. a) C C b C C s the ntercept o the consumpton uncton, how much consumpton wll be at zero ncome. We can thnk that, at zero ncome, the typcal consumer would consume out o hs assets. The slope b s the margnal propensty to consume whch shows how senstve consumpton s to changes n ncome. A one unt ncrease n ncome wll ncrease nvestment by b unts. We have omtted some varables, e.g. asset holdngs and expected uture ncome. We can thnk o changes n those varables as changng the ntercept o the consumpton uncton. b) Answers to exercses n Macroeconomcs by Nls Gottres 23
I d I I I shows what nvestment wll be the nterest rate s zero. The parameter d shows how senstve nvestment s to changes n the nterest rate. A one unt ncrease n the nterest rate wll reduce nvestment by d unts. c) In order to nd the equlbrum level o ncome or a gven nterest rate, we substtute the consumpton uncton and the nvestment uncton nto the goods market equlbrum condton: = C + b + I d. We then collect the -terms on the let hand sde (we subtract b on both sdes): b = C + I d. Then we rewrte the let hand sde: b = C + I d. Dvdng by (-b) we get the soluton or : = b + C I d. Ths s the equaton or the IS curve. We can llustrate ths relaton: Answers to exercses n Macroeconomcs by Nls Gottres 23
d b IS d) I there s an exogenous ncrease n the wllngness to nvest represented by I we can nd the eect on producton by rewrtng d b b b = C + I and takng the dervatve d di =. b Provded that < b <, the total eect on producton s larger than the change n nvestment. e) The eect o the nterest rate s ound by takng the dervatve wth respect to : d d d =. b An ncrease n the nterest rate by one unt reduces nvestment by d unts and then we have the multpler eect on that. The bgger the eect o nterest on nvestment (d) and the bgger the margnal propensty to consume (c) the larger s to total eect on ncome. ) In order to set the nterest rate at some target level, the central bank must adjust the money supply (the monetary base) to equal demand at that nterest rate. Hence the role o equaton (4) s to determne the money supply. 2. Answers to exercses n Macroeconomcs by Nls Gottres 23
a) I the central bank keeps the money supply constant, we must treat the nterest rate as endogenous. Thus we need to use all our equatons to solve or. To do ths we rst solve equaton (4) or the nterest rate by rewrtng: M = e P e M = P. Ths s the equaton or the LM curve. It shows that, or a gven money supply, hgher producton leads to a hgher nterest rate. The reason s that hgher ncome ncreases the demand or money so the nterest rate has to ncrease to keep demand or money equal to the exogenously gven supply. LM e Takng the dervatve we get the slope o the LM curve: d d = e The coecent e shows the eect o ncome on money demand and the coecent shows how senstve money demand s to the nterest rate. The stronger the eect o ncome on money demand (hgher e) the more the nterest rate wll ncrease. The more nterest-senstve money demand s (hgher ) the less the nterest rate needs to ncrease to brng equalty between demand and (gven) supply o money at a hgher level o producton. b) We substtute the expresson or the nterest rate (the LM relaton) nto the nvestment equaton: e M de d M I = I d I P = +. P Then we substtute ths and the consumpton uncton nto the goods market equlbrum condton: de d M = C + b + I + P Answers to exercses n Macroeconomcs by Nls Gottres 23
Collectng the -terms on the let hand sde de b C I + = + + d M P rewrtng de b C I + = + + d M P And dvdng by the expresson n parenthess, we get the soluton or or a gven money supply: C I de b + = + + dm P. c) = I de b + = I de b + d) When the nterest rate s kept constant, the multpler s. The derence b s the term de n the denomnator when M s kept constant. Producton ncreases less when the money supply s kept constant because the nterest rate ncreases and ths has a negatve eect on nvestment. The exogenous ncrease n nvestment rases demand and producton. An ncrease n producton wll ncrease the need or means o transactons. To keep the nterest rate constant, the central bank must ncrease the supply o money. I the central bank does not ncrease the money supply, there wll be a shortage o money and the nterest rate wll ncrease. The ncrease n the nterest rate has a negatve eect on nvestment and dampens the ncrease n producton. Ths s llustrated n the gure below where the orgnal values o the varables are denoted, those ater the shock wth an unchanged nterest rate are denoted 2 and those ater the shock wth a constant money supply are denoted 3. The exogenous ncrease n nvestment I > shts the IS curve to the rght. I the nterest rate s kept unchanged, producton ncreases to 2 and the central bank must ncrease the money supply so the LM curve shts to LM 2. Answers to exercses n Macroeconomcs by Nls Gottres 23
I the money supply s kept constant, the LM curve s unchanged, producton ncreases rom to 3 and the nterest rate rom to 3. LM LM 2 3 = 2 IS 2 3 2 IS 3. a) Agan we solve or the equlbrum level o ncome by the substtuton method. Frst we substtute the tax uncton nto the consumpton uncton: C = C + b + T t. Then we substtute or consumpton and nvestment n the goods market equlbrum condton and solve or : = C + b + T t + I d + G = C + b + bt bt + I d + G b + bt = C + bt + I d + G ( ) b + bt = C + bt + I d + G = C + bt + b bt I d + + G. b) = G b + bt The coecent t s the margnal tax.e. how much taxes ncrease when ncomes ncrease one unt. A hgh margnal tax reduces the multpler eect. Part o the ncrease n ncome s taxed away so the ncrease n consumpton wll be smaller and thereore the second and thrd round eects o the ncrease n demand wll be smaller. Answers to exercses n Macroeconomcs by Nls Gottres 23
c) b = T b + bt An ncrease n T shts down the tax schedule. Consumers have hgher dsposable ncome and how much they spend o that depends on b, the margnal propensty to consume. The drect eect on demand s b T and then we have a multpler eect on that. 4. Let us assume that the ncrease n ncome s evenly dstrbuted among the derent types o consumers. Snce 2 percent o the consumers spend everythng they have they spend 2 mllon. The others get 8 mllon o whch 4 s perceved as permanent and 4 s perceved as temporary. Accordng to our theory, they should spend the permanent part,.e. 4 mllon. Accordng to the theory n Chapter 4, a temporary ncrease n ncome should be put n the bank and you should only consume the nterest on ths amount. I we assume that the real nterest rate s 3 percent, the consumers wll spend 3 percent o 4 mllon.e..2 mllon. Thus the total ncrease n consumpton wll be 2 + 4 +.3 4 = 2 + 4 +.2 = 6.2 mllon. Thus the margnal propensty to consume s about.6. 5. Aggregate demand s determned by (, e e,, ) (,, ) = C r A + I r K. a) Wth the new technology, rms see new nvestment possbltes and consumers want to replace ther old cars. Thus, consumpton purchases and nvestment wll both ncrease and the IS curve wll sht out. b) I the central bank keeps the money supply constant, producton wll ncrease. c) I producton was orgnally on the natural level, the central bank wll be worred that wages wll ncrease so there wll be hgh nlaton. Thereore, t s lkely that the central bank wll rase the nterest rate. Answers to exercses n Macroeconomcs by Nls Gottres 23