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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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Is that derivation for $k(t+1)$ correct? Technically, you never reach the steady state, but only asimptotically as $t\rightarrow\infty$, but at infinity the $A(t)$ will also be infinite because it grows exponentially with time. I suspect that $k(t+1)$ should depend on $(1+g)$ instead of $A(t)$. Usually these models are expressed in units of effective labor, ...

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yes, the term that you showed for the ALDR non-stochastic steady state: $$\frac{ \beta_1 + \beta_2 }{1- \rho_1 -\rho_2}$$ is long-run multiplier or sometimes also called long run equilibrium coefficient (see Verbeek 2008 Guide to Modern Econometrics 4th ed. pp 340). As far as I can understand there is not much difference between the two concepts in the ...

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