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


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

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

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


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 ?

  • $\begingroup$ A reference or link to Wirl (1994) would be helpful - he had several papers in 1994: see ideas.repec.org/e/pwi178.html. $\endgroup$ Feb 24, 2017 at 18:32
  • $\begingroup$ Sorry, I was wrong for reference indeed. Thanks for the remark. I put the link. $\endgroup$ Feb 24, 2017 at 19:41
  • $\begingroup$ 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? $\endgroup$ Feb 25, 2017 at 12:58

1 Answer 1


"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.

  • $\begingroup$ Thanks Alecos, you are wright. Once I wrote the steady state equilibrium, I saw that a negative $\mu$ does not make sense. $\endgroup$ Mar 2, 2017 at 9:29

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.