How to elegantly transform a model in a recursive equation

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I would like to know if there is an elegant way to develop a model as :

$$ \begin{array}{l} C_t=cY_{t-1}+C_0,\ I_t= I_0 + \gamma (C_t - C_{t-1}), \ Y_t=C_t+I_t \end{array} $$

to obtain a resursive equation in $Y_t$ which can be put in a natural format to be solved by rsolve(). I can do it in a brute force way. The problem is to cope with the time index.

My problem is the following

Y = Function('Y')
C = Function('C')
I = Function('I')
C(t)=c*Y(t-1)+C_0
I(t)= I_0 - γ* (C(t) - C(t-1))
Y(t)=C(t)-I(t)
A(t)=C(t)-C(t-1)
show(A(t))

Until there no problems. But I do not know you to substite the value of A(t) in I(t)