State variable or jump variable

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.

• Is there a $K_t$ missing in the first law of motion? Jan 19 '21 at 9:05
• @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. Jan 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.