Deparmen of Applied Economics Johns Hopkins Universiy Economics 60 Macroeconomic Theory and Policy Final Exam Suggesed Soluions Professor Sanjay Chugh Spring 0 NAME: The Exam has a oal of four (4) problems and pages numbered one () hrough eigheen (8). Each problem s oal number of poins is shown below. Your soluions should consis of some appropriae combinaion of mahemaical analysis, graphical analysis, logical analysis, and economic inuiion, bu in no case do soluions need o be excepionally long. Your soluions should ge sraigh o he poin soluions wih irrelevan discussions and derivaions will be penalized. You are o answer all quesions in he spaces provided. You may use wo pages (double-sided) of noes. You may no use a calculaor. Problem / 5 Problem / 5 Problem 3 / 30 Problem 4 / 0 TOTAL / 00
Problem : Hyperbolic Impaience and Sock Prices (5 poins). In his problem you will sudy a sligh exension of he infinie-period economy from Chaper 8. Specifically, suppose he represenaive consumer has a lifeime uiliy funcion given by uc ( ) + γβuc ( ) + γβ uc ( ) + γβ uc ( ) +..., 3 + + + 3 in which, as usual, u(.) is he consumer s uiliy funcion in any period and β is a number beween zero and one ha measures he normal degree of consumer impaience. The number γ (he Greek leer gamma, which is he new feaure of he analysis here) is also a number beween zero and one, and i measures an addiional degree of consumer impaience, bu one ha ONLY applies beween period and period +. This laer aspec is refleced in he fac ha he facor γ is NOT successively raised o higher and higher powers as he summaion grows. The res of he framework is exacly as sudied in Chaper 8: a is he represenaive consumer s holdings of sock a he beginning of period, he nominal price of each uni of sock during period is S, and he nominal dividend paymen (per uni of sock) during period is D. Finally, he represenaive consumer s consumpion during period is c and he nominal price of consumpion during period is P. As usual, analogous noaion describes all hese variables in periods +, +, ec. The Lagrangian for he represenaive consumer s uiliy-maximizaion problem (saring from he perspecive of he beginning of period ) is uc ( ) + γβuc ( ) + γβ uc ( ) + γβ uc ( ) +... 3 + + + 3 + λ Y + ( S + D ) a Pc Sa + γβλ Y + ( S + D ) a P c S a + + + + + + + + + + + γβ λ+ Y+ ( S+ D+ ) a+ P+ c+ S+ a+ 3 + γβ λ+ 3 Y+ 3+ ( S+ 3+ D+ 3) a+ P+ 3c+ 3 S+ 3a+ 3 +... NOTE CAREFULLY WHERE THE ADDITIONAL IMPATIENCE FACTOR γ APPEARS IN THE LAGRANGIAN. (OVER) The idea here, which goes under he name hyperbolic impaience, is ha in he very shor run (i.e., beween period and period +), individuals degree of impaience may be differen from heir degree of impaience in he slighly longer shor run (i.e., beween period + and period +, say). Hyperbolic impaience is a phenomenon ha rouinely recurs in laboraory experimens in experimenal economics and psychology, and has many farreaching economic, financial, policy, and socieal implicaions.
Problem coninued a. (3 poins) Compue he firs-order condiions of he Lagrangian above wih respec o boh a and a +. (Noe: There is no need o compue firs-order condiions wih respec o any oher variables.) Soluion: The wo FOCs are λs + γβλ ( S + D ) = 0 + + + γβλ S + γβ λ ( S + D ) = 0 + + + + + b. (4 poins) Using he firs-order condiions you compued in par a, consruc wo disinc sock-pricing equaions, one for he price of sock in period, and one for he price of sock in period +. Your final expressions should be of he form S =... and S + =... (Noe: I s fine if your expressions here conain Lagrange mulipliers in hem.) Soluion: Simply rearranging he wo FOCs above and canceling he γ erm (along wih one β erm) in he second FOC, we have γβλ S = ( S + D ) + + + λ βλ S = ( S + D ) + + + + λ + For he quesions nex, observe ha he S expression and he S + expression are subly, bu imporanly, differen here. They would be idenical o each oher (oher han he fac ha he ime subscrips are differen, bu ha is as usual) if and only if γ =. If γ <, which is he case of hyperbolic impaience, hen sock prices are deermined in a somewha differen way in he very shor run compared o he longer shor run or medium run.
Problem coninued For he remainder of his problem, suppose i is known ha D + = D +, and ha S + =S +, and ha λ = λ + = λ +. c. (5 poins). Does he above informaion necessarily imply ha he economy is in a seadysae? Briefly and carefully explain why or why no; your response should make clear wha he definiion of a seady sae is. (Noe: To address his quesion, i s possible, hough no necessary, ha you may need o compue oher firs-order condiions besides he ones you have already compued above.) Soluion: No, none of hese saemens necessarily implies ha he economy is in a seady sae, which, recall, means ha all real variables become consan and never again change. There are wo ways of observing ha he above informaion does no imply he economy is in seady sae. Firs, he above saemens are all abou nominal variables, and in a seady sae i can be he case ha nominal variables coninue flucuaing over ime, even hough all real variables do no. Anoher way of arriving a he correc conclusion here is ha he saemens above only refer o periods, +, and +. In a seady-sae, (real) variables sele down o consan values forever, no jus for a few ime periods. d. (5 poins) Based on he above informaion and your sock-price expressions from par b, can you conclude ha he period- sock price (S ) is higher han S +, lower han S +, equal o S +, or is i impossible o deermine? Briefly and carefully explain he economics (i.e., he economic reasoning, no simply he mahemaics) of your finding. Soluion: You are given ha nominal sock prices, nominal dividends, and he Lagrange muliplier in period + and + are equal o each oher. Le s call hese common values S, D, and λ (ha is, S = S+ = S+ D= D+ = D, and + λ = λ+ = λ ). Insering hese common + βλ values in he period-+ sock price equaion, we have S = ( S+ D). Canceling erms, we λ have ha he nominal sock price in period + (and +) is S = βs+ βd (which we could of β course solve for he sock price as S = D if we needed o). β Now, using he common values of S, D, and he muliplier in he period- sock price equaion gives us S = γβ( S+ + D+ ) = γβ( S+ D) = γ ( β S+ β D). Noe ha he final erm in parenheses is nohing more han S, hence we have S = γ S. If γ <, hen clearly he sock-price in period is smaller han i is in period + (and period +). The economics of his is due o he hyperbolic impaience which makes consumers more impaien o purchase consumpion in he very shor run (period ) compared o he longer shor run. All else equal, his means ha in he very shor run, consumers do no care o save as much (due o he heir exreme impaience in he very shor run), which means heir demand for saving --- i.e., heir demand for sock is lower. Lower demand for sock means a lower price of sock, all else equal. 3
Problem coninued Now also suppose ha he uiliy funcion in every period is u(c) = ln c, and also ha he real ineres rae is zero in every period. e. (4 poins) Based on he uiliy funcion given, he fac ha r = 0, and he basic seup of he problem described above, consruc wo marginal raes of subsiuion (MRS): he MRS beween period- consumpion and period-+ consumpion, and he MRS beween period-+ consumpion and period-+ consumpion. Soluion: This only requires examining he lifeime uiliy funcion (he firs line of he Lagrangian above). By definiion, he MRS beween period consumpion and + consumpion u'( c) c+ is =, and he MRS beween period + consumpion and + consumpion is γβu'( c+ ) γβc γβu'( c+ ) u'( c+ ) c+ = =. Noe ha he form of he wo MRS funcions is differen: he γβ u'( c+ ) βu'( c+ ) βc+ hyperbolic impaience affecs he former MRS, bu no he laer MRS. f. (4 poins Harder) Based on he wo MRS funcions you compued in par e and on he fac ha r = 0 in every period, deermine which of he following wo consumpion growh raes c c OR + + is larger. Tha is, is he consumpion growh rae beween period and period + (he fracion on he lef) expeced o be larger han, smaller han, or equal o he consumpion growh rae beween period + and period + (he fracion on he righ), or is i impossible o deermine? Carefully explain your logic, and briefly explain he economics (i.e., he economic reasoning, no simply he mahemaics) of your finding. Soluion: The basic consumpion-savings opimaliy condiion saes ha he MRS beween wo consecuive ime periods is equaed o (+r). You are old here ha r = 0 always. Based on he wo MRS funcions consruced above, hen, i follows immediaely ha he consumpion growh rae beween period and + is smaller han he consumpion growh rae beween period + and period +. This follows because γ <. The economics is similar o above: hyperbolic impaience makes consumers consume much more in he very shor run (i.e., period ), which means ha he growh rae of consumpion beween period (already a very high consumpion period) and + will be low, compared o he similar comparison one period laer. c c + 4
Problem : The Cash-in-Advance (CIA) Framework (5 poins). A popular alernaive o he money in he uiliy funcion (MIU) framework is one in which money holdings direcly faciliae ransacions ha is, provide a medium of exchange role, one of he basic funcions of money ha he MIU framework capures in only a shorcu form. Suppose ha in any given ime period, here are wo ypes of consumpion goods, cash goods and credi goods. Cash goods, denoed by c, are goods whose purchase in period requires money, while credi goods, denoed by c, are goods whose purchase in period does no require money (i.e., hey can be bough on credi ). The marke nominal price of each ype of good is idenical, P. The represenaive consumer consumes boh cash and credi goods. Specifically, suppose he period- uiliy funcion is uc (, c, n ), wih n denoing he individual s labor during period. (As in Chaper, suppose ha oal hours available in any given ime period is 68, and he only possible uses of ime are labor or leisure.) The consumer s period- budge consrain is Pc + Pc + M + P B = Pw n + M + B, b Income is earned from labor supply (wih w denoing he marke deermined real wage in period, which is aken as given by he individual), and, for simpliciy, suppose here are no sock markes (hence one-period riskless bonds and money markes are he only wo asse markes). The consumer s bond and money holdings a he sar of period are B - and M -, and a he end of period are B and M. The individual s budge consrains for period +, +, are idenical o he above, wih he ime subscrips appropriaely updaed. As always, suppose he represenaive consumer s subjecive discoun facor beween any pair of consecuive ime periods is β (0,). In addiion o he budge consrain, in each ime period he represenaive consumer also has a cash in advance consrain, Pc = M. The cash-in-advance (CIA) consrain capures he idea ha in order o purchase some goods, a cerain amoun of money (or more generally, liquidiy such as checkable deposis) has o be held. In principle, he CIA consrain is an inequaliy consrain (specifically, Pc M ), bu in analyzing his problem, you are o assume ha i always holds wih equaliy, as wrien above. From he sandpoin of he analysis you will conduc, because he CIA is echnically an inequaliy consrain, you may NOT subsiue he CIA consrain ino he budge consrain. 5
Problem coninued The consumer s budge consrains and CIA consrains for period +, +, are idenical o he above, wih he ime subscrips appropriaely updaed. a. (4 poins) Se up an appropriae sequenial Lagrangian from he perspecive of he beginning of period. Define any new noaion you inroduce. Soluion: The Lagrangian is = 0 b ( ) ( ) β uc (, c, n) + λ Pwn+ M + B Pc Pc M PB + µ M Pc (noe he summaion operaor) in which λ denoes he muliplier on he period- budge consrain and µ denoes he muliplier on he period- CIA consrain. b. (6 poins) Based on he Lagrangian consruced in par a, derive he firs-order condiions (FOCs) wih respec o period s choices, c, c, n, B, and M. Define any new noaion you inroduce. Soluion: The firs-order condiions wih respec o c, c, n, B, M are, respecively: u λ P µ P = 0 u u λ P = 0 + λ Pw = 0 3 b λp + βλ = 0 + λ + µ + βλ = 0 + (The shorhand u sands for he parial derivaive of he uiliy funcion wih respec o he firs argumen, and similarly for u and u 3.) 6
Problem coninued c. (5 poins) Using he FOCs from above, derive he cash good/credi good opimaliy u( c, c, n ) condiion, which should have a final form f( i ) u( c, c, n ) =. Noe ha f(i ) is a funcion ha depends ONLY on he nominal ineres rae. You are o deermine f(i ) as par of his problem; here should be no oher variables or parameers a all on he righ-hand-side of he cash good/credi good opimaliy condiion you derive. Show clearly he imporan seps in he algebra. Soluion: Geing o he cash good/credi good opimaliy condiion requires working wih he FOCs wih respec o cash goods, credi goods, bonds, and money. λ The FOC wih respec o bonds implies βλ + =, which, when subsiued ino he FOC on + i λ money, yields = λ µ. Dividing hrough by λ, + i µ =, λ + i or, rewriing slighly, µ P = λ P + i i i = + i (Obviously, here are many ways o proceed hrough he algebra here; whichever roue sruck you as mos convenien is fine, he roue presened here is jus one suggesion.) Nex, from he FOCs on cash consumpion and credi consumpion, we have µ P = u u, so µ P u u u ha = =. Insering his in he las displayed expression above, we have he λp u u cash good/credi good opimaliy condiion in he requesed form, u u i = +, + i wih he nominal ineres rae appearing as he only variable/parameer on he righ-hand-side. 7
Problem coninued d. (5 poins) Suppose now ha he uiliy funcion is uc (, c, n ) = lnc + ln c + vn ( ), in which v(n ) is some unspecified funcion of labor. Taking ino accoun he fac ha he CIA consrain holds wih equaliy a he opimal choice, derive, based on his uiliy funcion and your work above, he real money demand funcion, M... P = where he erm on he righ-hand side is for you o deermine. Show clearly he imporan seps in he algebra. Also, qualiaively plo his relaionship in a diagram wih M/P on he horizonal axis and i on he verical axis, clearly labeling he wo axes. Soluion: The marginal uiliy funcions needed are obviously u = / c and u = / c. Wih he cash-in-advance consrain holding wih equaliy a he opimum, we can equivalenly wrie u = /( M / P), and hus can express he cash good/credi good opimaliy condiion as u c i = = +. Solving for real money balances, we have he real money demand h u M / P + i funcion M P i = c + + i which shows ha real money demand is decreasing in he nominal ineres rae (given ha i canno be sricly negaive). More echnically, compue he derivaive of M /P wih respec o i, which is always sricly negaive. Noice ha he money demand funcion is nohing bu he cash good/credi good opimaliy condiion solved for M/P. A furher, simpler, represenaion of he money demand funcion is available by recognizing ha c i for his uiliy funcion, he cash good/credi good opimaliy condiion is = + ; c + i subsiuing his ino he previous represenaion of he money demand funcion, we have M c P =, which is, obviously, a resaemen of he cash-in-advance consrain holding wih equaliy., 8
Problem coninued e. (5 poins) Recall ha in he MIU framework, he consumpion-money opimaliy condiion was marginal uiliy of real money holdings i =. marginal uiliy of consumpion + i Compare, in erms of economics, his opimaliy condiion o he opimaliy condiion you obained in par c above for he CIA framework by briefly commening on he similariies and differences beween he resuls prediced by he wo frameworks. Noe: his does no mean resae in words he mahemaics; raher, offer wo or hree houghs/criiques/ec. on how and why he wo frameworks do or no capure he same ideas, how and why he wo frameworks perhaps are or are no essenially idenical o each oher, and so on. Soluion: Each framework capures he idea ha money demand is a decreasing funcion of he nominal ineres rae, and each capures he idea of subsiuing away from cash-inensive aciviies (purchasing cash goods in he cash/credi model, and enjoying uiliy from real money balances in he MIU model) and ino non-cash aciviies as he nominal ineres rae rises. These are basic ideas in moneary analysis, and each framework ariculaes hem. An imporan difference beween he wo frameworks is ha in he cash/credi framework, cash goods are viewed as par of he resource fronier of he economy hey are a good ha mus be produced using whaever he economy s producion echnology is, hence a social planner in his economy would care abou cash goods. In conras, real money balances in he MIU framework are no par of he resource fronier of he economy a social planner in his economy would no care abou real money balances. This idea manifess iself mos clearly in he case of a zero nominal ineres rae (i.e., he Friedman Rule). Using he log funcional forms given above for ease of exposiion, in he c cash/credi model i = 0 implies c =. In he MIU model, i c = 0 implies = 0, or, a lile h M / P h M / P more informaively, =. In he former case, cash-inensive aciviy (i.e., purchase of c cash goods ) is bounded. In he laer case, cash-inensive aciviy ( consumpion of money balances) is unbounded: a social planner in his economy would choose o dump an infinie amoun of useless pieces of paper ino he economy in order o saiae he economy wih cash. 9
Problem 3: Elasiciy of Labor Supply and Fiscal Policy (30 poins). Consider he saic (i.e., one-period) consumpion-leisure framework. In quaniaive and policy applicaions ha use his framework, a commonly-used uiliy funcion is θ ( 68 ) / + ψ u(,) cl = lnc l, + / ψ in which c denoes consumpion, l denoes he number of hours (in a week) spen in leisure, and ψ and θ (he Greek leers psi and hea, respecively) are consans (even hough we will no assign any numerical value o hem) in he uiliy funcion. The represenaive individual has no conrol over eiher ψ or θ, and boh ψ > 0 and θ > 0. Labor is measured as n = 68 l, he real wage is denoed by w, and he labor-income ax rae is denoed by. The ake-home rae (he fracion of labor income ha an individual keeps) can hus be defined as S = (-). Finally, expressed in real erms, he represenaive individual s budge consrain is c= ( ) wn. Thus, his is exacly he consumpion-leisure framework sudied in Chaper, wih now a paricular funcional form for u(c,l). The consumpion-leisure opimaliy condiion for his problem (using he derivaives of he given uiliy funcion, which you do no need o verify) is / ψ θn c= S w. Solving his expression for n gives he labor supply funcion ψ S w n = θc. Finally, aking he naural logarihm of boh sides of his expression allows us o re-express he laer expression in logarihmic form: (which does no require verificaion). ln n = ψ ln w+ ψ ln S ψ lnc ψ lnθ In he analysis o follow, you will sar from eiher he logarihmic form of he labor supply funcion (he laer equaion) or he exac form of he labor supply funcion (he former equaion). 0
Problem 3 coninued Recall from basic microeconomics ha he elasiciy of a variable x wih respec o anoher variable y is defined as he percenage change in x induced by a one-percen change in y. As you sudied in basic microeconomics, elasiciies are especially useful measures of he sensiiviy of one variable o anoher because hey do no depend on he unis of measuremen of eiher variable. A convenien mehod for compuing an elasiciy (which you can ake o be rue here wihou proof) is ha he elasiciy of one variable (say, x) wih respec o anoher variable (say, y) is equal o he firs derivaive of he naural log of x wih respec o he naural log of y. Read well his saemen: i saes ha he elasiciy of he variable x wih respec o y can be compued ln x as. ln y a. (4 poins) Saring from he logarihmic form of he labor supply condiion provided above, compue he elasiciy of n wih respec o he real wage w. The resuling expression is he elasiciy of labor supply wih respec o he real wage for he given uiliy funcion. Clearly presen he seps and logic of your work. Soluion: The mos sraighforward way o proceed here was o make he following observaion(s): firs, imagine calling he enire expression ln n as x. Tha is, le s emporarily hink ha x= ln n. Similarly, imagine calling he enire expression ln w as y. Tha is, le s emporarily hink ha y= ln w. Wih hese emporary re-definiions, we can (emporarily) reexpress he logarihmic labor supply condiion as x = ψy+ ψ ln S ψ ln c ψ lnθ. Compuing he derivaive of he naural log of n wih respec o he naural log of w (which, according o he heurisic definiion of elasiciy given above, is he elasiciy n wih respec o w) now jus amouns o compuing he derivaive of x wih respec o y in he las expression. The elasiciy of labor supply wih respec o he real wage is hus simply ψ.
Problem 3 coninued Presiden Obama, his economic eam, and many (hough cerainly no all) members of Congress have implemened a variey of ax measures inended o boos labor-marke aciviy. Along he consumpion-labor dimension, wha his means is ha decreases in he labor-income ax rae have been pu ino effec, which equivalenly means ha increases in he ake-home rae S have been pu ino effec. b. (4 poins) Saring once again from he logarihmic form of he labor supply condiion provided above, compue he elasiciy of n wih respec o he ake-home rae S. The resuling expression is he ypical measuremen of he elasiciy of labor supply wih respec o he akehome rae for he given uiliy funcion. Clearly presen he seps and logic of your work. Soluion: Proceed in a very similar way as in par a above. This ime, le s emporarily rename x = ln n and y = ln S, which allows us o emporarily re-express he logarihmic labor supply condiion as x = ψ ln w+ ψy ψ ln c ψ lnθ. Compuing he derivaive of he naural log of n wih respec o he naural log of S (which, according o he heurisic definiion of elasiciy given above, is he elasiciy n wih respec o S) now jus amouns o compuing he derivaive of x wih respec o y in he las expression. The elasiciy of labor supply wih respec o he akehome rae is hus simply ψ (i.e., exacly he same as in par a). c. (4 poins) Based on he resul in par b, would a decrease in be expeced o increase labor supply, decrease labor supply, or leave labor supply unchanged? Clearly presen he seps and logic of your work and conclusion. Soluion: A decrease in he ax rae amouns o an equal (in size) increase in he ake-home rae S. We jus compued in par b ha an increase in he ake-home rae leads o an increase in labor supply. Which is he final conclusion we need here.
Problem 3 coninued d. (8 poins) Now sar from he exac form of he labor supply condiion provided above. Moreover, consider i in conjuncion wih he budge consrain (ha is, consider he pair of equaions, no jus he exac labor supply condiion in isolaion). Based on his pair of condiions, derive he soluion for he opimal choice of n. Soluion: Now we mus bring he exac form of he labor supply condiion ogeher wih he budge consrain. From he budge consrain, we can isolae he consumpion erm, c = Swn. Then inser his in he exac form of he labor supply funcion, which gives S w n = = θ S w n θn ψ For he purposes of par e below, we need no proceed furher, bu because you are asked o acually solve for n =, here are hree more algebraic seps needed (acually only wo seps if you combined he firs wo seps presened nex). Firs, pull he n on he righ-hand side ou of he square brackes: wih he use of he usual rules of exponens, his gives Second, muliply boh sides by n ψ, which gives /( + ψ ), which gives he final soluion requesed: ψ. ψ ψ + n = θ n = θ n= θ ψ n ψ. Then raise boh sides o he power ψ + ψ.. e. (4 poins) Based on he resul in par d, would a decrease in be expeced o increase labor supply, decrease labor supply, or leave labor supply unchanged? Clearly presen he seps and logic of your work and conclusion. Soluion: I is clear from he final (or even inermediae) expression in par d ha n does no depend a all on axes! This is clear because his expression does no conain S in i. Thus, a change in axes would no a all affec his version of labor supply. To undersand why (hough you did no need o conduc his par of he analysis), recall how we consruced wha we called he labor supply funcion in Chaper : we combined he consumpion-leisure opimaliy condiion wih he budge consrain, and raced ou combinaions of wages and (opimal choices of) leisure (labor). Here in par e, doing he same exac procedure shows us ha labor supply is independen of axes moreover, i is independen of real wages, as well. 3
Problem 3 coninued f. (6 poins Harder) Compare your conclusions in par c and par e abou how a change in ax policy may affec he quaniy of labor supply in he economy. Are he conclusions he same? If so, explain briefly he economics of why; if no, explain briefly he economics of why no; if i is impossible o ell, explain briefly he economics of why a comparison canno be made. Soluion: The conclusions in pars c and e are obviously differen. The difference is alluded o in he soluion o par e: in par e, we are looking a he full, macro opimal soluion (he combinaion of he consumpion-leisure opimaliy condiion and he budge consrain), whereas in par d, we are only considering, inuiively speaking, a slope or micro argumen, raher han boh a slope and a level. In erms of more formal economics, he analysis in par c is anamoun o analyzing he effecs of policy on jus he labor marke (why? because he analysis here reas consumpion as a consan). The analysis in par e insead is anamoun o analyzing joinly he effecs of policy on labor markes and goods markes. To he exen ha here are feedback effecs beween he wo markes, here is no reason o hink he answers from he analyses mus be he same. Loosely speaking, we could hink of he analysis in par c as a microeconomic analysis and he analysis in par e as a macroeconomic analysis. Wha his implies is ha one way (perhaps he mos imporan way) o undersand he difference beween microeconomic analysis and macroeconomic analysis is ha he laer rouinely considers feedback effecs across markes, whereas he former usually does no. 4
Problem 4: Financing Consrains and Labor Demand (0 poins). In our class discussion abou he way in which financing consrains affec firms profi maximizaion decisions, we focused on he effecs on firms physical capial invesmen. In realiy, mos firms spend wice as much on heir wage coss (i.e., heir labor coss) han on heir physical invesmen coss. (Tha is, for mos firms, roughly wo-hirds of heir oal coss are wages and salaries, while roughly one-hird of heir oal coss are devoed o mainaining or expanding heir physical capial.) For many firms, paymen of wages mus be made before he receip of revenues wihin any given period. (For example, imagine a firm ha has o pay is employees o build a compuer; he revenues from he sale of his compuer ypically don arrive for many weeks or monhs laer because of inheren lags in he shipping process, he reail process, ec.) For his reason, firms ypically need o borrow o pay for heir payroll coss. Bu, because of asymmeric informaion problems, lenders ypically require ha he firm pu up some financial collaeral o secure loans for his purpose. Here, you will analyze he consequences of financing consrains on firms wage paymens using a variaion of he acceleraor framework we sudied in class. For simpliciy, suppose ha he represenaive firm, which operaes in a wo-period economy, mus borrow in order o finance only period- wage coss; for some unspecified reason, suppose ha period- wage coss are no subjec o a financing consrain. As in our sudy of he acceleraor framework in class, he represenaive firm s wo-period discouned profi funcion is Pf( k, n) + Pk + ( S + D) a Pwn Pk Sa 0 Pf( k, n) Pk ( S + D) a Pwn Pk Sa + + + + i + i + i + i + i + i 3 and suppose now he financing consrain ha is relevan for firm profi-maximizaion is Pwn = S a. The noaion is as always: P denoes he nominal price of he oupu he firm produces and sells; S denoes he nominal price of sock; D denoes he nominal dividend paid by each uni of sock; n denoes he quaniy of labor he firm hires; w is he real wage; a 0, a, and a are, respecively, he firm s holdings of sock a he end of period 0, period, and period ; k, k, and k 3 are, respecively, he firm s ownership of physical capial a he end of period 0, period, and period ; i denoes he nominal ineres rae beween period and period ; and he producion funcion is denoed by f(.). Also as usual, subscrips on variables denoe he ime period of reference for ha variable. Finally, because his is a wo-period framework, we know a = 0 and k 3 = 0. (OVER) The commercial paper marke, abou which much has been discussed in he news media in he pas couple of years, is one ype of channel for such firm financing needs. 5
Problem 4 coninued a. (4 poins) Based on he informaion provided, consruc he Lagrangian for he firm s profi maximizaion problem. Define any new noaion you inroduce. Soluion: The Lagrangian for he firm s profi maximizaion problem is Pf( k, n) + Pk + ( S + D) a Pwn Pk Sa 0 Pf( k, n) Pk ( S + D) a Pwn Pk Sa + + + + i + i + i + i + i + i + λ 3 [ Sa Pwn] in which λ denoes he Lagrange muliplier on he financing consrain. b. (4 poins) Based on he Lagrangian you consruced in par a, compue he firs-order condiions wih respec o k and a. Soluion: The firs-order condiions are simply: Pf( k, n) P + i + i S + D S+ + λs = 0 + i k P + + = 0 6
r Problem 4 coninued c. (4 poins) Based on he Lagrangian you consruced in par a, compue he firs-order condiions wih respec o n and n. Soluion: The firs-order condiions are simply: Pf ( k, n) Pw λpw = 0 n Pf n( k, n) Pw = 0 + i + i Suppose ha, immediaely afer firm profi maximizing decisions have been made, he real reurn on STOCK, r STOCK, all of a sudden falls below r, he real reurn on riskless ( safe ) asses. Suppose ha before his shock occurred, i was he case ha r = r STOCK. d. (4 poins) Below is a graph of he invesmen (capial) marke in period. Does he adverse shock o r STOCK shif eiher he invesmen demand and/or he savings supply funcion? If so, explain how, in wha direcion, and why. Soluion: The invesmen demand funcion is unaffeced by he financing consrain (see he FOC on k above), hence exogenous changes in rstock have no effec on capial demand/invesmen demand. Furhermore, because nohing is said abou wheher financing fricions impinge on he savings supply side of he economy (i.e., on consumers consumpionsavings decisions), here is no basis for assering any shif of he savings funcion. Hence, here is no direc effec on he marke for physical capial. Naional Savings Invesmen I, S 7
Problem 4 coninued e. (4 poins) Below is a graph of he labor marke in period. Does he adverse shock o r STOCK shif eiher he labor demand and/or he labor supply funcion? If so, explain how, in wha direcion, and why. Soluion: No shif in labor supply because, as above, no saemens are made abou wheher financing fricions affec consumers behavior (which is wha would be required for a shif of he labor supply funcion). A fall in rstock will cause a RISE in λ, hence for a given level of w, he effecive marginal produc of period- labor FALLS, hence he labor demand curve shifs inwards. This effec arises because λ direcly appears in he period- FOC on labor above. Real wage in period Supply Demand n Period Labor Marke END OF EXAM 8