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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 $\bar{a}$?

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

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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 $a=1$.

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  • $\begingroup$ 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? $\endgroup$ – Giordano Feb 21 at 16:15
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    $\begingroup$ 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. $\endgroup$ – jpfeifer Feb 21 at 16:43

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