I have a problem to be maximized with the following program

$$max\int_{0}^{\infty}\left\{ u\left(C\right)+\beta S^{\chi}\right\} e^{-\rho t}dt$$ s.t

$$\dot{K}=AK-C$$ $$\dot{S}=\left(1-S\right)S-\gamma AK$$

where $K$ is physical capital $S$ is the natural capital. All capital letters represent aggregate variables. The economy enjoys the natural capital as a positive amenity and the natural capital is negatively affected by the physical capital accumulation by a rate of $\gamma$.

I am trying to express this program by variables per capita since I would like to have constant steady state values. Note that the population grows at rate $n$. ($L(t)=L(0)e^{nt}$ with $L(0)=1$) It is easy for physical capital.

I will have something as

$$\dot{k}=Ak-nk-c$$

However, for the natural capital, the logistic growth of the natural capital and the amenity $S^{\chi}$ complicate the issue.

Is there a way to express all the maximization program by variables per capita? I would really appreciate if you have some hints to give or some solutions. Or if it is not possible to express by per capita variables, I would be happy to know the reason. Thanks!

Take the differential equation for the natural resource. I will write $S \equiv sL$ where lowercase is the per capita magnitude. So

$$\dot{S}=\left(1-S\right)S-\gamma AK$$

$$\implies \frac {d(sL)}{dt} = (1-sL)sL - \gamma AkL$$

$$\implies \dot sL + s\dot L = sL - s^2L^2 -\gamma AkL$$

Divide by $L$ to get

$$ \dot s + sn = s - s^2L -\gamma Ak$$

and using $L=e^{nt}$ and re-arranging we get

$$ \dot s = (1-n) s - s^2e^{nt} -\gamma Ak$$

Setting this equal to zero, we have a quadratic polynomial in $s$

$$\dot s = 0 \implies e^{nt}s^2 - (1-n) s + \gamma Ak=0$$

We see that the leading term increases as $t$ increases. The only way to maintain equality with zero, and constant $s$, is if $k$ decreases as time passes (and this will be only temporary since $k$ cannot turn negative).

So it appears that you cannot have a "steady state in per capita terms" for all variables here, consumption, capital and the natural resource. Most probably you can maintain consumption and capital at constant steady state values while the per capita, at least, natural resource depletes to extinction (but you have to show that this is optimal with respect to intertemporal utility maximization).

If you assume that population size (N) is 1 and grows at the rate n=0, then would it not solve the problem? Defining, $s = S/N, k = K/N$

$\dot{S} = S -S^2 -\gamma AK \implies \frac{\dot{s}}{s} = 1 - sN - \gamma A\frac{k}{s} \implies \frac{\dot{s}}{s} = 1 - s - \gamma A\frac{k}{s} $

Likewise for the utility function I think.

  • Thanks for this. If the population size is 1 how can it grow at rate $n$? The size will change, no? – optimal control Aug 9 at 17:55
  • Yes. What I mean is that if we assume that N(t)=1, then even if population is growing (beyond 1), N(t) will disappear from your equation of motion. I admit my method is a bit crude. – erik Aug 9 at 17:56
  • I do not understand. N(t) is not a constant, it evolves over time. If it goes beyond one, how can it disappear? – optimal control Aug 9 at 18:03
  • By the way, you can have n = 0. In that case, level variables are already per capita with N(t)=1 – erik Aug 9 at 18:03
  • 1
    Yes I see your point now. Will edit my answer. – erik Aug 9 at 18:05

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