I have an optimal control problem where I have two control and one state variable.

$$max\int_{0}^{\infty}\left(u\left(c\right)-P_{M}M\right)e^{-\rho t}dt\tag{1}$$

where $P_{M}$ is the unit price of carbon abatement activity ($CO_2$ abatement at atmosphere) and $M$ is the control variable, which is the level of abatement that we choose optimally.

The state variable is


where $S$ is the environmental quality, $c$ is consumption, $R(S)$ is the regeneration of environment (like the production function in models with capital accumulation) and $\eta (M)$ is the carbon abatement function, which is supposed to be increasing and concave function. As it is logical, this function contributes to the increase of environmental quality.

I write the Hamiltonian ;


The FOC are ;




So, in this model, I have 2 control variables, consumption $c$ and abatement $M$ and a state variable $S$.

I have a doubt about writing the Jacobian Matrix, If I did not have the second control variable $M$, I would write a 2 dimension differential system of $\dot{c}$ and $\dot{S}$ but as I have a second control variable $M$, I am not sure if I can always describe the whole dynamics of the economy by two differential equations.

When I try to see the dynamics of the control variable $M$, with differentiating equation $(4)$ according to time, I have ;


According to the equation $(6)$ the dynamics of carbon abatement $M$ is represented by the dynamics of costate variable.

In this case, can I represent all dynamics describing this economy by only two differential equations which are $\dot{c}$ and $\dot{S}$ ?

Thanks in advance for suggestions and hints.


1 Answer 1


Differentiating $(3)$ with respect to time we get

$$u_{cc}\dot c = \dot \lambda \implies \frac {u_{cc}}{u_c} \dot c = \frac {\dot \lambda}{\lambda}$$

Inserting into $(6)$ we obtain

$$\dot M = -\frac {\eta_M}{\eta_{MM}}\frac {u_{cc}}{u_c} \dot c$$

So the fixed point of $M$ will happen under the same conditions that the fixed point of $c$ will. Also, the optimal time-evolution of $M$ is a scaled linear value of the optimal evolution of $c$. So it appears all aspects are captured if you "ignore" the $M$ variable.

A bit more formally, since your two decision variables are linearly dependent it means that the $3 \times 3$ Jacobian of the system will be singular at the fixed point. You will get a double eigenvalue and a single one, and you should be able to show what stability properties hold (or what is required for the desired stability notion to hold), in such a situation. Due to the existence of the double root, this requires a bit of a different mathematical approach.

  • $\begingroup$ Thanks for the clear answer. I was also thinking, as you have mentionned, the dynamics of $M$ are governed by the dynamics of consumption. Thanks for the explanation on the singularity at the fixed point. I did not think about that. $\endgroup$ Oct 30, 2015 at 11:14

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.