<p>Consider the following version of the Ak model:<span class="math-container">$$V^*(A_0,K_0)=max\sum\beta^t\sum P(A^t)\frac{c_t(A^t)^{1-\sigma}}{1-\sigma}$$</span> st<span class="math-container">$$ k_{t+1}(A^t)+c_t(A^t)\leq Ak_t(A^{t+1})$$</span> and non-negativities</p>
<p><span class="math-container">$A_t$</span> is an i.i.d. process with mean <span class="math-container">$1$</span> and <span class="math-container">$P(A_t)$</span> denotes the probability of a sequence <span class="math-container">$(A_0,A_1, ...,A_t)$</span>. We assume that σ &gt; 0 and β ∈ (0,1). Output is defined as <span class="math-container">$y_t = A_tk_t$</span>.</p>
<p>This might seem dumb but why is the Euler equation written as this: <span class="math-container">$$c_t(A^t)^{-\sigma}=\beta E_t[A_{t+1}c_{t+1}{(A^{t+1})}^{-\sigma}]$$</span></p>
<p>Then there is put another assumption that <span class="math-container">$k_{t+1}(A^t)=sA_tk_t(A^{t-1})$</span> Derive the savings rate <span class="math-container">$s$</span> in terms of the parameters of the model. The answer given is <span class="math-container">$s=[\beta E_t(A_{t+1}^{1-\sigma})]^\frac{1}{\sigma}$</span>. Which again, I'm not getting how it came to that :\</p>

CC BY-SA 4.0
2021-03-25T20:40:59.903Z
I am not sure if I understand the question, are you asking how is it derived, or more generally why does Euler equation equates present consumption to the expected future one in these intertemporal problems?
1muflon1
CC BY-SA 4.0
2021-03-25T20:41:46.543Z
I'm asking for the derivation behind the equation.
Maybeline Lee
CC BY-SA 4.0
2021-03-25T22:56:45.330Z
You can check the answer here https://economics.stackexchange.com/questions/42971/deriving-first-order-condition-of-hansen-and-singleton-1982/42972#42972. The idea is the same.
Alalalalaki
CC BY-SA 4.0
2021-03-25T23:05:33.110Z
Well, now I feel stupid :), it's been a long day. Thanks for the link @Alalalalaki I got it now
Maybeline Lee
<macroeconomics><self-study>
Stochastic AK model with uncertainty
Maybeline Lee
# Stochastic AK model with uncertainty

(Asked by: username_0 on 2021-03-25)

Consider the following version of the Ak model:$$V^\*(A\_0,K\_0)=max\sum\beta^t\sum P(A^t)\frac{c\_t(A^t)^{1-\sigma}}{1-\sigma}$$ st$$ k\_{t+1}(A^t)+c\_t(A^t)\leq Ak\_t(A^{t+1})$$ and non-negativities

$A\_t$ is an i.i.d. process with mean $1$ and $P(A\_t)$ denotes the probability of a sequence $(A\_0,A\_1, ...,A\_t)$. We assume that σ > 0 and β ∈ (0,1). Output is defined as $y\_t = A\_tk\_t$.

This might seem dumb but why is the Euler equation written as this: $$c\_t(A^t)^{-\sigma}=\beta E\_t[A\_{t+1}c\_{t+1}{(A^{t+1})}^{-\sigma}]$$

Then there is put another assumption that $k\_{t+1}(A^t)=sA\_tk\_t(A^{t-1})$ Derive the savings rate $s$ in terms of the parameters of the model. The answer given is $s=[\beta E\_t(A\_{t+1}^{1-\sigma})]^\frac{1}{\sigma}$. Which again, I'm not getting how it came to that :\

**Tags:** macroeconomics, self-study

> **Comments:**
> **username_1**: I am not sure if I understand the question, are you asking how is it derived, or more generally why does Euler equation equates present consumption to the expected future one in these intertemporal problems?  
> **username_0**: I'm asking for the derivation behind the equation.  
> **username_2**: You can check the answer here https://economics.stackexchange.com/questions/42971/deriving-first-order-condition-of-hansen-and-singleton-1982/42972#42972. The idea is the same.  
> **username_0**: Well, now I feel stupid :), it's been a long day. Thanks for the link @username_2 I got it now  

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## 0 answers

