A representative household maximizes

$E_t{\displaystyle \underset{j=0}{\overset{}{}}}{}^j[log(C_{t+j}-C_{t+j-1})-h_{t+j}],$

where

$C_{t+j}+K_{t+j}=w_{t+j}{h}_{t+j}+r_{t+j}{K}_{t+j-1}+{}_{t+j}-T.$

Discuss economic intuitions behind the term with $.$

Define Lagrangian.

What are the variables that are optimized or control variables?

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

Derive

$($a$)$ the labor supply condition; and

$($b$)$ the Euler equation with respect to consumption.

Discuss

$($a$)$ how the past consumption influences today's consumption;

$($b$)$ how the real interest rate influences today's consumption.

$($c$)$ how the real wage influences labor supply.