A representative household maximizes

$E_t{\displaystyle \underset{j=0}{\overset{}{}}}{}^{j}log{C}_{t+j},$

where

$C_{t+j}+K_{t+j}=Y+r_{t+j}{K}_{t+j-2}$

Note that the second term of the right$-$hand side of the equation is expressed by not $K_{t+j-1}$ but $K_{t+j-2}.$ In other words, the household receives returns from saving after two periods.

Define Lagrangian.

$($a$)$ What are the variables that are optimized or control variables?

$($b$)$ Derive the first$-$order condition$($s$).$

$($c$)$ Derive the Euler equation with respect to consumption.