1. Demographics and annuities

Death	Each instant, die with probability \(p\,dt\). Expected lifetime \(1/p\). Population normalized to 1.
Cohort size	\(N(s,t) = p\,e^{-p(t-s)}\), with \(\int_{-\infty}^t N(s,t)\,ds = 1\).
Annuities	Fair insurers: \(v\tfrac{1+\Delta r}{1-p\Delta} \approx v[1+(r+p)\Delta]\). Wealth earns \(r+p\) while alive and goes to the insurer at death.
Preferences	\(E_t\int_t^\infty \log c\,e^{-\theta(z-t)}dz = \int_t^\infty \log c(z)\,e^{-(p+\theta)(z-t)}dz\).

2. Household problem

Prices	\(R(t,z) = \exp\{-\int_t^z [r(\mu)+p]\,d\mu\}\)
Budget	\(\int_t^\infty [c-y]\,R(t,z)\,dz = v(t)\); \(\ h(t) = \int_t^\infty y(z)R(t,z)\,dz\)
Solution (log)	\(\boxed{c(t) = (\theta+p)\,[v(t)+h(t)]}\), because the weights \(e^{-(\theta+p)(z-t)}\) integrate to \(1/(\theta+p)\).
ODEs	\(\dot v = (r+p)v + y - c\), \(\ \dot h = (r+p)h - y\), \(\ \dot c = (r-\theta)c\). Boundaries: \(\lim R\,v = 0\) and \(\lim R\,y = 0\).

3. Aggregation

Labour income	\(y(s,t) = aY(t)e^{-\alpha(t-s)}\), with \(a = (p+\alpha)/p\).
Aggregates	\(C = (p+\theta)(H+V)\); \(\ \dot V = rV + Y - C\) (aggregate wealth grows at \(r\): the \(pV\) annuity income is a transfer from the dead); \(\ \dot H = (r+p+\alpha)H - Y\).
Key equation	\(\boxed{\dot C = (r+\alpha-\theta)\,C - (p+\alpha)(p+\theta)\,V}\). The last term is generational turnover.
Pure endowment	\(V=0\), \(C=Y\) ⇒ \(r = \theta-\alpha\). Individuals have \(\dot c/c = -\alpha = \dot y/y\), so the equilibrium is autarky.

4. Capital (Lesson 19)

Factor prices	\(r = \mathcal F'(K)\), \(\ Y = \mathcal F(K) - \mathcal F'(K)K\), \(\ V = K\), \(\ rK+Y = \mathcal F(K)\).
System	\(\dot C = [\mathcal F'(K)+\alpha-\theta]C - (p+\alpha)(p+\theta)K\), \(\quad \dot K = \mathcal F(K)-C\).
\(\dot K=0\)	\(C=\mathcal F(K)\): hump-shaped, peak at \(K_{GR}\). \(K\) falls above it.
\(\dot C=0\)	\(C = \dfrac{(p+\alpha)(p+\theta)K}{\mathcal F'(K)+\alpha-\theta}\): from the origin, increasing and convex, with an asymptote at \(\mathcal F'(\hat K)=\theta-\alpha\). \(C\) rises above it.
Steady state	\(\dfrac{\mathcal F(\bar K)}{\bar K} = \dfrac{(p+\alpha)(p+\theta)}{\mathcal F'(\bar K)+\alpha-\theta}\). Unique (falling LHS vs rising RHS). A saddle (\(\det J<0\)). Saddle-path slope \(\mathcal F'(\bar K)-\lambda_s\).
Selection	Off-saddle paths hit \(K=0\) or \(C=0\) in finite time (\(\dot C|_{C=0} = -(p+\alpha)(p+\theta)K<0\)). With \(p=0\), use the TVC instead.
Transition, \(K_0\lt \bar K\)	\(K, C, Y, H\) rise monotonically, and \(r\) falls. \(S(K) = (\theta+p)(K+h(K))\), which is nonlinear.
Comparative statics	\(\bar K\) falls with \(p\) and \(\theta\), and rises with \(\alpha\) (the RHS falls in \(\alpha\) iff \(\mathcal F'(\bar K)\lt \theta+p\), which always holds).
\(\alpha=0\)	\(\theta\lt \mathcal F'(\bar K)\lt \theta+p\) (lower bound from signs; upper from \(\mathcal F(K^{**})>\mathcal F'(K^{**})K^{**}\)). So \(\bar K\lt K_{GR}\): dynamically efficient.
\(\bar r<0\) iff	\((\alpha-\theta)\,\mathcal F(K_{GR})/K_{GR} > (p+\alpha)(p+\theta)\). Needs \(\alpha>\theta\) and small enough \(p\).
\(p\to0\)	Requires \(\alpha=0\). This is Ramsey: the \(\dot C=0\) locus is vertical at \(\mathcal F'(\bar K)=\theta\).

5. Fiscal policy (Lesson 20)

Government	\(\dot B = G-T+rB\). With \(\lim Be^{-\int r}=0\) ⇔ \(B(t) = \int_t^\infty (T-G)e^{-\int_t^z r}dz\).
Ricardian (\(\alpha=0\), PE)	Tax cut \(dT(t)<0\) repaid at \(t+s\): \(\boxed{dH(t) = -dT(t)\,[1-e^{-ps}]}\), \(dC = (p+\theta)dH>0\). Equivalence holds iff \(p=0\). In GE, \(r\) rises (endowment) or \(K\) is crowded out (capital).
Endowment + debt	\(C=Y-G\), \(V=B\): \(\ r = (p+\alpha)(p+\theta)\tfrac{B}{Y-G} + \theta-\alpha\). With \(b=B/(Y-G)\), \(k=(p+\alpha)(p+\theta)\), \(\sigma=(T-G)/(Y-G)\): \(\ \dot b = kb^2+(\theta-\alpha)b-\sigma = r(b)\,b-\sigma\).
\(T=G=0\)	Steady states \(0\) and \(\bar B = \tfrac{(\alpha-\theta)Y}{(p+\alpha)(p+\theta)}\) (positive iff \(\bar r<0\)). Slopes: \(\theta-\alpha\) at 0 and \(\alpha-\theta\) at \(\bar B\), so \(\bar B\) is unstable and has \(r=0\) (rolled-over debt). 2-period analogue: \(\bar B = \beta-\alpha = \) PAYG τ.
\(T\neq G\)	The smaller root is always stable (slope \(-\sqrt{\cdot}\)). Surplus: roots straddle 0, and the positive Ricardian root is unstable. Deficit: sustainable with \(b>0\) only if \(\alpha>\theta\), and only up to \(|\sigma|\le(\theta-\alpha)^2/4k\).
Capital + debt (\(\alpha=0\))	\([\mathcal F'-\theta]C^* = p(p+\theta)(B^*+K^*)\), \(C^*=\mathcal F(K^*)-G\), \(B^*=(T-G)/r^*\). Higher \(T\) ⇒ higher \(B^*\), lower \(K^*\), higher \(r^*\). A \(G\)-increase crowds in via \(B^*\) less than an equal \(T\)-cut, and lowers \(C\).
Balanced budget	\(B\) fixed, \(T=r B+G\): \([\mathcal F'-\theta](\mathcal F-G) = p(p+\theta)(B+K^*)\). Higher \(B\) or \(G\) ⇒ lower \(K^*\), higher \(r^*\). Effects vanish as \(p\to0\) (\(\mathcal F'(K^*)=\theta\)).