Use Lag Operator to find Lifetime Budget Constraint

Question

The budget constraint is

$c_t + tau_t + s_{t+1} =w_t(1-l_t) +(1+r_t)s_t$

And assume

$underset{t longrightarrow infty}{lim}  displaystyle{frac{s_t}{Pi_{i=1}^{t-1} (1+r_i)}} = 0$

Lag Operator $L$ is defined as $L cdot x_{t+1} = x_t$

How can I get lifetime budget constraint using the Lag Operator?

Many Thanks!

mark leeds · Answer

@T. G.: I think I obtained the expression for $s_{t+1}$ as a function of the other variables. I don't know what lifetime budget really means so I'll write the answer here and hope it's useful.

$s_{t+1} = sum_{t=0}^{infty} lambda_{t} (w_{t}(1-l_{t}) -c_{t} - tau_{t})$

where $lambda_{t} = prod_{i=0}^{t-1} (1 + r_{t-i})$.

I'm not certain if it's correct but intuitively, it looks like exponential smoothing of some kind of expression each period, except that the smoothing constant is not constant and is a function of the past interest rates ??

Use Lag Operator to find Lifetime Budget Constraint

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