Consider a two period model of consumption.
I'm confused by the fact that in the optimum condition it is the marginal utility of the current period that is discounted, not the marginal utility of the next period.
Could someone give me an intuition behind this result?
Below, the derivation is presented.
$$ \max_{c_{t},c_{t+1}}u_{t}\left(c_{t},c_{t+1}\right)$$
such that
$$c_{t}+\frac{1}{1+r_{t+1}}c_{t+1}=w_{t}+\frac{1}{1+r_{t+1}}w_{t+1}$$
FOC: (MU denotes marginal (aka instantaneous) utility)
$$\begin{cases} MU_{t}=\lambda\\ MU_{t+1}=\lambda\frac{1}{1+r_{t+1}}\\ c_{t}+\frac{1}{1+r_{t+1}}c_{t+1}=w_{t}+\frac{1}{1+r_{t+1}}w_{t+1} \end{cases}\Rightarrow\frac{1}{1+r_{t+1}}MU_{t}=MU_{t+1}$$