There are a total of two problems, each with multiple subparts.

Transcription

1 eparmen of Economcs Boson College Economcs 0 (Secon 05) acroeconomc Theory Problem Se Suggesed Soluons Professor Sanjay Chugh Fall 04 ue: ecember 9, 04 (no laer han :30pm) Insrucons: Clearly-wren (yped s srongly preferred, bu no requred) soluons mus be submed no laer han :30pm on he dae lsed above You mus subm your own ndependenly-wren soluons You are permed (n fac, encouraged) o work n groups no larger han hree members o hnk hrough ssues and deas, bu you mus subm your own ndependenly-wren soluons Under no crcumsances wll mulple verbam dencal soluons be consdered accepable If you do work wh ohers, each member of he group mus sae he oher members of he group wh whom he/she worked A person can only work n one group Your soluons, whch lkely requre some combnaon of mahemacal dervaons, economc reasonng, graphcal analyss, and pure logc, should be clearly, logcally, and horoughly presened; hey should no leave he reader (e, your gradng asssan and I) guessng abou wha you acually mean Your mehod of argumen(s) and approach o problems s as mporan as, f no more mporan han, your fnal answer Throughou, your analyss should be based on he frameworks, conceps, and mehods developed n class There are a oal of wo problems, each wh mulple subpars

2 Problem : Hrng, Job Openngs, and achng (60 pons) Le s reconsder he represenave frm s prof-maxmzaon framework Chaper 6 n a few ways Frs, le s hnk n erms of one perod, raher han he wo-perod analyss n Chaper 6 Second, suppose ha he physcal capal sock s k = (Noe ha no me subscrps are needed because he analyss ha follows s enrely whn one perod of me) Thrd, despe he seemng lack of mporance of physcal capal, here s sll dmnshng margnal produc of labor n he represenave frms producon funcon The producon funcon s n f n, n whch (0,) and n denoes labor demand Second, n addon o he noaon from Chaper 6 (n whch f() sands for he producon funcon, n denoes labor demand, -α (whch s srcly beween zero and one) denoes he mporance of labor n he producon process, and w denoes he real wage), we now also nroduce he followng: vac: he quany of job openngs (aka job vacances) FIN q : he probably ha any job openng (aka job vacancy) fnds a FIN suable employee By he defnons of probables, q [0,] (ha s, he probably s a number beween zero and one) Hence, he probably of no fndng a job s -q FIN (Noe: q FIN does no denoe a quany ) ω: (he Greek lowercase leer omega ) he cos ncurred by he frm o adverse each job openng (aka job vacancy) Thnk of hs as he me and expenses ha managers and he human resources (HR) deparmen would have o spend n order o aemp o hre a new employee (Refer o he prof funcon below o see hs more clearly) The represenave frm s prof funcon s A f n w n vac, n whch A > 0 represens oal facor producvy, and he job-hrng consran s FIN n q vac The job-hrng consran was no ncluded n he Chaper 6 framework, bu s mporan n descrbng he possbly of no beng able o hre a new employee for a job

3 The Lagrangan for he frm s opmzaon s FIN A n wn vac q vac n, n whch μ sands for he Lagrange mulpler on he job-hrng consran a (3 pons) Based on he Lagrangan gven above, compue he frs-order condon (FOC) wh respec o n splay he FOC clearly by drawng a box around Soluon: The FOC wh respec o n s A( ) n w 0 b (3 pons) Based on he Lagrangan gven above, compue he frs-order condon (FOC) wh respec o vac splay he FOC clearly by drawng a box around Soluon: The FOC wh respec o vac s FIN q 0 c ( pons) Based on he wo FOCs you compued above, consruc hs framework s labor demand funcon The fnal boxed expresson CANNOT nclude any Lagrange mulplers n (ha s, μ mus be elmnaed) Furhermore, he fnal boxed expresson should conan on he lef-hand sde only ω splay clearly he mahemacal seps/algebrac procedure by whch you oban he fnal expresson (CAUTION: ALL OF THE REST OF THE PARTS OF PROBLE ARE BASE ON THE LABOR EAN FUNCTION OBTAINE HERE IN PART C) Soluon: Isolang he erm from he FOC wh respec o vac yelds FIN q Inserng hs expresson for he mulpler no he FOC wh respec o n gves he job creaon condon A( ) FIN n w q Slghly re-expressng hs condon (afer mulplyng boh sdes by FIN q A( ) n w FIN q ), we have 3

4 d (6 pons) Sarng from he labor demand condon you obaned n par c, suppose for par d only ha q FIN = Wh q FIN =, how does your soluon n par c compare o he labor demand funcon n Chaper 6? escrbe BRIEFLY n boh mahemacal erms and n erms of economcs (Noe: economcs does no mean resang verbally he mahemacs) Soluon: Wh q FIN =, he labor demand funcon from par c shrnks down o A( ) n w Noe ha, excep for he recrung cos ω > 0, hs s que smlar o he labor demand funcon from Chaper 6 To noce hs even more clearly, rewre he expresson as A( ) n w e (6 pons) Now reurn o he case ha q FIN <, bu, for par e only, suppose he HRrelaed coss of adversng job openngs s ω = 0 Wh q FIN < and ω = 0, how does your soluon n par d compare o he labor demand funcon n Chaper 6? escrbe BRIEFLY n boh mahemacal erms and n erms of economcs (Noe: economcs does no mean resang verbally he mahemacs) Soluon: If ω = 0, he soluon o par c (or, equvalenly, he soluon o par d, doesn maer whch logcal pah you used) shows ha labor demand s characerzed by A( ) n w, whch s exacly he same as he labor demand funcon n Chaper 6 (noce ha he q FIN erm cancels ou once we have zero recrung coss ω = 0) The economc raonale s smply ha ω = 0 means ha here are no recrung coss For he remander of Problem, reurn o he case ha q FIN < and ω > 0 f (5 pons) Usng he expresson you obaned n par c, qualavely skech a dagram ha conans q FIN on he vercal axs and n on he horzonal axs (Noe: All ha maers for he qualave dagram s wheher he funcon s upward-slopng, downward-slopng, compleely horzonal, or compleely vercal) Soluon: The soluon for par c shows an upward-slopng relaonshp beween q FIN and n Noe ha q FIN s an exogenous varable, whle n s an endogenous varable, because q FIN s aken as gven by he represenave frm g (5 pons) Provde bref economc nerpreaon for he dagram drawn n par f (Noe: economc nerpreaon does no mean resang verbally he mahemacs) Soluon: The economc nerpreaon s ha, ceers parbus, he larger s he exogenous probably ha any gven job vacancy s able o mach wh and hre a searchng person, he more endogenous job vacances (hence, demand for labor) wll be posed 4

5 h (5 pons) How does an ncrease n oal facor producvy A qualavely affec he dagram drawn n par f? (Noe: All ha maers for hs qualave analyss s wheher he funcon shfs ouwards, shfs nwards, or doesn shf a all) Soluon: The economc nerpreaon s ha, ceers parbus, he larger s he margnal producvy of a new employee, he larger wll be labor demand (aka, job posngs) for any gven probably q FIN Hence, he enre funcon from par f shfs rghwards (5 pons) Provde bref economc nerpreaon for your analyss n par h (Noe: economc nerpreaon does no mean resang verbally he mahemacs) Soluon: The economc nerpreaon s already saed n he soluon o par h: ceers parbus, he larger s he margnal producvy of a new employee, he larger wll be labor demand (aka, job posngs) for any gven probably q FIN j (UCH HARER 0 pons) Usng he perfecly compeve dagram below, descrbe brefly and qualavely how real wages are deermned n he machng framework (Noe: hs enre Problem has nohng o do wh labor supply, bu noneheless s possble o hnk n erms of hs framework) Soluon: For a gven un of labor (on he horzonal axs) ha s smaller han he n a he nersecon of he (prncples of mcroeconomcs or prncples of macroeconomcs) labor supply curve and labor demand curve drawn below, here s a vercal gap beween he supply funcon and he demand funcon The labor supply funcon drawn IS he RS beween consumpon and labor (recall hs from Chaper ) The labor demand funcon drawn IS he margnal produc of labor (mpn) (recall hs from Chaper 6) The mplcaon s ha ANY REAL WAGE ha les nsde he GAP beween RS and mpn s a poenal real wage The search and machng framework says nohng more han hs ha s, does NOT nform us abou how any real wage comes abou 5

6 real wage Labor supply Labor demand labor 6

7 Problem : oney and oney Consder an exended verson of our nfneperod IU framework In addon o socks and nomnal bonds, suppose here are wo forms of money: and money (whch we wll denoe by ) and money (whch we wll denoe by ) boh drecly affec he represenave consumer s uly The perod- uly funcon s u c,, ln ln ln P P P P c r, whch, noe has hree argumens The Greek leer kappa (κ) n he uly funcon s a number beween zero and one, 0, over whch he represenave consumer has no conrol The perod- budge consran of he consumer s Pc B S a Y ( ) ( ) B ( S ) a, where denoes he nomnal neres rae on bonds held beween perod and (and hence on bonds held beween and ) and denoes he nomnal neres rae on money held beween perod and (and hence on money held beween and ) Thus, noe ha money poenally pays neres, n conras o money, whch pays zero neres As always, assume he represenave consumer maxmzes lfeme uly by opmally choosng consumpon and asses (e, n hs case choosng all four asses opmally) a (5 pons) Usng he funconal form for uly gven n hs problem, provde he lfeme sequenal Lagrange Soluon: The lfeme sequenal Lagrange s ln c ln ln ln c ln ln ln c ln ln P P P P P P Y ( S ) a ( ) ( ) B Pc Sa B Y ( S ) a ( ) ( ) B P c S a B Y ( S ) a ( ) ( ) B P c S a B b (5 pons) Based on he Lagrange funcon consruced n par a, consruc he fve frs-order condons: wh respec o c ; wh respec o a ; wh respec o B ; wh respec o ; and wh respec o splay he fve FOCs clearly by drawng a box around each 7

8 Soluon: The fve FOCs are, respecvely: P 0 c S ( S ( ) 0 / P 0 / P ) 0 / P ( ) 0 / P c (5 pons) Based on he FOCs n par b, compue he opmaly condon beween money and money Explan he mporan seps n your argumen In your fnal expresson, dsplay he margnal rae of subsuon beween real money and real money on he lef-hand sde Soluon: There were several ways o perform he algebrac seps n hs queson Below s presened one parcular pah Frs, hough, keep n mnd (hnkng back o he very begnnng of our sudes) ha he RS beween real money and real money s he rao of he margnal ules Gven he uly funcon n hs queson, he RS beween hem s / P / P / P / P or, afer cancellng erms, n more smplfed form, he RS s In he algebrac seps o follow, we ll confrm ha hs s he RS vde he FOC wh respec o money by he FOC wh respec o, whch gves ( ) 8

9 9 Nex, noce ha he FOC wh respec o B above allows us o wre Inserng hs expresson no he rgh-hand sde of he equaon mmedaely above gves us Afer groupng he λ erms on he rgh-hand sde, we have Afer cancellng he λ erms on he rgh-hand sde and wo more seps of algebra, he opmaly condon beween money and money s, n whch he lef-hand sde confrms he RS saed earler The rgh-hand sde,, s hus he relevan slope of he budge lne (whch, agan recallng he very begnnng of our sudes, s smply he relave prce) beween money and money d (5 pons) Skech n an ndfference curve/budge consran dagram he money / money opmaly condon from par c, wh clearly labeled axes and he slope of he approprae budge lne Soluon: The followng dagram porrays he money / money opmaly condon

10 slope = -/( ) * * n whch he pon ( *, * ) s he represenave consumer s opmal par of choces of he wo moneary nsrumens e (5 pons) In no more han 0 words, provde bref economc nerpreaon for he dagram drawn n par d (Noe: economc nerpreaon does no mean resang verbally he mahemacs) Soluon: The pon ( *, * ) s he represenave consumer s opmal par of choces of he wo moneary nsrumens, akng as gven he exogenous neres raes and f (5 pons) A sudden, unexplaned change n he value of would be nerpreable as whch of he followng: a preference shock, a echnology shock, or a moneary polcy shock? Brefly explan Soluon: The parameer, as seen n par c above, affecs he RS beween wo of he argumens o he uly funcon ( money and money) A change n hus affecs he slope of he ndfference curve, and hus s nerpreable as a preference shock (An nformal, alernave, nerpreaon of a change n κ s ha s a moneary polcy shock Srcly speakng, hough, n hs parcular queson, hs nerpreaon s no enrely correc because a moneary polcy shock s one ha affecs money supply Here, he shock affecs money demand) g (0 pons) Le ( c,, ) denoe he real money demand funcon for money Noe he hree argumens o he funcon () Usng he frs-order condons of he represenave consumer s Lagrangan, generae he funcon 0

11 ( c,, ) (e, solve for real money demand as a funcon of c,, and ) Brefly explan (economcally) why appears n hs money demand funcon (Noe: you mus deermne yourself whch are he relevan frs-order condons needed o creae hs money demand funcon draw on our approach from Chaper 4) In addon o clearly descrbng he algebra, dsplay he fnal -money demand funcon clearly by drawng a box around Soluon: Subsue he FOC on consumpon and bonds no he FOC on money o ge, afer several algebrac rearrangemens, c ( ) P Noe ha hs s very smlar o he usual money demand funcon obaned when uly s logarhmc (n our usual model we mplcly had 0 and 0 ) The neres rae appears n hs demand funcon smply because here s an neres benef of holdng hs asse, as opposed o no neres n money As he above money demand funcon shows, he larger s, he larger s money demand Thnk of as a savngs depos agans whch you can wre checks, and money as cash Cash earns you zero neres, whereas a savngs depos earns you some posve neres; on he oher hand, cash s acceped everywhere, bu checks agans your savngs depos are no acceped everywhere (e, savngs deposs are less lqud han cash)