Solutions to First Midterm

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1 rofessor Chrstano Economcs 3, Wnter 2004 Solutons to Frst Mdterm. Multple Choce. 2. (a) v. (b). (c) v. (d) v. (e). (f). (g) v. (a) The goods market s n equlbrum when total demand equals total producton,.e. = Z. n the standard S- model, Z = C d + d + G d = c 0 + c ( T ) + b + G: So usng the equlbrum condton, the equlbrum output as a functon of the exogenous parameters and of the rate of nterest, s: = c c 0 c T + + G b : (b) For a gven nterest rate, the total desred spendng curve s: Z( ) = c 0 + c ( T ) + b + G; so that the vertcal ntercept s c 0 c T + b + G; whle the slope s c : (n the gure, goods market equlbrum s ndcated by (), to remnd ourselves t s determned for a gven nterest rate.) (c) The nancal market s n equlbrum when money demand equals money supply,.e.: M d Solvng the equlbrum condton: M s = M s. = + L L for, = L + L M s ; where all varables on the rght hand sde are parameters but for the level of output. (n the gure, the money market equlbrum nterest rate s ndcated by ( ), to remnd ourselves that t s determned for a gven level of ncome). (d) The demand for money, expressed n terms of the nterest rate s: = L + L M d ; so that the vertcal ntercept s: L + L whle the slope s L.

2 (e). The S curve represents combnatons (; ) such that the goods market s n equlbrum. The S curve has a negatve slope because, as the nterest rate ncreases, desred nvestment decreases and so does goods market equlbrum output (wth the latter change beng larger n absolute value than the former, because of the multpler e ect). The curve represents combnatons (; ) such that nancal markets are n equlbrum. The curve has a postve slope because as ncome ncreases, money demand ncreases and bond demand decreases for a gven nterest rate. But snce money supply and bond supply are xed, equlbrum n nancal markets requres an ncrease n the nterest rate to ncrease bond demand and reduce money demand back to ther ntal equlbrum levels.. Above (below) the S there s excess supply (demand) n the goods market. For a gven ; the nterest rate s hgher (lower) than requred for = Z. Because desred nvestment s decreasng n the nterest rate, desred nvestment s low (hgh), and so > Z ( < Z). The stock of unntended nventory nvestment s postve (negatve).. Above (below) the there s excess supply (demand) n the money market. For a gven ; the nterest rate s hgher (lower) than requred for M s = M d. Because money demand s decreasng n the nterest rate (bonds are more attractve), real money demand s smaller (greater) than money supply, M s < M d. v. f G ncreases by G > 0; the S curve shfts rghtward. The magntude of the rghtward shft s gven by the change n equlbrum n the goods market for a gven nterest rate. n the standard S/ model, ths change s always bgger than the equlbrum change n, snce as ncreases, the equlbrum nterest rate must also ncrease to guarantee market clearng n nancal markets, nducng a decrease n nvestment demand. However, when calculatng the rghtward shft of the S-curve, we are not requrng nancal markets to be n equlbrum: The nterest rate s taken as gven. From the prevous answers: = c c 0 c T + + G b ; so that the rghtward shft s: G c for any value of the nterest rate (parallel shft). v. Dsequlbrum dynamcs nvolves two thngs: whch market varable adjusts and how fast. Output shfts slowly n the goods market and the nterest rate shfts extremely quckly n the nancal market. As a result of these assumptons along the adjustment path (; ) s always on the -curve,.e. nancal markets are always n equlbrum. Because of the expansonary scal shock, the S shfts rghtward by G c to S 0 ; ntersectng the at ( ; ) where < < + G c and > : Because of the slow adjustment n the goods market, the economy does not jump drectly to ( ; ): mmedately after the shock, Z = + G > : There s excess demand n the goods market and hence a decrease n nventores. Frms adjust ther producton upwards accordngly. As they do so, ncome ncreases, and ths n turn ncreases money demand, requrng nterest rate to ncrease at the same tme n order to mantan equlbrum n nancal markets Ths process of output and nterest rate adjustment contnues untl output and nterest rates reach ( ; ) : (Along the adjustment path, the rse n the nterest rate reduces desred nvestment demand, but there remans a (decreasng) excess aggregate desred demand as we are always at (; ) pars below the new S-curve.). 2

3 c 0 c T + + G b ntercept slope=c Z ntercept / L + L ( ) slope=-/l M s ( ) o 45 () Goods Market Money Market M d M S' c G "S-" Dagram S 3. (a) The equlbrum condton n the Keynesan Cross model s Solvng ths equaton for equlbrum yelds The multpler on nvestment s =! = Z c 0 + c T + + G = c c0 c T + + G c [c 0 c T +++G] c [c 0 c T ++G] (b) The equlbrum condton n the Keynesan Cross model wth t > 0 s t >0 = c :! = Z = c 0 + c t >0 T t t >0 + + G Solvng ths equaton for equlbrum yelds t >0 = c ( t ) c0 c T + + G 3

4 The multpler on nvestment s now t >0 = c ( t ) : Snce 0 < t < ; c ( t ) < c,.e. the nvestment multpler s smaller f t > 0 (and t s decreasng n t ). ntuton: Taxes reduce consumpton and hence demand. f taxes ncrease wth ncome (t > 0), they wll also decrease the e ect of a change n ncome on consumpton and hence demand. Ths wll dampen the multpler e ect of a change n autonomous expendture on equlbrum output. (c) The equlbrum condton n the Keynesan Cross model wth q > 0 (and t = 0) s q>0! = Z = c 0 + c q>0 T + + q + G Solvng ths equaton for equlbrum yelds q>0 = c q c0 c T + + G The multpler on nvestment s now q>0 = c q : Snce q > 0, c q > c (as long as q < c ),.e. the nvestment multpler s larger for q > 0 (and ncreasng n q). ntuton: f nvestment ncreases wth ncome (q > 0), the multpler process s ampl ed. The multpler e ect on equlbrum ncome of an ncrease n autonomous expendtures wll work through not only consumpton but also nvestment. Hence the nvestment multpler s larger for q > 0 (Accelerator E ect). (d) The magntude of the horzontal shft of the S curve s always equal to the product of the correspondng Keynesan Cross-Multpler and the change n autonomous expendture. Ths would be the e ect for a gven nterest rate. Equlbrum changes n output wll however nvolve changes n the nterest rate as nancal markets must also be n equlbrum. f b = 0;nvestment and hence aggregate demand does not depend on the nterest rate. Equlbrum output s solely determned n the goods market: The S curve s vertcal. Even though the equlbrum nterest rate rses n response to > 0 to restore equlbrum n nancal markets at a hgher ; the nvestment multpler s dentcal to the one of the Keynesan Cross model. f nvestment depends negatvely on the nterest rate f b > 0, the equlbrum response of output to wll be smaller than n the case b = 0;because hgher output requres a hgher nterest rate for the nancal markets to clear. (n the gure, ntal equlbrum s drawn at the same (; ) par to smplfy the exposton) 4

5 b= 0 S b=0 S b=0 b>0 b>0 b b>0 = = 0 c b= 0 "S-" Dagram S b>0 S b >0 4. (a) n decdng whether to nvest n a gven project, rms have to decde on what rate of return the project wll generate. Ths n turn depends on how actve the economy s. f rms expect the level of actvty n the economy to be hgh, then they wll expect a hgh rate of return n other words, they form ther expectaton of how hgh economc actvty wll be by lookng at current output. So, the reason q > 0 s that n ths case, a hgher causes rms to revse upward ther assessment of the rate of return on all projects. (b) Each rm has a bunch of potental projects layng around, each wth an expected rate of return. To nvest n a project, they have to convnce someone to gve them the cash. That person has other stu to do wth the cash (ths s measured by ), and so the rm manager won t even make an attempt at a project f ts rate of return s less than. For any rm, the nvestment demand curve s a downward sloped step functon. As you aggregate over the whole economy the steps become ner untl t s just a smooth downward sloped curve. So aggregate nvestment negatvely depends on the nterest rate,.e. b > 0: 5. (a) A decrease n the money supply (or a postve shock to the lqudty demand parameter L) would shft the curve upward. The new equlbrum n the goods and nancal market nvolves a hgher nterest rate ( > ) and lower output ( < ). The drecton of the adjustment 5

6 re ects the assumptons on the dsequlbrum dynamcs. ' "S-" Dagram S (b) A decrease n any component of autonomous expendture would shft the S curve leftward and the new equlbrum would nvolve both a lower nterest rate ( < ) and lower output ( < ). The drecton of the adjustment re ects the assumptons on the dsequlbrum dynamcs. S' "S-" Dagram S 6. n the S- model (a model of the short run), equlbrum output s determned by the demand sde of the economy (how much households, busnesses and governments want to buy). t s assumed that rms do whatever t takes to produce the output that demand requres. Now suppose that a postve technology shocks ncreases productvty n the producton sector. The technology shock does not a ect total demand (t does not appear anywhere n our desred consumpton, nvestment or government consumpton equatons). As productvty ncreases and aggregate demand s una ected, the number of workers shrnks, reducng employment (holdng hours per-worker xed). 7. Desred savng s S t = t C t T t = c0 + ( c)( t T ): As the loss of consumer con dence s unexpected by rms, = 0 ; so that,as a consequence of the fall n c 0 : S = S S 0 = c0 = $0: As the economy s n equlbrum n perod 0, total demand, whch falls by 0$ relatve to Z 0 ; s Z = Z 0 0$ = 0 0$ = 0$: So unntended nventory nvestment n perod one s u = Z = 0$: As the economy s n equlbrum n the perod 0 u = u u0 = u 0 = 0$. As desred nvestment s unchanged, the change n actual nvestment s a = 0$: 6

7 8. The equlbrum nterest rate s found by equatng money demand and money supply: M s = 00$ = M d = $00 (:05 ), so that = :05 or = 5%: Assumng that there are no assets other than money and bonds, wealth s nvested n ether of the two that s: W = M d + B d. From ths, and the fact that the equlbrum level of nomnal money s 00$; the equlbrum demand for bonds s 900$: 7

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