# How is the intertemporal budget constraint derived from all intratemporal constraints

Suppose an agent receives a fixed (exogenous) income every period, $$y_t$$, and can either spend it on $$c_t$$ or save it/borrow ($$s_{t+1}$$). Suppose also that the agent starts off with no savings or debt.

How do you get from this:

$$c_t + s_{t+1} = y_t + (1 + r_t) s_t$$

To this:

$$\sum_{t=0}^\infty c_t p_t = \sum_{t=0}^\infty y_t p_t$$

where $$p_t = \frac{1}{(1 + r_1)(1 + r_2)...(1 + r_{t})}$$

Let $$p_t = \dfrac{1}{\prod_{i = 0}^{t-1}(1+r_i)}$$.
Multiply $$s_{t+1} - (1+r_t) s_t = y_t - c_t$$ by $$p_{t+1}$$ to get: $$p_{t+1} s_{t+1} - p_{t} s_t = p_{t+1} y_t - p_{t+1} c_t$$ Now add over all $$t = 0, 1, \ldots, T$$ (and notice that we get a telescoping sum): $$p_{T+1} s_{T+1} = \sum_{t = 0}^T p_{t+1} y_t - \sum_{t = 0}^T p_{t+1} c_t.$$ This assumes that $$s_0 = 0$$.
Usually, one also imposes a no-ponzi condition ($$\lim_{T \to \infty} p_T s_{T} = 0$$). If so, (and if the limits of the sums are well defined, i.e. bounded), we can take limits for $$T \to \infty$$ of above identity to obtain: $$\sum_{t = 0}^\infty p_{t+1} c_t = \sum_{t = 0}^\infty p_{t+1} y_t.$$ So I guess your index on $$p$$ is one off.
Alternatively if the budget constraint is: $$s_{t+1} = (y_t - c_t + s_t)(1+r_t),$$ then you would get $$\sum_{t = 0}^\infty p_{t} c_t = \sum_{t = 0}^\infty p_{t} y_t.$$ The difference is that here you get your income and consume at the start of the period (before you receive interest) whereas in your specification, you first receive interest and only afterwards receive income and consume.
• You are right. My definition for $p_t$ was wrong. But thanks, I get the idea now! Commented Sep 12, 2021 at 15:10