<p>Differentiating $(3)$ with respect to time we get</p>

<p>$$u_{cc}\dot c = \dot \lambda \implies \frac {u_{cc}}{u_c} \dot c = \frac {\dot \lambda}{\lambda}$$</p>

<p>Inserting into $(6)$ we obtain</p>

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

<p>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.</p>

<p>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.</p>

CC BY-SA 3.0
2015-10-30T11:14:43.737Z
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.
optimal control
CC BY-SA 3.0
2015-10-29T19:10:46.487Z
2015-10-29T19:10:46.487Z
Alecos Papadopoulos
<p>I have an optimal control problem where I have two control and one state variable.</p>

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

<p>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.</p>

<p>The state variable is </p>

<p>$$\dot{S}=R\left(S\right)-c+\eta\left(M\right)\tag{2}$$</p>

<p>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.</p>

<p>I write the Hamiltonian ;</p>

<p>$$\mathcal{H}=u\left(c\right)-P_{M}M+\lambda\left[R\left(S\right)-c+\eta\left(M\right)\right]$$</p>

<p>The FOC are ;</p>

<p>$$u_{c}=\lambda\tag{3}$$</p>

<p>$$P_{M}=\lambda\eta_{M}\left(M\right)\tag{4}$$</p>

<p>$$\dot{\lambda}=\rho\lambda-\lambda\left(R_{S}\left(S\right)\right)\tag{5}$$</p>

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

<p>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. </p>

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

<p>$$\frac{\dot{\lambda}}{\lambda}+\frac{\eta_{MM}}{\eta_{M}}\dot{M}=0\tag{6}$$</p>

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

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

<p>Thanks in advance for suggestions and hints.</p>

<mathematical-economics><optimal-control>
Stability analysis and dimension of a dynamic control system
optimal control
# Stability analysis and dimension of a dynamic control system

(Asked by: username_0 on 2015-10-22)

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

$$\dot{S}=R\left(S\right)-c+\eta\left(M\right)\tag{2}$$

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 ;

$$\mathcal{H}=u\left(c\right)-P\_{M}M+\lambda\left[R\left(S\right)-c+\eta\left(M\right)\right]$$

The FOC are ;

$$u\_{c}=\lambda\tag{3}$$

$$P\_{M}=\lambda\eta\_{M}\left(M\right)\tag{4}$$

$$\dot{\lambda}=\rho\lambda-\lambda\left(R\_{S}\left(S\right)\right)\tag{5}$$

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 ;

$$\frac{\dot{\lambda}}{\lambda}+\frac{\eta\_{MM}}{\eta\_{M}}\dot{M}=0\tag{6}$$

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.

**Tags:** mathematical-economics, optimal-control

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## 1 answers

### Answer by username_1 on 2015-10-29. Score: 3 | Accepted answer

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.

> **Comments:**
> **username_0**: 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.  

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