Consider a small, open economy inhabited by a large number of individuals who live for an infinite number of periods and who can perfectly forecast the future. In this economy there is only one good, tradable at no cost, such that its domestic price is determined by the Law of One Price, P(t) 5 S(t)F, where PðtÞ is the domestic price of the good and PF is the foreign price of the good, which we will assume is equal to 1. Time is continuous and the economy is endowed with a constant and exogenous flow of the consumption good. There is perfect mobility of capital in the sense that agents can lend or borrow at the international interest rate of i > 0, which we assume as constant. The real money supply is defined as where M(t) is the nominal money supply. This economy finances an exogenous fiscal deficit, a. Show that the rate of growth of the real stock of money is equal to the difference between the rate of growth of money supply and inflation. b. Assume that the government of this country decides to fix the nominal exchange rate at level S, i.e., S(t) 5 S. Show that, if Rð0Þ is sufficiently large, it is possible to temporarily maintain this exchange rate. Also show that, while the nominal exchange rate SðtÞ remains fixed, the money supply will be constant. Calculate the real money supply. c. What happens with the international reserves while the nominal exchange rate remains constant? Based on the fiscal policy adopted by the government, explain why the fixed exchange rate regime is not sustainable. d. Assume that, when the fixed exchange rate regime ends, the international reserves will be null, i.e., R(t) 5 0. Show that after the end of the regime there is equilibrium with constant inflation and a constant real money supply. Compute inflation and the real money supply in this equilibrium. e. Considering that the fixed exchange rate regime is unsustainable, calculate the period T in which the regime is abandoned. Show that in period T the trajectory of price levels remains continuous, however, the real demand for money decreases abruptly. What happens with the level of international reserves in period T? What is the economic intuition for the result? f. Present illustrative graphs containing the trajectory of the stock of international reserves, the nominal money supply and the nominal exchange rate, both before and after period T. Explain each of these graphs.
Consider a small, open economy inhabited by a large number of individuals who live for an infinite number of periods and who can perfectly 
where M(t) is the nominal money supply. This economy finances an exogenous fiscal deficit,
a. Show that the rate of growth of the real stock of money is equal to the difference between the rate of growth of money supply and inflation.
b. Assume that the government of this country decides to fix the nominal exchange rate at level S, i.e., S(t) 5 S. Show that, if Rð0Þ is sufficiently large, it is possible to temporarily maintain this exchange rate. Also show that, while the nominal exchange rate SðtÞ remains fixed, the money supply will be constant. Calculate the real money supply.
c. What happens with the international reserves while the nominal exchange rate remains constant? Based on the fiscal policy adopted by the government, explain why the fixed exchange rate regime is not sustainable.
d. Assume that, when the fixed exchange rate regime ends, the international reserves will be null, i.e., R(t) 5 0. Show that after the end of the regime there is equilibrium with constant inflation and a constant real money supply. Compute inflation and the real money supply in this equilibrium.
e. Considering that the fixed exchange rate regime is unsustainable, calculate the period T in which the regime is abandoned. Show that in period T the trajectory of price levels remains continuous, however, the real
f. Present illustrative graphs containing the trajectory of the stock of international reserves, the nominal money supply and the nominal exchange rate, both before and after period T. Explain each of these graphs.
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