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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 complete 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. Comment. 83 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.

In this problem we change the policy regime in the model of Problem IX.1 so that becomes the instrument and equal to a positive constant, , as long as the monetary authority does not decide to change it. Now is endogenous. a) To characterize the movement over time of the economy, derive from the model a dynamic system in and . Draw the corresponding phase diagram and illustrate the path that the economy follows. Comment. Suppose that the economy has been in its steady state until time 0 b) Then an unanticipated downward shift in occurs, but after this shift everybody rightly expect to remain unchanged forever. Graphically illustrate by means of a phase diagram and time profiles what happens to and Comment. c) Assume instead that at time 0, the monetary authority credibly announces a downward shift in the instrument variable to take place at time 1 0 After this shift everybody rightly expect to remain unchanged forever. Graphically illustrate by means of a phase diagram and time profiles what happens to and for 0 Comment. d) Briefly discuss the model.

Consider the model and policy regime from Problem IX.2. a) Determine the short-term and long-term real interest rates in steady state and find an implicit solution for in steady state. How does in steady state depend on ?

Now, suppose that the economy has been in its steady state until time 0 b) Answer b) from problem IX.2 under the assumption that the unanticipated shift in the short-term rate is upward. Comment. c) Answer c) from problem IX.2 under the assumption that the anticipated shift in the short-term rate is upward. Comment. IX.4 Consider the following continuous time model which focuses on "very-short run" dynamics of a closed economy: = (( ) + ) (1) 0 0 1 0 0 = ( ) 0 0 (2) = (3) = 1 (4) 1 + = (5) = (6) where a dot over a variable denotes the derivative w.r.t. time and the superscript denotes subjective expectations. Further, = output, = real price of a consol paying one unit of output per time unit forever, = index of "level of confidence", = government spending on goods and services, = money supply, = output price, = nominal short-term interest rate, 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) - f) we assume that the central bank maintains the real money supply, at a given constant level, by letting the (nominal) money supply grow at a rate equal to the rate of inflation. a) Briefly interpret the equations (1) - (6) as well as and . From now on, suppose that expectations are rational 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 . Draw the corresponding 73 phase diagram, assuming that parameters are such that there exists a steady state with a nominal short-term interest rate Hint: it may be useful to find an expression for in terms of output and the real money supply. c) Illustrate the path that the economy follows for an arbitrary 0 0. Comment. Suppose that the economy has been in its steady state ("short-run equilibrium") until time 0 0 d) Then an unanticipated shift in the level of confidence to a value 0 occurs. But after this shift everybody rightly expects the new level of confidence to be maintained for a long time. Illustrate by a phase diagram what happens to and over time. Illustrate in another figure the time profiles of and for 0 Briefly explain in words. Hint: the following formula may be helpful: = 1 = 1 R e) What is the sign of the slope of the yield curve immediately after the shock? Comment. Imagine that at time 1 0 the economy has virtually settled down in the new steady state. f) For simplicity inflation is not endogenous in the model. As a crude representation of the operation ("behind the scene") of a Phillips curve, however, we imagine that at time 2 1 there is a jump down in the inflation rate to a negative level 0 . The consolidated government/central bank instantly lowers to the same level so that the real money supply does not change. Suppose that the market participants rightly expect these new circumstances to last for a long time. Illustrate by a phase diagram what happens to and over time. Illustrate in another figure the time profiles of and for 1. Whether the new steady state level of exceeds or does not exceed the steady state level before time 0 is ambiguous. Why? Imagine that at time 3 2 the economy has virtually settled down in the new steady state. Let the nominal interest rate in this steady state be denoted 00 Suppose 00 is practically equal to zero. The fiscal and monetary authorities find the situation unsatisfactory and decide a coordinated fiscal and monetary policy involving a shift of to 0 and a continuous adjustment of the money supply through open market operations so as to maintain the short-term nominal interest rate at the target level = 00 until the recession is over. g) Rewrite one of the basic equations of the model so as to indicate how the nominal money supply must move according to the new monetary policy rule. h) Within the model (with inflation staying at 0 for a relatively long time), is the coordinated fiscal and monetary policy more potent than a conventional expansionary fiscal policy in isolation (that is, with a monetary policy that, as above, maintains the real money supply)? Is it more potent than a conventional expansionary monetary policy in isolation (that is, with a "passive" fiscal policy as above)? Explain your answers. i) To the extent the stimulation of aggregate demand succeeds in raising economic activity, it is likely that (outside the model) inflation also gradually rises towards a more "normal level". Is this feature likely to counteract or support the effectiveness of the coordinated fiscal and monetary policy? Explain.

We consider a small open economy (SOE) satisfying (approximately): 1. Perfect mobility across borders of financial capital, but no mobility of labor. 2. Domestic and foreign financial claims are perfect substitutes (no uncertainty). 3. Domestic and foreign output goods are imperfect substitutes. Suppose the short-term behavior of the economy can be described by the following model in continuous time. Given the function ( ) 75 where 0 1 0 and 0 the model is: = (( ) + ) 0 = = ( ) 0 0 = 1 1 + = The endogenous variables are: output, long-term real interest rate, short-term nominal interest rate, money supply, real price of a long-term bond (a consol), short-term real interest rate and expected (forward-looking) rate of inflation, where is time. The superscript signifies an expected value. A dot over a variable denotes the time derivative. The variables and are exogenous and constant; their interpretation is as follows: nominal exchange rate, foreign price level, domestic price level, government spending on goods and services, and foreign short-term nominal interest rate. The parameter is constant. The initial value 0 of is predetermined. We assume that 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 coupled differential equations in and . c) Construct the corresponding phase diagram and illustrate the path the economy follows. Comment. d) Determine the long-term real interest rate in steady state. e) How does in steady state depend on and ? Suppose that the economy has been in its steady state until time 0 f) At time 0 an unanticipated upward shift in the foreign short-term nominal interest rate occurs. But after this shift everybody rightly expects the foreign short-term nominal interest rate to remain unchanged for a very long time. Illustrate by a phase diagram and by graphical time profiles what happens to , , and for 0

g) Assume instead that at time 0, people in the SOE become aware that a monetary tightening in the leading countries in the world economy is underway. As a crude representation of this, suppose the agents rightly expect an upward shift in the foreign short-term nominal interest rate to take place at time 1 0 After this shift everybody rightly expects the foreign short-term nominal interest rate to remain unchanged for a very long time. Illustrate by a phase diagram and by graphical time profiles what happens to , and for 0. Comment. h) Now imagine the scenario is somewhat different from that described in g). Until time 2 1 everything is as described in g). But at time 2 due to now foreseeable unemployment problems, the government of the SOE credibly announces an upward shift in to take place at time 3 2 After this shift everybody rightly expects to remain unchanged for a very long time. The size of the shift in is such as to reestablish, in the long run, an output level equal to that attained at time 1 Illustrate by a phase diagram and by graphical time profiles what happens to , , and for 2

Money in the long and short run (comparing Keynesian and classical views) Consider the Keynesian money demand hypothesis, = ( ) 0 0 (*) where 0 is the general price level (in terms of money), () is a real money demand function ("" for liquidity), is aggregate production per time unit, and 0 is the nominal interest rate on short-term bonds. Let time be continuous and define the short-term real interest rate by where (the inflation rate). Since asset markets move fast, it is natural to assume that the money market clears continuously, i.e., = ( ) for all 0 where is the money supply. Let the growth rate of any positive variable be denoted Abstracting from fluctuations around the trend, suppose that = and = both constant, and that the real interest rate is a constant, a) Briefly, interpret the signs of the partial derivatives of the money demand function in (*). b) Show that if the money demand function is specified as ( ) = () where 0 () 0 then a constant long-run inflation rate, is consistent with the model and that this satisfies a simple equation where also and enter. Relate to your empirical knowledge. 67 Recall that the (income) velocity of money is defined as Classical monetary theory (the Quantity Theory of Money) claims that for given monetary institutions the velocity of money is a constant and thus independent of the nominal interest rate. c) What is the prediction implied by the classical theory concerning the long-run inflation rate, given and ? Comment in relation to the above Keynesian result. We now consider a short time interval, say a month, where the level of the money supply is practically constant, apart from level shifts implemented by the central bank, through open market operations as part of its monetary policy. Suppose the money supply in this way shifts from to 0 d) According to the classical monetary theory (which is more or less shared by the "monetarists", like Milton Friedman, and also to some extent by the "new classical" theorists like Robert Lucas), which variable will respond in the short run and how? e) What is the likely further effect on aggregate demand, output, and employment as long as the rate of inflation does not respond very much? f) Now answer d) and e) from the point of view of Keynesian theory where and even are "sticky" in the short run?

a) Briefly describe different monetary transmission mechanisms. Make sure that both classical and Keynesian-style monetary transmission channels are included in the list. Briefly discuss.

b) To combat the sharply rising unemployment in the U.S., the Obama government decided to increase government spending substantially and the U.S. now has a huge government budget deficit. In response to this policy, economist Robert Barro from Harvard University declared that the policy was likely to raise expected future taxation considerably and as a result of this there would not be any stimulating effect on current aggregate demand and production. Briefly discuss. c) "According to Sidrauski's monetary Ramsey model, the long-run growth rate in the money supply is irrelevant for welfare." True or false? Why