# Stochastic AK model with uncertainty [duplicate]

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 :\

• 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
Mar 25 at 20:40
• I'm asking for the derivation behind the equation. Mar 25 at 20:41
• You can check the answer here economics.stackexchange.com/questions/42971/…. The idea is the same. Mar 25 at 22:56
• Well, now I feel stupid :), it's been a long day. Thanks for the link @Alalalalaki I got it now Mar 25 at 23:05