Constant to the power of t in steady state

I am wondering how to get the steady-state for the following Euler equation. I know that we can get rid of time in subscripts. However, here I have a constant (a) to the power of $$t$$. Does anyone know if there is a way to get rid of $$t$$ in power? Or can I just consider $$a^t$$ a new constant, say $$\bar{a}$$?

$$\frac{1}{c_{t}}=a^t \beta E_t \Big[(1+r_{t+1})\dfrac{1}{c_{t+1}}\Big]$$

If the variables are constant, everything in the equation is time-invariant, while $$a^t$$ will still grow or fall over time (unless $$a=1$$). This is a contradiction and no steady state exists, unless $$a=1$$.
• Thank you for your comment. This case is a bit confusing for me. I am explaining why: If I defined a new variable as $A_{h} \equiv a^{h}$, $h=t,t+1,...$ then the steady-state would be $1 = A \beta (1+r)$? Is this correct or Is my understanding wrong here? Feb 21, 2021 at 16:15
• No. Simply redefining the object that grows as $A$ does not allow you to assume that it is constant over time. If $A$ has a trend, then $r$ must have one as well. Feb 21, 2021 at 16:43