The Blanchard (1985) Model of Perpetual Youth

This section analyzes a way to relax the standard assumption of infinite

lifetimes in the Ramsey/Cass-Koopmans

growth model. The trick, introduced by

Blanchard Blanchard, 1985, is to assume that the economy

is populated by agents who face a constant probability of death.

Thus, an agent who has lived a thousand years is no more likely to die

in the next year than an agent who was born yesterday.

Time is measured continuously. If there is an instantaneous

probability $\pDies$ of dying, the probability (as viewed by a person

alive in period $t$ ) of still being alive (not dead) in period

$\tThen$ is

\begin{gathered}\begin{aligned} \Alive_{t}^{\tThen} & = e^{-\pDies(\tThen-t)} . \end{aligned}\end{gathered}

(1)

Exercise 1

Assuming the instantaneous utility function is $\uFunc(c) = \log c$ , and that the pure rate of time preference is $\timeRate$ ,

explain why the objective function at time {math}`t` of an individual

household in this model will be to maximize

```{math}
:label: eq:objfunc

\int_{t}^{\infty} (\log c_{\tThen}) e^{-(\timeRate+\pDies)(\tThen-t)} d\tThen.
```

where {math}`c_{\tThen}` is the consumer’s consumption at time {math}`\tThen`. For convenience,

you may wish to define

```{math}
:label: eq:thetahat

\begin{gathered}\begin{aligned}
        \hat{\timeRate} & =  \timeRate+\pDies. 
\end{aligned}\end{gathered}
```

Solution to Exercise 1

>
> A consumer with a pure time preference rate of zero will downweight
>
> the utility he receives conditional on being alive by the probability
>
> that he is still alive, yielding a discounted utility of
>
> ```{math}
> \begin{gathered}\begin{aligned}
>         \int_{t}^{\infty} (\log c_{\tThen}) \Alive_{t}^{\tThen} e^{-\timeRate (\tThen-t)}d\tThen & =  \int_{t}^{\infty} (\log c_{\tThen}) e^{-\timeRate (\tThen-t)} e^{-\pDies (\tThen-t)} \nonumber d\tThen \\
>          & =    \int_{t}^{\infty} (\log c_{\tThen}) e^{-(\timeRate+\pDies) (\tThen-t)} d\tThen
> \\       & =    \int_{t}^{\infty} (\log c_{\tThen}) e^{-\hat{\timeRate} (\tThen-t)} d\tThen
> \end{aligned}\end{gathered}
> ```
>
> where {math}`\hat{\timeRate} = \timeRate+\pDies`.
>
> A similar point holds in the discrete time model. A sensible thing to
>
> assume is that if you have died before {math}`t+1`, you get zero utility in
>
> {math}`t+1` and after. Thus, in the two-period context, if the probability
>
> of death between {math}`T-1` and {math}`T` was zero we would have
>
> ```{math}
> V_{T-1} = \max \uFunc(C_{T-1}) + \left(\frac{1}{1+\timeRate}\right) \uFunc(C_{T})
> ```
>
> while if there is a probability {math}`\pDies` of dying between {math}`T` and {math}`T+1`
>
> value would be:
>
> ```{math}
> :label: eq:approx
>
> \begin{gathered}\begin{aligned}
>    V_{T-1} & =  \max \uFunc(C_{T-1}) + (1-\pDies)\left(\frac{1}{1+\timeRate}\right) \uFunc(C_{T}) + \pDies*\beta*0 \nonumber 
> \\   & =  \max \uFunc(C_{T-1}) + \left(\frac{1-\pDies}{1+\timeRate}\right) \uFunc(C_{T}) \nonumber
> \\   & \approx  \max \uFunc(C_{T-1}) + \left(\frac{1}{1+\timeRate+\pDies}\right) \uFunc(C_{T})  
> \end{aligned}\end{gathered}
> ```
>
> But behavior in this case is virtually indistinguishable from the
>
> behavior that would be induced if the consumer had a time preference
>
> rate of {math}`\timeRate+\pDies`. In continuous time, the approximation in
>
> {eq}`eq:approx` becomes exact.

If the probability of death is constant, the expected remaining life

for an agent of any age is given by

```{math}
\int_{0}^{\infty} \pDies \tau e^{-\pDies \tau}d\tau = \pDies^{-1}.
```

which we will call the agent’s ‘horizon.’ For example, if the chances

of dying per year are {math}`\pDies=1/50`, then the agent’s horizon is 50 years.

We will assume that at every instant of time, a large cohort, whose size

is normalized to be {math}`\pDies`, is born.

Exercise 2

For an economy that has existed forever, explain why the

formula for the aggregate population at time {math}`t`, {math}`P_{t}`, will be

```{math}
:label: eq:totpop

\begin{gathered}\begin{aligned}
        P_{t} & =  \pDies \int_{-\infty}^{t} \Alive_{s}^{\tNow} ds 
\end{aligned}\end{gathered}
```

and show that this formula implies that the population is {math}`P_{t}=1`.

Solution to Exercise 2

>
> The aggregate population will be the sum of the still-alive persons
>
> from all past generations.
>
> The proportion of a population born at time {math}`s` that is still living
>
> at time {math}`\tThen` is {math}`\Alive_{s}^{\tThen}` by equation {eq}`eq:living`.
>
> Thus, the absolute size at time {math}`\tThen` of a cohort of size {math}`\pDies` born at
>
> time {math}`s` is {math}`\pDies \Alive_{s}^{\tThen} = \pDies e^{-\pDies (\tThen-s)}`. The economy’s
>
> total population will be the sum of the populations of all the cohorts
>
> that are currently living. Since the economy has existed forever,
>
> there will be remaining members of every cohort back to {math}`s=-\infty`.
>
> Thus, indexing each cohort by a time index {math}`s`, {eq}`eq:totpop` is
>
> simply the sum of the populations of all currently living members of
>
> every generation.
>
> Substituting the formula for {math}`\Alive`, the integral becomes
>
> ```{math}
> :label: eq:fmhint
>
> \begin{gathered}\begin{aligned}
>         P_{t} & =  \pDies \int_{-\infty}^{t} e^{-\pDies(t-s)} ds 
> \\        & =  \pDies \int_{t}^{\infty} e^{-\pDies(s-t)} ds 
> \\        & =  \pDies \int_{0}^{\infty} e^{-\pDies \tThen}d\tThen
> \\        & =  \pDies \pDies^{-1}  
> \\        & =  1 
> \end{aligned}\end{gathered}
> ```
>
> where the second line comes from a change of variables {math}`\tThen = s-t`
>
> and {eq}`eq:fmhint` follows from the hint.

Now some notation. We will define variable $x(s,t)$ as the value at

date $t$ of the variable $x$ for a consumer who was born at date $s$ .

Thus, $c(s,t)$ is consumption at $t$ of a consumer born at $s$ .

Suppose that the consumers in this economy do not have a bequest

motive. If they have positive assets at the instant when they die,

they are no happier than if they had zero assets. This means that if

someone were willing to pay them something while they are still alive

for the right to inherit their assets whenever they die, these consumers would

happily take that deal.

For a consumer with wealth $w(s,t)$ who has probability of dying $\pDies$ ,

the flow value of the right to inherit that wealth is $\pDies w(s,t)$ . We

will therefore assume that insurance companies exist that pay a

consumer with wealth $w(s,t)$ an amount $\pDies w(s,t)$ in exchange for the

right to receive that consumer’s wealth when he dies. (The insurance

company will make zero profits).

But notice that from the standpoint of the consumer, this is

equivalent to saying that the interest rate received on wealth is

higher by amount $\pDies$ .

Now suppose the marginal product of capital in this

perfectly-competitive economy is constant at $\rfree$ and suppose an agent

born in $s$ receives exogenous labor income in period $t$ of $y(s,t)$ .

This plus the insurance scheme implies that the agent’s dynamic budget

constraint is given by

\begin{gathered}\begin{aligned} \dot{w}(s,t) & = (\rfree+\pDies) w(s,t) + y(s,t)-c(s,t) . \end{aligned}\end{gathered}

(2)

Define the ‘effective’ interest rate as viewed by a consumer as $\hat{\rfree} = \rfree+\pDies$ .

Exercise 3

Write the current-value Hamiltonian for the

individual’s maximization problem and use it to show that the growth

rate of consumption in period {math}`t` for a consumer born at time {math}`s` is

given by

```{math}
\begin{gathered}\begin{aligned}
        \left(\frac{\dot{c}(s,t)}{c(s,t)}\right) & =  \hat{\rfree} - \hat{\timeRate}
\\  & =  \rfree-\timeRate.     
\end{aligned}\end{gathered}
```

Solution to Exercise 3

>
> The current-value Hamiltonian is written
>
> ```{math}
> \Ham(c,w,\lambda) = \log c(s,t) + \lambda(\hat{\rfree} w(s,t) + y(s,t) - c(s,t))
> ```
>
> so the first optimality condition {math}`\partial \Ham/\partial c(s,t) = 0` implies
>
> ```{math}
> \begin{gathered}\begin{aligned}
>         1/c(s,t) & =  \lambda
> \\  - \dot{c}(s,t)/c(s,t)^{2} & =  \dot{\lambda}       
> \end{aligned}\end{gathered}
> ```
>
> and the second optimization condtion implies
>
> ```{math}
> \begin{gathered}\begin{aligned}
>         \dot{\lambda} & =  \hat{\timeRate} \lambda - \partial \Ham/\partial w(s,t)  
> \\       & =  \hat{\timeRate} \lambda - \hat{\rfree} \lambda  
> \\      \left(\frac{\dot{\lambda}}{\lambda}\right) & =  \hat{\timeRate}-\hat{\rfree}
> \\  \left(\frac{-\dot{c}(s,t) c(s,t)}{c(s,t)^{2}}\right) & =  \hat{\timeRate}-\hat{\rfree}
> \\  \left(\frac{\dot{c}(s,t)}{c(s,t)}\right) & =  \hat{\rfree}-\hat{\timeRate} 
> \\   & =  \rfree-\timeRate     .
> \end{aligned}\end{gathered}
> ```

.

Exercise 4

Use the first order condition for consumption and the

intertemporal budget constraint implied by {eq}`eq:budconstr` to

show that the level of consumption in time {math}`t` for an individual born

at time {math}`s` is

```{math}
c(s,t) = \hat{\timeRate}(w(s,t)+h(s,t))
```

where recall that {math}`\hat{\timeRate}=\timeRate+\pDies` and human wealth is

```{math}
h(s,t) = \int_{t}^{\infty} \left(y(s,\tThen)/\hat{\mathcal{R}}_{t}^{\tThen}\right) d\tThen
```

where

```{math}
\begin{gathered}\begin{aligned}
  \hat{\mathcal{R}}_{t}^{\tThen} & =  e^{\int_{t}^{\tThen} (\rfree_{\mu}+\pDies) d\mu}
\end{aligned}\end{gathered}
```

(this is simply the compound discount factor necessary to take

account of time-varying interest rates; if interest rates are

constant at {math}`\rfree` it reduces to the usual term {math}`e^{-\rfree(\tThen-t)}`).

Solution to Exercise 4

>
> The IBC says that the PDV of consumption must equal wealth plus the PDV
>
> of future labor income:
>
> ```{math}
> \begin{gathered}\begin{aligned}
>         \int_{t}^{\infty} (c(s,\tThen)/\hat{\mathcal{R}}_{t}^{\tThen})  d\tThen & =  w(s,t) + \int_{t}^{\infty} y(\tThen,t)/\hat{\mathcal{R}}_{t}^{\tThen} d\tThen
> \\      \int_{t}^{\infty} (c(s,\tThen)/\hat{\mathcal{R}}_{t}^{\tThen}) d\tThen & =  w(s,t) + h(s,t).
> \end{aligned}\end{gathered}
> ```
>
> But if {math}`\dot{c}(s,t)/c(s,t) = \hat{\rfree}_{t}-\hat{\timeRate}` then
>
> ```{math}
> \begin{gathered}\begin{aligned}
>         c(s,\tThen) & =  c(s,t)e^{\int_{s}^{\tThen}\hat{\rfree}_\mu d\mu}e^{-\hat{\timeRate}(\tThen-t)}
> \end{aligned}\end{gathered}
> ```
>
> implying
>
> ```{math}
> \begin{gathered}\begin{aligned}
>         \int_{t}^{\infty} (c(s,\tThen)/\hat{\mathcal{R}}_{t}^{\tThen}) d\tThen & =  
> \int_{t}^{\infty} c(s,t) e^{\int_{s}^{\tThen}\hat{\rfree}_\mu  d\mu}e^{-\hat{\timeRate}(\tThen-t)}e^{-\int_{s}^{\tThen}\hat{\rfree}_\mu d\mu} d\tThen \nonumber
> \\         & =  c(s,t) \int_{t}^{\infty} e^{-\hat{\timeRate}(\tThen-t)} d\tThen
> \\   & =  c(s,t)/\hat{\timeRate}
> \end{aligned}\end{gathered}
> ```
>
> so that the IBC becomes:
>
> ```{math}
> \begin{gathered}\begin{aligned}
>         c(s,t)/\hat{\timeRate} & =  w(s,t)+h(s,t)  \\
>         c(s,t) & =  \hat{\timeRate} (w(s,t)+h(s,t)) .
> \end{aligned}\end{gathered}
> ```

Suppose we define upper-case variables as the aggregate value across all

generations currently living of the corresponding lower-case value, e.g.

aggregate consumption is

```{math}
\begin{gathered}\begin{aligned}
        C(t) & =  \int_{-\infty}^{t} \pDies c(\tThen,t) \Alive_{\tThen}^{t} d\tThen.
\end{aligned}\end{gathered}
```

Exercise 5

(Easy) Show that the aggregate level of consumption in

this economy is

```{math}
:label: eq:cagg

\begin{gathered}\begin{aligned}
        C(t) & =  \hat{\timeRate} (W(t) + H(t)) . 
\end{aligned}\end{gathered}
```

Solution to Exercise 5

\begin{gathered}\begin{aligned} C(t) & = \int_{-\infty}^{t} \pDies c(\tThen,t) \Alive_{\tThen}^{t} d\tThen \\ & = \int_{-\infty}^{t} \pDies \hat{\timeRate}(w(\tThen,t)+h(\tThen,t)) \Alive_{\tThen}^{t} d\tThen \\ & = \hat{\timeRate} (W(t)+H(t)) \end{aligned}\end{gathered}

(3)

from the definition of $W(t)$ and $H(t)$ .

Suppose all living agents in this economy receive the same noncapital

income, $y(s,t) = Y(t)$ . Since every member of the population has the

same income, and the size of the population is one, aggregate income

will also be $Y(t)$ . Aggregate human wealth at $t$ is therefore

\begin{gathered}\begin{aligned} H(t) & = \int_{t}^{\infty} (Y(\tThen)/\hat{\mathcal{R}}_{t}^{\tThen}) d\tThen \end{aligned}\end{gathered}

(4)

where

\begin{gathered}\begin{aligned} \dot{H}(t) & = \hat{\rfree}_{t} H(t) - Y(t) . \end{aligned}\end{gathered}

(5)

The differential equation for aggregate wealth can be

shown to be

\begin{gathered}\begin{aligned} \dot{W}(t) & = w(t,t) - \pDies W(t) + \int_{-\infty}^{t} \dot{w}(\tThen,t) \pDies e^{-\pDies(t-\tThen)} d\tThen, \end{aligned}\end{gathered}

(6)

where $w(t,t)=0$ is the wealth of newly born generations, $\pDies W(t)$ is

the wealth of those who are dying at the moment, and the last term is

the change in wealth for those who neither die nor are born in this

period. But (2) implies that

\begin{gathered}\begin{aligned} \int_{-\infty}^{t} \dot{w}(\tThen,t) \pDies e^{-\pDies(t-\tThen)} d\tThen & = (\rfree+\pDies) W(t) + Y(t)-C(t) \end{aligned}\end{gathered}

(7)

so (6) becomes

\begin{gathered}\begin{aligned} \dot{W}(t) & = \rfree W(t) + Y(t)-C(t). \end{aligned}\end{gathered}

(8)

Collecting, writing out $\hat{\rfree}=\rfree+\pDies$ and $\hat{\timeRate}=\timeRate+\pDies$ , and

dropping the $(t)$ arguments gives us the following equations for

aggregate variables:

\begin{gathered}\begin{aligned} C & = (\pDies+\timeRate) (H+W) \\ \dot{H} & = (\rfree+\pDies) H - Y \\ \dot{W} & = \rfree W + Y - C % \end{aligned}\end{gathered}

(9)

Use these equations to show that in this economy

\begin{gathered}\begin{aligned} \dot{C} & = (\rfree-\timeRate) C - \pDies(\pDies+\timeRate) W \end{aligned}\end{gathered}

(10)

Hint: Differentiate (9) and substitute out for $H$ by solving

(9) for $H$ .

Answer:
Time differentiate (9) and substitute for $\dot{H}$ ,
$\dot{W}$ , and $H$ to get
$\begin{gathered}\begin{aligned} \dot{C} & = (\pDies+\timeRate)\left[(\rfree+\pDies) H - Y + \rfree W + Y - C\right] \\ & = (\pDies+\timeRate)\left[(\rfree+\pDies) H + \rfree W \right] - C (\pDies+\timeRate) \\ & = (\pDies+\timeRate)\left[(\rfree+\pDies) \left(\frac{C}{\pDies+\timeRate}-W\right)+r W\right] - C (\pDies+\timeRate) \\ & = (\rfree+\pDies) C + (\pDies+\timeRate)\left[ rW - (\rfree+\pDies) W \right] - C(\pDies+\timeRate) \\ & = (\rfree-\timeRate) C - (\pDies+\timeRate)\pDies W. \end{aligned}\end{gathered}$
(11)

Now assume there is a standard production function

$\FFunc(K)=K^{\alpha}-\delta K$ and assume perfect competition so that the

net interest rate $r$ is equal to the net marginal product of capital,

\rfree = \FFunc^{\prime}(K) = \alpha K^{\alpha-1}-\delta

(12)

and the aggregate capital stock at time $t$ is the same as aggregate

nonhuman wealth, $K(t)=W(t)$ . The aggregate accumulation equation is

just the usual

\begin{gathered}\begin{aligned} \dot{K} & = K^{\alpha}-\delta K -C. \end{aligned}\end{gathered}

(13)

Exercise 6

Show that the following equation

\begin{gathered}\begin{aligned} \pDies(\pDies+\timeRate) K & = C(\alpha K^{\alpha-1}-\delta-\timeRate) \end{aligned}\end{gathered}

(14)

describes the $\dot{C}=0$ locus. Use this equation to show that

\begin{gathered}\begin{aligned} \lim_{C \rightarrow 0} K & = 0 \\ \lim_{C \rightarrow \infty} K & = ((\timeRate+\delta)/\alpha)^{1/{\alpha-1}}. \end{aligned}\end{gathered}

(15)

Solution to Exercise 6

Answer:
Rewriting the $\dot{C}$ equation as a function of $K$ yields
$\begin{gathered}\begin{aligned} \dot{C} & = (\alpha K^{\alpha-1} - \timeRate - \delta)C - (\pDies+\timeRate)\pDies K. \end{aligned}\end{gathered}$
(16)
The $\dot{C}=0$ locus is therefore given by
$\begin{gathered}\begin{aligned} 0 & = (\alpha K^{\alpha-1}-\timeRate-\delta) - \pDies(\pDies+\timeRate) K/C \\ \pDies(\pDies+\timeRate) K/C & = (\alpha K^{\alpha-1}-\delta-\timeRate) \\ \pDies(\pDies+\timeRate) K & = C(\alpha K^{\alpha-1}-\delta-\timeRate) \end{aligned}\end{gathered}$
(17)
In the limit as $C \rightarrow 0$ this expression approaches
$\pDies(\pDies+\timeRate) K = 0$
(18)
which can be true only if $\lim_{\{C \rightarrow 0\}} K = 0$ .
On the other hand, as $C \rightarrow \infty$ we know that $K$
must remain finite (because the DBC does not allow infinite
accumulation of $K$ – infinite $K$ would imply infinite depreciation which
could never be paid for by a production function with diminishing marginal returns), which means that
$\begin{gathered}\begin{aligned} \lim_{\{C \rightarrow \infty\}} \alpha K^{\alpha-1} & = (\timeRate+\delta) \\ K^{*} & = ((\timeRate+\delta)/\alpha)^{1/{\alpha-1}}. \end{aligned}\end{gathered}$
(19)

Solution to Exercise 7

Answer:
Using (13), the $\dot{K}=0$ locus is
$\begin{gathered}\begin{aligned} C & = K^{\alpha}-\delta K \end{aligned}\end{gathered}$
(20)
which yields the usual hump-shaped $\dot{K}=0$ locus.
Rewriting the $\dot{C}$ equation as a function of $K$ yields
$\begin{gathered}\begin{aligned} \dot{C} & = (\alpha K^{\alpha-1} - \timeRate - \delta)C - (\pDies+\timeRate)\pDies K. \end{aligned}\end{gathered}$
(21)
The $\dot{C}=0$ locus is therefore given by
$\begin{gathered}\begin{aligned} 0 & = (\alpha K^{\alpha-1}-\timeRate-\delta) - \pDies(\pDies+\timeRate) K/C \\ \pDies(\pDies+\timeRate) K/C & = (\alpha K^{\alpha-1}-\delta-\timeRate) \\ \pDies(\pDies+\timeRate) K & = C(\alpha K^{\alpha-1}-\delta-\timeRate) \end{aligned}\end{gathered}$
(22)
In the limit as $C \rightarrow 0$ this expression approaches
$\pDies(\pDies+\timeRate) K = 0$
(23)
which can be true only if $\lim_{\{C \rightarrow 0\}} K = 0$ .
On the other hand, as $C \rightarrow \infty$ we know that $K$
must remain finite (because the DBC does not allow infinite
accumulation of $K$ ), which means that
$\begin{gathered}\begin{aligned} \lim_{\{C \rightarrow \infty\}} \alpha K^{\alpha-1} & = (\timeRate+\delta) \\ K^{*} & = ((\timeRate+\delta)/\alpha)^{1/{(\alpha-1)}}. \end{aligned}\end{gathered}$
(24)
In the infinite horizon economy we have $\pDies = 0$ and so the
steady-state interest rate would be the $K^{*}$ where $\alpha > K^{\alpha-1}-\delta = \timeRate$ . But since $K/C$ and $\pDies(\pDies+\timeRate)$ are
strictly positive, in this finite-horizon economy we would have
$\dot{C}/C < 0$ at $K=K^{*}$ . It is clear therefore that in order
for (17) to hold we will need $\alpha K^{\alpha-1}$
to be larger than it is at $K^{*}$ , which is to say we need a higher
steady-state interest rate, and thus we need a lower steady-state
capital stock, which is depicted in the figure as $\bar{K}$ .
This makes sense because the finite-horizon consumers in this economy
discount the future more than the representative agent does, because
they die but a representative agent does not.
These results are combined in the figure, which shows that the intersection
of the $\dot{C}=0$ locus intersects the $\dot{K}=0$ locus at point $A$
which corresponds to a lower level of the capital stock than in the
infinite horizon model.

Now consider the introduction of a government that finances spending

either by lump-sum taxes or by debt. Its dynamic budget constraint is

\dot{D} = \rfree D + G - \TaxLev

(25)

where $D$ is government debt, $G$ is government spending, and

$\TaxLev$ is a lump-sum per capita tax. Defining

\begin{gathered}\begin{aligned} \mathcal{R}_t^{s} = e^{\int_t^{s}\rfree_v dv} \end{aligned}\end{gathered}

(26)

as the compound interest factor between time $t$ and time $s$ , the

government is also required to satisfy the transversality condition

\lim_{t \rightarrow \infty} D_t/\mathcal{R}_t^{s} = 0.

(27)

Consider the following fiscal policy experiment. Until time $t$ there

has been no government ( $G_s = D_s = \TaxLev_s=0~\forall~s<t$ ). At date

$t$ the government issues a quantity $D$ of debt and announces that

future lump sum taxes will be imposed in amounts exactly large enough

to pay the interest on this debt (so subsequently, $\dot{D}=0$

forever). The government rebates the proceeds of its sale of debt to

the public as a per-capita lump sum of $D$ per person. The government

will never engage in any spending (aside from paying interest on the

debt). Define the new variables

\begin{gathered}\begin{aligned} \mathcal{W} & = K + D \\ \mathcal{Y} & = Y - \TaxLev \\ \mathcal{H} & = \int_{t}^{\infty} \mathcal{Y}/\hat{\mathcal{R}}_{t}^{s} ds \end{aligned}\end{gathered}

(28)

Solution to Exercise 8

Answer:
The effect of the government policy is twofold. On the one hand,
the distribution of government bonds increases the consumers’
wealth $W$ by an amount equal to the value of the bonds received,
resulting in a new definition of wealth $\mathcal{W}$ which
includes the bonds. On the other hand, the higher value of taxes
off to infinity reduces the consumers’ human wealth by an amount
equal to the present discounted value of the taxes.
The change in (redefined) wealth is now net income $Y-\timeRate$
minus consumption.

Solution to Exercise 9

Answer:
Time differentiating (29) yields
$\begin{gathered}\begin{aligned} \dot{C} & = (\pDies+\timeRate) (\dot{\mathcal{H}} + \dot{\mathcal{W}}) \\ & = (\pDies+\timeRate)(\rfree \mathcal{W}+(\rfree+\pDies)\mathcal{H}-C) \end{aligned}\end{gathered}$
(31)
Now solve (29) for $\mathcal{H}$ ,
$\begin{gathered}\begin{aligned} \mathcal{H} & = \left(\frac{C}{\pDies+ \timeRate}\right)-\mathcal{W} \end{aligned}\end{gathered}$
(32)
and substitute into (31) to obtain
$\begin{gathered}\begin{aligned} \dot{C} & = (\pDies+\timeRate)(\rfree \mathcal{W}+(\rfree+\pDies)(C(\pDies+\timeRate)^{-1} - \mathcal{W})-C) \\ & = (\rfree+\pDies)C - (\pDies+\timeRate)C- \pDies(\pDies+\timeRate)\mathcal{W} \\ & = (\rfree-\timeRate)C - \pDies(\pDies+\timeRate)(K+D). \end{aligned}\end{gathered}$
(33)
Since we are assuming that $D$ is a constant (after the fiscal
experiment), any combination of $C$ and $W$ that would have been on
the $\dot{C}=0$ locus before the policy shift now has a value $\dot{C} = -\pDies(\pDies+\timeRate)D$ . This means that the $C$ that would restore
$\dot{C}=0$ must be a larger $C$ , which says that the $\dot{C}=0$
locus shifts up (or, equivalently, to the left). Thus, the new
equilibrium will be at a lower value of $K$ and a higher interest
rate.
The phase diagram shows that the new equilibrium point $A'$ is to the left
of the original equilibrium. This is because at a given level of the aggregate
capital stock, consumers spend more because $\mathcal{W}>W$ . Thus,
Ricardian equivalence does not hold in this model, because a tax cut today
financed by a future perpetual tax is a transfer of resources from future
consumers to today’s consumers, and there are no altruistic links that
make current consumers offset this by saving more on behalf of future
generations.
The next figure shows the path of consumption per capita in this
economy. Prior to time 0, the economy was in its steady-state
equilibrium at the level of consumption $C_0$ corresponding to the
equilibrium labeled $A$ in the phase diagram. At time 0, the fiscal
policy is carried out. The fiscal policy immediately increases
consumption because it amounts to a transfer of resources from future
to current generations. However, the higher level of consumption runs
down the capital stock per capita, and so over time consumption asypmtotically
approaches a new, lower equilibrium level of consumption $C'$ .

Blanchard shows that if the model is changed so that each

consumer’s income declines exponentially at rate $\gamma$ after

birth, the result is equivalent to assuming that future labor income

is discounted at an interest rate that is higher by $\gamma$ . He

further shows that the equations of motion of the model change to

\begin{gathered}\begin{aligned} \dot{C} & = (\alpha K^{\alpha-1}-\delta +\gamma-\timeRate)C-(\pDies+\gamma)(\pDies+\timeRate)K \\ \dot{K} & = K^{\alpha}-\delta K - C. % \end{aligned}\end{gathered}

(34)

Note that it is possible to rewrite the $\dot{C}=0$ locus as

\begin{gathered}\begin{aligned} (\alpha K^{\alpha-1}-\delta +\gamma-\timeRate)C & = (\pDies+\gamma)(\pDies+\timeRate)K \end{aligned}\end{gathered}

(35)

Solution to Exercise 10

Answer:
As $C$ goes to infinity on the LHS of (35), the only way
the equation can continue to hold is if
$\begin{gathered}\begin{aligned} \lim_{C \rightarrow \infty} (\alpha K^{\alpha-1}-\delta +\gamma-\timeRate) & = 0 \end{aligned}\end{gathered}$
(37)
which implies (36).
The new phase diagram shows the $\dot{C}=0$ locus intersecting the
$\dot{K}=0$ locus to the right of the maximum of the $\dot{K}=0$ locus.
This is implied by the fact that the net interest rate is negative, which
means that the net interest rate could be increased by reducing
the capital stock.
The reason this is an interesting case is that in this case it is
possible for the economy to be in a condition of dynamic inefficiency,
just as in the 2-period OLG models discussed early in the class. The idea
is to think of declining labor income as a way to generate a ‘life cycle’
saving motive. In such a case the fiscal experiment examined above
is interesting because it could rescue an economy with too much capital
from a state of dynamic inefficiency.

References¶

Blanchard, O. J. (1985). Debt, Deficits, and Finite Horizons. Journal of Political Economy, 93(2), 223–247. 10.1086/261297