Deparmen of Eonomis Boson College Eonomis 0 (Seion 05) Maroeonomi Theory Praie Problem Se 7 Suggesed Soluions Professor Sanjay Chugh Fall 03. Lags in Labor Hiring. Raher han supposing ha he represenaive firm a he beginning of period an deide how muh labor i would like o hire for use in period, suppose ha labor used in period mus be hosen in period -. (Tha is, suppose n is a sok (aka sae) variable.) As usual, apial for use in produion in period mus be purhased in period - beause of he ime o build surrounding apial goods. Wih his lag in labor hiring, onsru he lifeime (in he wo-period model) profi funion of he firm, and show ha he real ineres rae now is a relevan prie for labor as well as apial goods. Provide brief eonomi inuiion. (Hin: Make as lose an analogy wih our model of firm ownership of apial as you an in pariular, hink of workers in his model as being owned (onraually obligaed o) firms.) Soluion: Wih employees being onraually bound o ( owned by ) firms, he period- nominal profis of a firm are given by PR = P f ( k, n) + Pk + Pwn Pk + Pwn +, in whih labor used in produion in period, n, is hosen in period - (and hus labor used in produion in period +, n +, is hosen in period. In analogy wih our model wih only apial pre-deermined, he employees of a firm are a valuable asse, wih oal marke value Pwn -- noie ha his erm eners posiively in period profis, raher han negaively wih non-pre-deermined labor. Wha eners negaively in period profis here is he purhase of period + labor, namely he erm Pwn +. In he wo period model, disouned nominal profis of he firm are herefore P f( k, n ) Pk Pwn Pk Pwn PR = Pf ( k, n) + Pk + Pwn Pk Pwn + + + + i + i + i + i + i 3 3 i i i i i The usual zero-erminal-asses ondiion in his ase means ha k 3 = 0 and n 3 = 0 (he laer, again, beause labor should be hough of as an asse here). Fousing aenion on he hoie of n (sine n was hosen in period -), he firs-order ondiion of he lifeime profi funion wih respe o n is
P f ( k, n ) Pw n Pw + + i + + i = 0. This expression an be rearranged o yield (using he exa Fisher equaion) ( + r) w = fn( k, n) + w. If he real wage were equal o one in eah period, his ondiion would redue o r = f (, ) n k n, whih would be almos idenial o he ondiion we derived in lass regarding apial demand (exep of ourse in ha ase f k is he relevan marginal produ raher han f n ). The expression r = f (, ) n k n shows ha if firms mus hoose labor for period in period, he real ineres rae beween period and period is a relevan prie o onsider whih makes sense beause here is now an ineres opporuniy os assoiaed wih hiring labor (ie, invesmen in hiring). However, in general of ourse w and w are no one, hene he above ondiion is no exaly he same as he apial demand ondiion. In he apial demand ondiion, he real prie of apial goods is he same as he real prie of onsumpion (whih is one ) noe he disussion on p. 70-7 of he Leure Noes desribing ha beause apial goods and onsumpion goods are assumed o be he same goods (ie, ompuers an be viewed as boh onsumpion goods and apial goods), he dollar prie of eah in our heoreial model is he same. The same is no rue of labor he nominal prie of labor is W, whih in general is differen from P.. Preferene Shoks in he Consumpion-Savings Model. In he wo-period onsumpion-savings model (in whih he represenaive onsumer has no onrol over his real labor inome y and y ), suppose he represenaive onsumer s uiliy funion is u(, B ), where, as usual, denoes onsumpion in period, denoes onsumpion in period, and B is a preferene parameer. a. Use an indifferene-urve/budge-onsrain diagram o illusrae he effe of an inrease in B on he onsumer s opimal hoie of period- onsumpion. Soluion: An inrease in B means eah uni of period- onsumpion delivers more uiliy o he onsumer. Thus, in uiliy erms, period- onsumpion has now beome more valuable relaive o period- onsumpion, implying ha in order o say on a given indifferene urve he onsumer now needs o give up fewer unis of in order o ge one more uni of. In a diagram wih on he verial axis and on he horizonal axis, his is represened by a flaening of he indifferene map. Beause he LBC is unaffeed, he flaening of he indifferene map means ha he new opimal hoie feaures smaller period- onsumpion and hene larger period- onsumpion, as shown in he aompanying diagram. As drawn, onsumpion in period is smaller han real inome in period, bu ha is irrelevan.
New opimal hoie Iniial opimal hoie Indifferene urves wih high value of B y Indifferene urves wih low value of B y b. Illusrae he effe of an inrease in B on he privae savings funion. Provide eonomi inerpreaion for he resul you find. Soluion: We an dedue he effe on privae savings in period using he diagram in par a above. The real ineres rae has no hanged (in oher words, he slope of he LBC has no hanged), ye he represenaive onsumer s savings in period has inreased. This follows direly from he observaion ha inome y is onsan while onsumpion in period falls. This resul would be rue for any hoie of he real ineres rae (in oher words, no maer he slope of he LBC), hene he privae savings funion shifs ouwards, as shown below. 3
s(r) s'(r) r shif ou due o rise in B Period- Savings. In he monhs preeding he U.S. invasion of Iraq, daa shows ha onsumers dereased heir onsumpion and inreased heir savings. Is an inrease in B and he effes you analyzed in pars a and b above onsisen wih he idea ha onsumpion fell and savings inreased beause of a looming war? If so, explain why; if no, explain why no. Soluion: Yes, hese effes are onsisen wih developmens in onsumpion and savings behavior in he U.S. leading up o he invasion of Iraq. An inerpreaion we an give using he model here is ha onsumers believed fuure maroeonomi ondiions would be beer han urren (i.e., jus before he war) maroeonomi ondiions, hene a fall in onsumpion in he presen (period ) aompanied by a (expeed) rise in onsumpion in he fuure (period ). Wih B pre-muliplying onsumpion in he uiliy funion (in he ase here, period- onsumpion), he erm B an be inerpreed as a measure of onsumer onfidene : a rise in B signals ha onsumers are shifing heir preferenes owards onsumpion (in ha period). So here, we migh inerpre evens as onsumers being more onfiden abou he fuure han he presen, hene hey pospone some onsumpion unil he fuure. d. Using a Lagrangian and assuming he uiliy funion is u (, B ) = ln( ) + ln( B ), show how he represenaive onsumer s MRS (and hene opimal hoies of onsumpion over ime) depends on B. Soluion: Seing up he Lagrangian in he wo-period model as always, we have 4
y ln( ) + ln( B ) + λ y+ r r, + + in whih for simpliiy we have assumed he iniial asses equal zero beause i does no a all affe he onsumpion-savings opimaliy ondiion (verify his yourself). The FOCs on and are, respeively, λ = 0 B λ = 0 B + r In he FOC on, noe ha he B erm ends up aneling ou (beause, reall, he derivaive of an expression suh as ln( x ) is /( x) = / x). Combining hese wo FOCs as usual hen yields ha a he opimal hoie, / r / = +, he lef-hand-side of whih is he ineremporal MRS, as always. Noe ha i is independen of he preferene shifer B, whih urns ou o be a speial feaure of he log uiliy funion. e. How would your analysis in pars a and b hange if he onsumer s uiliy funion were u( D, ) (insead of u(, B ) ) and you were old ha he value D dereased? ( D is simply some oher measure of preferene shoks.) Soluion: Here, we reurn o a general uiliy speifiaion, no neessarily log. Wih he uiliy funion wrien as u( D, ) and a derease in D, he analysis above is ompleely unhanged. The fall in D makes onsumpion in period less valuable in uiliy erms relaive o period- onsumpion, whih means ha in order o obain one more uni of period- onsumpion while remaining on he same indifferene urve he onsumer mus give up more unis of period- onsumpion han he had o before he fall in D. Bu in a diagram wih on he verial axis and on he horizonal axis, his simply means ha he indifferene urves beome flaer, jus as in par a. 5
New opimal hoie Iniial opimal hoie Indifferene urves wih low value of D y Indifferene urves wih high value of D y This exerise auions you o hink abou he underlying eonomis speifially, how he onsumer s marginal rae of subsiuion (refer o Chaper ) is affeed when analyzing preferene shoks. We anno make a blanke saemen suh as he indifferene map flaens when he measure of he preferene shok inreases beause i depends on exaly how we inrodue he preferene shok ino our heoreial model. Here in par d we inrodued he preferene shok by aahing i o period- onsumpion, whereas earlier we inrodued he preferene shok by aahing i o period- onsumpion. 3. Ineremporal Consumpion-Leisure Model A Numerial Look. Consider he ineremporal onsumpion-savings model. Suppose he lifeime uiliy funion is given by v( B, l, B, l) = u( B, l) + u( B, l), whih is a sligh modifiaion of he uiliy funion presened in Chaper 5. The modifiaion is ha preferene shifers B and B ener he lifeime uiliy funion, wih B he preferene shifer in period one and B he preferene shifer in period wo. In eah of he wo periods he funion u akes he form u( B, l) = B + l. Noe he subsrips -- =, depending on whih period we are onsidering. Labor ax raes, real wages, he real ineres rae beween period one and period wo, and he preferene realizaions are given by: = 0.5, = 0., w = 0., w = 0.5, r = 0.5, B =, B =.. Finally, he iniial asses of he onsumer are zero. Soluion: Noe ha you needed o ompue he marginal uiliy funions. For he given lifeime uiliy funion, he marginal uiliy funions are, for =, : 6
v B = ; vl = l a. Consru he marginal rae of subsiuion funions beween onsumpion and leisure in eah of period one and period wo (Hin: hese expressions will be funions of onsumpion and leisure you are no being asked o solve for any numerial values ye). How does he preferene shifer affe his inraemporal margin? Soluion: As by now is rouine, he onsumpion-leisure marginal rae of subsiuion funion is MRSl = v / l v. Wih he given funions, he marginal rae of subsiuion funion in period, where is eiher or, is hus MRSl (, ) l =. B l Again, noe ha his funion is he MRS funion for period =,. From his funion i is lear ha a rise in B lowers his MRS, meaning a rise in B flaens he indifferene map over onsumpion and leisure wihin a given period. b. Consru he marginal rae of subsiuion funion beween period-one onsumpion and period-wo onsumpion. (Hin: Again, you are no being asked o solve for any numerial values ye.) How do he preferene shifers affe his ineremporal margin? Soluion: Again as by now should be rouine, he ineremporal MRS funion is given by MRS = v / v. Noe he subsrips: v denoes he marginal uiliy funion wih respe o period-one onsumpion, and v denoes he marginal uiliy funion wih respe o period-wo onsumpion. Using he given v funion, we have B MRS (, ) =. B The raio of B values aross he wo periods affes he slope of he indifferene map beween period-one and period-wo onsumpion. The larger is he raio B / B, he seeper is he indifferene map aross onsumpion in he wo periods he inerpreaion of his is ha he larger is B relaive o B, he more onfiden (reall our inerpreaion of B from lass) onsumers are abou he presen (period one) han hey are abou he fuure (period wo), hene he more period-wo onsumpion hey are willing o give up for a given inrease in period-one onsumpion (whih is our usual inerpreaion of he slope of an indifferene urve wih ploed on he horizonal axis and ploed on he verial axis). 7
. Using he expressions you developed in pars a and b along wih he lifeime budge onsrain (expressed in real erms ) and he given numerial values, solve numerially for he opimal hoies of onsumpion in eah of he wo periods and of leisure in he wo periods. (Hin: You need o se up and solve he appropriae Lagrangian.) (Noe: he ompuaions here are messy and he final answers do no neessarily work ou niely. To preserve some numerial auray, arry ou your ompuaions o a leas four deimal plaes.) Soluion: The LBC in real erms is ( ) (68 ) ( ) (68 ) w + = w l + l. (0.) + r + r This expression follows readily from expression (34) on p. 60 of he Leure Noes (i s probably a good idea o derive his from expression (34) if you don see i immediaely), wih zero iniial asses imposed. This LBC involves he four unknowns,,, l, and l, whih are he variables you are asked o solve for. We need hree oher expressions involving hese variables hese hree are he wo onsumpion-leisure opimaliy ondiions (one for eah of period one and period wo) and he one onsumpion-savings opimaliy ondiion. By now you should know how hese opimaliy ondiions an be obained by formulaing he appropriae Lagrangian for ease of exposiion he Lagrangian is omied here. Suffie i o say i is simply he above onsumpion-leisure and onsumpion-savings opimaliy ondiions ha emerge from he Lagrangian. The onsumpion-leisure opimaliy ondiions for period one and period wo and he onsumpion-savings opimaliy ondiion are, respeively, MRS l = = ( ) w, (0.) B l = =, (0.3) MRSl ( ) w B l MRS B = = + r. (0.4) B By now you should know he inerpreaion of hese opimaliy ondiions: hey simply represen he angeny beween a relevan budge onsrain and a relevan indifferene urve. Equaions (0.), (0.), (0.3), and (0.4) are now four equaions in he four unknowns,, l and l, so we an solve wih some algebrai effor. Le s deide o express he unknowns, l, and l all in erms of. One we have done his, we an subsiue ino he LBC and solve for. From (0.4), we ge ha 8
from (0.), we ge ha B = ( + r) ; (0.5) B l = ; (0.6) ( ) w B and from (0.3) we similarly ge ha l =. (0.7) ( ) w B In (0.7), we need o subsiue ou using (0.5) (beause, reall, we are rying o express he unknowns in erms of ), giving us + r l =. (0.8) ( ) w B Now, subsiue ino he LBC using (0.5), (0.6), and (0.8). Doing so and olleing all he resuling erms involving on he lef-hand-side (you should perform hese algebrai seps yourself.) gives us B + r 68( ) w + ( + r) + + 68( ) w B ( ) wb ( ) wb = +, (0.9) + r in whih he only unknown, as desired, is. Insering all of he given numerial values, * * we finally find ha = 4.33. Then using (0.5), (0.6), and (0.8) we find = 6.5437, * l = 4.6754, and l * = 36.37. The individual hus works 68 4.6754 = 5.346 hours per week in he firs period and 68 36.37 = 3.678 hours per week in he seond period. d. Based on your answer in par, how muh (in real erms) does he onsumer save in period one? Wha is he asse posiion ha he onsumer begins period wo wih? Soluion: Reall ha real privae savings (inlusive of axes is) inome minus ax paymens minus onsumpion. Given he soluion above, oal real inome in period one is (68 l) w = 5.0649, of whih he amoun paid in axes is (68 l ) w = 0.7597. Disposable inome (gross inome less axes) in period one is hus 5.0649 0.7597 = 4.305. Subraing period-one onsumpion, we have ha real savings in period one is 4.305 4.33 = 0.89. Beause he onsumer began period one wih zero asses, a he end of period one his real asse posiion is hus 0.89. (Then, wih posiive asses o begin period wo, he individual is able o onsume more han his inome in period wo perform his alulaion o verify his for yourself.) e. Suppose B were insead higher, a.6. How are your soluions in pars and d affeed? Provide brief inerpreaion in erms of onsumer onfidene. 9
Soluion: Examining he soluion (0.9), we see ha B eners he soluion in only one plae. I is easy o onlude from (0.9) ha a higher value of B will lead o a lower * value of opimal period-one onsumpion. Speifially, = 3.993, whih hen implies * = 8.4476, l * = 38.405, and l = 3.994. Wih B higher relaive o B (and wih he pariular way B eners he uiliy funion, speifially, muliplying ), he onsumer is more onfiden abou he eonomi sae in he fuure (period wo) han in he presen (period one). He hus works and onsumes less in period one, and works and onsumes more in period wo due o he rise in B. Savings in period one rises o ( ) w(68 l) =.0839, onsisen wih he inreased desire o pospone onsumpion unil period wo. 0