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Uncovered interest parity in discrete time. Starting with discrete time (period analysis), let Japan be the domestic country and define: = nominal exchange rate (U per $), = domestic price level, = foreign price level, = nominal (short-term) interest rate, = foreign (short-term) nominal interest rate. The uncovered interest parity (UIP) condition is 1 + = 1 (1 + )+1 a) Interpret. b) Show that for both and +1 "small", + +1

a) The expectations theory of the term structure predicts that the longterm interest rate tends to be higher than the short-term interest rate." True or false? Comment. b) We consider a Blanchard-and-Fischer-style dynamic short-run model of a small open economy with fixed prices (zero inflation), floating exchange rate, and no risk premium. It takes time for output to adjust to changes in output demand. The central bank pursues the policy = 0 + 1 where is the short-term nominal interest rate, is aggregate output, 0 and 1 are constants and 1 0 Suppose the economy has been in steady state until time 0 Then the central bank unexpectedly changes 0 to a lower constant, 00 After 0 everybody rightly expects the monetary policy to remain = 00 + 1. "There will be no exchange rate overshooting phenomenon generated by this policy shift." True or false?

Consider a small open economy satisfying (approximately) the conditions 1-3 in Problem IX.5. Suppose the short-term behavior of the economy can be described by the following model in continuous time. Given the function ( ) where 0 1 0 and 0 the model is: = ( ) = ( ) 0 = ( ) 0 0 = + The endogenous variables are: = output demand, = output, = real interest rate, = real exchange rate, = nominal interest rate, = nominal exchange rate, = expected (forward-looking) rate of inflation, all at time ; the superscript denotes expectation. The variables , and are exogenous and constant; their interpretation is as follows: = money supply, = domestic price level, = foreign price level, and = foreign nominal interest rate. The parameter is constant. The initial value 0 of is given. Expectations are rational and speculative bubbles never occur. a) Briefly interpret the model. b) To characterize the movement over time of the economy, derive two differential equations in and . c) Construct the corresponding phase diagram and illustrate the path that the economy follows. Comment. Suppose that the economy has been in steady state until time 0 d) At time 0 an unanticipated tightening of monetary policy (downward shift in ) occurs. After 0 everybody rightly expects the money supply to remain at the new lower level, 0 forever. Illustrate by a phase diagram and a separate figure with time profiles what happens to and for 0 Comment. e) Assume instead that at time 0, due to foreseeable overheating problems everybody become aware that the monetary authority will at time 1 0 carry into effect a shift in money supply to the level 0 Illustrate by a phase diagram and a separate figure with time profiles what happens to , and for 0.

Consider a small open economy satisfying (approximately) the conditions 1-3 in Problem IX.5. Suppose the short-term behavior of the economy can be described by the following model in continuous time: = (( ) ) 0 0 1 0 0 = ( ) 0 0 = + The endogenous variables are: = output, = real interest rate, = real exchange rate, = nominal interest rate, = nominal exchange rate, = expected (forward-looking) rate of inflation, all at time ; the superscript denotes expectation. The variables and are exogenous and constant; their interpretation is as follows: = money supply, = domestic price level, = foreign price level and = foreign nominal interest rate. The parameter is constant. The initial value 0 of is given. Expectations are rational and speculative bubbles never occur. a) Briefly interpret the first three equations of the model. b) Derive two differential equations in and that characterize the movement over time of the economy. c) Construct the corresponding phase diagram and illustrate the path that the economy follows for 0. Comment. Suppose that the economy has been in steady state until time 0 d) At time 0 an unanticipated upward shift in occurs. After 0 everybody rightly expects the money supply to remain at the new higher level, 0 forever. Illustrate by a phase diagram and a separate figure with time profiles what happens to and for 0 Explain in detail by words the economic intuition behind what happens.

Assume instead that at time 0 everybody become aware that the monetary authority will at time 1 0 carry into effect a shift in money supply to the level 0 Illustrate by a phase diagram and a separate figure with time profiles what happens to and for 0.

In this problem we consider short-run aspects of a small open economy (SOE) satisfying: (i) Perfect mobility across borders of financial capital, but no mobility of labor. (ii) Domestic and foreign financial claims are perfect substitutes. (iii) Domestic and foreign output goods are imperfect substitutes. More specifically, it is assumed that at least outside the zero lower bound on the interest rate: = (( ) + ) (1) 0 0 1 0 0 0 = ( ) 0 0 (2) = + (3) (4) (5) Time is continuous. The endogenous variables are: = output, = nominal interest rate, = nominal exchange rate, = expected (forward-looking) rate of inflation, all at time ; the superscript denotes expectation. The variables and are exogenous and constant; their interpretation is as follows: = a demand shift parameter, = government spending on goods and services, = money supply, = domestic price level, = foreign price level, and = nominal world market interest rate. The parameter is constant. The initial value 0 of is given. Expectations are rational. We assume speculative bubbles never occur. a) Briefly interpret the model. 81 b) Derive two key differential equations and construct a phase diagram portraying the dynamics of the economy in () space. Indicate the path that the economy follows for 0. Comment. Suppose that the SOE is initially in steady state. Then, unexpectedly, a recession in the leading economies in the world comes about and gives rise to an offsetting monetary policy in these countries. As a crude representation of these events vis-a-vis our SOE we "translate" them into two unanticipated parameter shifts occurring at time 0 0: a shift in the demand shift parameter to 0 and a shift in the world interest rate to 0 , where 0 is close to zero. c) Suggest an interpretation of the fall in the demand shift parameter. d) Assume that after time 0 the public in the SOE rightly expects that the mentioned two new parameter values will remain in force for a long time and that no policy change in the SOE will occur. Under these circumstances illustrate how the SOE evolves for 0, using a phase diagram as well as a figure with time profiles of and presupposing that the sign of the steady-state effect on is dominated by the influence from the fall in the world interest rate. Explain the economic intuition. e) As an alternative scenario imagine that at time 1 0 the monetary and fiscal authorities in the SOE find the situation unsatisfactory and contemplate monetary and fiscal measures to stimulate economic activity. It is soon realized, however, that neither conventional monetary policy (upward shift in ) nor conventional fiscal policy (upward shift in ) will be very effective. Give plausible reasons for this unfavorable outlook. f) May a combination of conventional monetary and fiscal policy work? Why or why not? g) If international coordination of fiscal policy is possible, can this then improve the outlook? Why or why not?

In this problem we consider a model of a closed economy in the "very short run". Time is continuous. The assumptions are: = (( ) + ) (1) 0 0 1 0 0 = ( ) 0 0 (2) = 1 (3) 1 + = (4) (5) = (6) = 0 + 1 1 0 (7) where a dot over a variable denotes the derivative w.r.t. time and the superscript indicates expected value (until further notice a subjective expectation). Further, = output, = real long-term interest rate, = real price of a consol paying one unit of output per time unit forever, = nominal short-term interest rate, = level of "confidence", = government spending on goods and services, = money supply, = output price, and = rate of inflation. The tax revenue function is implicit in the demand function (). The variables and are exogenous positive constants. The initial values 0 and 0 are historically given. In questions a) - e) it is assumed that exogenous variables and initial conditions are such that 0 for all 0. a) Briefly interpret the model. Make sure you explain why the real longterm interest rate can be written as in (3). From now on we assume perfect foresight and that speculative bubbles do not arise. b) To characterize the movement of the economy over time, derive from the model a dynamic system in and . Comment on what the role of equation (2) is in the model. c) Draw the corresponding phase diagram, assuming that parameters are such that there exists a steady state with a nominal short-term interest rate Illustrate the path that the economy follows for an arbitrary 0 0.

Now suppose that the economy is already in its steady state ("short-run equilibrium"). Let the steady state values of and be denoted and respectively. d) Find an analytical expression for the effect on of a unit increase in (the "spending multiplier"). Unexpectedly an adverse demand shock occurs at time 1 0 (that is, a shift of confidence level from to 0 ). We assume that after this shift everybody rightly expect the new level of confidence to be maintained for a long time. e) Illustrate by a phase diagram what happens to and over time, presupposing that a steady state with 0 still exists. Illustrate in another figure the time profiles of and for 1 Explain the intuition in words. Hint: in the absence of bubbles the following formula holds: = 1 = 1 R f) Briefly evaluate the model.

Uncovered interest parity in discrete time. Starting with discrete time (period analysis), let Japan be the domestic country and define: = nominal exchange rate (U per $), = domestic price level, = foreign price level, = nominal (short-term) interest rate, = foreign (short-term) nominal interest rate. The uncovered interest parity (UIP) condition is 1 + = 1 (1 + )+1 a) Interpret. b) Show that for both and +1 "small", + +1 Hint: You may use that +1 = +1 + 1 and that the product of two small numbers is "very small"

a) The expectations theory of the term structure predicts that the longterm interest rate tends to be higher than the short-term interest rate." True or false? Comment. b) We consider a Blanchard-and-Fischer-style dynamic short-run model of a small open economy with fixed prices (zero inflation), floating exchange rate, and no risk premium. It takes time for output to adjust to changes in output demand. The central bank pursues the policy = 0 + 1 where is the short-term nominal interest rate, is aggregate output, 0 and 1 are constants and 1 0 Suppose the economy has been in steady state until time 0 Then the central bank unexpectedly changes 0 to a lower constant, 00 After 0 everybody rightly expects the monetary policy to remain = 00 + 1. "There will be no exchange rate overshooting phenomenon generated by this policy shift." True or false?

We consider a small open economy (SOE) satisfying the following conditions: 1. Perfect mobility across borders of financial capital, but no mobility of labor. 2. Domestic and foreign bonds are perfect substitutes and command the same expected rate of return. 3. Domestic and foreign output goods are imperfect substitutes. 4. Nominal prices are sluggish and follow an exogenous constant inflation path. Aggregate output demand is = ( ) + ( ) + ( ) + ( ) + (*) where T and T = + ( ) 0 0 1 0 (1 0 ) + (1 0 ) (1 0 ) + + 1 0 1 + 0 0 1 = 0 Notation: is output demand, is after-tax income, is output, is the long-term real interest rate, and is the real exchange rate, where is a given and constant nominal exchange rate, and is the domestic 85 price level while is the foreign price level; measures "fiscal tightness", is government spending on goods and services, T is net tax revenue, and ( ) is a tax function. We assume that the domestic (forward-looking) inflation rate, is constant and equals the foreign (forward-looking) inflation rate, Hence, is a constant and so is the real exchange rate, from now denoted The dynamics of the economy is described by the following equations: = (( ) + ) 0 0 0 given, (1) = ( ) 0 0 (2) = 1 (3) 1 + = (4) (5) = 0 (6) where is the domestic short-term nominal interest rate, the foreign shortterm nominal interest rate, the domestic short-term real interest rate, the money supply, and the real price of a long-term bond (a consol). The superscript indicates an subjective expectation. The variables and are exogenous positive constants, . The initial values 0 and 0 are given. Expectations are rational and there are never speculative bubbles. The parameters are such that the speed of adjustment towards steady state is high. a) Briefly interpret (2), (3), and (4) of the model. b) To characterize the movement over time of the economy, derive two differential equations in and respectively. c) Construct the corresponding phase diagram and illustrate the path the economy follows for an arbitrary 0 0. Comment. d) Let the steady-state values of the long-term real interest rate and output be denoted and respectively. Find these values. Finally, derive the spending and tax multipliers, and Suppose that the economy has been in its steady state until time 0

e) Suppose the government is dissatisfied with the level of employment and at time 0 0 decides (unexpectedly) to increase to a higher level, 0 Suppose further that, owing to the automatic budget reaction under high unemployment, people rightly expect this higher level of spending to be maintained for quite some time without a rise in Under the simplifying assumption that the new spending level is permanent, illustrate by a new phase diagram what happens for 0 f) Assume instead that at time 0, the government credibly announces an upward shift in the level of government spending from to 0 to take place at time 1 0 Illustrate by a phase diagram and by graphical time profiles what happens to and for 0 Comment. Hint: the answer to this question may be easier than one might immediately think. g) Briefly evaluate the model.