Advanced Macro: Numerical Methods
Simplest consumption/savings model:
F.O.C. reads as: \[\beta E_t \left[ \frac{U^{\prime}(c_{t+1})}{U^{\prime}(c_t)} \overline{r} \right] - 1 \leq 0 \perp c_t \leq w_t\] and \[0 \leq \beta E_t \left[ \frac{U^{\prime}(c_{t+1})}{U^{\prime}(c_t)} \overline{r} \right] - 1 \perp 0 \leq c_t \]
First one reads: if my marginal utility of consumption today is higher than expected mg. utility of cons. tomorrow, I’d like to consume more, but I can’t because, consumption is bounded by income (and no-borrowing constraint).
Second one reads: only way I could tolerate higher utility in the present, than in the future, would be if I want dissave more than I can, or equivalently, consume less than zero. This is never happening.
Consider the new keynesian model we have seen in the introduction:
The model satisfies the same specification with:
These are not real first order conditions as they are not derived from a maximization program
It is possible to add a zero-lower bound constraint by replacing IRS with: \[ \alpha_{\pi} \pi_t + \alpha_{y} y_t \leq i_t \perp 0 \leq i_t\]