5.2. The derivation of the LM curve assumes that M is exogenous. But suppose instead that the Federal Reserve has some target interest rate i and that it adjusts M to keep i always equal to /.

(a) With this policy, what is the slope of the "LM curve" (that is, the set of combinations of i and Y that cause money demand and supply to be equal)?

(b) With this policy, what is the slope of the AD curve?

5.3. The government budget in the standard Keynesian modeL

(a) The balanced budget multiplier. (See Haavelmo, 1945.) Suppose that planned expenditure is given by (5.5), E = C(Y ~ ) + I(i - <)+ G.

(z) How do equal increases in G and T affect the position of the /5 curve? Specifically, what is the effect on for a given level of i ?

(zz) How do equal increases in G and T affect the position of the AD curve? Specifically, what is the effect on for a given level of P?

(b) Automatic stabilizers. Suppose that tax revenues, , instead of being exogenous, are a function of income: T = T{Y), T{Y) > 0. With this change, find how an increase in ( ) affects the following:

(() The slope of the IS curve.

(zz) The effects of changes in G and M on for a given P.

5.4. The liquidity trap and the Pigou effect. Assume that the nominal interest rate is so low that the opportunity cost of holding money is negligible. Suppose thai as a result people are indifferent concerning the division of their wealth between money and other assets, and that they are therefore willing to change their money holdings without any change in the interest rate.

(a) The liquidity trap. (Keynes, 1936.) In this situation, what is the slope of the AD curve? If prices are completely flexible (so the AS curve is vertical), is aggregate demand irrelevant to output?

(b) The Pigou effect. (Pigou, 1943.) Suppose that, in addition, planned expenditure depends on real wealth as well as the variables in (5.4). Since the publics holdings of high-powered money are one component of wealth, a fall in the price level increases real wealth. If prices are completely flexible (so the AS curve is vertical), is aggregate demand irrelevant to output?

5.5. The Mundell effect. (Mundell, 1963.) In the IS-LM model, how does a fall in expected inflation, tt", affect z, , and z - ?

5.6. The multiplier-accelerator. (Samuelson, 1939.) Consider the following model of income determination. (1) Consumption depends on the previous periods income: C, = a -\-bY, ]. (2) The desired capital stock (or inventory stock) is proportional to the previous periods output: K* = cTf-i. (3) Investment equals the difference between the desired capital stock and the stock inherited from the previous period: I, = K* ~ Kt-i = K* - cYt-2. (4) Government purchases are constant: Gf = G. (5) = Ct -i- It -i- Gt.

(a) Express in terms of ( 1, Yt-z, and the parameters of the model.

(b) Suppose b = 0.9 and = 0.5. Suppose there is a one-time disturbance to government purchases; specifically, suppose that G is equal to G - 1 in

period f and is equal to G in all other periods. How does this shock affect output over time?

5.7. (This follows Mankiw and Summers, 1986.) Suppose thai the demand for real money balances depends on the interest rate, /, and on disposable income - ; in other words, suppose that the correct way to write the LM equation is M/P = LU, Y - T).

(a) With this change to the IS-LM-AS model, can one tell whether a tax cut (that is, a fall in ) increases or decreases output? Assume a closed economy.

(b) Redo part (a) assuming an open economy under the assumptions that the exchange rate is floating, exchange-rate expectations are static, and capital is perfectly mobile.

(c) Redo part (b) assuming a fixed exchange rate.

5.8. Describe how each of the following changes affect income, the exchange rate, and net exports at a given price level under: (1) a floating exchange rate and perfect capital mobility, (2) a fixed exchange rate and perfect capital mobility, and (3) a floating exchange rate and imperfect capital mobility. Assume static exchange-rate expectations, and assume that planned expenditure is given by the expression in n. 8.

(a) The demand for money at a given / and falls.

(b) The foreign interest rate rises.

(c) The country adopts protectionist policies, so that net exports at a given real exchange rate are higher than before.

5.9. Exchange-market intervention. Suppose that the central bank intervenes in the foreign exchange market by purchasing foreign currency for dollars, and that it sterilizes this intervention by selling bonds for dollars to keep the money stock unchanged. With this intervention, NX and CF must sum to a positive amount rather than to zero (see equation [5.21]).

(a) What are the effects of this intervention on output, the exchange rate, and the price level under a floating exchange rate, static exchange-rate expectations, and imperfect capital mobility?

(b) How, if at all, do the results in part {a) change if capital is perfectly mobile?

5.10. The algebra of exchange-rate overshooting. Consider a simplified open-economy model: m - p = hy ~ ki,y = b{e - p) - a(t - p), i = a, p = . The variables y,m,p, and £ are the logs of output, money, the price level, and the exchange rate, respectively; i is the nominal interest rate, and p is inflation. All variables are expressed as deviations from their usual values; p* and i * are normalized to zero, and are therefore omitted. The main changes from our usual model are that price adjustment takes a particularly simple form and that the equations are linear. h,k, b, a, and are all positive.

Assume that initially y=i=p = m= p = 0. Now suppose that there is a permanent increase in m.

(a) Show that once prices have adjusted fully (so p = 0), = i =0 and p = e = m.

(b) Show that there are parameter values such that at the time of the increase m m, £ jumps immediately to exactly m and then remains constant-so that there is neither overshooting nor undershooting.2

5.11. Consider the model of aggregate demand in an open economy with imperfect capital mobility in Section 5.3, without the simpliftcation assumed in equation (5.22). In addition to our usual assumptions, assume JVAp*/ > £fp*/p, nx,-c > 0, nxy < 0, and £v - NXy < 1.

(a) Derive an expression for the slope of the 15** curve (that is, the combinations of I and Y associated with the (j, Y, e) combinations that solve [5.12] and [5.21]).

(b) Does £ rise, fall, or remain constant as we move down the IS** curve?

(c) Is it still true that greater capital mobility (that is, a larger value of Cf(*)) makes the IS** curve flatter?

5.12. The analysis of Case 1 in Section 5.4 assumes that employment is determined by labor demand. A more realistic assumption may be that employment at a given real wage equals the minimum of demand and supply; this is known as the short-side rule.

(a) Draw diagrams showing the situation in the labor market under this assumption when

(/) P is at the level that generates the maximum possible output, (ii) P IS above the level that generates the maximum possible output.

(b) With this assumption, what does the aggregate supply curve look like?

5.13. Consider the model ot aggregate supply in Case 2 ot Section 5.4. Suppose that aggregate demand at P equals Y. Show the resulting situation in the labor market.

5.14. Suppose that the production funcUon is Y = AF{I) (where F(*) > 0, f "(•) < 0, and A > 0), and that A falls. How does this negative technology shock affect the AS curve under each of the models of aggregate supply in Section 5.4?

5.15. Destabilizing price flexibility. (De Long and Summers, 1986b.) Consider the following closed-economy variant of the model in Problem 5.10: = -a{i-p), m - p = -ki.p = . Assume a > Q,k > 0, > 0, and < 1.

(a) Assume that initially y = i= p = m= p = 0. Now suppose that at some time-time 0 for convenience-there is a permanent drop in m to some lower leveL m.

2The result that there are parameter values such that the exchange rate neither overshoots nor undershoots m response to a monetary disturbance implies that, except in unusual cases, there are perturbations of these parameter values that lead to each result. Showmg this is complicated, however, and is therefore omitted.