# Calculating the equilibrium of a time series system

Hi there math experts.

I would like to calculate the equilibrium of two linear equations. However, they're part of a time series, where $a_{-1}$ defines the lagged value of $a$. I don't know how to translate that into to linear equations from which I can calculate the equilibrium.

The system I have is:

$$\log(C)=\log(C_{-1})+0.4\left(\log(Y)-\log(Y_{-1})\right)+0.407\left(\log(Y_{-1}^{0.9}\cdot W_{-1}^{0.1})-\log(C_{-1})\right)\\ W=W_{-1}+Y-C$$ where $C$ and $W$ are to be determined (endogene) and $Y$ is constantly maintained at 0.001.

How do I calculated the steady state of the system?

At equilibrium, $X_{-1}=X$ for every $X$ in $\{C,Y,W\}$, then the equations become $Y^{0.9}W^{0.1}=C$ and $Y=C$. Hence the equilibrium is $$W=C=Y.$$