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.
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$\begingroup$ Is there a $K_t$ missing in the first law of motion? $\endgroup$– BayesianJan 19 '21 at 9:05
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$\begingroup$ @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. $\endgroup$– BensonJan 19 '21 at 12:00
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 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.
