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I'm doing some self-studying using the publicly available preliminary exams from UC Davis and need help understanding a statement given in the solutions.

This particular problem is from the 2015 August Macro Prelim:

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The answer to this question is:

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My question: How does one get from the Euler Equation $$\frac{k_{t+1}}{c_t} = \alpha \beta E\left[ \frac{y_{t+1}}{c_{t+1}} \right]$$ that $c_t = (1-\alpha \beta)y_t$ and $k_{t+1} = \alpha \beta y_t$? It's not obvious to me that rearranging that equation in any manner yields the provided result.

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  • $\begingroup$ By typing up your question, rather than presenting it as an image, you make your question more accessible to vision impaired readers and make it a more discoverable resource through search. $\endgroup$
    – BKay
    Commented Jan 4, 2023 at 19:56
  • $\begingroup$ Thanks - I made an edit to include the Euler equation in question. I was not sure whether including the whole question was necessary since the solutions seemed to suggest not, but wanted to hedge against the chance other details were indeed relevant. If they turn out to be, I will further edit. $\endgroup$
    – qwerty
    Commented Jan 4, 2023 at 20:19

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Perhaps $c_t = y_t - k_{t+1}$ and $k_{t+1}$ under steady state is $\alpha \beta y_{t+1}$. You can think of the $c_{t}$ as coming from the accounting identity of C = Y - I in the long run.

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  • $\begingroup$ I think utilizing the budget constraint makes sense, but I'm still a bit confused on the timing aspect. Since $z$ is still stochastic, why wouldn't the steady state for $k$ include an expectation over $z$? $\endgroup$
    – qwerty
    Commented Jan 8, 2023 at 20:01
  • $\begingroup$ This is independent of the stochastic process under the current functional forms. Also you aren’t solving the expectations bit just yet. Only when you get the policy functions and think about the complete system do you need to think about expectations. $\endgroup$
    – EndLoop
    Commented Jan 9, 2023 at 21:35

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