# Consumption smoothing via Euler equation

A representative cnsumer maximizes their lifetime utility function: $$U=\sum_{t=0}^{\infty}\beta^{t}\text{ln}\left(C_{t}\right)$$ defined over consumption. They supply one unit of labour inelastically, each period, and obtain a known sequence of wages, $$w_{t}=w$$, each period. The agent faces budget constraint: $$P_{t}C_{t}+a_{t}\leq w_{t}+\left(1+r\right)a_{t-1}$$ where $$a_{t}$$ represents the quantity of bond holdings purchased at time $$r,$$ on which a return of $$r$$ units is obtained per $$a_{t}$$ and $$P_{t}$$ represents a change in the price of consumption. The Euler equation is: $$\frac{1}{C_{t}}=\frac{\beta}{C_{t+1}}\left(\frac{P_{t}}{P_{t+1}}\right)\left(1+r_{t}\right)$$ Imagine now, that at period $$T$$, wages drop to say 0 (i.e. the agent does not receive any labour incme). Intuitively, given a discount factor and positive interest rates, consumptions are linked across periods through the Euler equation. Is there an obvious way to see consumption smoothing in this case? For instance, imagine a case where the individual was consuming all wealth in each period (i.e. maximizing static utility only). Where/how can it be discerned that in the lifetime utility case, the individual will have higher consumption in period $$T$$ relative to the static utility case?