Deterministic Consumption-Saving Problem

Macro

CRRA Example

The Budget Constraint would be:

$$\sum _{t=0}{R}^{-t}{c}_t=R{a}_0+\sum R^{-t}{y}_t-\underset{T\to \infty }{\lim }{R}^{-T}{a}_{T+1}$$

The smoothing motive refers to the tendency of consumers to prefer a stable consumption pattern over time, rather than experiencing significant fluctuations in their consumption levels.

Euler Equation with Borrowing Constraint

We apply a sequence of multiplier ${\mu }_t$ here,

thus,

$$\mathcal{L}=$$