# Simplfying Euler equation under expectations

I am working through a textbook that derives the following first order (Euler) equation between periods $$t$$ and $$t+1$$. We end up with: $$-C_{j,t}^{-\sigma}+\beta E_{t}\left\{ \left(\frac{C_{j,t+1}^{-\sigma}}{P_{t+1}}\right)\left[\left(1-\delta\right)P_{t+1}+R_{t+1}\right]\right\} =0$$ The variables carry the usual meanings, as $$P,R$$ and $$C$$ represent prices, interest rates and consumption respectively, and $$t$$ indexes time. Next period prices and interest rates are unknown, leading to the expectations operator. The author then simplifies this to: $$\left(\frac{E_{t}C_{j,t+1}}{C_{j,t}}\right)^{\sigma}=\beta\left[\left(1-\delta\right)+E_{t}\left(\frac{R_{t+1}}{P_{t+1}}\right)\right]$$ This seems to me sloppy use of the expectations operator without any additional assumptions given. For instance, simplying the first expression, one gets: \begin{align*} C_{j,t}^{-\sigma} & =\beta E_{t}\left\{ \left(\frac{C_{j,t+1}^{-\sigma}}{P_{t+1}}\right)\left[\left(1-\delta\right)P_{t+1}+R_{t+1}\right]\right\} \\ & =\beta E_{t}\left(\left(\frac{C_{j,t+1}^{-\sigma}}{P_{t+1}}\right)\left(1-\delta\right)P_{t+1}\right)+\beta E_{t}\left(\left(\frac{C_{j,t+1}^{-\sigma}}{P_{t+1}}\right)R_{t+1}\right)\\ & =\beta\left(1-\delta\right)E_{t}\left(\frac{C_{j,t+1}^{-\sigma}}{P_{t+1}}\right)+\beta E_{t}\left(\left(\frac{C_{j,t+1}^{-\sigma}}{P_{t+1}}\right)R_{t+1}\right) \end{align*} From here, there is no more room for simplification unless one makes restrictive (and unrealistic) assumptions on the nature of the random variables at $$t+1.$$ In particular, consumption tomorrow will depend on relative prices and interest rates, so we cannot use the fact that $$E[XY]=E[X]E[Y].$$ How can we simplify the above expression to obtain what the author has done? Thank you for your time.

• This is the first time I have seen this weird simplification, Would be interesting to know what textbook you are referring to. Alecos answer, as good as it is, shows you what assumptions you would need to make to get there. But, as Alecos says, it doesn't tell you whether you should make these assumptions, or even if they are sensible. That's your call.
– BrsG
Sep 7, 2022 at 16:47

A formal assumption and twice ignoring Jensen's inequality can lead to the approximation.

FORMAL ASSUMPTION

$${\rm Cov} \left[C_{j,t+1}^{-\sigma}, \,\frac{R_{t+1}}{P_{t+1}}\right] =0.$$ Technically, this does not mean that we assume that the two components are independent, only that their covariance is zero. They may exhibit non-linear dependence. The fact that we are looking at a non-linear function of consumption (reciprocal and raised to a power) may be seen as a supporting "argument".

TWICE IGNORING JENSEN'S INEQUALITY

...by approximating $$E_t\left[\frac{1}{C_{j,t+1}^{\sigma}}\right] \approx \frac{1}{\left[E_tC_{j,t+1}\right]^{\sigma}}.$$

Depending on the value of $$\sigma$$, the one blind-eye may tend to offset the effect of the other blind-eye, or intensify it.

Accept these steps at your own peril.

• Thank you so much, very well explained. Sep 7, 2022 at 13:11