<p>I'm assuming pollution is a bad in consumer utility (as is typically seen) or otherwise relevant to some agent's payoffs -- if it's not, then <span class="math-container">$P_t$</span> is irrelevant and you might as well ignore it.</p>
<p><span class="math-container">$P_t$</span> <strong>is</strong> a state variable: it's something which describes the history of the system, and is necessary to compute utility. See page 117, Definition 3.1 in <a href="http://adp.princeton.edu/Papers/Powell-ClearingtheJungleofStochasticOptimizationOctober2014.pdf" rel="nofollow noreferrer">these notes</a> for a more detailed definition of state variables (&quot;utility function&quot; = &quot;contribution function&quot;), the section has some more interesting discussion as well.</p>
<p>If pollution appeared in utility and you were solving a planner's problem, I think there would be no confusion since you would need <span class="math-container">$P_t$</span> to calculate the decision rule. Where your problem gets a bit different is that the agents don't use it to compute their decision rules. Mechanically, that makes your calculations a bit simpler: optimize agents' objectives over potential values of <span class="math-container">$K_t$</span> only, since the decision rule is going to be &quot;flat&quot; over <span class="math-container">$P_t$</span>.</p>
<p>That an eigenvalue is bigger than one suggests that pollution or capital in this economy will &quot;explode&quot;. But that doesn't bear on whether <span class="math-container">$P_t$</span> is a state variable or not.</p>

CC BY-SA 4.0
2021-01-20T10:13:28.390Z
Thanks for your detailed explanation!!
Benson
CC BY-SA 4.0
2021-01-19T15:26:47.577Z
2021-01-19T15:26:47.577Z
penGuinKeeper
<p>In my OLG model, I have two law of motions for capital and pollution
<span class="math-container">\begin{align*}
&amp;K_{t+1} = s_t\\
&amp;P_{t+1} = (1-\delta)P_t+\rho K_t
\end{align*}</span>
where <span class="math-container">$K$</span> is capital, <span class="math-container">$s$</span> is the saving from the last generation and <span class="math-container">$\rho, \delta$</span> are exogenous parameters. Pollution in my model is just a by-product of the use of capital. Firms and households do not take it into account when making decisions in each period. I know that <span class="math-container">$K$</span> is a predetermined (state) variable, but I am not sure whether pollution (<span class="math-container">$P$</span>) is a state variable. In my numerical simulation, one of the absolute value of the eigenvalues is greater than one and the other one is smaller than one in my two-dimensional system.
Originally I think both <span class="math-container">$K,P$</span> are state variables. Any comments to my misunderstanding will be appreciated.</p>

CC BY-SA 4.0
2021-01-19T09:05:20.907Z
Is there a $K_t$ missing in the first law of motion?
Bayesian
CC BY-SA 4.0
2021-01-19T12:00:25.587Z
@Bayesian I assume that the depreciation rate of capital is 1 for simplicity. In fact, saving is a function of $K_t$. My model follows the standard setting of OLG model.
Benson
<olg>
State variable or jump variable
Benson
# State variable or jump variable

(Asked by: username_0 on 2021-01-19)

In my OLG model, I have two law of motions for capital and pollution
\begin{align\*}
&K\_{t+1} = s\_t\\
&P\_{t+1} = (1-\delta)P\_t+\rho K\_t
\end{align\*}
where $K$ is capital, $s$ is the saving from the last generation and $\rho, \delta$ are exogenous parameters. Pollution in my model is just a by-product of the use of capital. Firms and households do not take it into account when making decisions in each period. I know that $K$ is a predetermined (state) variable, but I am not sure whether pollution ($P$) is a state variable. In my numerical simulation, one of the absolute value of the eigenvalues is greater than one and the other one is smaller than one in my two-dimensional system.
Originally I think both $K,P$ are state variables. Any comments to my misunderstanding will be appreciated.

**Tags:** olg
> **Comments:**
> **username_1**: Is there a $K_t$ missing in the first law of motion?  
> **username_0**: @Bayesian I assume that the depreciation rate of capital is 1 for simplicity. In fact, saving is a function of $K_t$. My model follows the standard setting of OLG model.  

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## 1 answers

### Answer by username_2 on 2021-01-19. Score: 2 | Accepted answer

I'm assuming pollution is a bad in consumer utility (as is typically seen) or otherwise relevant to some agent's payoffs -- if it's not, then $P\_t$ is irrelevant and you might as well ignore it.

$P\_t$ **is** a state variable: it's something which describes the history of the system, and is necessary to compute utility. See page 117, Definition 3.1 in [these notes](http://adp.princeton.edu/Papers/Powell-ClearingtheJungleofStochasticOptimizationOctober2014.pdf) for a more detailed definition of state variables ("utility function" = "contribution function"), the section has some more interesting discussion as well.

If pollution appeared in utility and you were solving a planner's problem, I think there would be no confusion since you would need $P\_t$ to calculate the decision rule. Where your problem gets a bit different is that the agents don't use it to compute their decision rules. Mechanically, that makes your calculations a bit simpler: optimize agents' objectives over potential values of $K\_t$ only, since the decision rule is going to be "flat" over $P\_t$.

That an eigenvalue is bigger than one suggests that pollution or capital in this economy will "explode". But that doesn't bear on whether $P\_t$ is a state variable or not.

> **Comments:**
> **username_0**: Thanks for your detailed explanation!!  

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