\(L=\mathbb{R}^\infty\) (one good per date \(t=1,2,\dots\)); agents \(I=\{0,1,2,\dots\}\) indexed by birth date. Generation \(i\ge1\): \(u^i=v^i(x^i_i,x^i_{i+1})\), endowment \((e^i_i,e^i_{i+1})\gg0\). Initial old: \(u^0=x^0_1\), endowment \(e^0_1\).
Feasibility at each date: \(x^{t-1}_t+x^t_t=e^{t-1}_t+e^t_t\equiv\bar e_t\).
Budget: \(p_ix_i+p_{i+1}x_{i+1}=p_ie^i_i+p_{i+1}e^i_{i+1}\). Unique CE = no trade \(x^i=e^i\), by forward induction from the initial old.
Prices: \(\dfrac{p_{t+1}}{p_t}=\dfrac{1}{1+r_t}=\dfrac{v^t_2(e^t_t,e^t_{t+1})}{v^t_1(e^t_t,e^t_{t+1})}\), and \(p_t=\prod_{s\lt t}(1+r_s)^{-1}\) with \(p_1=1\).
Log example \(v=(1-\beta)\log c_y+\beta\log c_o\), endowments \((1-\alpha,\alpha)\): \(\displaystyle 1+\bar r=\frac{(1-\beta)\alpha}{\beta(1-\alpha)},\quad \bar r=\frac{\alpha-\beta}{\beta(1-\alpha)}\). Savings: \(s(r)=\beta(1-\alpha)-\dfrac{(1-\beta)\alpha}{1+r}\), increasing in \(r\) and \(\beta\), decreasing in \(\alpha\).
Symmetric: \(c_y+c_o=1\). Best (for the young): \((c_y^*,c_o^*)=(1-\beta,\beta)\). CE: \((1-\alpha,\alpha)\). All \(t\ge1\) prefer \(c^*\) unless \(\alpha=\beta\); the initial old prefer \(c^*\) iff \(\beta>\alpha\).
| Case | Parameters | \(\bar r\) | \(p\cdot\sum\bar x^i\) | Conclusion |
|---|---|---|---|---|
| 1 | \(\beta>\alpha\) | \(<0\) | \(\infty\) | CE Pareto-dominated by \(c^*\) |
| 2 | \(\beta=\alpha\) | \(0\) | \(\infty\) | CE \(=c^*\), PO |
| 3 | \(\beta\lt \alpha\) | \(>0\) | \((1+\bar r)/\bar r\) | Not ranked; CE is PO (1WT) |
Tax young \(\tau\), give to current old: after-tax endowments \((1-\alpha-\tau,\ \alpha+\tau)\), initial old \(\alpha+\tau\). The CE is still no trade. \(\tau=\beta-\alpha\) implements \(c^*\). If \(\beta>\alpha\) (\(\bar r<0\)), every \(\tau\in(0,\bar\tau)\) is a Pareto improvement, where \(\bar\tau>\beta-\alpha\) solves \(v(1-\alpha-\bar\tau,\alpha+\bar\tau)=v(1-\alpha,\alpha)\). If \(\beta\le\alpha\), any \(\tau>0\) hurts generations \(t\ge1\).
Theorem. LNS + \(p\cdot\sum_i\bar x^i\lt \infty\) ⇒ the CE is PO. The proof needs finiteness to sum \(p\cdot x^i\ge p\cdot\bar x^i\) (strict for one \(i'\)) into \(\sum p\cdot x^i>\sum p\cdot\bar x^i\). Otherwise it reads "\(\infty>\infty\)".
Double infinity (Shell 1971): finitely many goods ⇒ value \(=\sum_\ell p_\ell\bar e_\ell\lt \infty\); finitely many agents (monotone \(u\)) ⇒ \(p\cdot\bar x^i\le p\cdot e^i\lt \infty\), finite sum. Only infinitely many agents and goods can break it.
Finite value is sufficient, not necessary (Case 2). In Case 1, every agent's improved bundle costs more at CE prices, by \(p_t(\beta-\alpha)\big(\tfrac{1}{1+\bar r}-1\big)\).
\(N_t=(1+n)^t\); \(e^t_t=(1+g)^t(1-\alpha)\), \(e^t_{t+1}=(1+g)^t\alpha\). Savings \(s_t=(1+g)^ts(r)\). Still no trade, and \(\bar r\) is unchanged by \(g\) and \(n\).
Let \(G\equiv(1+g)(1+n)\) and \(W\equiv(1-\alpha)G+\alpha\). Normalized feasibility: \(G\hat c_y+\hat c_o=W\). Best: \(\hat c_y^*=(1-\beta)W/G\), \(\hat c_o^*=\beta W\).
Inefficiency condition: \(c^*\) Pareto-dominates the CE \(\iff\hat c^*_o>\alpha\iff \boxed{1+\bar r<(1+g)(1+n)}\) (roughly \(\bar r\lt n+g\)). This holds \(\iff\) the value of the aggregate endowment \(W\sum_t(G/(1+\bar r))^{t-1}\) is infinite.
PAYG: \(\tau_t=\hat\tau(1+g)^t\); each old person gets \(\hat\tau G\) (normalized). Optimal \(\hat\tau=\beta(1-\alpha)-\dfrac{\alpha(1-\beta)}{G}\), which is positive iff \(G>1+\bar r\). Post-reform \(1+r=G\) (golden rule / biological interest rate).
Note III form (\(g=0\)): \(c_y+\dfrac{c_o}{1+n}=(1-\alpha)+\dfrac{\alpha}{1+n}\); PAYG gives \(c_o=\alpha+(1+n)\tau\). A permanent fall in \(n\) with fixed \(\tau\) rotates the line around autarky. The current old and all future generations lose.
(Note III notation: young α, old 1−α, PAYG tax θ.) HH: \(s_t+c^t_t=e^t_t-\tau^t_t\), \(c^t_{t+1}=(1+r_t)s_t+e^t_{t+1}-\tau^t_{t+1}\). Gov: \(B_t+g_t=(1+r_{t-1})B_{t-1}+\tau^t_t+\tau^{t-1}_t\). Walras: \(B_t+s_t=(1+r_{t-1})(B_{t-1}+s_{t-1})\), so \(B_0+s_0=0\Rightarrow B_t+s_t=0\).
PAYG (\(\tau^t_t=\theta,\ \tau^t_{t+1}=-\theta\)): \(B_t=s_t=0\); PV of taxes \(=\theta\bar r/(1+\bar r)\).
Privatized: \(\tilde\tau^0_1=-\theta\), \(\tilde\tau^t_t=\theta\bar r/(1+\bar r)\), \(\tilde\tau^t_{t+1}=0\). Lifetime budgets unchanged ⇒ \(\tilde c=c\), \(\tilde r=\bar r\); \(\tilde s_t=\theta/(1+\bar r)\), \(\tilde B_t=-\theta/(1+\bar r)\) for all \(t\ge1\). Implicit liability → explicit, rolled-over debt.
Ricardian equivalence, generation by generation: timing changes are neutral iff each generation's PV of taxes is preserved. Shifting PV across generations is not neutral.
Capital (Diamond): \(w_t=F_L(K_t,1)\), \(r_t=F_K(K_{t+1},1)-\delta\), asset clearing \(B_t+s_t=K_{t+1}\); with no government, \(K_{t+1}=s\big(F_K(K_{t+1},1)-\delta,\,F_L(K_t,1),\dots\big)\). PAYG lowers saving (\(\partial s^*/\partial\tau<0\)) and, if \(0\lt \partial s^*/\partial K<1\), lowers the stable steady-state \(K\).
Cobb–Douglas, only old-age consumption: \(K_{t+1}=(1-\alpha)K_t^\alpha-\tau\), \(K^*=(1-\alpha)^{1/(1-\alpha)}\); generation 1: \(\left.\dfrac{dc^1_2}{d\tau}\right|_{0}=\delta-\dfrac{\alpha^2}{1-\alpha}\). It can gain via a higher return on capital, but that does not mean all generations can gain when \(r>0\).
Heterogeneity (types \(\alpha_n\), log): \(c_y(\alpha,r)/c_y(\bar\alpha,r)=\dfrac{1+(1-\alpha)r}{1+(1-\bar\alpha)r}\); \(\bar r=\dfrac{\bar\alpha-\beta}{\beta(1-\bar\alpha)}\), so \(\bar r<0\iff\bar\alpha\lt \beta\). A uniform PAYG tax need not help every type.
\(N\)-period lives: if all \(r_t<0\), so that \(p_t\) is increasing, then shifting \(\tau\) from ages \(1..Y\) to \(Y\tau/(N-Y)\) at ages \(Y{+}1..N\) has positive value \(P^t(\tau,Y)>0\) for all generations ⇒ Pareto improvement (pure exchange only).