So far the OLG economy had no way to move goods through time except trade between generations. Now we add a storable, productive asset: capital. One generation is born each period, of constant size one (no population growth). There is a constant-returns production function \(F(K,L)\), and capital depreciates at rate \(\delta\) III p.15:
$$K_{t+1} = I_t + (1-\delta)K_t, \qquad g_t + I_t + C_t = F(K_t, L_t), \qquad C_t = c^t_t + c^{t-1}_t .$$Labour is supplied inelastically by the young, so \(L_t = 1\) throughout.
A competitive firm rents capital at rental rate \(v_t\) and hires labour at wage \(w_t\). With constant returns and price-taking,
$$w_t = F_L(K_t, L_t), \qquad v_t = F_K(K_t, L_t).$$Now track one unit of goods saved by a young household at date \(t\). There are two ways to carry it to \(t+1\):
| Asset bought at \(t\) | Payoff at \(t+1\) |
|---|---|
| Bond (government debt or a private loan) | \(1 + r_t\) |
| One unit of capital, installed for production at \(t+1\) | rental \(v_{t+1}\) + undepreciated capital \(1-\delta\) |
If both are held in equilibrium, no-arbitrage requires equal payoffs: \(1 + r_t = v_{t+1} + 1 - \delta\), i.e. \(v_{t+1} = r_t + \delta\). With \(L_{t+1}=1\):
The interest rate between \(t\) and \(t+1\) depends on next period's capital stock. That stock is exactly what today's young save.
This timing is what makes the equilibrium law of motion implicit (§5). Savers' return depends on how much they save in total.
Generation \(t\) earns \(w_t\) when young, pays \(\tau^t_t\) when young and \(\tau^t_{t+1}\) when old, and saves \(s_t\):
$$s_t + c^t_t = w_t - \tau^t_t, \qquad c^t_{t+1} = (1+r_t)s_t - \tau^t_{t+1}.$$Eliminating \(s_t\) gives the lifetime budget constraint III p.15:
$$c^t_t + \frac{c^t_{t+1}}{1+r_t} = w_t - \tau^t_t - \frac{\tau^t_{t+1}}{1+r_t} \;\equiv\; W_t .$$Optimal saving depends on three things: the interest rate, first-period disposable income, and lifetime wealth \(W_t\). Alvarez writes
$$s_t = s\!\left(r_t,\; w_t - \tau^t_t,\; w_t - \tau^t_t - \frac{\tau^t_{t+1}}{1+r_t}\right).$$The second argument matters because saving is disposable first-period income minus first-period consumption: \(s = (w-\tau^t_t) - c_y(r, W)\).
With a government that issues assets \(B_t\) (\(B_t<0\) is debt) and buys \(g_t\), the government budget is \(B_t + g_t = (1+r_{t-1})B_{t-1} + \tau^t_t + \tau^{t-1}_t\), as in III p.3. Household saving now has two uses: lending to the government and holding capital.
If household budgets, the government budget and goods-market clearing hold at every date, and \(B_0 + s_0 = K_1\), then $$B_t + s_t = K_{t+1}\quad\text{for all } t\ge 1 .$$
The young are the only savers. Whatever they don't lend to the government must end up as next period's capital. This is the same accounting as in Lesson 14, with \(K_{t+1}\) replacing zero on the right-hand side.
With no taxes, government spending or debt (\(\tau=g=B=0\)), asset-market clearing becomes \(s_t = K_{t+1}\). Substituting factor prices III p.16:
$$K_{t+1} = s\big(F_K(K_{t+1},1) - \delta,\; F_L(K_t,1),\; F_L(K_t,1)\big).$$Given \(K_1>0\), owned by the initial old, this equation pins down \(K_2\), then \(K_3\), and so on. It is implicit because \(K_{t+1}\) appears on both sides through \(r_t\). By the implicit function theorem it defines a map \(K_{t+1} = s^*(K_t)\) locally, provided
$$1 - s_r\,F_{KK}(K_{t+1},1) \neq 0 .$$Since \(F_{KK}<0\), this holds automatically whenever saving is non-decreasing in \(r\) (\(s_r\ge 0\)). The slides' "local conditions in the problem set" amount to this denominator being positive.
Introduce PAYG: \(\tau^t_t = \tau\), \(\tau^t_{t+1} = -\tau\). Each old person receives the tax paid by the one young person alive. Holding \(r_t\) and \(w_t\) fixed, lifetime wealth changes by III p.17
$$\frac{\partial W_t}{\partial \tau} = -\Big(1 - \frac{1}{1+r_t}\Big) = -\frac{r_t}{1+r_t} < 0 \quad\text{when } r_t>0 .$$The household pays \(\tau\) today and gets back \(\tau\) tomorrow, which is a zero net return. Saving privately would have earned \(r_t>0\).
If old-age consumption is a normal good and \(r_t>0\), then at fixed prices \(\partial s_t/\partial\tau < -\tfrac{1}{1+r_t} < 0\).
The economics: PAYG directly replaces private saving for retirement, since the transfer \(\tau\) arrives when old anyway, and the wealth loss reduces old-age consumption further. Both forces cut \(s\).
In general equilibrium, write the law of motion with the policy as \(K_{t+1} = s^*(K_t,\tau)\). Applying the implicit function theorem to \(K' - s\big(F_K(K',1)-\delta,\,\cdot\,\big) = 0\) gives
$$\frac{\partial s^*}{\partial\tau} = \frac{s_\tau}{1 - s_r F_{KK}} < 0,$$using the claim and the positive denominator from §5. A steady state solves \(K^* = s^*(K^*,\tau)\). Differentiate both sides with respect to \(\tau\):
$$\frac{dK^*}{d\tau} = \frac{\partial s^*}{\partial K}\frac{dK^*}{d\tau} + \frac{\partial s^*}{\partial \tau} \quad\Longrightarrow\quad \boxed{\frac{dK^*}{d\tau} = \frac{\partial s^*/\partial\tau}{1 - \partial s^*/\partial K}} .$$Graphically, a higher \(\tau\) shifts the map \(s^*(\cdot,\tau)\) down. Where the map crosses the 45° line from above, the crossing moves left. Where it crosses from below, the crossing moves right. The figure below shows both cases at once.
Let \(F(K,L) = K^\alpha L^{1-\alpha}\) and suppose consumers care only about old-age consumption III p.18. Then the young save everything they have after tax, \(s_t = w_t - \tau\), and saving does not depend on \(r_t\). That makes the law of motion explicit:
Generation 0 (initial old) owns \(K_1\), whose return is already fixed, and receives \(\tau\) from the young. It gains \(\tau\).
Generation 1 earns \(w_1 = K^*\), pays \(\tau\) and saves \(K_2 = K^* - \tau\). When old it receives the gross return on its capital plus the transfer from generation 2:
$$c^1_2(\tau) = (K^*-\tau)\big[1 - \delta + \alpha (K^*-\tau)^{\alpha-1}\big] + \tau .$$Differentiate and evaluate at \(\tau=0\):
So for \(\delta>0\), generation 1 gains when \(\alpha\) is small enough, namely when \(\alpha^2/(1-\alpha) < \delta\) III p.18.
In the pure-endowment economy, social security only reshuffles goods; nothing about production changes. With capital, taxing the young lowers \(K_2\), which raises the marginal product of capital \(\alpha K_2^{\alpha-1}\). Generation 1 is the owner of \(K_2\). It loses \(1+r\) per unit of foregone saving but gets \(\tau\) back from generation 2, and every remaining unit earns more. The term \(-\alpha(\alpha-1)K_2^{\alpha-1} = \alpha(1-\alpha)K_2^{\alpha-1}>0\) in step 1 is exactly this price effect. It is a pecuniary transfer from generation 2, whose wage \(w_2 = (1-\alpha)K_2^\alpha\) is lower.
Let \(K(\tau)\) be the stable steady state of \(K = (1-\alpha)K^\alpha - \tau\). A steady-state generation consumes \(c_{ss}(\tau) = K(\tau)\big[1-\delta+\alpha K(\tau)^{\alpha-1}\big] + \tau\).
Steady-state generations gain from PAYG if and only if \(r^*<0\). That is the dynamic-inefficiency condition of Lesson 12, with no population growth. The gain to generation 1 is a transition effect. When \(r^*>0\), the CE has finite aggregate value and the First Welfare Theorem applies (Lesson 12). So no feasible reallocation, PAYG included, can make every generation better off: someone, here the generations far in the future, must lose III p.19.
These follow the format of Alvarez's exams: true/false items with short justifications, then long derivations broken into small steps. Attempt each part before opening the solution.
For each statement, say whether it is true or false and give a justification of one to four lines.
False. It depends on parameters. The steady-state interest rate is \(\bar r = \frac{\alpha}{\beta(1-\alpha)}-\delta\) (Exercise 15.2). If \(\bar r<0\), capital is above the golden rule, and permanently cutting \(K\) raises consumption at every date: a Pareto improvement. Capital does not open the missing markets between living and unborn generations. When \(\bar r<0\), the value of the aggregate endowment is again infinite (Lesson 12).
False. Saving is \(s_t=\beta w_t\), independent of \(r_t\). With log utility and no income when old, the income and substitution effects of \(r\) cancel exactly. This fails as soon as there is old-age income: with PAYG, \(s_t=\beta(w_t-\tau)-(1-\beta)\tau/(1+r_t)\), which does rise with \(r_t\) (Exercise 15.2(i)).
False. \(\bar K=[\beta(1-\alpha)A]^{1/(1-\alpha)}\) does rise with \(A\), but \(F_K(\bar K,1)=\alpha A\bar K^{\alpha-1}=\frac{\alpha}{\beta(1-\alpha)}\). Capital rises exactly enough to keep the marginal product unchanged, so \(\bar r\) does not depend on \(A\). Only preferences (\(\beta\)), the capital share (\(\alpha\)) and depreciation (\(\delta\)) decide whether the economy is dynamically inefficient.
False. The first part can happen: \(r(0)=f'(k_0)-\delta<0\) when \(k_0\) is large. But there is a single agent with finite wealth, so the value of aggregate consumption is finite, and the First Welfare Theorem applies. The agent simply runs capital down. A negative interest rate signals inefficiency only in combination with infinitely many agents and goods, as in OLG.
True. With \(\bar r>0\) the equilibrium is Pareto optimal, so nobody can gain without someone losing. In the Cobb–Douglas example of §7, the initial old gain \(\tau\), and generation 1 gains whenever \(\delta>\alpha^2/(1-\alpha)\), through a higher return on scarcer capital. Steady-state generations lose: \(c_{ss}'(0)=-\bar r/(1-\alpha)<0\).
False. \(dK^*/d\tau = s^*_\tau/(1-s^*_K)\), with \(s^*_\tau<0\). At a stable steady state (\(0<s^*_K<1\)) capital falls. At an unstable one (\(s^*_K>1\)) the denominator is negative and the steady state rises. That case is never reached from nearby initial conditions, but the word "always" is wrong.
False. It is implied by Walras' law. Add the young and old budgets and the government budget, then use goods-market clearing and \(F=w+vK\) (constant returns, or profits paid out). This gives \(B_t+s_t-K_{t+1}=(1+r_{t-1})(B_{t-1}+s_{t-1}-K_t)\), which is zero for all \(t\) once it is zero at the start (§4).
Consider a two-period OLG economy with no heterogeneity. Generation \(t\) has utility \((1-\beta)\log c^t_t+\beta\log c^t_{t+1}\). Each generation has one unit of labour, supplied inelastically when young, and each cohort has size one. The initial old own the capital stock \(K_1>0\). Capital evolves as \(K_{t+1}=I_t+(1-\delta)K_t\), and feasibility is \(I_t+c^t_t+c^{t-1}_t=F(K_t,L_t)\) with \(F(K,L)=AK^\alpha L^{1-\alpha}\), \(A>0\), \(\alpha\in(0,1)\). Firms hire labour at wage \(w_t\) and rent capital at rental rate \(v_t\). Young agents face \(c^t_t+s_t=w_t\) and, when old, \(c^t_{t+1}=s_t(1+r_t)\).
Compute \(F_K\) at the steady state from (f). With no population growth, steady-state consumption is \(F(K,1)-\delta K\). Ask what happens to it if \(K\) is permanently reduced.
(a) The firm maximizes \(F(K,L)-v_tK-w_tL\): \(\;v_t=\alpha AK_t^{\alpha-1}L_t^{1-\alpha}\) and \(w_t=(1-\alpha)AK_t^{\alpha}L_t^{-\alpha}\).
(b) \(L_t=1\): one cohort of size one supplies one unit each.
(c) Maximize \((1-\beta)\log(w_t-s_t)+\beta\log[s_t(1+r_t)]\). The FOC \(\frac{1-\beta}{w_t-s_t}=\frac{\beta}{s_t}\) gives \(s_t=\beta w_t\). The \(1+r_t\) term drops out because it enters \(\log c^t_{t+1}\) additively.
(d) No arbitrage between bonds and capital: \(1+r_t=v_{t+1}+1-\delta\). Note the date: the rental rate is earned at \(t+1\), when the capital saved at \(t\) is used.
(e) \(s_t=K_{t+1}\). The young are the only savers and there are no government bonds, so their saving is next period's capital (§4 with \(B\equiv0\)).
(f) \(K_{t+1}=\beta w_t=\beta(1-\alpha)AK_t^{\alpha}\). The map is concave, infinitely steep at 0, and crosses the 45° line once, at \(\bar K=[\beta(1-\alpha)A]^{1/(1-\alpha)}\). Its slope there is \(\alpha<1\), so the path converges monotonically from any \(K_1>0\).
(g) It depends on parameters. At the steady state, \(A\bar K^{\alpha-1}=1/(\beta(1-\alpha))\), so $$\bar r=F_K(\bar K,1)-\delta=\frac{\alpha}{\beta(1-\alpha)}-\delta .$$ If \(\bar r\ge0\), the steady state is efficient. If \(\bar r<0\), i.e. \(\alpha\lt \delta\beta(1-\alpha)\), it is dynamically inefficient. To see this, at some date \(T\) cut \(K_{T+1}\) permanently by a small \(\varepsilon\). Consumption at \(T\) rises by \(\varepsilon\). At every later date, steady-state consumption \(F(K,1)-\delta K\) changes by \(-(F_K-\delta)\varepsilon=-\bar r\varepsilon>0\). Total consumption is higher at every date from \(T\) on. Hand the extra to the old at each date, so every generation from \(T-1\) on gains and no one loses.
On the official solution. The official answer explains the failure of the First Welfare Theorem by "missing markets between generations". The sharper statement, as in Lesson 12, is that when \(\bar r<0\) the prices \(p_t\) do not fall, so the value of the aggregate endowment is infinite. Some student review notes describe the endowment OLG inefficiency as "over-accumulation of capital". That language is right only here, where there is capital.
(h) Let cohort size be \(N_t=(1+n)^t\). Then \(K_{t+1}=N_ts_t\), so per-worker capital satisfies \((1+n)k_{t+1}=\beta(1-\alpha)Ak_t^\alpha\). At the steady state \(A\bar k^{\alpha-1}=\frac{1+n}{\beta(1-\alpha)}\), so \(\bar r=\frac{\alpha(1+n)}{\beta(1-\alpha)}-\delta\). The golden rule maximizes \(f(k)-(n+\delta)k\), i.e. \(r=n\). So the economy is inefficient iff \(\bar r<n\), i.e. \(\frac{\alpha(1+n)}{\beta(1-\alpha)}<1+n-(1-\delta)\). With \(\delta=1\), this is \(\alpha(1+n)\lt \beta(1-\alpha)(1+n)\), i.e. \(\alpha\lt \beta(1-\alpha)\), whatever \(n\) is.
(i) Lifetime wealth is \(W_t=w_t-\tau+\frac{\tau}{1+r_t}\), and log utility gives \(c^t_t=(1-\beta)W_t\). So $$s_t=(w_t-\tau)-c^t_t=\beta(w_t-\tau)-\frac{(1-\beta)\tau}{1+r_t},$$ which is increasing in \(r_t\): a higher \(r\) lowers the present value of the transfer, so the young must save more. Using \(1+r_t=\alpha AK_{t+1}^{\alpha-1}+1-\delta\) and \(s_t=K_{t+1}\), $$K_{t+1}=\beta\big[(1-\alpha)AK_t^\alpha-\tau\big]-\frac{(1-\beta)\tau}{1-\delta+\alpha AK_{t+1}^{\alpha-1}} .$$ \(K_{t+1}\) appears on both sides: this is the implicit law of motion of §5.
Generation \(t\) has a general utility \(u(c^t_t,c^t_{t+1})\), works one unit when young, and faces \(c^t_t+s_t=w_t-\tau^t_t\) and \(c^t_{t+1}=s_t(1+r_t)-\tau^t_{t+1}\). Technology \(F\) has constant returns with \(F_{KK}<0\), and capital depreciates at rate \(\delta\). There is no government spending or debt (\(g_t=0\), \(B_t=0\)). PAYG is \(\tau^t_t=\tau\), \(\tau^t_{t+1}=-\tau\), and assume \(r_t>0\).
(a) A unit saved at \(t\) can buy a bond paying \(1+r_t\) at \(t+1\), or a unit of capital that is used at \(t+1\). The capital earns the rental \(v_{t+1}=F_K(K_{t+1},1)\) plus the undepreciated \(1-\delta\). If both are held, the payoffs must be equal: \(r_t=F_K(K_{t+1},1)-\delta\). The capital stock is \(K_{t+1}\), not \(K_t\). Both 2025 solution sets slip on this index.
(b) The FOC is \(u_1/u_2=1+r_t\). Together with the lifetime budget \(c^t_t+\frac{c^t_{t+1}}{1+r_t}=W_t\equiv w_t-\tau^t_t-\frac{\tau^t_{t+1}}{1+r_t}\), it determines \(c^t_t=c_y(r_t,W_t)\). Then \(s_t=(w_t-\tau^t_t)-c_y(r_t,W_t)\), a function of the three listed arguments.
(c) With \(\tau^t_{t+1}=-\tau\), \(W=w-\frac{r}{1+r}\tau\). Raising the tax from \(\tau\) to \(\tau'\) changes wealth by \(\Delta W=-\frac{r}{1+r}(\tau'-\tau)<0\), using \(r>0\). Differencing the lifetime budget, \(\Delta c_y+\frac{\Delta c_o}{1+r}=\Delta W\). Since \(c_o\) is not inferior and \(\Delta W<0\), \(\Delta c_o\le0\), so \(\Delta c_y\ge\Delta W\). Then $$\Delta s=-(\tau'-\tau)-\Delta c_y\le-(\tau'-\tau)+\tfrac{r}{1+r}(\tau'-\tau)=-\tfrac{\tau'-\tau}{1+r}<0 .$$ Each unit of tax lowers saving by at least \(1/(1+r)\).
(d) Define \(\phi(K',K,\tau)=K'-s\big(F_K(K',1)-\delta,\;F_L(K,1)-\tau,\;F_L(K,1)-\tau+\tfrac{\tau}{1+F_K(K',1)-\delta}\big)\). An equilibrium has \(\phi=0\). At \(\tau=0\), \(\partial\phi/\partial K'=1-s_1F_{KK}\), where \(s_1\) is the derivative with respect to \(r\). This is strictly positive because \(s_1>0\) and \(F_{KK}<0\). By the implicit function theorem, \(\phi=0\) defines \(K'=s^*(K,\tau)\) locally. Wording note: the problem set says this assumption means "the substitution effect is not strong enough". In fact \(\partial s/\partial r>0\) means the substitution effect dominates the income effect.
(e) Differentiate \(\phi=0\) with respect to \(\tau\) at \(\tau=0\). The terms in \(\tau\,dr/d\tau\) vanish, leaving $$\frac{\partial s^*}{\partial\tau}=\frac{-s_2-s_3\frac{r}{1+r}}{1-s_1F_{KK}}=\frac{\partial s/\partial\tau\big|_{r\text{ fixed}}}{1-s_1F_{KK}}<0 .$$ The numerator is negative by (c), the denominator positive by (d). The indirect effect works through the interest rate: lower \(K_{t+1}\) raises \(r_t\), which raises saving since \(s_1>0\). That dampens the fall but cannot reverse it. Watch the sign of the \(s_3\) term: \(\frac{\partial}{\partial\tau}\big(-\tau+\frac{\tau}{1+r}\big)=-\frac{r}{1+r}\). One 2025 solution set writes \(+s_3\frac{r}{1+r}\).
(f) Differentiate \(K^*=s^*(K^*,\tau)\): \(\frac{dK^*}{d\tau}=\frac{s^*_\tau}{1-s^*_K}<0\). Graphically, the map shifts down, and since it crosses the 45° line from above, the crossing moves left.
(g) Now \(1-s^*_K<0\), so \(dK^*/d\tau>0\): the map crosses from below and shifting it down moves the crossing right. But a steady state with slope above one is unstable, since deviations grow. The perverse comparative static applies only at a steady state the economy would never settle in. You can see both cases in the first figure of this lesson.
Modify the model of Exercise 15.3: \(F(K,L)=K^\alpha L^{1-\alpha}\), utility is \(c^t_{t+1}\) (only old-age consumption matters), and \(g_t=B_t=0\). The economy starts at the no-social-security steady state \(K_1=K^*\). At date 1, PAYG \((\tau,-\tau)\) is announced.
(a) Young-age consumption is worthless to the agent, so it saves everything after tax: \(s_t=w_t-\tau^t_t\). With \(B_t=0\), \(K_{t+1}=s_t\), and \(w_t=F_L(K_t,1)=(1-\alpha)K_t^\alpha\).
(b) \(K^*=(1-\alpha)K^{*\alpha}\) gives \(K^*=(1-\alpha)^{1/(1-\alpha)}\) and \(K^{*\,\alpha-1}=1/(1-\alpha)\). Then \(r^*=\alpha K^{*\,\alpha-1}-\delta=\frac{\alpha}{1-\alpha}-\delta\).
(c) \(K_1=K^*\) is predetermined, so \(w_1=(1-\alpha)K^{*\alpha}=K^*\) regardless of \(\tau\). Generation 1 saves \(K_2=K^*-\tau\), earns \(1+r_1=1-\delta+\alpha K_2^{\alpha-1}\) on it, and receives \(\tau\) from generation 2.
(d) Write \(c^1_2=(1-\delta)K_2+\alpha K_2^\alpha+\tau\) with \(dK_2/d\tau=-1\). Then \(\frac{dc^1_2}{d\tau}=-(1-\delta)-\alpha^2K_2^{\alpha-1}+1=\delta-\alpha^2K_2^{\alpha-1}\). At \(\tau=0\) this is \(\delta-\frac{\alpha^2}{1-\alpha}\). It is positive iff \(\alpha^2+\delta\alpha-\delta<0\), i.e. \(\alpha\lt \hat\alpha=\frac{-\delta+\sqrt{\delta^2+4\delta}}{2}\).
(e) We need \(\delta>\alpha^2/(1-\alpha)\) for the gain and \(r^*>0\iff\delta\lt \alpha/(1-\alpha)\). The interval is non-empty because \(\alpha^2\lt \alpha\). Note: neither 2025 solution set checks this. "Small enough \(\alpha\)" alone is not sufficient: as \(\alpha\to0\), \(r^*\to-\delta<0\), and the example slides into the dynamically inefficient case, where a gain is no surprise.
(f) No. With \(r^*>0\) the pre-reform equilibrium is Pareto optimal, so a gain for some generations must be paid for by losses elsewhere. Directly, differentiate the steady state \(K=(1-\alpha)K^\alpha-\tau\): \(K'(0)=-1/(1-\alpha)\). Steady-state consumption is \(c_{ss}=K(1-\delta)+\alpha K^\alpha+\tau\), so $$c_{ss}'(0)=\big[1-\delta+\alpha^2K^{*\,\alpha-1}\big]K'(0)+1=1-\frac{1-\delta+\alpha^2/(1-\alpha)}{1-\alpha}=\frac{\delta-\alpha/(1-\alpha)}{1-\alpha}=-\frac{r^*}{1-\alpha}<0 .$$
(g) Generation 1's principal is returned exactly by the transfer, so only the return on its smaller saving changes. Taxing the young lowers \(K_2\), which raises \(F_K\). Generation 1's wage is already fixed, so it captures the higher return without bearing the lower wage. The cost falls on generation 2, whose wage \((1-\alpha)K_2^\alpha\) is lower. This is a general-equilibrium price effect. It cannot arise in the endowment economy, where social security moves goods but changes no productive opportunity.
Suppose households own capital directly, rent it to firms, and sell the undepreciated stock to the next young generation. Show that the return on this strategy equals \(F_K(K_{t+1},1)+1-\delta\). Conclude that if the government also issued bonds paying \(1+r_t>F_K(K_{t+1},1)+1-\delta\), no one would hold capital. What would that imply for \(K_{t+1}\) and for \(F_K\)?
Buy one unit of capital at \(t\). At \(t+1\), rent it for \(v_{t+1}=F_K(K_{t+1},1)\), then sell the \(1-\delta\) that remains. Installed capital and goods trade one for one. The gross return is \(F_K(K_{t+1},1)+1-\delta\). If bonds paid strictly more, every saver would hold only bonds and \(K_{t+1}=0\). But then \(F_K(0,1)=\infty\) under Inada conditions, which contradicts the inequality. So in an equilibrium with \(K_{t+1}>0\) the returns must be equal, which is the content of \(v_{t+1}=r_t+\delta\).
Redo the Walras' law argument of §4 when \(F\) has decreasing returns, with profits \(\pi_t=F(K_t,1)-w_t-v_tK_t>0\) paid to the young. Show that the conclusion \(B_t+s_t=K_{t+1}\) survives once household budgets include \(\pi_t\).
The young budget becomes \(s_t+c^t_t=w_t+\pi_t-\tau^t_t\). In step 3, \(F(K_t,1)=w_t+\pi_t+(r_{t-1}+\delta)K_t\). The extra \(\pi_t\) appears on both sides and cancels. Everything else is unchanged, so the same recursion and induction apply. Walras' law needs every unit of output to be someone's income, which is the role of \(\sum_i\theta^i_j=1\) in Lesson 2.
The claim in §6 assumed \(r_t>0\). Suppose instead \(r_t<0\), so PAYG raises lifetime wealth. Show that if both \(c_y\) and \(c_o\) are normal, saving still falls with \(\tau\) at fixed prices.
From the lifetime budget \(c_y+c_o/(1+r)=W\), normality of \(c_y\) gives \(\partial c_o/\partial W\le1+r\).
Again \(\partial s/\partial\tau=\frac{1}{1+r}\big(c_{o,W}\cdot(-\tfrac{r}{1+r})-1\big)\). With \(r<0\) the first term is positive, but \(c_{o,W}\le1+r\) implies \(c_{o,W}\cdot\frac{-r}{1+r}\le-r\). So the bracket is at most \(-r-1=-(1+r)<0\), and saving falls. The transfer received when old replaces private saving one for one; the wealth effect can offset only part of that.