Theory of Income · Part IV: Perpetual Youth (Blanchard–Yaari) · Lesson 18 of 25

Perpetual Youth II: Aggregation and the Endowment Economy

Covers Blanchard–Yaari slides 11–21 · About 40 min · Prerequisite: Lesson 17
By the end you can:
  1. aggregate individual behaviour into \(C=(p+\theta)(H+V)\), \(\dot V = rV+Y-C\) and \(\dot H=(r+p+\alpha)H-Y\), and explain every term;
  2. derive the aggregate consumption equation \(\dot C=(r+\alpha-\theta)C-(p+\alpha)(p+\theta)V\) line by line;
  3. solve the pure-endowment equilibrium, \(r=\theta-\alpha\) with autarky, and answer the slide p.21 questions, including when the equilibrium is Pareto-dominated.
Primary source Blanchard, O. J. (1985), "Debt, Deficits, and Finite Horizons," JPE 93(2): 223–247, §II (aggregation) and the start of §III. Annuities as in Yaari (1965), REStud 32(2). Slides: BY slides 11–21.
Notation alert

1. Aggregating consumption is immediate

From Lesson 17, each agent consumes \(c(s,t) = (p+\theta)\big(h(s,t)+v(s,t)\big)\). The MPC \(p+\theta\) is the same for every cohort, so integrating over cohorts with weights \(N(s,t)\) gives BY slide 11 $$C(t) = (p+\theta)\big(H(t)+V(t)\big),\qquad H(t)\equiv\int_{-\infty}^t N(s,t)h(s,t)\,ds,\quad V(t)\equiv\int_{-\infty}^tN(s,t)v(s,t)\,ds .$$ This is the payoff from log utility plus age-independent mortality. A common linear consumption function aggregates exactly, so the distribution of wealth across cohorts doesn't matter for \(C\). It is not yet a closed system, though: we still need the laws of motion of \(H\) and \(V\).

2. Labour income declines with age

To let agents have a life-cycle reason to save, Blanchard assumes each agent's labour endowment declines exponentially with age at rate \(\alpha\ge0\) BY slide 12: $$y(s,t) = a\,Y(t)\,e^{-\alpha(t-s)},$$ where \(Y(t)\) is aggregate labour income and \(a>0\) is a constant that makes the individual incomes add up.

Derivation: \(a=(p+\alpha)/p\)
  1. Aggregate income is the sum over cohorts: \(Y(t) = \int_{-\infty}^t N(s,t)y(s,t)\,ds = \int_{-\infty}^t p e^{-p(t-s)}\,aY(t)e^{-\alpha(t-s)}\,ds\).
  2. Pull out \(aY(t)\) and integrate: \(\int_{-\infty}^t pe^{-(p+\alpha)(t-s)}ds = \dfrac{p}{p+\alpha}\).
  3. So \(Y(t) = aY(t)\dfrac{p}{p+\alpha}\), which gives \(\boxed{a = \dfrac{p+\alpha}{p}}\).
∎

A newborn earns \(y(t,t) = aY = (1+\alpha/p)\,Y\), more than average income, and its income then falls below average as it ages. With \(\alpha=0\), \(a=1\) and everyone earns \(Y\).

3. Aggregate human wealth

Derivation: \(H(t)=\int_t^\infty Y(z)e^{-\int_t^z(r+p+\alpha)}dz\) BY slides 13–14
  1. Individual human wealth. Substitute the income profile into \(h(s,t)=\int_t^\infty y(s,z)R(t,z)dz\) and split \(e^{-\alpha(z-s)} = e^{-\alpha(z-t)}e^{-\alpha(t-s)}\): $$h(s,t) = a\,e^{-\alpha(t-s)}\underbrace{\int_t^\infty Y(z)\,e^{-\alpha(z-t)}R(t,z)\,dz}_{\equiv\ \tilde H(t)\ \text{(same for every cohort)}} .$$ An older agent has proportionally less human wealth: the factor \(e^{-\alpha(t-s)}\) is the agent's current income relative to a newborn's.
  2. Aggregate. \(H(t) = \int_{-\infty}^t pe^{-p(t-s)}\,a e^{-\alpha(t-s)}ds\cdot\tilde H(t) = a\dfrac{p}{p+\alpha}\tilde H(t) = \tilde H(t)\), using \(a = (p+\alpha)/p\).
  3. Substitute \(R\). With \(R(t,z) = e^{-\int_t^z(r+p)}\), $$H(t) = \int_t^\infty Y(z)\exp\left(-\int_t^z\big[r(\mu)+p+\alpha\big]d\mu\right)dz .$$
  4. Differentiate in \(t\) (Leibniz, as in Lesson 17 §6.2): $$\boxed{\dot H(t) = \big[r(t)+p+\alpha\big]H(t) - Y(t)},\qquad \lim_{z\to\infty}H(z)e^{-\int_t^z(r+p+\alpha)}=0 .$$
∎
Intuition: why future income is discounted at \(r+p+\alpha\)

\(H(t)\) is the human wealth of the people alive at \(t\), not of everyone who will ever live. Of the aggregate labour income \(Y(z)\) earned at a future date \(z\), only part goes to today's agents. A fraction is earned by people not yet born, who get a disproportionate share because newborns have the highest income. Today's agents' share of future income shrinks at rate \(p\) (they die) plus \(\alpha\) (their income falls relative to newcomers'). This extra discounting is the heart of the model: it is why government debt, which shifts taxes onto the unborn, can be "net wealth" for people alive today (Lesson 20).

4. Aggregate non-human wealth grows at \(r\), not \(r+p\)

Derivation: \(\dot V = rV+Y-C\) BY slide 15
  1. Definition. \(V(t) = \int_{-\infty}^t v(s,t)\,pe^{-p(t-s)}\,ds\).
  2. Leibniz rule. The upper limit \(t\) contributes the integrand at \(s=t\), namely \(v(t,t)\,p\). Differentiating inside the integral contributes two terms, one from \(\partial_t v\) and one from \(\partial_t\big(pe^{-p(t-s)}\big) = -p\cdot pe^{-p(t-s)}\): $$\dot V(t) = p\,v(t,t) - p\,V(t) + \int_{-\infty}^t \frac{\partial v(s,t)}{\partial t}\,pe^{-p(t-s)}\,ds .$$
  3. Newborns have no financial wealth. \(v(t,t)=0\): there are no bequests, so the OLG has no altruism.
  4. Individual budget. \(\partial_tv(s,t) = [r(t)+p]v(s,t) + y(s,t) - c(s,t)\). Integrating against \(N(s,t)\) gives \((r+p)V + Y - C\).
  5. Combine. \(\dot V = 0 - pV + (r+p)V + Y - C\), so $$\boxed{\dot V(t) = r(t)V(t) + Y(t) - C(t)} .$$
∎

Each survivor's wealth grows at \(r+p\), yet aggregate wealth grows at \(r\). The difference, \(pV\), is the annuity transfer: wealth of the dying that insurers pass to the living. It is a pure redistribution within the population, so it nets out. Society as a whole can only earn \(r\).

5. The aggregate consumption equation

Collecting the pieces BY slide 16: $$C = (p+\theta)(H+V),\qquad \dot V = rV + Y - C,\qquad \dot H = (r+p+\alpha)H - Y .$$ Eliminating \(H\) gives a two-equation system in \((C,V)\).

Algebra: \(\dot C = (r+\alpha-\theta)C - (p+\alpha)(p+\theta)V\) BY slide 17
  1. Differentiate the consumption function. \(\dot C = (p+\theta)(\dot H + \dot V)\).
  2. Substitute the ODEs. \(\dot H + \dot V = (r+p+\alpha)H - Y + rV + Y - C = r(H+V) + (p+\alpha)H - C\). Labour income cancels.
  3. Use \(C=(p+\theta)(H+V)\) on the first term. \(\dot C = rC + (p+\theta)(p+\alpha)H - (p+\theta)C = (r-p-\theta)C + (p+\alpha)(p+\theta)H\).
  4. Replace \(H\). Add and subtract \((p+\alpha)(p+\theta)V\): \((p+\alpha)(p+\theta)H = (p+\alpha)(p+\theta)(H+V) - (p+\alpha)(p+\theta)V = (p+\alpha)C - (p+\alpha)(p+\theta)V\).
  5. Collect. \(\dot C = (r-p-\theta+p+\alpha)C - (p+\alpha)(p+\theta)V\), so $$\boxed{\dot C(t) = \big[r(t)+\alpha-\theta\big]C(t) - (p+\alpha)(p+\theta)V(t)},\qquad \dot V(t) = r(t)V(t)+Y(t)-C(t).$$
∎
Reading the \(\dot C\) equation

If everyone alive today lived forever (no turnover), aggregate consumption would grow like an individual's, at \(r-\theta\). Two corrections apply:

Boundary conditions

For agents to have finite human wealth we need \(\lim_{T\to\infty}Y(T)e^{-\int_t^T[r+\alpha+p]} = 0\) BY slide 18. Otherwise agents could borrow unbounded amounts. Because the effective discount rate is \(r+p+\alpha\), equilibrium interest rates can be negative (down to \(-(p+\alpha)\) with constant \(Y\)) without anyone's wealth becoming infinite. Contrast the representative-agent model, where \(r>0\) is needed with constant income.

6. General equilibrium I: the pure endowment economy

Slide 19 lists three closures of the model BY slide 19: (i) a pure endowment, \(Y=C\) and \(V=0\); (ii) capital, \(V=K\) (Lesson 19); (iii) government debt, \(V=B+K\) (Lesson 20). Here we solve (i).

Result: the endowment equilibrium is autarky at \(r=\theta-\alpha\) BY slide 20

Suppose aggregate labour income is constant, \(Y(t)=Y\), there is no storage, and the only assets are private IOUs, so \(V(t)=\int N(s,t)v(s,t)ds = 0\) (every lender is matched by a borrower). Then the equilibrium interest rate is $$r(t) = \theta-\alpha\quad\text{for all }t,$$ and, if every agent alive at \(t=0\) starts with \(v(s,0)=0\), individual consumption equals individual income: \(c(s,t)=y(s,t)\) and \(v(s,t)=0\). Nobody trades.

Proof
  1. Goods market. \(C(t) = Y\) for all \(t\), so \(\dot C = 0\).
  2. Asset market. \(V(t) = 0\). Plug into the \(\dot C\) equation: \(0 = (r+\alpha-\theta)Y - 0\). Since \(Y>0\), \(r = \theta-\alpha\).
  3. Individuals. \(\dot c(s,t)/c(s,t) = r-\theta = -\alpha\), the same rate at which income falls.
  4. Levels. A newborn has \(v=0\) and \(h(t,t) = aY\int_t^\infty e^{-(r+p+\alpha)(z-t)}dz = aY/(r+p+\alpha) = aY/(\theta+p)\). So \(c(t,t) = (\theta+p)h(t,t) = aY = y(t,t)\). Consumption starts at income and declines at the same rate, so \(c(s,t) = y(s,t)\) forever and \(v(s,t)=0\). The same argument applies to anyone alive at 0 with \(v(s,0)=0\).
  5. Consistency. \(V=0\) and \(C=\int Ny = Y\). ✓
∎

This is the continuous-time analogue of Note II's no-trade result (Lesson 8). The interest rate adjusts until no one wants to trade. And just as in Note II, where \(r̄<0\) iff \(\alpha<\beta\) (the old are poorly endowed), here \(r<0\) iff \(\alpha>\theta\): income falls steeply with age, everyone wants to save for old age, and there is no asset to save in.

See it: the interest rate that stops trade

Fix a constant \(r\) and look at a newborn's plan, with \(Y=1\). Then \(y(x) = ae^{-\alpha x}\) at age \(x\), \(h(\text{birth}) = a/(r+p+\alpha)\), \(c(x) = (\theta+p)h\,e^{(r-\theta)x}\), and \(v(x) = c(x)/(\theta+p) - y(x)/(r+p+\alpha)\). Integrating \(v\) against the age distribution \(pe^{-px}\) gives the wealth the population would like to hold at this \(r\) (in steady state): $$\frac{V^d(r)}{Y} = \frac{r+\alpha-\theta}{(r+p+\alpha)(p+\theta-r)},\qquad -(p+\alpha)<r<p+\theta .$$ Market clearing \(V^d(r)=0\) holds exactly at \(r=\theta-\alpha\).

Interactive: age profiles at a given interest rate (\(Y=1\)). Left: labour income \(y\) (black) and consumption \(c\) (blue) by age. Right: financial wealth \(v\) by age (orange). Try: press "set r = θ − α". The two curves on the left coincide and wealth is zero at every age: autarky. Raise \(r\) and the young save (\(v>0\)), so aggregate desired wealth \(V^d>0\), more than the zero supply of assets. Lower \(r\) and the young borrow against their high early income, so \(V^d<0\). With \(\alpha>\theta\), the market-clearing rate is negative.

7. Exercises

The slide questions on BY slide 21 are Exercises 18.4–18.5. No past exam has asked about Blanchard–Yaari directly, so the rest follow the formats Alvarez does use: true/false with a short justification, and guided derivations with hints and point weights. Work with general parameters throughout.

Exercise 18.1 Exam-inspired True or false? Justify. (7 × 10 points)
  1. In the pure-endowment perpetual-youth economy (with no initial financial wealth), the competitive equilibrium has no trade. exam-inspired
    Show solution

    True. The equilibrium rate is \(r=\theta-\alpha\). Then every agent's desired consumption grows at \(r-\theta=-\alpha\), the same rate at which its income declines. Together with the budget constraint and \(v(s,0)=0\), this gives \(c(s,t)=y(s,t)\) and \(v(s,t)=0\). This mirrors the no-trade result of Note II (Lesson 8).

  2. In the endowment economy with flat income profiles (\(\alpha=0\)), finite lives push the interest rate below the rate of time preference. exam-inspired
    Show solution

    False. \(r=\theta-\alpha=\theta\) for every \(p\). Finite horizons alone create no motive to trade when income is flat. The higher impatience \(\theta+p\) is exactly offset by the annuity return \(r+p\).

  3. Aggregate financial wealth \(V\) accumulates at the annuity rate \(r+p\). exam-inspired
    Show solution

    False. \(\dot V = rV+Y-C\). The extra \(pV\) that survivors earn is a transfer from the wealth of those who die, so it nets out across the population. Newborns enter with \(v(t,t)=0\).

  4. Aggregate consumption depends on how financial and human wealth are distributed across cohorts. exam-inspired
    Show solution

    False. With log utility every agent has the same MPC \(\theta+p\), independent of age and prices. So \(C=(\theta+p)(H+V)\) depends only on aggregate wealth. This linearity with a common slope is the Gorman-type condition (Lesson 25) that makes aggregation exact.

  5. The aggregate consumption Euler equation is \(\dot C=(r-\theta)C\), as in the representative-agent model. exam-inspired
    Show solution

    False unless \(p=\alpha=0\). In fact \(\dot C=(r+\alpha-\theta)C-(p+\alpha)(p+\theta)V\). The last term is generational turnover: dying agents hold wealth, and newborns who replace them hold none. The \(+\alpha\) arises because newborns earn more than the average, so aggregate consumption of the alive is propped up relative to the individual Euler equation.

  6. The equilibrium interest rate in the endowment economy can be negative, and when it is, there is a feasible allocation that makes every generation better off. exam-inspired
    Show solution

    True. \(r=\theta-\alpha<0\) iff \(\alpha>\theta\), allowed because the bound is \(r>-(p+\alpha)\). Then slightly flattening every age profile of consumption raises the utility of every cohort, both those alive at 0 and those yet to be born (Exercise 18.5(d)). This is the continuous-time counterpart of "negative \(r\) ⇒ not Pareto optimal" in the two-period OLG.

  7. \(r=\theta\) can be an equilibrium of the endowment economy only if labour income does not decline with age. exam-inspired
    Show solution

    True. With \(C=Y\) and \(V=0\), \(\dot C=0\) forces \(r=\theta-\alpha\), so \(r=\theta\) requires \(\alpha=0\). With \(\alpha>0\), at \(r=\theta\) everyone wants to save while young (\(V^d>0\)), but no asset is in positive net supply. It takes capital or government debt to support a higher rate.

Exercise 18.2 Slides pp.12–14 Income profile and aggregate human wealth (40 points)

Labour income is \(y(s,t)=aY(t)e^{-\alpha(t-s)}\), and \(R(t,z)=\exp(-\int_t^z[r(\mu)+p]d\mu)\).

  1. (10) Show that consistency with aggregate income requires \(a=(p+\alpha)/p\). At what age does an agent earn exactly the average \(Y(t)\)?
  2. (15) Show that aggregate human wealth \(H(t)=\int_{-\infty}^t h(s,t)N(s,t)ds\) equals \(\int_t^\infty Y(z)\exp\big(-\int_t^z[r(\mu)+p+\alpha]d\mu\big)dz\). [Hint: write \(h(s,t)=a\big[\int_t^\infty Y(z)e^{-\alpha(z-t)}R(t,z)dz\big]e^{-\alpha(t-s)}\), then integrate over cohorts.]
  3. (10) Differentiate to obtain \(\dot H=(r+p+\alpha)H-Y\), and state the boundary condition.
  4. (5) Interpret the discount rate \(r+p+\alpha\).
Show solution

(a) \(Y=\int_{-\infty}^tpe^{-p(t-s)}aYe^{-\alpha(t-s)}ds = aYp/(p+\alpha)\), so \(a=(p+\alpha)/p\). Income equals \(Y\) when \(ae^{-\alpha x}=1\): at age \(x=\ln(1+\alpha/p)/\alpha\).
(b) The bracket is common to all cohorts. Integrating \(e^{-\alpha(t-s)}\) against \(pe^{-p(t-s)}\) gives \(p/(p+\alpha)\), which cancels \(a\). So \(H(t)=\int_t^\infty Y(z)e^{-\alpha(z-t)}R(t,z)dz\), which is the expression shown.
(c) By Leibniz: \(\dot H = -Y(t)+(r(t)+p+\alpha)H(t)\). The boundary condition is \(\lim_{z\to\infty}H(z)e^{-\int_t^z(r+p+\alpha)}=0\).
(d) \(H\) is the human wealth of those alive now. Their claim on future aggregate income shrinks at rate \(p\) (they die) plus α (their income falls relative to newborns'), on top of discounting at \(r\).

Exercise 18.3 Slides pp.15–17 The aggregate system (40 points)
  1. (15) Differentiate \(V(t)=\int_{-\infty}^tv(s,t)pe^{-p(t-s)}ds\) and use \(v(t,t)=0\) and the individual budget constraint to show \(\dot V=rV+Y-C\).
  2. (20) Starting from \(C=(\theta+p)(H+V)\), show that \(\dot C=(r+\alpha-\theta)C-(p+\alpha)(p+\theta)V\). [Hint: substitute \(\dot H\) and \(\dot V\), then replace \(H\) by \(C/(\theta+p)-V\).]
  3. (5) Which term disappears when \(p=\alpha=0\), and why does that make sense?
Show solution

(a) \(\dot V = pv(t,t) - pV + \int_{-\infty}^t\dot v(s,t)pe^{-p(t-s)}ds\). Substituting \(\dot v=(r+p)v+y-c\) gives \(\dot V = 0 - pV + (r+p)V + Y - C = rV+Y-C\).
(b) \(\dot C=(\theta+p)[(r+p+\alpha)H-Y+rV+Y-C] = (\theta+p)r(H+V)+(\theta+p)(p+\alpha)H-(\theta+p)C\). The first term is \(rC\). Replacing \(H=C/(\theta+p)-V\) in the second term gives \((p+\alpha)C-(p+\alpha)(p+\theta)V\). Collecting terms: \(\dot C=(r+\alpha-\theta)C-(p+\alpha)(p+\theta)V\).
(c) The turnover term \(-(p+\alpha)(p+\theta)V\) and the \(+\alpha\) both vanish, leaving \(\dot C=(r-\theta)C\). With infinitely lived agents and flat income, there are no newborns to dilute wealth, so the aggregate behaves like one agent.

Exercise 18.4 Slides pp.20–21 The endowment equilibrium (50 points)

In the pure endowment economy, \(C=Y\) is constant and assets are only claims between agents.

  1. (10) Argue that \(V(t)=0\), and derive the equilibrium interest rate.
  2. (10) Show that if \(v(s,0)=0\) for every cohort, the equilibrium is autarky.
  3. (10) What did we assume about the distribution of wealth at \(t=0\)? Suppose instead \(V(0)=0\) but \(v(s,0)\neq0\) for some cohorts. Is the interest rate the same? Show that individual positions shrink at rate α. [Hint: at this \(r\), \((\theta+p)h(s,t)=y(s,t)\).]
  4. (10) Find the equilibrium interest rate and individual consumption when \(\alpha=0\), and when \(p=0\). Why must \(\alpha=0\) if \(p=0\)?
  5. (10) Can \(r(t)=\theta\) be an equilibrium? Explain using the sign of desired aggregate saving at \(r=\theta\).
Show solution

(a) There is no capital and no government, so every asset is someone's liability, and \(\int N v\,ds=0\). With \(C=Y\) constant, \(0=\dot C=(r+\alpha-\theta)Y\), so \(r=\theta-\alpha\).
(b) At \(r=\theta-\alpha\), \(h(s,t)=y(s,t)/(r+p+\alpha)=y(s,t)/(\theta+p)\), so \(c=(\theta+p)(v+h)=(\theta+p)v+y\). Then \(\dot v=(r+p)v+y-c=(r-\theta)v=-\alpha v\). With \(v(s,0)=0\), \(v\equiv0\) and \(c=y\): autarky.
(c) For the interest rate only \(V(0)=0\) matters, since the \(\dot C\) equation involves aggregates only. By (b), \(\dot v=-\alpha v\), so any initial positions decay at rate α and \(V\) stays 0.
(d) \(\alpha=0\): \(r=\theta\), \(a=1\), and everyone consumes \(Y\) at every age. \(p=0\): no births or deaths. A declining income profile would make aggregate income fall at rate α, and \(a=(p+\alpha)/p\) is undefined, so stationarity requires \(\alpha=0\). Then \(r=\theta\), the Ramsey endowment economy.
(e) Only if \(\alpha=0\). At \(r=\theta\), individual consumption is flat but income falls with age, so every agent saves when young and \(V^d>0\) (Exercise 18.6). With no asset in positive net supply, that excess demand for saving pushes \(r\) down to \(\theta-\alpha\).

Exercise 18.5 Slides p.21 Exam-inspired Is the flat allocation \(c(s,t)=Y\) Pareto superior? (60 points)

Consider the allocation in which everyone alive consumes \(Y\) at every date.

  1. (10) Show it is feasible.
  2. (15) Compute the lifetime utility of a cohort born at \(s\ge0\) under the equilibrium (\(c=aYe^{-\alpha x}\)) and under the flat allocation. Find the condition under which newborns prefer the flat allocation.
  3. (15) Show that agents already alive at \(t=0\) gain more from switching the older they are. Conclude when the flat allocation Pareto-dominates the equilibrium, and show this requires \(\alpha>\theta\).
  4. (20) Show that a small flattening of the profile, \(c = \frac{p+\gamma}{p}Ye^{-\gamma(t-s)}\) with \(\gamma\) slightly below \(\alpha\), Pareto-improves if and only if \(\alpha>\theta\), i.e. \(r<0\). Relate this to Note II's condition for inefficiency.
Hint

\(\int_0^\infty xe^{-(\theta+p)x}dx = 1/(\theta+p)^2\). For (d), differentiate the newborn's utility with respect to γ and evaluate at γ = α.

Show solution

(a) \(\int_{-\infty}^tN(s,t)\,Y\,ds = Y\). ✓

(b) Equilibrium: \(U^{eq} = \int_0^\infty\log\big(aYe^{-\alpha x}\big)e^{-(\theta+p)x}dx = \dfrac{\log(aY)}{\theta+p} - \dfrac{\alpha}{(\theta+p)^2}\). Flat: \(U^{flat} = \dfrac{\log Y}{\theta+p}\). Newborns prefer flat iff \(\log a < \dfrac{\alpha}{\theta+p}\), i.e. $$\log\Big(1+\frac{\alpha}{p}\Big) < \frac{\alpha}{\theta+p}.$$

(c) An agent of age \(A\) at \(t=0\) has remaining equilibrium consumption \(aYe^{-\alpha A}e^{-\alpha x}\). The gap \(U^{flat}-U^{eq} = \big[\alpha A - \log a\big]/(\theta+p) + \alpha/(\theta+p)^2\) is increasing in \(A\), since older agents are poorer in equilibrium. So newborns are the binding group, and the flat allocation dominates iff the inequality in (b) holds. Let \(f(\alpha) = \alpha/(\theta+p) - \log(1+\alpha/p)\). Then \(f(0)=0\) and \(f'(\alpha) = \frac1{\theta+p} - \frac1{p+\alpha}\), negative for \(\alpha<\theta\) and positive for \(\alpha>\theta\). So \(f>0\) requires \(\alpha>\theta\) (i.e. \(r<0\)), and holds once α is large enough.

(d) The γ-family is feasible: \(\int pe^{-px}\frac{p+\gamma}{p}e^{-\gamma x}dx = 1\). The newborn's utility is \(\frac{\log((p+\gamma)Y/p)}{\theta+p} - \frac{\gamma}{(\theta+p)^2}\), with derivative \(\frac{1}{\theta+p}\big[\frac1{p+\gamma} - \frac1{\theta+p}\big]\). At \(\gamma=\alpha\) this is negative iff \(\alpha>\theta\), so lowering γ slightly raises newborns' utility exactly when \(r=\theta-\alpha<0\). For an agent of age \(A\) the derivative has an extra \(-A/(\theta+p)\), so older agents gain even more. A small flattening therefore Pareto-improves iff \(r<0\). This matches Note II (Lesson 10): with zero growth, a negative equilibrium interest rate means that transfers from young to old can make everyone better off. If \(\alpha\le\theta\), no member of the family dominates. Flattening hurts newborns, and steepening hurts sufficiently old agents.

Exercise 18.6 Proof Desired wealth \(V^d(r)\) (30 points)

Hold \(r\) constant and suppose every cohort follows the plan of a newborn facing that \(r\) forever.

  1. (15) Show that a newborn's wealth at age \(x\) is \(v(x) = \frac{c_0e^{(r-\theta)x}}{\theta+p} - \frac{ae^{-\alpha x}}{r+p+\alpha}Y\), with \(c_0 = \frac{(\theta+p)aY}{r+p+\alpha}\).
  2. (10) Integrate over the age distribution to show \(\dfrac{V^d}{Y} = \dfrac{r+\alpha-\theta}{(r+p+\alpha)(p+\theta-r)}\) for \(r<p+\theta\).
  3. (5) Conclude that \(r=\theta-\alpha\) is the only rate at which the population wants zero net financial wealth, and that desired wealth is positive above it.
Show solution

(a) At age \(x\), \(h(x)=aYe^{-\alpha x}/(r+p+\alpha)\) and \(c(x)=c_0e^{(r-\theta)x}\), with \(c_0=(\theta+p)h(0)\) because \(v(0)=0\). Then \(v=c/(\theta+p)-h\).
(b) \(\int_0^\infty pe^{-px}v(x)dx = \frac{paY}{(r+p+\alpha)(p+\theta-r)} - \frac{paY}{(r+p+\alpha)(p+\alpha)}\). Using \(pa=p+\alpha\), this is \(\frac{Y}{r+p+\alpha}\big[\frac{p+\alpha}{p+\theta-r}-1\big] = \frac{(r+\alpha-\theta)Y}{(r+p+\alpha)(p+\theta-r)}\).
(c) The denominator is positive on \(-(p+\alpha)<r<p+\theta\), so \(V^d\) has the sign of \(r-(\theta-\alpha)\).

Exercise 18.7 Exam-inspired Where do the \(p\)'s go? (30 points)

Explain in words, with reference to the equations, why:

  1. (10) individual wealth earns \(r+p\), while aggregate wealth earns \(r\);
  2. (10) individual human wealth is discounted at \(r+p\), while aggregate human wealth is discounted at \(r+p+\alpha\);
  3. (10) \(p\) disappears from the individual Euler equation but appears in the aggregate one.
Show solution

(i) The extra \(p\) is a transfer from the dying to survivors via annuities. It nets out in the aggregate (§4).
(ii) An individual's future income is discounted for interest and for its own death. The aggregate \(H\) belongs to the current population, whose share of future aggregate income shrinks at \(p+\alpha\): they die, and their income falls relative to newborns' (§3).
(iii) For an individual, higher impatience and higher return cancel. In the aggregate, turnover matters: newborns arriving with zero financial wealth replace wealthy dying agents, which gives the \(-(p+\alpha)(p+\theta)V\) term (§5).

8. Check your understanding

1. Why does aggregate non-human wealth grow at \(r\) even though each agent's grows at \(r+p\)?
  • Newborns bring new financial wealth into the economy
  • Insurance companies keep the premium p as a profit
  • The pV term is a transfer from the dead to survivors
  • Only half of the population holds any financial assets
Leibniz gives \(\dot V = p v(t,t) - pV + (r+p)V + Y - C\). Newborns bring \(v(t,t)=0\), and the \(-pV\) (wealth lost by the dying) cancels the \(+pV\) paid to survivors.
2. In the pure endowment economy, what determines the sign of the equilibrium interest rate?
  • Whether income falls faster than the discount rate θ
  • Whether the death rate p is larger than the discount rate
  • Whether aggregate labour income Y is growing over time
  • Whether newborns earn more than the average agent earns
\(r=\theta-\alpha\), so \(r<0\) iff \(\alpha>\theta\). The death rate \(p\) does not appear at all. Newborns always earn more than average when \(\alpha>0\), but that alone doesn't fix the sign.
3. Which term would vanish from \(\dot C\) if newborns inherited the wealth of the dying?
  • The term (r + α − θ)C involving the interest rate
  • The term −(p + α)(p + θ)V involving wealth
  • Neither term, since bequests are already in the model
  • Both terms, since consumption would then be constant
The \(V\) term captures the replacement of wealthy dying agents by wealth-less newborns. With full intergenerational transfers (dynastic altruism, as in Barro), that turnover effect disappears. With \(\alpha=0\) we'd be back to \(\dot C=(r-\theta)C\), and Ricardian equivalence would return.
4. Why does aggregate human wealth \(H\) discount future income at \(r+p+\alpha\)?
  • Labour income is riskier than returns on financial assets
  • Agents are more impatient about labour than about capital
  • The aggregate interest rate exceeds the individual rate
  • Current agents' share of future income shrinks over time
\(H\) values future income accruing to people alive today. Their share of \(Y(z)\) falls at rate \(p\) (death) plus \(\alpha\) (their income declines relative to high-earning newborns).

Key takeaways

💬 Stuck or curious? Ask me (your Claude teacher) anything. Try "redo the Leibniz step for V̇ with a general cohort density" or "how does the flat-allocation argument compare with Note II's best symmetric allocation?"
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