The two-period OLG model of Part II has a sharp young/old split, which makes dynamics with capital and debt awkward: every policy question turns into an infinite sequence of cohort-specific budgets. Blanchard's "perpetual youth" model keeps the essential OLG ingredient, namely that people alive today are not the same as people alive tomorrow, so new cohorts keep arriving with no assets. It drops the age structure that makes OLG intractable.
The trick is that every agent faces the same constant hazard of death, whatever their age. Nobody is "young" or "old" in terms of their horizon BY slide 3. That is unrealistic but tractable: with log utility (and, in Lesson 18, an exponential income profile) individual decisions aggregate exactly. We get a two-equation system in aggregate consumption and wealth that looks like the Ramsey model, plus one extra term that captures the turnover of generations.
Time is continuous. In any interval of length \(\Delta\), each living agent dies with probability \(p\Delta\) (to first order), independently of age, with \(p\in(0,\infty)\) BY slide 2.
Two consequences follow. (i) The probability of surviving \(x\) more years is \(e^{-px}\) for anyone, whatever their age: the exponential distribution is memoryless. (ii) Expected remaining lifetime is therefore the same for everyone: $$\int_0^\infty e^{-px}\,dx = \frac1p .$$ With \(p = 0.02\), everyone alive, at every age, expects 50 more years. That is the "perpetual youth."
With an uncertain lifetime and no bequest motive, holding wealth at the moment of death is pure waste. Following Yaari (1965), assume competitive insurance companies sell actuarially fair annuities. You hand the company your wealth. Each instant it pays you a premium while you live, and when you die it keeps what you held. With negative wealth the contract runs the other way: you pay a premium, and your debt is cancelled when you die, which is life insurance BY slide 3.
Every instant, a fraction \(p\) of wealth holders die. Their wealth doesn't vanish: the insurer redistributes it to survivors in proportion to their holdings. A survivor's return is therefore \(r\) plus \(p\). For the economy as a whole, though, no wealth is created; it is only moved from the dead to the living. That is why, in Lesson 18, aggregate wealth grows at \(r\) while each individual's grows at \(r+p\).
An agent at \(t\) maximizes expected discounted utility, getting utility only while alive. The expectation is only over the date of death \(T\) BY slide 4: $$E_t\left[\int_t^{T} u(c(z))\,e^{-\theta(z-t)}\,dz\right], \qquad \Pr(T > z\mid T>t) = e^{-p(z-t)} .$$ One util at \(t+\Delta\) is worth \(1/(1+\Delta\theta)\) utils at \(t\), so \(\theta\in(0,\infty)\) is the pure rate of time preference.
So a mortal agent behaves like an immortal one with a higher discount rate, \(\theta+p\). But the same agent also earns a higher return, \(r+p\). Section 7 shows that the two \(p\)'s cancel in the Euler equation but not in the consumption function.
Define the intertemporal price \(R(t,z)\), the price at \(t\) of one unit of the good delivered at \(z\) conditional on being alive (the annuity market prices exactly this). Let \(v(t)\) be non-human (financial) wealth and \(y(z)\) labour income. The present-value budget is BY slide 5 $$\int_t^\infty \big[c(z)-y(z)\big]R(t,z)\,dz = v(t).$$ Define human wealth \(h(t)\equiv\int_t^\infty y(z)R(t,z)\,dz\), the market value of future labour income. The budget becomes \(\int_t^\infty c(z)R(t,z)\,dz = v(t) + h(t)\): the present value of consumption equals total wealth.
With \(u=\log\), the optimal plan satisfies $$c(t) = (\theta+p)\big(v(t)+h(t)\big).$$ The marginal propensity to consume out of total wealth is \(\theta+p\). It is the same for every agent, whatever their age and whatever the path of interest rates.
The proof uses a lemma you already know from Cobb–Douglas demand BY slide 6.
The solution to \(\max_{c_1,\dots,c_n}\sum_{i=1}^n a_i\log c_i\) subject to \(\sum_{i=1}^n R_ic_i = W\) (with \(a_i>0\)) is $$R_i c_i = \frac{a_i}{\sum_{k=1}^n a_k}\,W \qquad\text{for each } i.$$ The expenditure share on good \(i\) depends only on its preference weight.
The slide writes the objective as "\(a_1\log c_1 + a_1\log c_2+\dots\)" and the budget as "\(R_1c_1+R_1c_2+\dots\)". The second subscripts should be \(a_2\) and \(R_2\) (and so on), as in the lemma above.
Define the instantaneous rate \(r(\mu)\) by writing BY slide 7 $$R(t,z) = \exp\left(-\int_t^z \big[r(\mu)+p\big]\,d\mu\right).$$ Why this form? Take the discrete case with periods of length Δ and \(p=0\) BY slide 8. The price of a good \((z-t)/\Delta\) periods ahead is the inverse of the compounded gross return: $$R(t,z)^{-1} = \prod_{i=1}^{(z-t)/\Delta}\big(1+r_{t+i}\Delta\big)\quad\Rightarrow\quad -\log R(t,z) = \sum_{i=1}^{(z-t)/\Delta}\log\big(1+r_{t+i}\Delta\big) = \sum_{i}\big[r_{t+i}\Delta + o(\Delta)\big].$$ The sum is a Riemann sum, so as \(\Delta\downarrow0\), \(\log R(t,z)\to-\int_t^z r(\mu)d\mu\). With mortality, the per-period gross return on annuities is \((1+r\Delta)/(1-p\Delta)\). Its log is \((r+p)\Delta + o(\Delta)\), which adds the \(p\).
A fact we will use repeatedly: \(\dfrac{\partial R(t,z)}{\partial t} = \big[r(t)+p\big]R(t,z)\). Moving the start date forward removes one instant of discounting.
The ODE alone does not pin down the budget; it holds for any \(v(T)\). Letting \(T\to\infty\) in Step 1, the sequential constraint is equivalent to the present-value constraint if and only if BY slide 10 $$\lim_{T\to\infty} R(t,T)\,v(T) = \lim_{T\to\infty} v(T)\,e^{-\int_t^T (r(\mu)+p)d\mu} = 0 .$$
| If the limit is… | then… | because… |
|---|---|---|
| \(>0\) | the plan is not optimal | the agent "dies holding" wealth of positive present value. Spending that amount on extra consumption at any finite date is affordable and raises utility. |
| \(<0\) | the plan is not feasible | debt is rolled over forever and never repaid in present value (a Ponzi scheme). No lender would accept it, and it violates the PV budget. |
| \(=0\) | ODE + limit ⇔ PV budget | this is the condition we impose. |
Human wealth must also be finite, which requires \(\lim_{T\to\infty}y(T)e^{-\int_t^T(r+p)} = 0\) (in fact the integral must converge). If human wealth were infinite, the agent could borrow without bound against it and the problem would have no solution. With constant \(y\) and \(r\), finiteness needs \(r+p>0\). So negative interest rates are fine as long as \(r>-p\), a point that matters in Lessons 18–20.
Death makes the agent more impatient (discount \(\theta+p\)), but annuities raise the return by exactly the same \(p\) (return \(r+p\)). In the Euler equation only the gap between return and impatience matters, \((r+p)-(\theta+p) = r-\theta\), so \(p\) cancels. The level is different: the MPC out of wealth is \(\theta+p\), not \(\theta\), because a shorter expected horizon means total wealth is spread over fewer expected years. Even so, when all agents face the same \(r\), the cross-section of consumption growth is identical for everyone. Differences between cohorts come only from their different wealth levels, and newborns arrive with \(v=0\).
Take constant \(r\) and constant labour income \(y\) (this is \(\alpha=0\) in the next lesson's notation) and an agent born with \(v=0\). Then \(h = y/(r+p)\), initial consumption is \(c_0 = (\theta+p)y/(r+p)\), and \(c(x) = c_0e^{(r-\theta)x}\) at age \(x\). Since \(h\) is constant, \(v(x) = c(x)/(\theta+p) - h\), or $$\frac{v(x)}{y} = \frac{e^{(r-\theta)x}-1}{r+p}.$$
No past exam has asked about Blanchard–Yaari directly, so these exercises follow the formats Alvarez does use: true/false with a short justification, and guided derivations with hints and point weights. Answer every part with general parameters before opening the solutions.
Give a verdict and a justification of one to four lines for each statement.
False. The Euler equation is \(\dot c/c = r-\theta\), and \(p\) cancels. Death raises the effective discount rate to \(\theta+p\), but fair annuities raise the return to \(r+p\) by exactly the same amount. A higher \(p\) does raise the MPC, \(\theta+p\), so the level of consumption out of a given wealth changes, but not its growth rate.
False. Death arrives at a constant hazard \(p\), so remaining lifetime is exponential with mean \(1/p\) at every age (memorylessness). That is exactly why the model has "no young and old": everyone alive has the same horizon and the same MPC.
True (Yaari 1965). In every state the agent cares about (being alive), an annuity pays \(r+p\), while a plain bond pays only \(r\). The bond's only advantage is a payout at death, which the agent doesn't value. Borrowers likewise borrow through annuity-type contracts, which builds life insurance into their debt.
False. The fraction is \(\theta+p\). The utility weights \(e^{-(\theta+p)(z-t)}\) integrate to \(1/(\theta+p)\), and with log utility the share of wealth spent on date-\(t\) consumption equals date \(t\)'s share of total weight. The death rate enters the MPC even though it drops out of the Euler equation.
False. The requirement is that human wealth be finite: \(h=\int_t^\infty y\,e^{-(r+p)(z-t)}dz<\infty\), i.e. \(r+p>0\). So \(r\in(-p,0)\) is allowed. With income declining at rate α, the bound loosens to \(r>-(p+\alpha)\).
False. A strictly positive limit means wealth is left unspent "at infinity". The plan is feasible but not optimal, because consuming that wealth earlier raises utility. A strictly negative limit is the infeasible case: debt is rolled over forever, a Ponzi scheme.
False. With a constant \(r\), the MPC becomes \((r+p)-(r-\theta)/\sigma\), and with a time-varying \(r\) it depends on the whole future rate path (Exercise 17.7). Only log utility makes the MPC independent of prices, which is what makes aggregation trivial in Lesson 18.
(a) \([N(s,t+\Delta)-N(s,t)]/\Delta = -pN(s,t)\), so as \(\Delta\downarrow 0\), \(\partial_tN = -pN\). Therefore \(N(s,t)=N(s,s)e^{-p(t-s)} = pe^{-p(t-s)}\).
(b) \(\int_{-\infty}^t pe^{-p(t-s)}ds = 1\). People older than \(A\) were born before \(t-A\): \(\int_{-\infty}^{t-A}pe^{-p(t-s)}ds = e^{-pA}\).
(c) Age \(x=t-s\) has density \(pe^{-px}\), so the mean age is \(\int_0^\infty xpe^{-px}dx = 1/p\). Remaining lifetime is exponential with rate \(p\) at every age, so its mean is also \(1/p\). The age distribution in a stationary population with constant hazard mirrors the remaining-lifetime distribution.
(a) Per unit of agents, the insurer invests \(v\) and receives \(v(1+\Delta r)\). It pays the fraction \(1-p\Delta\) that survive, each \(v(1+\Delta r)/(1-p\Delta)\), so total payouts equal total receipts. Expanding, \(\frac{1+\Delta r}{1-p\Delta} = 1+(r+p)\Delta+o(\Delta)\).
(b) With \(v<0\) the agent is a borrower whose debt grows at \(r+p\). It pays interest \(r\) plus a premium \(p\), and the premium funds repayment to lenders for those who die. At death the debt is cancelled: built-in life insurance, so lenders face no default risk.
(c) A bond pays \(r\) and leaves wealth at death, which the agent doesn't value. An annuity pays \(r+p>r\) in every state the agent cares about. It dominates, so wealth is held only as annuities.
(a) The FOC is \(a_i/c_i = \lambda R_i\), so \(R_ic_i = a_i/\lambda\). Summing gives \(W=\sum_k a_k/\lambda\), so \(\lambda=\sum_ka_k/W\) and \(R_ic_i = Wa_i/\sum_ka_k\).
(b) Only the ratios \(a_i/\sum_ka_k\) appear.
(c) The budget is \(\int_t^\infty c(z)R(t,z)dz = v(t)+h(t)\equiv W\). The weights integrate to \(\int_t^\infty e^{-(\theta+p)(z-t)}dz = 1/(\theta+p)\). Date \(t\) has weight density 1 and \(R(t,t)=1\), so expenditure on \(c(t)\) per unit of time is \(W\cdot 1/(1/(\theta+p))\). That gives \(c(t) = (\theta+p)W\).
Show that if \(v(\cdot)\) satisfies \(\dot v = (r+p)v + y - c\) and \(\lim_{T\to\infty}R(t,T)v(T)=0\), then \(\int_t^\infty[c-y]R(t,z)dz = v(t)\). [Hint: compute \(\frac{d}{dz}\big[R(t,z)v(z)\big]\).]
\(\partial_zR(t,z) = -(r(z)+p)R(t,z)\), so \(\frac{d}{dz}[R(t,z)v(z)] = R(t,z)[\dot v - (r+p)v] = R(t,z)[y(z)-c(z)]\). Integrating from \(t\) to \(T\), \(R(t,T)v(T) - v(t) = \int_t^T[y-c]R\,dz\). Letting \(T\to\infty\) and using the boundary condition gives \(v(t) = \int_t^\infty[c-y]R\,dz\). ∎ The converse is §6.
An agent is born at age 0 with \(v=0\) and earns a constant \(y>0\) for life. The interest rate is constant at \(r>-p\). Answer in terms of \(y,r,\theta,p\).
For (b), \(c(x) = c_0e^{(r-\theta)x}\) and \(h\) is constant, so \(v(x) = c(x)/(\theta+p) - h\).
(a) \(h(x)=\int_x^\infty ye^{-(r+p)(z-x)}dz = y/(r+p)\), independent of age because income and rates are constant. Then \(c_0=(\theta+p)h=y(\theta+p)/(r+p)\).
(b) \(c(x)=c_0e^{(r-\theta)x}=(\theta+p)he^{(r-\theta)x}\), so \(v(x)=c(x)/(\theta+p)-h = h(e^{(r-\theta)x}-1)\).
(c) If \(r>\theta\), \(v(x)>0\) for \(x>0\): a saver whose consumption starts below income (\(c_0<y\)) and rises. If \(r<\theta\), the agent is a borrower whose consumption starts above income and falls. Solving \(c_0e^{(r-\theta)x^*}=y\) gives \(x^* = \ln(y/c_0)/(r-\theta) = \ln[(r+p)/(\theta+p)]/(r-\theta)\). Numerator and denominator always have the same sign, so \(x^*>0\) for every \(r\ne\theta\).
(d) \(e^{-(r+p)x}h(e^{(r-\theta)x}-1) = h(e^{-(\theta+p)x}-e^{-(r+p)x})\to0\), since \(\theta+p>0\) and \(r+p>0\).
(e) The life-insurance component. While alive, the borrower pays the premium \(p\) on its debt, and the insurer uses the pooled premiums to repay the lenders of those who die.
Replace \(\log c\) with \(c^{1-\sigma}/(1-\sigma)\), \(\sigma>0\).
(a) The FOC is \(e^{-(\theta+p)(z-t)}c(z)^{-\sigma}=\lambda R(t,z)\). Taking logs and differentiating in \(z\): \(-(\theta+p)-\sigma\dot c/c = -(r+p)\), so \(\dot c/c=(r-\theta)/\sigma\). Yes, \(p\) cancels.
(b) \(c(z)=c(t)e^{(r-\theta)(z-t)/\sigma}\). Substituting into \(\int_t^\infty c(z)e^{-(r+p)(z-t)}dz=v+h\) gives \(c(t)/[(r+p)-(r-\theta)/\sigma] = v+h\). The MPC is positive iff \((r+p)>(r-\theta)/\sigma\), i.e. the PV of consumption converges. At \(\sigma=1\) it reduces to \(\theta+p\).
(c) With log utility the MPC is \(\theta+p\) for everyone, independent of prices. So individual consumption functions add up to \(C=(\theta+p)(H+V)\) (Lesson 18). With σ≠1 the MPC depends on the future rate path, so the aggregate equation would involve those paths too.