Teoria y Politica Monetaria, Instructor: Marco Del Negro

Problem Set 8

Introduce domestic bonds in the open economy Sidrausky model studied in class. Specifically, assume that the objective function of the representative agent is:

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

subject to the budget constraint:

p_t c_t+M_t+1+B_t+1+e_tA_t+1=p_t y_t+M_t+e_tA_t(1+r^w)+B_t(1+R_t)+T_t

where A_t represents the amount of financial activities ($ denominated) held by the household at the beginning of period t, and B_t represents the amount of government bonds (Peso denominated) held by the household at the beginning of period t. The foreign interest rate is constant and equal to $r^w=\frac1{\beta}$.
The budget constraint of the government is now:

M^s_t+1-M^s_t+B^s_t+1-B^s_t-B^s_tR_t=T_t

where B^s_t represents the amount of government bonds (Peso denominated) issued by the government at the beginning of period t. Note that the government holds no international reserves by assumption.
Assume that: i) the Law of One Price holds and foreign prices are constant and equal to 1 ( $p^w_t=1 \mbox{, all }t$), ii) perfect international capital mobility, iii) exchange rates are flexible. Further assume that:

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

a) Find the first order conditions with respect to M_t+1, A_t+1, and B_t+1.
b) Show that the following arbitrage relationship holds:

$$(1+R_t)=(1+r^w_t)\frac{e_t}{e_{t-1}}$$

c) Focus on the stationary equilibrium, that is, where:

$$A_t=A_0 \mbox{, all }t.$$

i) Find equilibrium consumption.
ii) Find the equilibrium nominal interest rate R.
iii) Find the rate of depreciation of the exchange rate.
iv) Find a formula for the level of the exchange rate.