<blockquote>
  <p><em>"Then, is it possible to say that $\mu$ could take a negative value as natural capital represents a cost for capital accumulation ?"</em></p>
</blockquote>

<p>No. $\mu$ can be thought of as the shadow price of natural resources. Being a "price", it has to be non-negative.
Note that if we set $\mu=0$ the problem reverts back to the standard model, which has an intuitive explanation: if "we don't care" about natural resources, their "price" is zero.</p>

<p>To see this from another route, for the standard CRRA utility function</p>

<p>$$u(c) = \frac {c^{1-\theta}-1}{1-\theta}$$</p>

<p>we would get</p>

<p>$$\frac {\dot c}{c} = \frac{1}{\theta}[A-\rho] - \frac{1}{\theta}\frac{\mu}{\lambda}\gamma A$$</p>

<p>So if $\mu &lt;0$ we would obtain a <em>higher</em> consumption growth rate, which doesn't sound very plausible if by  $\mu &lt;0$ we would want to express that "we care" about natural resources and so we would <em>lower</em> capital accumulation that hurts them, and consequently <em>lower</em> the consumption growth rate.</p>

CC BY-SA 3.0
2017-03-02T09:29:46.867Z
Thanks Alecos, you are wright. Once I wrote the steady state equilibrium, I saw that a negative $\mu$ does not make sense.
optimal control
CC BY-SA 3.0
2017-02-25T14:59:52.787Z
2017-02-25T14:59:52.787Z
Alecos Papadopoulos
<p>I think about a dynamic problem where social planner maximizes the following utility ;</p>

<p>$$\underset{c\left(t\right)}{max}\int_{0}^{\infty}u\left(c\left(t\right)\right)e^{-\rho t}$$</p>

<p>subject to two constraints</p>

<p>$$\dot{K}\left(t\right)=AK\left(t\right)-c\left(t\right)$$</p>

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

<p>where $S(t)$ and $K(t)$ hold for natural capital (seas, lakes, forests etc.) and physical capital respectively. $c(t)$ represents consumption.</p>

<p>In this $AK$ model, physical capital generates some wastes which are harmful to natural resource stock with a constant parameter $\gamma$. 
This feature is close to <a href="https://ideas.repec.org/a/eee/dyncon/v28y2004i6p1149-1157.html" rel="nofollow noreferrer">Wirl (2004)</a> in environmental economics literature. </p>

<p>The Hamiltonian of the problem is </p>

<p>$$\mathcal{H}=u\left(c\left(t\right)\right)+\lambda\left(t\right)\left(AK\left(t\right)-c\left(t\right)\right)+\mu\left(t\right)\left(\left(1-S\left(t\right)\right)S\left(t\right)-\gamma AK\left(t\right)\right)$$</p>

<p>$\lambda$ and $\mu$ are co-state variables for physical and natural capital.</p>

<p>Dynamics of co-state variables are </p>

<p>$$\dot{\lambda}=\rho\lambda+\mu\gamma A-\lambda A$$</p>

<p>$$\dot{\mu}=\rho\mu-\mu\left(1-2S\right)$$</p>

<p>In fact, in this model, it is easy to remark that natural capital don't have any <strong><em>amenity value</em></strong> don't provide any positive utility and it is just like a "sink". </p>

<p>As the social planner just takes into account the negative effect of capital accumulation (it creates wastes), I think natural capital enters in this model as a "cost". </p>

<p>Then, is it possible to say that $\mu$ could take a negative value as natural capital represents a cost for capital accumulation ? </p>

CC BY-SA 3.0
2017-02-24T18:32:20.410Z
A reference or link to Wirl (1994) would be helpful - he had several papers  in 1994: see https://ideas.repec.org/e/pwi178.html.
Adam Bailey
CC BY-SA 3.0
2017-02-24T19:41:05.653Z
Sorry, I was wrong for reference indeed. Thanks for the remark. I put the link.
optimal control
CC BY-SA 3.0
2017-02-25T12:58:03.077Z
I think the model needs a bit more specification.  Presumably it's required that $S(t) \geq 0$?  If so, what happens if $S(t)$ approaches zero?  Is $AK(t)$ then constrained so as not to take $S(t)$ below zero?
Adam Bailey
<environmental-economics><economic-growth><dynamic-optimization>
Dynamic Optimization : Resource Stock as a Sink
optimal control
# Dynamic Optimization : Resource Stock as a Sink

(Asked by: username_0 on 2017-02-24)

I think about a dynamic problem where social planner maximizes the following utility ;

$$\underset{c\left(t\right)}{max}\int\_{0}^{\infty}u\left(c\left(t\right)\right)e^{-\rho t}$$

subject to two constraints

$$\dot{K}\left(t\right)=AK\left(t\right)-c\left(t\right)$$

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

where $S(t)$ and $K(t)$ hold for natural capital (seas, lakes, forests etc.) and physical capital respectively. $c(t)$ represents consumption.

In this $AK$ model, physical capital generates some wastes which are harmful to natural resource stock with a constant parameter $\gamma$.
This feature is close to [Wirl (2004)](https://ideas.repec.org/a/eee/dyncon/v28y2004i6p1149-1157.html) in environmental economics literature.

The Hamiltonian of the problem is

$$\mathcal{H}=u\left(c\left(t\right)\right)+\lambda\left(t\right)\left(AK\left(t\right)-c\left(t\right)\right)+\mu\left(t\right)\left(\left(1-S\left(t\right)\right)S\left(t\right)-\gamma AK\left(t\right)\right)$$

$\lambda$ and $\mu$ are co-state variables for physical and natural capital.

Dynamics of co-state variables are

$$\dot{\lambda}=\rho\lambda+\mu\gamma A-\lambda A$$

$$\dot{\mu}=\rho\mu-\mu\left(1-2S\right)$$

In fact, in this model, it is easy to remark that natural capital don't have any ***amenity value*** don't provide any positive utility and it is just like a "sink".

As the social planner just takes into account the negative effect of capital accumulation (it creates wastes), I think natural capital enters in this model as a "cost".

Then, is it possible to say that $\mu$ could take a negative value as natural capital represents a cost for capital accumulation ?

**Tags:** environmental-economics, economic-growth, dynamic-optimization
> **Comments:**
> **username_1**: A reference or link to Wirl (1994) would be helpful - he had several papers  in 1994: see https://ideas.repec.org/e/pwi178.html.  
> **username_0**: Sorry, I was wrong for reference indeed. Thanks for the remark. I put the link.  
> **username_1**: I think the model needs a bit more specification.  Presumably it's required that $S(t) \geq 0$?  If so, what happens if $S(t)$ approaches zero?  Is $AK(t)$ then constrained so as not to take $S(t)$ below zero?  

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## 1 answers

### Answer by username_2 on 2017-02-25. Score: 1 | Accepted answer

> *"Then, is it possible to say that $\mu$ could take a negative value as natural capital represents a cost for capital accumulation ?"*

No. $\mu$ can be thought of as the shadow price of natural resources. Being a "price", it has to be non-negative.
Note that if we set $\mu=0$ the problem reverts back to the standard model, which has an intuitive explanation: if "we don't care" about natural resources, their "price" is zero.

To see this from another route, for the standard CRRA utility function

$$u(c) = \frac {c^{1-\theta}-1}{1-\theta}$$

we would get

$$\frac {\dot c}{c} = \frac{1}{\theta}[A-\rho] - \frac{1}{\theta}\frac{\mu}{\lambda}\gamma A$$

So if $\mu <0$ we would obtain a *higher* consumption growth rate, which doesn't sound very plausible if by $\mu <0$ we would want to express that "we care" about natural resources and so we would *lower* capital accumulation that hurts them, and consequently *lower* the consumption growth rate.

> **Comments:**
> **username_0**: Thanks Alecos, you are wright. Once I wrote the steady state equilibrium, I saw that a negative $\mu$ does not make sense.  

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