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

<p>$$\dot{S}=\left(1-S\right)S-\gamma AK$$</p>

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

<p>$$\implies \dot sL + s\dot L = sL - s^2L^2 -\gamma AkL$$</p>

<p>Divide by $L$ to get</p>

<p>$$ \dot s + sn = s - s^2L -\gamma Ak$$</p>

<p>and using $L=e^{nt}$ and re-arranging we get</p>

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

<p>Setting this equal to zero, we have a quadratic polynomial in $s$</p>

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

<p>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). </p>

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

CC BY-SA 4.0
2018-08-10T11:51:20.677Z
2018-08-10T11:51:20.677Z
Alecos Papadopoulos
<p>If you assume that population size (N) is 1 and grows at the rate n=0, then would it not solve the problem? Defining, $s = S/N, k = K/N$</p>

<p>$\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} $</p>

<p>Likewise for the utility function I think. </p>

CC BY-SA 4.0
2018-08-09T17:55:01.343Z
Thanks for this. If the population size is 1 how can it grow at rate $n$? The size will change, no?
optimal control
CC BY-SA 4.0
2018-08-09T17:56:33.903Z
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
CC BY-SA 4.0
2018-08-09T18:03:04.750Z
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
CC BY-SA 4.0
2018-08-09T18:03:42.697Z
By the way, you can have n = 0. In that case, level variables are already per capita with N(t)=1
erik
CC BY-SA 4.0
2018-08-09T18:05:20.677Z
If there is no population growth, I agree but in the case where n is not zero, I do not understand your method.
optimal control
CC BY-SA 4.0
2018-08-09T18:05:23.180Z
Yes I see your point now. Will edit my answer.
erik
CC BY-SA 4.0
2018-08-09T17:27:05.613Z
2018-08-09T18:06:08.943Z
2018-08-09T18:06:08.943Z
erik
<p>I have a problem to be maximized with the following program</p>

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

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

<p>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 $\gamma$.</p>

<p>I am trying to express this program by <strong><em>variables per capita</em></strong> 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. </p>

<p>I will have something as </p>

<p>$$\dot{k}=Ak-nk-c$$</p>

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

<p>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! </p>

<economic-growth><environmental-economics>
A maximization program expressed by per capita variales - Ramsey model
optimal control
# A maximization program expressed by per capita variales - Ramsey model

(Asked by: username_0 on 2018-08-09)

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!

**Tags:** economic-growth, environmental-economics

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## 2 answers

### Answer by username_1 on 2018-08-10. Score: 1

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).


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### Answer by username_2 on 2018-08-09. Score: 0

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.

> **Comments:**
> **username_0**: Thanks for this. If the population size is 1 how can it grow at rate $n$? The size will change, no?  
> **username_2**: 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.  
> **username_0**: I do not understand. N(t) is not a constant, it evolves over time. If it goes beyond one, how can it disappear?  
> **username_2**: By the way, you can have n = 0. In that case, level variables are already per capita with N(t)=1  
> **username_0**: If there is no population growth, I agree but in the case where n is not zero, I do not understand your method.  
> **username_2**: Yes I see your point now. Will edit my answer.  

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