.e. \(s(\bar r)=0\). This gives \(1+\bar r=\frac{(1-\beta)\alpha}{\beta(1-\alpha)}\), the same as without growth. Population growth only multiplies zero saving by \(N_t\). Productivity growth scales the endowment along a ray, and homothetic (log) preferences have the same MRS everywhere on a ray.
  4. (d) \(N_t\hat c_y(1+g)^t+N_{t-1}\hat c_o(1+g)^{t-1}=N_t(1-\alpha)(1+g)^t+N_{t-1}\alpha(1+g)^{t-1}\). Divide by \(N_{t-1}(1+g)^{t-1}\): \(\hat c_yG+\hat c_o=W\). Maximizing \((1-\beta)\log\hat c_y+\beta\log\hat c_o\) on this line gives expenditure shares \(1-\beta\) and \(\beta\): \(\hat c^*_y=(1-\beta)W/G\) and \(\hat c^*_o=\beta W\).
  5. (e) Generations \(t\ge1\): the CE point \((1-\alpha,\alpha)\) is on the same line, so they weakly prefer \(\hat c^*\), strictly unless the two coincide. Initial old: they consume \(\alpha\) in the CE and \(\hat c^*_o\) under the alternative, so they gain iff \(\beta[(1-\alpha)G+\alpha]>\alpha\iff G>\frac{\alpha(1-\beta)}{\beta(1-\alpha)}=1+\bar r\). So dominance holds iff \(1+\bar r<(1+g)(1+n)\).
  6. (f) Revenue at \(t\) is \(N_t\hat\tau(1+g)^t\), shared among \(N_{t-1}\) old. Each receives \(\hat\tau(1+n)(1+g)^t\), or \(\hat\tau G\) in units of \((1+g)^{t-1}\). Set the young's consumption \(1-\alpha-\hat\tau=\hat c^*_y\). This gives \(\hat\tau=\beta(1-\alpha)-\frac{\alpha(1-\beta)}{G}\). Check the old: \(\alpha+\hat\tau G=\beta(1-\alpha)G+\beta\alpha=\hat c^*_o\) ✓. \(\hat\tau>0\) iff \(G>1+\bar r\), as in (e).
  7. (g) The new equilibrium again has no private trade at the after-tax endowments, so \(1+r\) is the MRS at \(\hat c^*\): \(\frac{(1-\beta)\hat c^*_o}{\beta\hat c^*_y}=\frac{(1-\beta)\beta W}{\beta(1-\beta)W/G}=G\).
Exercise 13.3 Exam-inspired Proof

Show that in the growing economy the value of the aggregate endowment at CE prices is finite iff \(1+\bar r>(1+g)(1+n)\). Use this to give a one-line proof that the CE is Pareto optimal when \(1+\bar r>G\). Then explain why it does not contradict Exercise 13.2(e) when \(1+\bar r<G\).

Show solution

The aggregate endowment at \(t\) is \(N_t(1-\alpha)(1+g)^t+N_{t-1}\alpha(1+g)^{t-1}=N_{t-1}(1+g)^{t-1}W\). With \(N_0=1\) this equals \(G^{t-1}W\), and \(p_t=(1+\bar r)^{-(t-1)}\). So the value is \(W\sum_{t\ge1}(G/(1+\bar r))^{t-1}\), a geometric series that is finite iff \(G<1+\bar r\). One-line proof: LNS holds and the value is finite, so the 1WT of Lesson 12 (Exercise 12.2) applies. When \(1+\bar r<G\) the value is infinite, the theorem is silent, and Exercise 13.2(e) exhibits a Pareto improvement. No contradiction.

Exercise 13.4 Problem set 2025, P6 (Q1–Q5) Slides N3 pp.25–26

Population growth only: \(N_{t+1}=(1+n)N_t\) with \(N_0=1\), endowments \((1-\alpha,\alpha)\) for every generation and \(e^0_1=\alpha\), preferences \((1-\beta)\log c_y+\beta\log c_o\). (This problem uses Note II's convention: the young get \(1-\alpha\).)

  1. (5 pts) Write market clearing at date \(t\), in aggregate and per young person.
  2. (5 pts) Argue that there is no trade in equilibrium.
  3. (10 pts) Derive per-capita savings \(s(r)\) and the equilibrium \(\bar r\). Does \(\bar r\) depend on \(n\)?
  4. (10 pts) Find the best symmetric allocation \((c^*_y,c^*_o)\) subject to \(c_y+\frac{c_o}{1+n}=(1-\alpha)+\frac{\alpha}{1+n}\).
  5. (15 pts) Find the PAYG tax per young \(\tau^*\) that implements it, given that each old person receives \((1+n)\tau\). Sign \(\partial\tau^*/\partial n\), and show that \(\tau^*>0\iff n>\bar r\). Explain the dependence on \(n\) in two lines.
Show solution
  1. (a) \(N_tc^t_t+N_{t-1}c^{t-1}_t=N_t(1-\alpha)+N_{t-1}\alpha\). Dividing by \(N_t\): \(c^t_t+\frac{c^{t-1}_t}{1+n}=(1-\alpha)+\frac{\alpha}{1+n}\).
  2. (b) Within a generation agents are identical, so they cannot trade with each other. Across generations, the initial old eat their endowment, and the forward induction of Lesson 8 does the rest.
  3. (c) \(s(r)=\beta(1-\alpha)-\frac{(1-\beta)\alpha}{1+r}\), and aggregate saving is \(N_ts(r)\). Setting it to zero gives \(\bar r=\frac{\alpha-\beta}{\beta(1-\alpha)}\), which is independent of \(n\).
  4. (d) On the line, the price of \(c_o\) is \(1/(1+n)\). The FOC is \(\frac{\beta}{1-\beta}\frac{c_y}{c_o}=\frac{1}{1+n}\). Log shares give \(c^*_y=(1-\beta)\big[(1-\alpha)+\frac{\alpha}{1+n}\big]\) and \(c^*_o=\beta[(1+n)(1-\alpha)+\alpha]\).
  5. (e) Set \(1-\alpha-\tau=c^*_y\). Then \(\tau^*=\beta(1-\alpha)-\frac{(1-\beta)\alpha}{1+n}\). Check: \(\alpha+(1+n)\tau^*=\beta(1+n)(1-\alpha)+\beta\alpha=c^*_o\) ✓. \(\partial\tau^*/\partial n=\frac{(1-\beta)\alpha}{(1+n)^2}>0\). And \(\tau^*>0\iff1+n>\frac{(1-\beta)\alpha}{\beta(1-\alpha)}=1+\bar r\iff n>\bar r\): social security is needed exactly when the economy is dynamically inefficient relative to \(n\). Why it rises with \(n\): \(n\) is PAYG's implicit return. Higher \(n\) lowers the present value (at rate \(n\)) of the old-age endowment \(\alpha/(1+n)\). With log utility the young consume a fixed share \(1-\beta\) of that smaller wealth, so the tax that gets them there must be larger. (The circulating solutions' "substitution beats income" story is loose; this wealth channel is the clean reason.)
Exercise 13.5 Problem set 2025, P6 Q6 Slides N3 p.27

Continue Exercise 13.4 and assume \(\beta>\alpha\), so the no-transfer CE has \(\bar r<0\). The system with \(\tau^*\) is in place when, at date \(T\), population growth falls permanently to \(n'=0\). The tax per young stays at \(\tau^*\).

  1. (10 pts) In \((c_y,c_o)\) space, draw the two feasible symmetric-allocation lines (for \(n\) and for \(n'=0\)), the no-social-security equilibrium, the best symmetric allocation for \(n\), and the new allocation at \(n'=0\). Which point lies on both lines?
  2. (10 pts) What happens to the welfare of the old at \(T\) and of current and future young?
  3. (10 pts) Show that \(\tau^*-(\beta-\alpha)=\frac{\alpha(1-\beta)n}{1+n}>0\). Interpret this relative to the best tax for the new regime. Would cutting the tax to \(\beta-\alpha\) at \(T+1\) be a Pareto improvement?
  4. (5 pts) Suppose instead \(\beta<\alpha\), but \(n\) was large enough that \(\tau^*>0\). After the fall, is PAYG still worth having for generations born after \(T\)? Who would lose if it were abolished?
Show solution
  1. (a) The \(n\) line is \(c_o=(1+n)(1-\alpha)+\alpha-(1+n)c_y\), with slope \(-(1+n)\). The \(n'=0\) line is \(c_y+c_o=1\), with slope \(-1\). Both pass through the endowment point \((1-\alpha,\alpha)\), which is the no-social-security CE. The best allocation for \(n\) is the tangency \((c^*_y,c^*_o)\) on the steeper line, to the left of the endowment. The new allocation is \((c^*_y,\alpha+\tau^*)\), which lies on \(c_y+c_o=1\) because \(c^*_y=1-\alpha-\tau^*\). It sits directly below the old optimum.
  2. (b) The old at \(T\) receive \(\tau^*\) instead of \((1+n)\tau^*\), so they are worse off. Every current and future young pays the same tax and gets a lower benefit, so they are worse off too: same \(c_y\), lower \(c_o\).
  3. (c) \(\tau^*-(\beta-\alpha)=\alpha-\alpha\beta-\frac{(1-\beta)\alpha}{1+n}=\alpha(1-\beta)\big(1-\frac{1}{1+n}\big)=\frac{\alpha(1-\beta)n}{1+n}\). With \(n'=0\), the best PAYG tax is \(\beta-\alpha\) (Lesson 11), so the inherited tax overshoots: \(c_y=1-\alpha-\tau^*<1-\beta\). Cutting it to \(\beta-\alpha\) raises the welfare of generations born from \(T+1\) on. It is not a Pareto improvement, though, because the generation that is old at \(T+1\) paid \(\tau^*\) when young and would now receive only \(\beta-\alpha\). The cut redistributes across generations.
  4. (d) With \(\beta<\alpha\) the no-growth economy has \(\bar r>0=n'\), so it is efficient without PAYG. Any \(\tau>0\) now lowers every future generation's welfare relative to \(\tau=0\), because it moves them along \(c_y+c_o=1\) away from their preferred point. Abolishing would still hurt the generation that is old when it happens, since they paid in. Once \(G\le1+\bar r\), PAYG is a pure transfer from future generations to the current old.

Ambiguity in the problem set: the question also mentions a "constant transfer to the old". Holding the old's transfer at \((1+n)\tau^*\) instead would force a higher tax on the young, who would then bear the loss. The circulating solutions use the fixed-tax reading, as here.

Exercise 13.6 Problem set 3 (2022), P4 (stochastic OLG: \(r<g\) but efficient)

Two-period lives, mass one per cohort. The young have labour income \(w_t=(1-\delta)Y_t\). One tree, owned by the initial old, pays \(d_t=\delta Y_t\). Output grows randomly: \(Y_{t+1}=Y_tz_{t+1}\), \(Y_1=1\), with \(z\) i.i.d. and positive. Preferences are \((1-\beta)\log c^t_t+\beta E_t\log c^t_{t+1}\). The young can buy shares of the tree at ex-dividend price \(q^e_t\), and one-period riskless discount bonds at price \(q^b_t\).

  1. (5 pts) What are the equilibrium holdings of trees and bonds? One line each.
  2. (10 pts) Write the young's two FOCs. Guess \(q^e_t=Q^eY_t\) and show that \(Q^e=\beta(1-\delta)\). Hint: with log utility the young consume a share \(1-\beta\) of income.
  3. (10 pts) Let \(K\equiv1+\frac{\delta}{\beta(1-\delta)}\). Show that the expected gross return on the tree is \(R^e=K\,E[z]\) and the riskless gross rate is \(R^b=K/E[1/z]\).
  4. (5 pts) Let \(\log z\sim N(\mu,\sigma^2)\) and define growth by \(e^g\equiv E[z]=e^{\mu+\sigma^2/2}\). Show that \(R^b/e^g=Ke^{-\sigma^2}\), so the riskless rate is below growth when \(\sigma^2\) is large.
  5. (10 pts) Is the value of aggregate consumption finite? Hint: aggregate consumption is \(Y_t=d_t/\delta\), so its value is \(1/\delta\) times the value of the tree's dividends.
  6. (10 pts) Is the CE Pareto optimal? Can social security make every generation better off? Does the comparison of the riskless rate with \(g\) matter? Three lines.
Show solution
  1. (a) Trees: the young hold the one tree (the old sell it). Bonds: zero, because members of a cohort are identical and the old cannot be counterparties.
  2. (b) Tree: \((1-\beta)\frac{q^e_t}{c^t_t}=\beta E_t\big[\frac{q^e_{t+1}+d_{t+1}}{c^t_{t+1}}\big]\). Bond: \((1-\beta)\frac{q^b_t}{c^t_t}=\beta E_t\big[\frac{1}{c^t_{t+1}}\big]\). In equilibrium \(c^t_{t+1}=q^e_{t+1}+d_{t+1}\), so the tree FOC's right side is \(\beta\), and \(c^t_t=w_t-q^e_t\). Hence \((1-\beta)q^e_t=\beta(w_t-q^e_t)\), so \(q^e_t=\beta w_t=\beta(1-\delta)Y_t\).
  3. (c) \(q^e_{t+1}+d_{t+1}=Y_{t+1}[\beta(1-\delta)+\delta]\), so \(R^e=E\big[\frac{Y_{t+1}(\beta(1-\delta)+\delta)}{\beta(1-\delta)Y_t}\big]=K\,E[z]\). From the bond FOC with \(c^t_t=(1-\beta)(1-\delta)Y_t\): \(q^b_t=\frac{\beta(1-\delta)}{\beta(1-\delta)+\delta}E[1/z]=E[1/z]/K\). So \(R^b=1/q^b=K/E[1/z]\).
  4. (d) \(E[1/z]=e^{-\mu+\sigma^2/2}=e^{-g+\sigma^2}\), so \(R^b=Ke^{g-\sigma^2}\) and \(R^b/e^g=Ke^{-\sigma^2}\), which is below 1 for large \(\sigma^2\). The equity premium is \(R^e/R^b=e^{\sigma^2}\).
  5. (e) Yes. The value of the dividend stream is the tree's cum-dividend value at \(t=1\), \(d_1+q^e_1=[\delta+\beta(1-\delta)]Y_1\). The bubble term vanishes, because the expected discounted resale value shrinks by the factor \(\frac{\beta(1-\delta)}{\beta(1-\delta)+\delta}<1\) each period. So the value of aggregate consumption is \(\frac{1}{\delta}[\delta+\beta(1-\delta)]Y_1=\big[1+\frac{\beta(1-\delta)}{\delta}\big]Y_1<\infty\). Error in the circulating solution: it writes the tree's value as \(Y_1+q^e_1\) instead of \(d_1+q^e_1=\delta Y_1+q^e_1\). The conclusion (finite) is unaffected.
  6. (f) Finite value plus LNS means the 1WT applies, so the CE is PO and no social security scheme can improve every generation. The riskless rate being below \(g\) is irrelevant. What matters is the return on the claim to aggregate consumption, the tree, with \(R^e=Ke^g>e^g\). The policy lesson: "\(r<g\)" for safe assets is not, by itself, evidence of dynamic inefficiency. With US numbers (riskless ≈ growth ≈ 3%, equity premium ≈ 5%), the relevant return comfortably exceeds growth.

Key takeaways

💬 Stuck or curious? Ask me (your Claude teacher) anything. Try "how would the analysis change if only the old-age endowment grew?" or "connect \(r\lt g\) here to the recent debate on government debt when \(r\lt g\) (Blanchard 2019)."
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