# Solving a system of difference equations in an exponential steady-state growth setting

I have a system of difference equations in an exponential steady-state balanced growth setting; all variables grow at the same steady-state rate, say, $g$. I use the notations as follows: Variable $X$ at time $t$ is denoted by $X(t)$. As any variable grows at the rate of $g$, $X(t)$ can be expressed as $X(t)=X(0)e^{tg}$, where $X(0)$ is the initial value of $X$.

Now, my equations system is this:

X(t)=Y(t-a)
Z(t)=X(t-b)
Y(t)=pZ(t-c)


Variables with time notation are endogenous variables: $X$, $Y$, $Z$.

Variables without time notation are exogenous: $a$, $b$, $c$, $p$ and also the initial values of the endogenous variables, $X(0)$, $Y(0)$, $Z(0)$ are exogenous.

You can interpret these equations as saying that, as for the first equation, $X$ at any time is equal to $Y$ of $a$ periods ago.

I would like to do two things using Mathematica:

First, find a solution of the three endogenous variables $X(t)$, $Y(t)$, and $Z(t)$ expressed with exogenous parameters given in the model and with the growth rate $g$.

Second, plugging successively one equation into another yields what can be called 'characteristic equation' of the system. I would like to generate the simplest expression of this characteristic equation. I know it will be consisted of $g$ and the other exogenous parameters. You can verify this by manually solving the model. (In this sense, in this model, $g$ is endogenously determined by the exogenous parameters.)

Any help will be greatly appreciated!

• Can you provide a physical example where these equations systems arise, I may have solved similar systems but need to check the underlying process....not that it will help greatly, but just interested to know. Sep 3 '14 at 1:05
• This system comes from a growth theory in economics. Say, Y is investment, X is produced output, Z is sales revenue.
– ppp
Sep 3 '14 at 1:17
• Investment Y comes out as output X a periods later. Produced output X is sold b periods later thereby generating sales revenue Z. A fraction p of sales revenue Z is invested again as Y
– ppp
Sep 3 '14 at 1:25
• No this system is out of my "zone"...but look forward to good answers to your query Sep 3 '14 at 1:34
• Can you, just as a quick experiment, try this: Assuming[rho > 0, Simplify[RSolve[{x[t] == y[t - 1], z[t] == x[t - 2], y[t] == rho y[t - 3]}, {x[t], y[t], z[t]}, t]]] If you add in the six values for initial conditions of x,y,z then all the C[ i ] should disappear and get something like the exponentials you are describing. If all a,b,c,rho are abstract variables then I don't think this will give you any help.
– Bill
Sep 3 '14 at 5:50