8. Perpetual Youth Model

• Consider an endowment economy in a continuous time setting.
• Assume that there is a unit mass of population at any point of time.
• Assume that each individual in the economy, denoted by his birth time \(s\), has life length \(z\) that follows an exponential distribution (i.i.d. across agents) with parameter \(p\), i.e. \(z\sim\exp(p)\) or the p.d.f \(f(\cdot)\) and c.d.f \(F(\cdot)\) satisfies $$f(z)=\begin{cases}p\cdot e^{-pz}&z\ge0\\0&z<0\end{cases}\qquad F(z)=\begin{cases}1-e^{-pz}&z\ge0\\0&z<0\end{cases}$$
• The expected life length is $$\begin{aligned}\mathbb E[z]&=\int_0^\infty z\cdot p\cdot e^{-pz}dz=\underbrace{\left[z\cdot(-e^{-pz})\right]_0^\infty}_{=0-0=0}-\int_0^\infty(-e^{-pz})dz\\&=\int_0^\infty e^{-pz}dz=\left[-\frac1p e^{-pz}\right]_0^\infty=\frac1p\end{aligned}$$
• The probability of being alive at time \(t\ge s\) is \(\mathbb P(z\ge t-s)=1-F(t-s)=e^{-p(t-s)}\). So, since there is a continuum of individuals, by law of large numbers, at time \(t\ge s\) there is \(e^{-p(t-s)}\) proportion of individuals born at \(s\) are still alive.
• \(p\) can be interpreted as the rate of death (hazard rate) by the following argument: the conditional probability of death at \(t\) is $$\frac{\mathbb P(\text{die at }t)}{\mathbb P(\text{still alive before }t)}=\frac{\mathbb P(\text{life length is }t-s)}{\mathbb P(\text{life length not shorter than }t-s)}=\frac{f(t-s)}{1-F(t-s)}=\frac{p\cdot e^{-p(t-s)}}{e^{-p(t-s)}}=p$$ which is the probability of dying in the next moment conditional on being alive now.

• At any moment, there is exactly the same amount of individuals born to replace the dead individuals, which makes the population constant at 1. At each time point \(s\), the number of dying people is \(p\cdot ds\), so the number of newly born individuals is exactly \(p\cdot ds\).
• Verify that the population is constant at 1: $$\begin{aligned}\text{total population at }t&=\int_{-\infty}^t\underbrace{p\cdot ds}_{\text{cohort born at }s}\cdot\underbrace{e^{-p(t-s)}}_{\text{proportion of alive cohort }s}\\&=\int_{-\infty}^t pe^{-p(t-s)}ds=\left[e^{-p(t-s)}\right]_{-\infty}^t=1-0=1\end{aligned}$$

• The risk free interest rate is \(r\).
• Consider a fair bond, which only repays to an agent if he saves previously and is still alive. The accumulated wealth of a 1-unit investment after \(\Delta t\) is \(1+r\Delta t\); the effective (alive) size of population for the repayment after \(\Delta t\) is \(1-p\Delta t\); so the effective net rate of return is \(r+p\) because $$\frac{1+r\Delta t}{1-p\Delta t}=\frac{1-p\Delta t+p\Delta t+r\Delta t}{1-p\Delta t}=1+\frac{(p+r)\Delta t}{1-p\Delta t}\Rightarrow\lim_{\Delta t\to0}\frac{1+r\Delta t}{1-p\Delta t}=1+(p+r)\Delta t$$ which means that the effective rate of return for the alive people is always \(p+r\).
• The monetary discounting factor for the period from \(s\) to \(t\) is denoted by \(R(s,t)\), which is the period \(s\) value of 1 unit of money in period \(t\), i.e. $$R(s,t)=e^{\int_s^t-(r(\mu)+p)d\mu}$$
• Assume that each individual has preferences (for consumption \(c\)) represented by \(u(c(z))\), with a utility discounting factor \(\theta\), i.e. in the continuous model, the discounting factor is \(e^{-\theta t}\approx\frac{1}{1+\theta t}\) for consumption \(t\) length of time later.
• Each individual has an endowment of \(y(z)\) when his life length is \(z\).

8.2 Individual's Maximization Problem

8.2.1 The problem

Every individual (maximizing at time \(s\)) solves the remaining lifetime utility maximization problem: $$\begin{aligned}&\max_{c(t)}\mathbb E\left[\int_s^\infty u(c(t))\cdot e^{-\theta(t-s)}dt\right]\\\Leftrightarrow{}&\max_{c(t)}\int_s^\infty u(c(t))\cdot e^{-\theta(t-s)}\cdot\underbrace{e^{-p(t-s)}}_{=1-F(t-s)}dt\\\Leftrightarrow{}&\max_{c(t)}\int_s^\infty u(c(t))\cdot e^{-(\theta+p)(t-s)}R(s,t)dt\\&\text{s.t. }v(s)=\int_s^\infty(c(t)-y(t))R(s,t)dt\end{aligned}$$ where \(v(s)\) is the non-human wealth (not from endowment) of cohort \(s\) when they were born.

From now on, we will assume a particular utility functional form, i.e. \(u(c)=\ln c\). So, the maximization problem can be rewritten as $$\max_{c(t)}\int_s^\infty\ln c(t)\cdot e^{-(\theta+p)(t-s)}dt\quad\text{s.t.}\quad\left(\int_s^\infty c(t)R(s,t)dt\right)=\underbrace{\left(\int_s^\infty y(t)R(s,t)dt\right)}_{\equiv h(s),\text{ human wealth}}+v(s)\tag{8.1}$$ where \(h(t)\) is defined as human wealth (total discounted endowment income).

8.2.2 The solution

The Lagrangian is $$\mathcal L=\left(\int_s^\infty\ln c(t)\cdot e^{-(\theta+p)(t-s)}dt\right)+\lambda\left[h(s)+v(s)-\left(\int_s^\infty c(t)R(s,t)dt\right)\right]$$ The first-order condition for \(c(t)\) is $$\frac{1}{c(t)}e^{-(\theta+p)(t-s)}=\lambda R(s,t)\Rightarrow c(t)=\frac{e^{-(\theta+p)(t-s)}}{\lambda R(s,t)}\tag{8.2}$$ Plug (8.2) into the budget constraint (8.1): $$\begin{aligned}\left(\int_s^\infty\frac{e^{-(\theta+p)(t-s)}}{\lambda R(s,t)}R(s,t)dt\right)&=h(s)+v(s)\\\Rightarrow\frac1\lambda\int_s^\infty e^{-(\theta+p)(t-s)}dt&=h(s)+v(s)\\\Rightarrow\frac1\lambda\left[-\frac{1}{\theta+p}e^{-(\theta+p)(t-s)}\right]_s^\infty&=h(s)+v(s)\\\Rightarrow\frac1\lambda\frac{1}{\theta+p}&=h(s)+v(s)\\\Rightarrow\lambda&=\frac{1}{(\theta+p)(h(s)+v(s))}\\\Rightarrow c(t)&=(\theta+p)(h(s)+v(s))\frac{e^{-(\theta+p)(t-s)}}{R(s,t)}\tag{8.3}\end{aligned}$$ Evaluated at \(t=s\) (where \(R(s,s)=1\)), $$c(s)=(\theta+p)(h(s)+v(s))\tag{8.4}$$ Since the individual faces exactly the same problem at any point of time (i.e. he will always re-optimize instead of using previous strategy), the \(c(t)\) in (8.3), the period \(t\) consumption decided earlier at period \(s\), is useless. Thus, the optimal consumption at any time \(t\), denoted by \(c^\star(t)\), is given by (8.4) evaluated at \(t\), i.e. $$c^\star(t)=(\theta+p)(h(t)+v(t))\tag{8.5}$$ which is independent of the individual's birth period \(s\).

8.3.1 Define aggregate variables

From the solution in (8.5), we have obtained the optimal consumption for each individual at time \(t\). We can also derive the expression for the social aggregate levels.

• Denote the number of cohort \(s\) (born at \(s\)) alive at \(t\) by \(N(s,t)\). Notice that the total size of cohort \(s\) at birth is \(p\), and the proportion remaining at \(t\) is \(1-F(t-s)=e^{-p(t-s)}\), so $$N(s,t)\equiv p\cdot e^{-p(t-s)}$$
• Denote the human income of cohort \(s\) (born at \(s\)) at \(t\) by \(h(s,t)\). By the Leibniz integral rule, we can obtain the following dynamic equation $$\begin{aligned}h(s,t)&=\int_t^\infty y(s,z)R(t,z)dz\\\Rightarrow\frac{\partial h(s,t)}{\partial t}&=-y(s,t)R(t,t)+\int_t^\infty y(s,z)\frac{\partial}{\partial t}R(t,z)dz\\&=-y(s,t)+\int_t^\infty y(s,z)e^{\int_t^z-(r(\mu)+p)d\mu}(r(t)+p)dz\\&=-y(s,t)+(r(t)+p)\int_t^\infty y(s,z)R(t,z)dz\\\Rightarrow\frac{\partial h(s,t)}{\partial t}&=-y(s,t)+(r(t)+p)h(s,t)\end{aligned}\tag{8.6}$$
• Denote the non-human income of cohort \(s\) (born at \(s\)) at \(t\) by \(v(s,t)\). By the Leibniz integral rule, $$\begin{aligned}v(s,t)&=\int_t^\infty(c(s,z)-y(s,z))R(t,z)dz\\\Rightarrow\frac{\partial v(s,t)}{\partial t}&=-(c(s,t)-y(s,t))\underbrace{R(t,t)}_{=1}+\int_t^\infty(c(s,z)-y(s,z))\frac{\partial}{\partial t}R(t,z)dz\\&=y(s,t)-c(s,t)+(r(t)+p)v(s,t)\end{aligned}\tag{8.7}$$
• Denote the consumption of cohort \(s\) (born at \(s\)) at \(t\) by \(c(s,t)\), which is always optimally chosen \(c(s,t)=c^\star(s,t)\). Based on the results in (8.6) and (8.7), we can obtain the following dynamic equation $$\begin{aligned}c(s,t)&=(\theta+p)(h(s,t)+v(s,t))\\\Rightarrow\frac{\partial c(s,t)}{\partial t}&=(\theta+p)\left(\frac{dh(s,t)}{dt}+\frac{dv(s,t)}{dt}\right)\\&=(\theta+p)\{[-y(s,t)+(r(t)+p)h(s,t)]+[y(s,t)-c(s,t)+(r(t)+p)v(s,t)]\}\\&=(\theta+p)[(r(t)+p)(h(s,t)+v(s,t))-c(s,t)]\\&=(\theta+p)\left[(r(t)+p)\frac{c(s,t)}{\theta+p}-c(s,t)\right]\\\Rightarrow\frac{\partial c(s,t)}{\partial t}&=(r(t)-\theta)c(s,t)\end{aligned}\tag{8.8}$$

• Define the time \(t\) social aggregate human income \(H(t)\) as the sum of human income across all past cohorts up to time \(t\), i.e. \(H(t)=\int_{-\infty}^t h(s,t)N(s,t)ds\). By the Leibniz integral rule, $$\begin{aligned}\frac{dH(t)}{dt}&=h(t,t)N(t,t)+\int_{-\infty}^t\frac{\partial}{\partial t}[h(s,t)N(s,t)]ds\\\Rightarrow\dot H(t)&=h(t,t)\cdot p+\int_{-\infty}^t(-y(s,t)+(r(t)+p)h(s,t))N(s,t)ds-p\int_{-\infty}^t h(s,t)pe^{-p(t-s)}ds\\&=h(t,t)\cdot p-\underbrace{\int_{-\infty}^t y(s,t)N(s,t)ds}_{\equiv Y(t),\text{ total income}}+\int_{-\infty}^t(r(t)+p)h(s,t)N(s,t)ds-pH(t)\\\Rightarrow\dot H(t)&=h(t,t)\cdot p-Y(t)+r(t)H(t)\end{aligned}\tag{8.9}$$ where \(Y(t)\equiv\int_{-\infty}^t y(s,t)N(s,t)ds\) is the total income of all living cohorts at time \(t\).
• Define the time \(t\) social aggregate non-human income \(V(t)=\int_{-\infty}^t v(s,t)N(s,t)ds\). By the Leibniz integral rule (assume that initial non-human wealth is 0), $$\begin{aligned}\frac{dV(t)}{dt}&=\underbrace{v(t,t)}_{=0\text{ by assumption}}N(t,t)+\int_{-\infty}^t\frac{\partial}{\partial t}v(s,t)N(s,t)ds\\\Rightarrow\dot V(t)&=\underbrace{\int_{-\infty}^t y(s,t)N(s,t)ds}_{\equiv Y(t),\text{ total income}}-\underbrace{\int_{-\infty}^t c(s,t)N(s,t)ds}_{\equiv C(t),\text{ total consumption}}+r(t)\int_{-\infty}^t v(s,t)N(s,t)ds\\\Rightarrow\dot V(t)&=Y(t)-C(t)+r(t)V(t)\end{aligned}\tag{8.10}$$

• Define the time \(t\) social aggregate consumption \(C(t)\) as the sum of consumption across all past cohorts up to time \(t\), i.e. \(C(t)=\int_{-\infty}^t c(s,t)N(s,t)ds\). From the solution (8.5) to the individual problem, the optimally selected social consumption level \(C(t)\) satisfies $$\begin{aligned}C(t)&=\int_{-\infty}^t c(s,t)N(s,t)ds=\int_{-\infty}^t(\theta+p)(h(s,t)+v(s,t))N(s,t)ds\\&=(\theta+p)\left(\int_{-\infty}^t h(s,t)N(s,t)ds+\int_{-\infty}^t v(s,t)N(s,t)ds\right)=(\theta+p)(H(t)+V(t))\end{aligned}$$ Based on the results in (8.9) and (8.10), we can obtain the dynamic equation $$\begin{aligned}\dot C(t)&=(\theta+p)(\dot H(t)+\dot V(t))\\&=(\theta+p)[(r(t)+p)\cdots]\\\Rightarrow\dot C(t)&=p(\theta+p)h(t,t)+(r(t)-\theta-p)C(t)\end{aligned}$$

8.3.2 Summarize the system of equations

From the discussion above, we have obtained the following system of equations: - For human wealth: individual \(h(s,t)=\int_t^\infty y(s,z)R(t,z)dz\), \(\dfrac{\partial h(s,t)}{\partial t}=-y(s,t)+(r(t)+p)h(s,t)\); aggregate \(H(t)=\int_{-\infty}^t h(s,t)N(s,t)ds\), \(\dot H(t)=h(t,t)\cdot p-Y(t)+r(t)H(t)\). - For non-human wealth: individual \(v(s,t)=\int_t^\infty(c(s,z)-y(s,z))R(t,z)dz\), \(\dfrac{\partial v(s,t)}{\partial t}=y(s,t)-c(s,t)+(r(t)+p)v(s,t)\); aggregate \(V(t)=\int_{-\infty}^t v(s,t)N(s,t)ds\), \(\dot V(t)=Y(t)-C(t)+r(t)V(t)\). - For consumption: individual \(c(s,t)=(\theta+p)(h(s,t)+v(s,t))\), \(\dfrac{\partial c(s,t)}{\partial t}=(r(t)-\theta)c(s,t)\); aggregate \(C(t)=(\theta+p)(H(t)+V(t))\), \(\dot C(t)=p(\theta+p)h(t,t)+(r(t)-\theta-p)C(t)\).

8.3.3 An example of decaying income

Suppose that the income \(y(s,t)\) is decaying at rate \(\alpha\) with age, then guess that $$y(s,t)=m\cdot Y(t)\cdot e^{-\alpha(t-s)}$$ which is verified by the following: by definition of \(Y(t)\), $$\begin{aligned}Y(t)&=\int_{-\infty}^t y(s,t)N(s,t)ds=\int_{-\infty}^t m\cdot Y(t)\cdot e^{-\alpha(t-s)}p\cdot e^{-p(t-s)}ds\\&=m\cdot Y(t)\cdot p\cdot\left[\frac{1}{p+\alpha}e^{-(p+\alpha)(t-s)}\right]_{-\infty}^t=m\cdot Y(t)\cdot\frac{p}{p+\alpha}\\\Rightarrow m&=\frac{p+\alpha}{p}\\\Rightarrow y(s,t)&=\frac{p+\alpha}{p}\cdot Y(t)\cdot e^{-\alpha(t-s)}\end{aligned}\tag{8.11}$$ Plug (8.11) into the system of equations in subsection 8.3.2 to obtain the aggregate dynamic equations (e.g. for aggregate human wealth \(H(t)=\int_t^\infty Y(z)e^{\int_t^z-(r(\mu)+p+\alpha)d\mu}dz\), \(\dot H(t)=(p+\alpha+r(t))H(t)-Y(t)\); for aggregate consumption \(\dot C(t)=(r(t)+\alpha-\theta)C(t)-(\theta+p)(p+\alpha)V(t)\)).

In this subsection, we will continue to impose the decaying income assumption in subsection 8.3.3, which gives us the following system of equations: - Aggregate non-human wealth: \(\dot V(t)=Y(t)-C(t)+r(t)V(t)\) (8.12). - Aggregate consumption: \(\dot C(t)=(r(t)+\alpha-\theta)C(t)-(\theta+p)(p+\alpha)V(t)\) (8.13).

Use these conditions to consider the following four specific economies:

8.4.1 Pure endowment economy

Assume the aggregate income \(Y(t)=Y\) is constant, and the aggregate consumption is also a constant that equals aggregate income so that there is zero net saving, i.e. \(C(t)=Y(t)=Y\) for \(\forall t\). - Aggregate non-human wealth: \(\dot V(t)=\underbrace{Y(t)-C(t)}_{=0}+r(t)V(t)=r(t)V(t)\Rightarrow\dot V(t)=0\) and \(V(t)=0\) for \(\forall t\) (income and consumption break even at all time, and if \(V(t)\ne0\) will lead to infinite wealth or debt, which cannot be optimal). - The interest rate \(r(t)\) that implements zero aggregate saving: \(0=\dot C(t)=(r(t)+\alpha-\theta)C(t)-(\theta+p)(p+\alpha)\underbrace{V(t)}_{=0}\Rightarrow r(t)=\theta-\alpha\). - Individual consumption: \(\dfrac{\partial c(s,t)}{\partial t}=(r(t)-\theta)c(s,t)\), plug in \(r(t)=\theta-\alpha\) to have \(\dfrac{\partial c(s,t)}{\partial t}=-\alpha c(s,t)\), which is negative. So, consumption is decreasing over time also at rate \(\alpha\) (the same as individual income decaying rate).

8.4.2 Production and capital accumulation economy

Assume that the neoclassical production technology $$\text{Total Aggregate Income}=F\left(K(t),\underbrace{L(t)}_{=1}\right)$$ is available to the economy on aggregate level, i.e. production is implemented on the society level and individuals receive capital income and labor income. Here, \(K(t)\) is the aggregate capital which follows the law of motion $$\dot K(t)=\underbrace{F(K(t),1)}_{\equiv f(K(t))}-\delta K(t)-C(t)$$ where \(\delta\in(0,1)\) is the depreciation rate. Capital is exactly the aggregate non-human wealth, so \(V(t)=K(t)\).

Factor incomes: the production can be conducted by a firm, whose profit maximization problem yields the result that marginal product equals marginal cost (factor price), and all products are split by labor and capital. - Capital: marginal net (of depreciation) product is \(F_K(K(t),1)-\delta\), and marginal cost is \(r(t)\), so \(F_K(K(t),1)-\delta=r(t)\Rightarrow r(t)=f'(K(t))\) (8.14). - Labor: denote wage by \(w(t)\), then \(\underbrace{w(t)\cdot1}_{\text{total labor income}}+\underbrace{r(t)\cdot K(t)}_{\text{total capital income}}=\underbrace{f(K(t))}_{\text{total net product}}\Rightarrow w(t)=f(K(t))-K(t)f'(K(t))\). Notice that \(w(t)\) corresponds to the total aggregate income \(Y(t)\) in the general case.

Substitute the conditions \(V(t)=K(t)\), \(r(t)=f'(K(t))\), \(Y(t)=f(K(t))-K(t)f'(K(t))\) into (8.13) and (8.12): $$\dot C(t)=(f'(K(t))+\alpha-\theta)C(t)-(\theta+p)(p+\alpha)K(t)\tag{8.15}$$ $$\dot K(t)=f(K(t))-K(t)f'(K(t))-C(t)+f'(K(t))K(t)=f(K(t))-C(t)\tag{8.16}$$ Steady state: the two state variables are \(C(t)\) and \(K(t)\). (8.15) and (8.16) only involve \(C(t)\) and \(K(t)\). - \(\dot C(t)=0\): \((f'(K(t))+\alpha-\theta)C(t)=(\theta+p)(p+\alpha)K(t)\Rightarrow C(t)=\dfrac{(\theta+p)(p+\alpha)K(t)}{f'(K(t))+\alpha-\theta}\) (8.17). - \(\dot K(t)=0\): \(C(t)=f(K(t))\) (8.18).

From (8.17) and (8.18), the only solution (steady state capital) \(\bar K\) satisfies $$f(\bar K)=\frac{(\theta+p)(p+\alpha)\bar K}{f'(\bar K)+\alpha-\theta}\tag{8.19}$$

Figure 1 (phase diagram in \((K,C)\) space, paraphrased): the vertical axis is \(C\), the horizontal axis \(K\). The "steady \(c\) locus" \(\dot C(t)=0\) corresponds to (8.17); the "steady \(k\) locus" \(\dot K(t)=0\) corresponds to (8.18) (the production function, hump-shaped). They intersect at the steady state \((\bar K,\bar C)\), with a saddle path converging to it; arrows in the four quadrants indicate the dynamics.

Note that from (8.19), the LHS \(f(\bar K)\) is unchanged in \(p\), \(\alpha\) and \(\theta\), while the RHS of (8.19) is increasing in \(p\), increasing in \(\theta\) and decreasing in \(\alpha\) because $$\begin{aligned}\frac{\partial f(\bar K)}{\partial\alpha}&=\frac{(\theta+p)\bar K(f'(\bar K)+\alpha-\theta)-(\theta+p)(p+\alpha)\bar K}{(f'(\bar K)+\alpha-\theta)^2}\\&=\frac{(\theta+p)\bar K}{(f'(\bar K)+\alpha-\theta)^2}[(f'(\bar K)+\alpha-\theta)-(p+\alpha)]\\&=\underbrace{\frac{(\theta+p)\bar K}{(f'(\bar K)+\alpha-\theta)^2}}_{>0}\underbrace{(f'(\bar K)-\theta-p)}_{\text{assumed to be}<0}\end{aligned}$$ So, the steady state \(\bar K\) is: - decreasing in \(p\): intuitively, higher \(p\) means dying more quickly, so everyone will be more impatient and save less; - decreasing in \(\theta\): intuitively, higher \(\theta\) directly means less patience, so everyone will save less; - increasing in \(\alpha\): intuitively, higher \(\alpha\) means that income decays more quickly, so individuals have stronger incentive to save.

8.4.3 Pure endowment economy with government

Assume that there is a government that charges an aggregate tax \(T(t)\), has expenditure \(G(t)\), and has cumulative government debt \(B(t)\) at time \(t\). Then, the present value of government must be zero to make the government budget balance condition hold at each point \(t\). - Lifetime government budget constraint: \(B(t)=\int_t^\infty(T(z)-G(z))e^{\int_t^z-r(\mu)d\mu}dz\). - Sequential government budget constraint (by Leibniz integral rule): $$\begin{aligned}\dot B(t)&=-(T(t)-G(t))R(t,t)+\int_t^\infty(T(z)-G(z))\frac{\partial}{\partial t}e^{\int_t^z-r(\mu)d\mu}dz\\&=G(t)-T(t)+r(t)\int_t^\infty(T(z)-G(z))e^{\int_t^z-r(\mu)d\mu}dz\\\Rightarrow\dot B(t)&=G(t)-T(t)+r(t)B(t)\end{aligned}\tag{8.20}$$

For simplicity, now assume that \(\alpha=0\), i.e. income is not decaying. Also assume that the sum of consumption and government expenditure is equal to income, i.e. \(C(t)+G(t)=Y(t)\), and government debt is the non-human wealth held of individuals, i.e. \(V(t)=B(t)\). Then, by (8.13) and (8.20), we have the following system of two equations: $$\dot C(t)=(r(t)+\alpha-\theta)C(t)-(\theta+p)(p+\alpha)V(t)\Rightarrow\dot C(t)=(r(t)+\alpha-\theta)(Y(t)-G(t))-(\theta+p)(p+\alpha)B(t)\tag{8.21}$$ $$\dot B(t)=G(t)-T(t)+r(t)B(t)\tag{8.22}$$ Evaluate (8.21) at steady state \(\dot C(t)=0\): $$r(t)=\frac{(\theta+p)(p+\alpha)B(t)}{Y(t)-G(t)}+\theta-\alpha\tag{8.23}$$ Plug (8.23) into (8.22): $$\dot B(t)=\frac{(\theta+p)(p+\alpha)B^2(t)}{Y(t)-G(t)}+(\theta-\alpha)B(t)+G(t)-T(t)$$ which is the law of motion of debt under steady state consumption. To discuss the steady states of debt and the stableness, we now remove the government expenditure and taxation (but keep the debt rolling over), i.e. \(G(t)=T(t)=0\) for \(\forall t\): $$\dot B(t)=\frac{(\theta+p)(p+\alpha)B^2(t)}{Y(t)}+(\theta-\alpha)B(t)$$ The steady state \(\dot B(t)=0\) requires that $$\frac{(\theta+p)(p+\alpha)B^2(t)}{Y(t)}=(\alpha-\theta)B(t)\Rightarrow\bar B_1=0\text{ and }\bar B_2=\frac{(\alpha-\theta)Y(t)}{(\theta+p)(p+\alpha)}$$ where \(\bar B_1\) and \(\bar B_2\) are the two steady state debt levels under steady state consumption. - When \(\theta<\alpha\): \(\bar B_2>\bar B_1=0\). Figure 2 (paraphrased): the \(\dot B\)-vs-\(B\) parabola opening upward, with roots \(\bar B_1=0\) and \(\bar B_2>0\); the arrows show \(\bar B_1=0\) is stable and \(\bar B_2\) is unstable. - When \(\theta>\alpha\): \(\bar B_2<\bar B_1=0\). Figure 3 (paraphrased): the parabola with roots \(\bar B_2<0\) and \(\bar B_1=0\); \(\bar B_2\) is stable and \(\bar B_1=0\) is unstable. (Here "stable" means that any deviation from the steady state will revert and the system will finally return to the steady state.)

8.4.4 Production and capital accumulation economy with government

Assume zero income decaying rate, i.e. \(\alpha=0\). And assume that government debt \(B(t)\) and accumulated capital \(K(t)\) constitutes the aggregate non-human income \(V(t)\), i.e. \(B(t)+K(t)=V(t)\) for \(\forall t\). Finally, assume constant government expenditure and taxation, i.e. \(G(t)=G\) and \(T(t)=T\). Then, we have the following system of equations: - From (8.13), setting \(\alpha=0\) and \(V(t)=B(t)+K(t)\), evaluated at the consumption steady state \(\dot C(t)=0\): $$C(t)=\frac{(\theta+p)p(B(t)+K(t))}{r(t)-\theta}\tag{8.24}$$ - From (8.16) (adding an additional resources usage \(G(t)\) and setting \(G(t)=G\)), evaluated at the capital steady state \(\dot K(t)=0\): $$\dot K(t)=f(K(t))-C(t)-G\Rightarrow f(K(t))=C(t)+G\tag{8.25}$$ - From (8.20) (the derivation of which doesn't involve \(K(t)\), setting \(G(t)=G\) and \(T(t)=T\)), evaluated at the government debt steady state \(\dot B(t)=0\): $$\dot B(t)=r(t)B(t)+G-T\Rightarrow B(t)=\frac{T-G}{r(t)}\tag{8.26}$$ Denote the steady state variables by \(\bar C\), \(\bar K\) and \(\bar B\). The steady state conditions (8.24), (8.25) and (8.26) plus the profit maximization condition (8.14) (i.e. \(r(t)=f'(K(t))\)) give us $$\begin{cases}\bar C(f'(\bar K)-\theta)=(\theta+p)p(\bar B+\bar K)\\f(\bar K)=\bar C+G\\\bar B f'(\bar K)=T-G\end{cases}$$ In this system of three steady state equations, we have three unknowns \(\bar C\), \(\bar K\) and \(\bar B\), so we can explicitly solve them. In particular, we can do the following rearranging to reduce them to one equation for one unknown \(\bar K\) in (8.27) to pin down \(\bar K\): $$f(\bar K)=\bar C+G\Rightarrow f(\bar K)=\frac{(\theta+p)p(\bar B+\bar K)}{f'(\bar K)-\theta}+G\Rightarrow f(\bar K)=\frac{(\theta+p)p\left(\frac{T-G}{f'(\bar K)}+\bar K\right)}{f'(\bar K)-\theta}+G\tag{8.27}$$ Alternatively, if we started by assuming \(G(t)=G\) and \(B(t)=B\), then analogously, we would have the following three steady state conditions $$\begin{cases}\bar C(f'(\bar K)-\theta)=(\theta+p)p(B+\bar K)\\f(\bar K)=\bar C+G\\B f'(\bar K)=\bar T-G\end{cases}$$ where we have three equations with three unknowns \(\bar C\), \(\bar K\) and \(\bar T\), and we can also rearrange to explicitly solve them.