# Dynamic Optimization : Resource Stock as a Sink

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) 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 ?

• A reference or link to Wirl (1994) would be helpful - he had several papers in 1994: see ideas.repec.org/e/pwi178.html. – Adam Bailey Feb 24 '17 at 18:32
• Sorry, I was wrong for reference indeed. Thanks for the remark. I put the link. – optimal control Feb 24 '17 at 19:41
• 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 Feb 25 '17 at 12:58

"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.

• Thanks Alecos, you are wright. Once I wrote the steady state equilibrium, I saw that a negative $\mu$ does not make sense. – optimal control Mar 2 '17 at 9:29