Marco Del Negro, Teoria y Politica Monetaria, Spring 2000

Problem Set 4

Consider an economy where the government budget constraint is:
$M_{t+1}^s+B_{t+1}^s=M_t^s+B_t^s(1+R_t)+p_tg_t-p_t\tau_t$
where B_t^s is the supply of nominal (one period) bonds, R_t is the it nominal interest rate, g_t is government spending (in units of the consumption good), and $\tau_t$ are taxes (in units of the consumption good).
The objective function of the household is:

$$max_{\{M_{t+1},c_t\}_{t=0}^{\infty}} \sum_{t=0}^\infty \beta^t u(c_t,\frac{M_t}{p_t})$$

subject to the budget constraints:
$M_{t+1} +B_{t+1}+p_t c_{t}\leq M_t+B_t(1+R_t)+p_t y_{t}-p_t\tau_t\\ B_{t+1} \geq -B$
where B is some positive number.
a) find the equilibrium under the following assumptions:

$$y_t=y \mbox{, }g_t=g \mbox{, }B_t=0, M^s_t=M^s_0(1+\mu)^t \mbox{, all } t$$

b) Consider now the case in which taxes are ``distortive" (because of collection costs like in the model of Barro JPE 1979), in the sense that higher tax rates decrease disposable output. In particular, assume that the intertemporal budget constraint of the household is now:

$$M_{t+1} +B_{t+1}+p_t c_{t}\leq M_t+B_t(1+R_t)+p_t y^*_{t}-p_t\tau_t$$

where

$$y^*_t \equiv y_t -\phi \frac{\tau^2_t}{y_t}$$

where the term $\phi \frac{\tau^2_t}{y_t}$ indicates collection costs.
Find the equilibrium under the same set of assumptions as in point a).