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3

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 $a=1$.

3

I don't know the paper nor the notation, so I am just guessing here. I gues $N(a,t)$ is the number of agents of age $a$ at time period $t$. Let's follow the number of age $B$ accross generations: \begin{align*} N(B,t) &= e^f N(B, t- B),\\ &= e^{2f} N(B, t - 2 B),\\ &= \ldots,\\ &= e^{f t/B} N(B,0),\\ &= e^{\eta t} N(B, 0). \end{align*} ... 2 Let's guess that the value function is of the form a + b \ln(k). Then substituting for V(k) = a + b \ln(k) in the Bellman equation gives: a + b \ln(k) = \max_{k'}\left(\ln(k^\alpha - k') + \beta(a + b \ln(k')\right) $$The first order condition is given by:$$ \begin{align*} &\frac{-1}{k^\alpha - k'} + \beta b \frac{1}{k'} = 0,\\ \to & k' = \...

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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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