14.02 Principles of Macroeconomics Problem Set 2 Fall 2005 ***Solution***

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1 4.02 Principle of Macroeconomic Problem Set 2 Fall 2005 ***Solution*** Pote: Weneay, September 2, 2005 Due: Weneay, September 28, 2005 Pleae write your name AND your TA name on your problem et. Thank! Exercie I. True/Fale/Uncertain? Explain. A ecreae of the reerve ratio ϑ prouce the ame effect on the interet rate a an expanionary open market operation. Uncertain. For a given proportion of money that people want to hol in currency (c), a ecreae in ϑ increae the money multiplier an therefore the money upply. The LM curve hift own, a it woul o if there were a monetary expanion. However, you may alo think that a ecreae in ϑ increae the proportion of money that people want to hol in currency (c). Thi coul be jutifie by the lower warranty on their epoit. In thi cae, the combine effect of a ecreae in ϑ an an increae in c ha an uncertain effect on the money upply an a clear comparion with a monetary expanion cannot be mae. 2. The afer an more efficient the banking ector i coniere to be, the higher the money multiplier. True. For any level of ϑ < afer an more efficient financial market make it poible to hol a lower amount of money for a given amount of tranaction. The proportion of money hel in currency c ecreae an the money multiplier increae. (Pleae ee c + ϑ( c) alo the Focu box on page 70 an 7 of the textbook for a icuion of thi iue.). The bon that offer the highet interet rate ha to be the mot expenive one. Fale. Accoring to the pricing formula for bon (here the example for a one year bon that $00 pay $00 next year) $P =, the higher the interet rate, the lower the price of the ( + i) bon.

2 4. An increae in conumer confience hift the LM curve own an the IS curve to the right. Fale. An increae in conumer confience correpon to an increae in the value of the parameter c 0. Autonomou pening increae an the IS curve move right. However, the money market i not affecte (beyon the effect on output), an the LM tay where it wa (ince the effect on output i reflecte in a movement along the LM curve). 5. A monetary expanion i more effective in changing the interet rate when money eman i very enitive to the interet rate. Fale. The enitivity of money eman to the interet rate i repreente by the lope of the money eman curve. The flatter the curve i, the more enitive the eman i ince a mall change in the interet rate caue a big change in the amount of money emane. When the money eman curve i flat, the ame change in money upply ha a maller impact on the interet rate than when the money eman curve i teep. Intuitively, the interet rate ha to change le to make money eman equal the new money upply ince money eman i very enitive to the change in interet rate. Exercie II. The Money Market Aume that the following i true about the economy: Money Deman: M = $ Y ( 0. 4 i) There i a one-year-bon that promie a payment of $2, where i i the interet rate on the bon. Nominal Income: $ Y = 00 The only bank i the central bank. Money Supply: M = 220. What i the money eman when the interet rate i 4%? When i=20%? For which interet rate i the money eman lower? Explain why. M M = 00 *( ) = 00*0.6 = 96 ( ) = 00* ' = 00* = The higher the interet rate i, the higher the cot of keeping money intea of bon (the opportunity cot of money). Money eman i thu lower at the higher interet rate.

3 2. What i the price of the bon when i = 4%? When i = 20%? Explain. If i=4% 0.04 = 2 P P.04P = 2 =.54 P If i=20% 0.2 = 2 P P.2P = 2 = 0 P $2 i the face value of the one year bon or, in other wor, the amount of money pai face value when the bon expire. The general formula for one year bon i$ P =. + i The price of the bon ecreae when the interet rate increae.. Draw a graph for money upply an the eman an calculate the equilibrium i. i Money M = M 220 = 00* 0. 4 i 220 = i 00 i = 220 i = 0.2 i = 20% ( )

4 4. Suppoe the central bank increae the money upply by 0. Graph an calculate the new equilibrium i. i Money ( 0. ) 0 = 00* 4 i 0 = i 00 i = 0 i = 0. i = 0% 5. Keep M = 0. To what amount woul nominal GDP have to change to have a money market equilibrium level conitent with an interet rate i equal to 20%? Explain. M = M 0 = $ Y * ( ) 0 = $ Y * 0.2 $ Y = 650 In orer to have a money market equilibrium with i = 20% an M = 0, Y houl be quite high: $650. Thi epen on the enitivity of the money eman to the interet rate. Kept contant the money upply an ouble the interet rate, equilibrium income increae by 50%.

5 Exercie III. Money Multiplier Keep the ame money eman an the nominal income a initially given in Exercie II. Now imagine that there i a banking ector collecting epoit. The central bank require a reerve ratio of ϑ = 50%. People want to keep one thir of their money eman a currency, an the ret a epoit. The upply of central bank money i H = 220. ) Calculate the money multiplier, CU, D, R, H (the eman for central bank money). Compare the equilibrium i with the one obtaine in the previou exercie in part. Explain. Money multiplier: M = M cm + ϑ c M = H c + c ϑ ( ) ( ) [ c + ϑ ( c) ] M = H [ c + ϑ ( c) ]*00*( 0. 4 i) = H 2 *00*(0.4 i ) = *( 0.4 i ) = i = 0 0 = 00i i = 0. i = 0% M = ( ) = M = 0 0 CU = = 0 2 D = *0 = R = ϑ D = * *0 = 0 2 H = CU + R = = * 2 = 2 =.5 The equilibrium interet rate i = 0. i only half of the one foune in part of the previou exercie. The reaon i that the preence of a banking ector lea to a bigger multiplication of central bank money through the money multiplier. The money upply increae, an therefore the equilibrium interet rate (the rewar that ha to be given for holing bon intea of currency) i lower.

6 2) Suppoe the central bank money upply i ecreae by 22. Calculate the new equilibrium i. 2 *00*(0.4 i ) = * ( 0.4 i ) = i = = 00i i = 0. i = % ) Keep H = 98. What happen to i if the nominal GDP ecreae by 275? Explain. 2 *825*(0.4 i ) = *( 0.4 i ) = i = 297 = 825i i = 4% A ecreae in nominal GDP lower money eman, which for a given level of money upply lea to a ecreae in the interet rate. The interet rate ha to fall to give people an incentive to hol money intea of bon, uch that it raie money eman to the level where it i equal to money upply. 4) How can the central bank retore $ Y = 00 keeping part ) of thi exercie? The Central ank can change the reerve ratio a follow: ( ) 98 x * 00* = x * 00*0.6 = 98 x * 96 = 98 2 x = = c + ϑ ( c) = +ϑ 2 = 2 + 4ϑ ϑ = 4 H an i at the ame level ue in

7 Exercie IV. IS-LM Moel Aume C = *Y D I = * Y 400i T = 200 G = 00 H = 200 P CU = c = 0.4 M ϑ = 0.25 M / P = 2.5Y 7000 ( ) i ) Derive the IS relation. ( Y 200) Y Y = * i + Y = Y 400i Y = i i = 600 Y / ( ) 800 2) Derive the LM relation. M = M M M = P P H = c + ϑ c $ YL i H ( ( )) ( ) = ( c + ϑ ( c) ) YL( i) P 200 = 0.55* 2.5Y 7000i i = 2. 5Y Y = i i = Y / ( ) ( ) 2800

8 ) Solve for equilibrium real output, interet rate, C an I an graph the IS-LM iagram. i IS LM Y Y = i Y = i i = i 600 i = i = i = 40.4% Y = C = * ( ) C = I = * * 0.4 I = 295.6

9 4) The government ecie to increae it eficit: G increae by 90 to 90. Draw a graph an ecribe what happen in the economy. Fin equilibrium Y, i, C, an I. i IS IS LM Y The IS curve hift to the right. Y = i i = i 600 i = i = Y = C = * ( ) C = I = * * I = 0.7 5) What happen to our moel if the invetment i alo a function of ipoable income Y D, intea of Y. Dicu thi aumption an fin the equilibrium i an Y before an after the fical expanion. Auming the invetment epen on ipoable income we allow for the taxe to affect the equilibrium outcome in two channel: conumption an invetment. Since taxe alo lower invetment now, in aition to conumption, the IS curve i further to the left than in the moel where taxe only affect conumption. ( Y 200) ( Y 200) Y = * i + Y = Y 400i Y = i i = 520 Y / ( ) 800

10 Y = i Y = i i = i 600 i = i = 8.2% Y = After Y = i Y = i i = i 600 i = i = 0.42 Y = ) What happen in the tanar etting, with I=I(Y,i), not I=I( Y D,i), if the fical expanion i riven through a ecreae in taxe T from 200 to 0? Y = Y 400i Y = i Y = i i = i 600 i = i = 0.49 Y = ) Compare your anwer to part 6 with that to part 4. Even if G = T = 90, the fical expanion prouce ifferent reult if it i implemente through an increae of government pening or through a ecreae in taxe. The effect of the ecreae in taxe i mitigate by the fact that one ollar le of taxe oe not mean one ollar more of conumption, but jut 50 cent (ince part of ipoable income i ave). Intea one ollar more of public expene G increae the autonomou pening by exactly one ollar.