15 Theories of Investment Expenditures

Transcription

1 Economics 314 Coursebook, 2014 Jeffrey Parker 15 Theories of Invesmen Expendiures Chaper 15 Conens A. Topics and Tools... 2 B. How Firms Inves... 2 Wha invesmen is and wha i is no... 2 Financing of invesmen... 3 The Modigliani-Miller heorem... 6 Sources of finance and credi availabiliy effecs... 7 C. Invesmen and he Business Cycle... 8 Keynes and he business cycle... 8 The acceleraor heory of invesmen The muliplier-acceleraor model D. Undersanding Romer s Chaper The firm s profi funcion User cos of capial Adjusmen-cos model Tobin s q Average and marginal q Dynamic analysis E. Empirical Sudies of Invesmen Empirical evidence on he acceleraion principle Empirical esing of he neoclassical model Empirical specificaion of adjusmen coss and q models Empirical resuls using average q Alernaive sraegies for esimaing he adjusmen-cos model Using ax reforms as naural experimens Imperfec capial markes: Cash flow and relaed variables Effecs of irreversibiliy, uncerainy, and non-convex adjusmen coss F. Works Cied in Tex... 48

2 A. Topics and Tools Invesmen expendiures play a key role in many heories of he business cycle, including Keynes s heory. Macroeconomic heoriss have agreed on a basic framework ha models he invesmen sraegy of a profi-maximizing firm. However, empirical evidence has failed o provide subsanial suppor for his model, which has been a source of frusraion for hose involved in invesmen modeling. In his chaper, we will invesigae some of he deails of he invesmen process, including how firms raise he funds hey use for invesmen, how models of he various kinds of invesmen differ, and how financial markes affec invesmen expendiures. The Romer ex develops he model in he form known as he q heory of invesmen. This is based on a measure of he desirabiliy of invesmen known as Tobin s q. The q heory is easily reconciled wih oher approaches o invesmen, all of which lead o he same basic resul. B. How Firms Inves Wha invesmen is and wha i is no The erm invesmen means somehing differen o macroeconomiss han i does o mos of he res of he world. For example, if you ask your banker abou invesmen, she will probably sar alking abou socks and muual funds ha she would like you o purchase, or new kinds of deposi accouns ha her bank offers. To an economis, hese purchases of financial asses are no invesmen. From an aggregae poin of view, financial asses do no represen real ne wealh for he economy as a whole; insead, hey reflec credi relaionships wihin he economy. Financial asses such as loans and bank accouns represen conracs o pay ineres and repay principal on borrowed money. Socks represen parial ownership of a corporaion, implying a righ o voe on he governance of he corporaion and o receive dividends as deermined by he direcors ha he shareholders elec. In eiher case, he financial asse of one individual in he economy is offse by a financial liabiliy of anoher person or corporaion. Thus, when we aggregae he wealh of all members of he economy, hese asses and liabiliies cancel ou and financial asses disappear. Thus, if you inves in a financial asse, someone else is disinvesing a he same ime, so aggregae, or social, capial does no rise. Macroeconomiss reserve he erm invesmen for ransacions ha increase he magniude of real aggregae capial in he economy. This includes mainly he pur- 15 2

3 chase (or producion) of new real durable asses such as facories and machines. 1 The caegory of invesmen ha receives he mos aenion is business fixed invesmen, which is he purchase of new srucures and equipmen by business firms for producion purposes. However, here are wo oher caegories, and hey are also imporan. Invenory invesmen consiss of increases in socks of unsold goods or unused inpu maerials. This kind of invesmen is quie differen from business fixed invesmen because invenory capial normally has a very shor life span. When invenories decrease from one period o he nex, as someimes happens even a an aggregae level, invenory invesmen is negaive. Anoher unique feaure of invenory invesmen is ha i ofen occurs uninenionally. Unsold producs are couned as invenory invesmen wheher he firm bough hem inending o build up is invenory or simply ended up selling less han i expeced o sell. Residenial invesmen consiss of purchases of new housing unis, wheher by firms or households. As discussed in he consumpion chaper, new home purchases by households are couned as invesmen, wih a monhly renal flow of housing services couned under consumpion of services. Like invenory invesmen, residenial invesmen ends o behave quie differenly in some ways han business fixed invesmen. Three oher caegories of invesmen are worhy of menion, hough none of hem is caegorized as invesmen in he U.S. Naional Income and Produc Accouns. One is invesmen in human capial. Educaion could easily be classified as invesmen, bu he U.S. income accounans do no do so. A second is consumer purchases of durable goods. Refrigeraors and auomobiles have all he characerisics of capial and are so classified when purchased by firms, bu in he Unied Saes hey are classified under consumpion raher han invesmen. Finally, governmen invesmen in roads, bridges, buildings, and oher durable asses are classified as governmen spending raher han invesmen in he Unied Saes. The heories discussed in Romer s ex apply mainly o business fixed invesmen. To supplemen hese heories, Secion D below will presen prominen heories of invenory and residenial invesmen. These heories are also examined briefly in Mankiw s ex. Financing of invesmen Where do/should firms ge he funds wih which o make invesmen expendiures? The answers o his quesion are he subjec maer of finance, which is one of he main disciplines augh in academic schools of business. There are hree main sources of invesmen funds for firms: (1) inernal funding using accumulaed profis, 1 Noe ha we exclude resale ransacions involving exising asses for he same reason we exclude purely financial ransacions he invesmen by he buyer is exacly offse by he disinvesmen of he seller. 15 3

4 (2) borrowing eiher from banks or hrough he issue of financial asses such as (longerm) bonds or (shor-erm) commercial paper, and (3) issuing new shares of sock new equiy. 2 Each of hese funding mehods imposes explici and/or implici coss on he firm. If he firm borrows in order o finance is invesmen, i pays an explici ineres cos. If i uses inernal funds for invesmen, i is forgoing oher uses of hose funds. Had he firm no used he inernal funds for new capial, i could have earned ineres on he funds by lending hem or purchasing financial asses. 3 Thus, he implici cos of each dollar of inernally funded invesmen is (a leas) he ineres on forgone lending. In a perfec capial marke, where all borrowers and lenders pay and receive a uniform ineres rae, he explici ineres cos of loan-financed invesmen equals he implici forgone-ineres cos of self-financed invesmen, so he cos is he same wheher he firm finances hrough borrowing or inernally. Issuing new shares of sock creaes coss for hose who own exising shares. Since he new shares represen claims on he firm s fuure profis, hey dilue he claims of exising shareholders in direc proporion o he amoun of new sock issued. If here are 1000 shares in he firm iniially ousanding and he firm issues 1000 addiional ones, hen each exising share s claim on profis is reduced by half, from 1/1000 o 1/2000. If he expeced profis of he firm fail o rise (as hey should if he invesmen is a wise one), he price of each share will fall o half is original value. Thus, owners of he firm may incur a cos in erms of reduced marke value of heir shares and diluion of heir share of fuure profis unless he proceeds of he sock issue are used in ways ha raise profis commensuraely. There is a fundamenal difference beween financing invesmen hrough borrowing and financing eiher wih accumulaed cash or by issuing new sock. Borrowing creaes a legal obligaion o repay (wih ineres) ha is no presen when invesmen is financed inernally or wih equiy. Suppose ha a firm s invesmen does no pay off. If i has financed he invesmen wih inernal funds, hen hose funds are simply los. If new sock was issued, hen all sockholders will see a reducion in share values and perhaps a low or zero dividend yield. However, here is no legal obligaion in eiher case. In conras, a firm whose invesmen fizzles is sill legally obligaed o pay ineres and principal on any deb incurred. Failure o make he required paymens resuls in bankrupcy of he firm. Thus, deb financing enails a bankrupcy risk ha is no presen wih eiher inernal or equiy finance. 2 There are also many hybrid financial insrumens such as preferred sock, converible bonds, and he like. We shall no concern ourselves wih hese. 3 Alernaively, accumulaed profis ha are used for invesmen could have been disribued direcly o shareholders as dividends. The value of such dividends may also be par of he opporuniy cos of invesmen. 15 4

5 From he sandpoin of he lender, credi ransacions are very convenien. Whereas an owner of equiy mus be consanly vigilan abou he company s performance and profis, a bond-holder need only worry if he firm s performance deerioraes o he poin ha bankrupcy hreaens. The decision abou how much of he firm s capial sock should be financed by borrowing vs. equiy or cash is someimes called he leverage or gearing decision. A firm is said o be highly levered or highly geared if i has a lo of deb relaive o he amoun of is equiy. Recall ha levers and gears are simple machines ha allow a small amoun of effor o move large objecs. As he following example shows, high leverage allows he owners of a firm o make a small invesmen (in he financial sense) of heir own money bu o conrol a large volume of asses. Consider he gains or losses of he owners of wo firms, LL (for low leverage) and HL (high leverage). Firm LL has no deb ousanding and finances all of is invesmen hrough accumulaed earnings or by issuing new equiy. Suppose ha LL has $1000 of capial backed by 1000 ousanding shares and ha i earns eiher $150 or $50 in profi, depending on wheher i has a good or bad year. Each share represens a claim on 1/1000 of ha profi, so he owners of he LL sand o earn a profi of $0.15 or $0.05 on each share invesed. If good and bad years happen wih equal frequency, he expeced reurn is $0.10 per share (a 10% reurn) and he range of oucomes around he mean is ±$0.05. Now consider HL, which has $1000 in capial backed by 500 shares and $500 in borrowing a a 10% ineres rae. The owners of he firm have now only pu up $500 of heir own money, bu hey sill conrol he enire $1000 worh of asses and have claim on he enire profi of he company (afer ineres is paid). The lever of debfinance gives hem he same economic clou as he LL owners wih half as much personal invesmen. The firm now pays $50 in ineres each year, so profi is eiher $100 or $0 depending on wheher i has a good or bad year. Since each share represens a claim on 1/500 of oal profi, he owners of HL will earn $0.20 per share in a good year and zero in a bad one. Alhough he mean reurn is sill $0.10 (10%), he range has now doubled o ±$0.10. Thus, high leverage allows owners of he firms o increase he variaion of heir reurns (and heir poenial profis or losses) for each dollar invesed. Taking leverage o he exreme, consider firm RHL (for ridiculously high leverage) ha has only 1 share ousanding and $999 of deb. The owner of RHL has a oal invesmen of $1 in he firm, bu conrols $1000 of asses. RHL mus pay $99.90 in ineres each period, so i makes a profi is $50.10 in a good year and loses $49.90 in a bad year. The mean reurn is sill $0.10, bu he range is an asounding ±$50.00! 15 5

6 The Modigliani-Miller heorem You would expec ha he leverage decision of a firm would affec he firm s araciveness o poenial buyers of is sock and o poenial lenders and, in pracice, invesors ofen do pay aenion o leverage raios. However, in a perfec capial marke where everyone has full informaion abou he probabiliies of good and bad years and where everyone borrows and lends a he same ineres rae, he value of he firm and he araciveness of is equiy urn ou o be oally independen of how i is financed. Franco Modigliani and Meron Miller demonsraed his mos remarkable resul in a famous 1958 paper. 4 The Modigliani-Miller heorem demonsraes ha under condiions of perfec capial markes, he cos of invesmen o firms is he same regardless of which of he hree mehods of finance i chooses. To undersand he logic of he Modigliani-Miller heorem, we reurn o our example firms from above. Suppose ha you have $100 o inves and are considering wheher o buy shares in LL or in HL. A firs glance, an invesmen in HL appears much riskier. However, his risk is fully diversifiable; you can unwind he high leverage of HL by combining your invesmen in HL wih a simulaneous purchase of bonds, whose reurn does no depend on HL s profi. Suppose ha you inves he enire $100 in 100 shares of LL. This earns you $15 in dividends in a good year and $5 in dividends in a bad one. 5 Alernaively, you could inves $50 in 50 shares of HL and $50 in bonds paying 10% ineres (perhaps even he same bonds issued by HL!). The bonds pay you $5 every year, while he shares in HL pay $10 in a good year and nohing in a bad one. Thus, you ge $15 in a good year and $5 in a bad year, which is he same reurn in boh good and bad years ha you go from invesing $100 in LL shares. This example shows ha individual invesors can choose he riskiness of heir porfolios independenly of he financial decisions of he firms. A low-risk porfolio can eiher consis of los of low-risk shares or of a few high-risk shares balanced by bonds. Furher, high leverage increases boh he supply of bonds (because firms are borrowing more) and he demand for bonds (when invesors wan more bonds o balance off he high-risk equiies), hence here will be no effec on marke ineres raes. The Modigliani-Miller heorem shows ha under ideal condiions he decision abou how much o inves is independen of he decision abou how o finance ha invesmen, since he value of he firm is he same regardless of wheher he firm is- 4 See Modigliani and Miller (1958). These wo economiss were awarded he Nobel Prize in economics for his (and oher) work in 1985 and 1990, respecively. 5 We are assuming ha he firms pay ou all profis in dividends. Modigliani and Miller (1961) showed ha he firm s dividend policy does no affec our fundamenal resul. 15 6

7 sues bonds (becoming highly levered) or uses accumulaed profi or he proceeds from issuing new equiy. This independence allows macroeconomiss o focus only on he firm s invesmen decision, leaving analysis of he decision abou how o raise he required funds o specialiss in finance. Of course he assumpions underlying he Modigliani-Miller heorem, like hose of mos macroeconomic heories, are unlikely o be compleely fulfilled. The world is full of informaion asymmeries and oher capial-marke imperfecions ha lead o some imporan excepions o he Modigliani-Miller resul. Sources of finance and credi availabiliy effecs One imporan excepion o Modigliani-Miller resuls from differences beween he financing opions of small and large firms. While informaion abou he prospecive profi oucomes (and, herefore, abou he crediworhiness) of large firms is likely o be widely known, ousiders may know lile abou smaller firms in he economy. Everyone knows Microsof s repuaion and many analyss and invesmen advisors rack Microsof s performance on a week-o-week basis. However, i would be difficul for a poenial invesor o ge similar informaion abou a iny sarup company. Since invesors will no usually inves in firms hey know nohing abou, small companies mus ofen rely on borrowing from banks or aracing invesmen from specialized venure capialiss raher han raising invesmen funds by selling bonds or shares direcly o he public. Banks are less relucan han he general public o lend o small firms for several reasons. Firs, banks may have access o deailed informaion abou hese firms ransacions hrough records of heir checking accouns and of oher financial ransacions in which he bank has paricipaed. These records allow banks o verify informaion ha he firms provide abou heir financial performance. Second, many banks have deparmens ha specialize in small-business lending, so hey have professional loan officers who have exensive experience in evaluaing small firms financial prospecs. Finally, banks are large insiuions ha have large amouns of (heir deposiors ) money o lend. By lending o a large number of relaively small firms, hey are able o diversify heir risk more effecively han mos individual invesors can. If small firms are dependen on he banking sysem for credi, hen hey may be especially sensiive o condiions in he banking secor. I is widely believed ha a ighening of moneary policy by he cenral bank (he Federal Reserve Sysem in he Unied Saes) causes commercial banks o reduce he volume of heir lending. While ineres raes for large firms ypically go up somewha as a resul of moneary ighening, hese firms usually sill have access o funds hrough financial markes. Small firms however may find heir financial ap dried up compleely due o a so-called credi crunch in he banking sysem. 15 7

8 As we shall sudy laer, he banking sysem is quie sensiive o he availabiliy of reserves ha are supplied by he cenral bank. When he cenral bank conracs he supply of reserves relaive o he demand for hem, banks mus reduce lending below he level hey would oherwise choose. When his happens, banks ypically respond wih some combinaion of (1) raising ineres raes on loans and (2) reducing credi lines and refusing o lend o marginal cusomers. 6 Thus, small firms may be more sensiive o changes in credi condiions han large firms. This asymmery lies a he hear of a debae abou wheher here is a credi channel of influence of moneary policy on he economy. Proponens of he credi channel have found some evidence ha igh credi affecs invesmen by small firms more han ha of large firms. 7 C. Invesmen and he Business Cycle Economiss have long recognized ha invesmen ends o be he mos volaile of he componens of expendiure over he business cycle. Of course, srong correlaions beween invesmen and oupu only mean ha he wo variables end o move ogeher; hey do no allow us o deermine he direcion(s) of causaliy. For ha, we need he framework of economic heory wih which o inerpre he daa. Some economiss have inferred from he high volailiy of invesmen ha flucuaions in invesmen demand are a major cause of business cycles. Ohers have argued ha he wide variaion in invesmen over he cycle reflecs consumpion smoohing: invesmen ges squeezed ou as households aemp o mainain heir consumpion expendiures a a high level during recessions. Keynes and he business cycle A saisfacory heory of he business cycle was a pressing need in he 1930s, when he Grea Depression infliced widespread economic suffering on Europe and America. John Maynard Keynes aemped o fill ha need wih The General Theory 6 In a perfecly compeiive marke, we would expec his raioning of credi o be done enirely by raising ineres raes (he price of credi) o a level ha would equae supply and demand. However, credi markes are plagued by an informaion problem known as adverse selecion, which implies ha increasing he ineres rae on loans will mean ha relaively safe borrowers will sop borrowing bu very risky borrowers (who are likely o defaul anyway) will be araced. Thus, if banks raised ineres raes enough o reduce demand o equaliy wih supply, hey would probably end up wih a subsanially riskier pool of borrowers and a higher defaul rae. 7 A survey aricle on he credi-channel lieraure is Bernanke and Gerler (1995). 15 8

9 of Employmen, Ineres and Money, which he wroe in Alhough he ambiguiies in The General Theory have proved sufficien o susain a huge lieraure aemping o inerpre Keynes, one of he poins on which mos scholars agree is ha Keynes believed ha flucuaions in invesmen were he primary source of cyclical flucuaions. Keynes began by rejecing he classical assumpion ha he economy auomaically revers o full employmen quickly and reliably. Under condiions where markes do no clear, he argued, a shorage of aggregae demand may preven he economy from producing a full capaciy. Since invesmen is he componen of aggregae demand ha falls mos srongly in business-cycle downurns, i was a naural candidae for Keynes in his search for he causes of hese declines in demand. The underlying principles of Keynes s heory of invesmen do no differ much from he heories ha we sudy oday. He used somewha differen erminology and ignored some of he subleies ha subsequen heoreical work has filled in, bu his basic framework was similar boh o ha of classical economiss and o he framework we use oday. This heory assers ha invesmen is he resul of firms balancing he expeced reurn on new capial we call i he marginal produc of capial; he called he marginal efficiency of capial wih he cos of capial, which depends primarily on he real ineres rae. However, Keynes and classical economiss emphasized differen kinds of flucuaions wihin his similar framework. Classical (and ofen modern) economiss usually emphasized he effec ha changes in real ineres raes have on invesmen. This effec occurs as firms move up and down on heir downward-sloping invesmendemand curves. Keynes believed ha he large flucuaions in invesmen were due o shifs in he invesmen-demand curve iself raher han o movemens along he curve. According o Keynes, he invesmen-demand curve is volaile because i depends on firms expecaions of he profiabiliy of invesmen. Keynes hough ha he animal spiris of invesors ended o flucuae wildly in waves of opimism and pessimism. He viewed he business cycle as a sequence of conagious spells of overopimism and over-pessimism. During an economic boom, businesspeople projec he rapid expansion of he economy (and of he demand for heir producs) o coninue. They respond o hese favorable projecions of fuure demand by increasing heir producion capaciy hrough high levels of invesmen in new capial. This high spending hen fuels he expansion, raising demand for he producs of oher firms and encouraging heir opimism. (Recall ha oupu is deermined by aggregae demand in Keynes s sysem.) Because hese opimisic expecaions evenually run ahead of he economy s abiliy o susain he expansion, disappoinmen is ineviable. When he economy begins o urn downward, many firms find ha hey have subsanial excess capaciy, 15 9

10 boh because demand is now falling and because heir high raes of invesmen have lef hem wih he capaciy o produce an unrealisically high volume of oupu. Firms faced wih his excess capaciy sop invesing, which lowers aggregae demand and accenuaes he downward pressure on he economy. As demand and oupu decline, firms become even more pessimisic, keeping invesmen near zero during he conracion phase of he cycle. The cycle evenually sars back upward when firms in some indusries find heir capial socks depreciaed o he exen ha hey need o buy some new capial goods o susain heir (low) curren levels of producion. This iniial rickle of invesmen sars aggregae demand on he road o recovery. Opimism gradually begins o replace pessimism, and he expansion phase of he nex cycle begins. 8 The acceleraor heory of invesmen Among he earlies empirical invesmen models was he acceleraion principle, or acceleraor. 9 In modern exbooks, he acceleraor model survives as a heory of invenory invesmen, as discussed on page 481 of Mankiw s ex. The acceleraor is a simple model ha incorporaes he kind of feedback from curren oupu o invesmen ha Keynes saw occurring hrough he effec of curren oupu on invesors expecaions. The acceleraor model begins wih an assumpion ha firms desired capialoupu raio is roughly consan. 10 This implies ha he desired capial sock for any * period is proporional o he level of oupu in, K = σy, where σ (he lower-case Greek leer sigma) is he desired capial-oupu raio. Suppose ha firms inves in period in order o bring heir capial socks o he desired level K + 1 * in period + 1. * * Then, if depreciaion is zero for simpliciy, I = K+ 1 K. Bu since K = K, ha means ha I = σ (Y + 1 Y ). Thus, he simples acceleraor model predics ha invesmen is proporional o he increase in oupu in he coming period. Firms, of course, do no observe fuure oupu wih cerainy, so he Y + 1 erm mus be inerpreed as an expecaion. The dependence of invesmen on expecaions is boh realisic and cenral o Keynes s ideas. However, since we canno observe expecaions of firms abou fuure oupu, his feaure of he acceleraor model posed problems for hose who wished o implemen i. The mos common way of resolving his difficuly was o assume ha firms expec he change in oupu in he 8 A relaively lucid inerpreaion of Keynes s business cycle heory is presened in Hicks (1950). 9 The origins of he principle go back a leas o Carver (1903). The name acceleraion principle seems o have been coined by Clark (1917). 10 Since we are focused on shor-run business-cycle flucuaions here, i is reasonable o ignore changes in K/Y ha may be associaed wih long-run advances in echnology

11 coming period o be equal o he change in he curren period. In mahemaical erms, hey assume ha E (Y + 1 Y ) = Y Y 1. While modern heoriss who are accusomed o using raional expecaions will find faul wih his myopic heory of expecaions, i reflecs quie reasonably wha Keynes hough was happening in he 1930s ha firms observed a rise or decline in oupu and exrapolaed ha change ino he fuure in deermining heir invesmen spending. Since he capial-oupu raio in mos economies is larger han one (ofen hree or more in advanced economies), moderae expeced changes in oupu are capable of riggering relaively large changes in invesmen in he acceleraor model. This is one of he reasons ha his heory gained grea populariy afer he Grea Depression as a model of invesmen. The muliplier-acceleraor model The acceleraor model can be combined wih Keynes s heory of he consumpion muliplier o produce a simple model of cyclical behavior. The muliplier, which Keynes acually borrowed from R.F. Kahn, was among he cleares conceps in The General Theory. Assuming ha oupu is deermined by aggregae demand, which includes consumpion and invesmen, he muliplier shows ha changes in consumpion will amplify he effec of any change in invesmen on oal oupu and income. Suppose ha invesmen increases for some exogenous reason. This will raise aggregae spending and cause oupu o rise. Since oupu equals income in he economy, aggregae income rises as he producers of he new invesmen goods enjoy higher sales and incomes. According o Keynes s fundamenal psychological law, hese people will spend par bu no all of he increase in heir incomes. This leads o a secondary (bu smaller) increase in aggregae demand, which raises he incomes of hose who produce he producs ha he firs wave of new consumers buys. As heir incomes go up, hey will in urn increase consumpion spending, bu by less han heir incomes rose. Thus, an increase in invesmen ses off a never-ending sequence of ever-smaller increases in consumpion demand ha augmen or muliply he effec of invesmen on income. By simple algebra, he sum of hese effecs, or he Keynesian muliplier, can be shown o converge in he limi o 1/(1 MPC). Keynesian economiss generally reckoned he MPC o be well in excess of one-half, so he muliplier was hough o more han double he effec of invesmen on oupu. 11 When he muliplier is combined wih he acceleraor, he resuling model is capable of ineresing dynamics. This is paricularly rue if eiher he consumpion func- 11 As a realiy check, i is imporan o remember ha he muliplier only works wih his degree of simpliciy in a world in which rises in aggregae demand cause producers o raise oupu one-for-one in response, i.e., when he aggregae-supply curve is horizonal. To he exen ha producers raise prices raher han producing more (or if he increase in demand causes a rise in ineres raes), much or all of he muliplier effec may be neuralized

12 ion or he invesmen funcion has a lagged response o changes in income. 12 Suppose ha boh consumpion and invesmen have a one-period lag in heir response o income changes, so ha C = (1 s)y 1 and I = σ(y 1 Y 2 ). We close he model by assuming ha Y = C + I, which reflecs boh he fac ha we are ignoring he oher componens of aggregae demand (governmen spending and ne expors) and he Keynesian assumpion ha oupu is deermined by demand. In his model, i is ofen assumed ha Y, C, and I represen he deviaion of he respecive variables from heir rend growh pahs raher han he oal levels of he variables hemselves. We will adop ha inerpreaion here. To examine he dynamic behavior of real oupu, we can subsiue for C and I o ge Y = (1 s + σ)y 1 σy 2. This expresses he cyclical componen of real oupu as a second-order difference equaion. The dynamic behavior of Y depends on he 2 parameers s and σ. There are four possible cases. Case I: If σ< (1 s ), hen he value of Y declines seadily over ime. If Y is inerpreed as he deviaion of oupu from is rend, hen Case I implies ha oupu will reurn smoohly o is rend 2 growh pah. Case II: If (1 s ) <σ< 1, hen he sysem will reurn o is rend pah, bu no monoonically. Insead, i will overshoo he rend pah, urn iself around, and hen overshoo again. The paern of convergence will be as a damped sine wave, oscillaing firs above hen below he growh pah. As long as σ < 1, he oscillaions will become gradually smaller and evenually damp ou. Case III: If 2 1 <σ< (1 + s ), hen he sysem will again oscillae, bu he oscillaions will grow 2 ever larger wih ime insead of gradually damping ou. Case IV: If σ> (1 + s ), hen oupu explodes monoonically away from is rend pah. Case II appears o hold some promise as an explanaion for why an economy migh have business cycles. Cases III and IV predic ha he economy will explode away from is rend growh pah. The explosive cases seem unrealisic, hough Hicks buil a heory of business cycles around an explosive muliplier-acceleraor model coupled wih floor and ceiling consrains ha bounded oupu in a range around he growh pah The original mahemaical analysis of he muliplier-acceleraor model was Samuelson (1939). Samuelson was a rue pioneer of he use of mahemaics in economic analysis. His Foundaions of Economic Analysis, originally published in 1947, was he firs deailed saemen of economic heory in mahemaical erms. Samuelson s famous Economics ex dominaed he marke for decades. He won he Nobel Prize for economic science in 1970, he second year in which he prize was awarded. The presenaion of he muliplier-acceleraor model used here is based on Chaper 17 of Allen (1968). 13 See Hicks (1950). Hicks argued ha capaciy consrains would preven he economy from moving oo far above is growh pah and ha he nonnegaiviy consrain on gross inves

13 The muliplier-acceleraor model is no longer used much as a heory of business cycles, hough he acceleraor occasionally crops up in analyses of paricular regions or indusries. There are several reasons for his. The firs is ha macroeconomiss have become very skepical of he aggregae-demand-based heory of oupu deerminaion embodied in he model. As you know, modern approaches o business cycles emphasize he join deerminaion of oupu by demand and supply facors, wih ineres raes and prices playing an imporan role. Moreover, he grea variaion in lenghs and severiy of business cycles over ime and across counries argues agains an endogenous explanaion of he cycle such as ha provided by he muliplier-acceleraor model. In such a model, each business cycle should be he same lengh and, depending on he formulaion, perhaps of he same magniude as well. Modern macroeconomic heory usually assumes ha he business cycle resuls from repeaed random disurbances o he economy by posiive or negaive shocks, ogeher wih a sable convergence mechanism such as ha of Case I (or possibly Case II) above. Depending on he iming and magniude of shocks, i is possible o have boh shor and long business cycles and boh severe and mild ones in such a model. D. Undersanding Romer s Chaper 9 Because invesmen is essenially a dynamic problem, Romer s Chaper 9 uses some sophisicaed mehods of dynamic modeling. Alhough Romer presens hese mehods in a fairly undersandable form, some addiional inuiive inerpreaion may faciliae your undersanding. This secion aemps o provide ha inerpreaion. The firm s profi funcion Romer hrows us a curve on he firs page of Chaper 9 by using a highly absrac funcional represenaion of he firm s profi funcion. Firs, hink carefully abou wha Romer means by he funcion π(k, X 1, X 2,..., X n ). Profis are equal o π(k, X 1, X 2,..., X n ) r K K, where r K is he nominal renal price he firm pays o use one uni of capial. Thus, he π funcion mus measure profi before subracing ou he cos of capial. To have a convenien ile, we shall call his operaing profi. men would preven i from moving oo far below i. When he economy hi he floor and ceiling ha bound he growh pah, i would be sen back oward he pah

14 Wha would he operaing profi funcion of a ypical firm look like? Think abou he simples possible case: a compeiive firm ha produces using only capial and labor. Such a firm has revenue equal o PQ, where P is he compeiive marke price of is produc and Q is he amoun i produces. The firm s producion is consrained by is producion funcion, so Q = F(K, L). In addiion o capial coss, he firm incurs labor coss equal o wl, where w is he marke wage. Thus, he firm s profi excluding capial cos (operaing profi, as we are calling i) is P F(K, L) wl = π(k, P, w). For he compeiive firm, he Xs in Romer s profi funcion are P and w, he prices of oupu and of he oher inpu, labor. 14 For a firm wih monopoly power, P is no exogenous o he firm bu insead is affeced by he amoun of oupu he firm wans o sell. For such a firm, P would be replaced in he lis of Xs in he π funcion by he deerminans of he posiion of demand curve for he firm s produc, such as consumer incomes and he prices of subsiues and complemens. Maximizaion of profi involves seing he derivaive of π(k, X 1, X 2,..., X n ) r K K wih respec o he amoun of capial inpu equal o zero. Romer s equaion (9.1) follows from seing π/ K = 0. For he compeiive firm we discussed above, π K (K, X 1, X 2,..., X n ) = [PF(K, L) wl]/ K = PF K (K, L). Thus, equaion (9.1) implies PF K (K, L) = r K. This equaion is nohing more han he sandard profi-maximizaion condiion: marginal revenue produc equals marginal facor cos. I can be rewrien as F K (K, L) = r K /P, which says ha a compeiive firm maximizes profi where he marginal produc of capial equals he real renal rae on capial. We can draw he lef-hand side as a decreasing funcion of K since he marginal produc of capial is assumed o decrease as more capial is employed (given he level of labor). This means ha he MPK curve can be inerpreed as he real demand curve for capial, wih he firm rening capial up o he level where he marginal produc equals he real renal price. User cos of capial Firms usually own mos of he capial ha hey use raher han rening i. We can sill hink of he cos of capial services as a renal price, bu we mus figure ou wha a profi-maximizing owner of capial would charge (herself) for using one year s 14 Noice ha L is no included. The amoun of labor is a choice variable for he firm a every momen in ime; only variables in he environmen ha are exogenous o he firm are pu ino he π funcion. For a given producion funcion we would eliminae L from he profi funcion by calculaing a profi-maximizing labor demand funcion L*(w, P) and subsiuing in for L in he expression above so ha π(k, P, w) = P F [K, L*(w, P)] w L*(w, P)

15 worh of capial services. This price is wha we call he user cos of capial, which is (in equilibrium) equal o he renal price of capial. Dale Jorgenson developed he heory of he user cos of capial in he 1960s. 15 Consider he long-run problem of a firm ha produces oupu in coninuous ime according o he producion funcion Q() = F(K(), L()). I sells is oupu a price P() and pays w() for labor a ime. The firm buys new capial goods (invess) a rae K () and pays p K () for each uni of capial. Capial is assumed o depreciae a a consan rae δ. The presen value of his firm s (infiniely long) lifeime flow of profis is 0 ( [ ] K ) r V(0) = e PF () K ()), L () wl () () p () K () d. (1) The expression in large parenheses is he firm s ne cash flow a ime, which is he difference beween is revenues and is curren expendiures for labor and new capial goods. 16 The firm s objecive is o choose pahs for K() and L() in order o maximize V(0), given he pahs of w and P. Since equaion (1) conains boh he level and he change in K, he maximizaion problem mus be done using he calculus of variaions, which is a echnique you do no need o learn ye. Jorgenson was able o demonsrae ha a firm ha maximizes he presen value of is ne cash flow V(0) chooses exacly he same pah for K and L as a firm ha maximizes momen-by-momen he level of profi a each momen in ime, provided ha he price aached o capial services is he user cos of capial. In oher words, a firm ha maximizes he presen value of is long-run ne cash flow would se MPK = r K ()/P() a each momen in ime, where r K () is given by Romer s equaion (9.4). Noe ha his is he same condiion ha we derived earlier for a firm ha rens is capial. I is reassuring ha a firm ha owns is capial would choose he same amoun of capial as one ha rens from someone else in a compeiive marke. Inuiively, he expression for he user cos of capial ells us ha he implici ren ha he firm charges iself for using is own capial mus include compensaion for hree cos componens. The firm mus compensae iself for (1) he ineres i could earn if i sold he capial and bough bonds wih he proceeds, (2) he fracion δ of he capial ha wears ou during he period, and (3) any decrease in he marke price of 15 Jorgenson is a disinguished ex-reedie who now eaches a Harvard. The developmen of he user cos of capial was among his firs major conribuions o economic research. 16 Noe ha we use he erm ne cash flow raher han profis. The brackeed expression deducs oulays a ime on new capial goods raher han he cos of he services of he capial goods ha are used in producing ime oupu. We reserve he erm profi o refer o revenue less he laer concep of cos and use ne cash flow o denoe he difference beween ime revenues and acual cash oulays

16 capial ha occurs during he period. A decline in he price of capial goods means ha he poenial resale value of he capial is falling, which makes using capial more cosly. If capial-good prices are rising, his cos componen is negaive and reduces he user cos of capial. Adjusmen-cos model Jorgenson s dynamic model of invesmen describes he opimal invesmen behavior of a firm ha operaes in an environmen where i can adjus is capial sock up or down very quickly in order o say on is opimal pah. This is no a problem as long as he desired capial pah is smooh, so ha he level of invesmen K () does no become oo large in eiher a posiive or negaive direcion. However, if here were a sudden leap in rk ( ), perhaps due o a sudden change in he ineres rae, hen he firm would wan o change is capial sock by some finie amoun insanly in order o raise or lower he MPK. A discree increase in K() implies an infinie rae 17 of invesmen a he momen of he change, K () =. An infinie rae of invesmen is obviously implausible, so how can we revise our model in order o preclude such behavior? We do his by inroducing coss of rapid adjusmen of he capial sock. The more quickly he firm adjuss is capial sock (i.e., he higher he rae of invesmen) he higher he cos i is assumed o incur. We C I 0 0 C 0 = 0, represen adjusmen coss mahemaically as ( ), wih C [ ] =, [ ] and C [ I] 0 >. The easies way o hink of he adjusmen-cos funcion (and he form of he funcion ha we mos ofen use) is as a parabola opening upward from is verex a he origin. 18 The firm incurs posiive adjusmen coss when i changes is capial sock eiher upward or downward, and hose coss rise a an increasing rae as he amoun of ne invesmen ges furher from zero Think abou he ime pah of K(). The rae of invesmen is equal o he slope of he ime pah wih respec o ime. If K() jumps upward a momen 0, hen he slope of he ime pah is verical a 0. Since he slope of a verical line is infinie, his implies ha K ( 0 ) =. 18 This form of he adjusmen-cos funcion has srong implicaions for he behavior of invesmen ha are no fully consisen wih observed behavior. The sric convexiy of he adjusmen-cos funcion implies ha firms always incur lower adjusmen coss if hey spread an invesmen projec over more years. If his were he case, hen we would expec o see coninuous, gradual invesmen by firms raher han large, discree facory addiions. Since mos invesmen in srucures is lumpy i may be more appropriae o recognize he presence of fixed adjusmen coss in addiion o he variable coss modeled here. This is discussed in Romer s Secion 9.8 and in he empirical discussion below. 19 We have assumed ha he depreciaion rae is zero. Relaxing ha assumpion makes he adjusmen-cos funcion a lile awkward. Wih posiive depreciaion, he cos funcion we 15 16

17 The form of he profi funcion ha Romer describes on page 409 may require some clarificaion. He posis a model wih N idenical firms. Each firm has a real operaing profi funcion given by π[k()] κ(), where κ() is he firm s capial sock and K() = N κ() is he aggregae indusry-wide capial sock. The π funcion represens he profi he firm earns per uni of is capial. This is no he same π funcion ha Romer used earlier in he chaper. The assumpion of consan reurns o scale implies ha he firm s real profi can be represened in his muliplicaive manner. The π funcion is downward-sloping because he indusry faces a downward-sloping demand curve, so ha as he indusry s capial sock and oupu increase, profi per uni of oupu decreases. The funcion π[k()]κ() gives he firm s real operaing coss as a funcion of is own capial sock and he aggregae indusry sock. Romer s equaion (9.6) is direcly analogous o our equaion (1) wih four differences. Firs, Romer has π[k()]κ() in place of revenue less labor coss. Second, adjusmen coss are included. Third, he expression is in real raher han nominal erms, meaning ha we do no need he price of capial in fron of he invesmen variable (and ha he π[k()]κ() funcion mus implicily express revenue less labor coss in real raher han nominal erms). Finally, Romer has no subsiued for he I( ) variable as κ () in he inegral expression, so he maximizaion has o be done invoking he consrain raher han unconsrained. You need no worry abou he mahemaics of maximizing (9.6). Romer does i firs in discree ime, hen in coninuous ime. In discree ime, he maximizaion is done by he mehod of Lagrange mulipliers; in coninuous ime i is done as a coninuous-ime Hamilonian. This is essenially he same mehod ha we used o maximize uiliy in he Ramsey growh model. The key variable ha comes ou of his maximizaion problem is he Lagrange muliplier (in discree ime) or cosae variable (in coninuous ime) q(). You may recall from our inroducion of he mehod of Lagrange mulipliers (in Chaper 3) ha he value of he muliplier can be inerpreed as he marginal gain from releasing he consrain by one uni. In he presen case, he consrain is ha capial growh is accomplished only hrough (cosly) invesmen. Thus, he marginal gain from releasing he consrain is he increase in profi ha would occur if he firm could obain one addiional uni of capial wihou incurring invesmen or adjusmen coss. Tobin s q James Tobin, anoher Nobel-prize winner, formulaed an invesmen heory based on financial markes. Tobin argued ha firms invesmen level should depend on he raio of he presen value of insalled capial o he replacemen cos of capial. have assumed implies ha zero adjusmen coss occur where he firm exacly replaces is depreciaing capial

18 This raio is Tobin s q. The q heory of invesmen argues ha firms will wan o increase heir capial when q > 1 and decrease heir capial sock when q < 1. If q > 1, a firm can buy one dollar s worh of capial (a replacemen cos) and earn profis ha have presen value in excess of one dollar. Under hose condiions, firms increase profis by invesing in more capial, so we expec invesmen o be high. If q < 1, hen he presen value of he profis earned by insalling new capial are less han he cos of he capial, so more invesmen lowers profi. We expec invesmen o be near zero if q < 1. When q < 1, someone seeking o ener a paricular indusry can acquire he necessary capial asses more cheaply by buying an exising firm han by building a new one wih new capial. This is rue because he value of insalled capial (i.e., he cos of buying an exising firm) is less han he replacemen cos (he cos of building a new firm). Romer s analysis shows ha Tobin s q is exacly he cosae variable (or Lagrange muliplier) q. The key o undersanding he connecion beween he cosae variable and Tobin s marke inerpreaion of q is Romer s equaion (9.24). This equaion shows ha q() is equal o he presen value (as of ime ) of he sream of real profis per uni of capial ha will be earned from ime ino he infinie fuure. Since a prospecive buyer of a share in a firm has a claim on his sream of profis, she will be willing o pay exacly his presen value of he sream for each uni of capial she implicily buys when she buys shares in he firm. Because we are normalizing he real cos of new capial a one, q will hus equal he raio of he marke value of a firm s sock (qκ) o he replacemen cos of is capial (κ). If q > 1, hen firms can sell a share of new sock for more han a dollar, buy a dollar s worh of capial, and pocke he difference as profi. Hence invesmen will be high when q > 1. When we solve he model for he opimal rae of invesmen, i urns ou o be an increasing funcion of q, K ( ) = f( q ( )), wih f > 1 and f (1) = 0. If he adjusmen cos funcion is quadraic, as we suggesed earlier, hen he f funcion is linear and invesmen is a linear funcion of q. Average and marginal q The cosae variable q() measures he value o he firm of a marginal uni of capial relaive o is replacemen cos. In he real world, i is very difficul o ge daa on marginal q, bu somewha easier o esimae average q. The reason ha average q is easier o measure is ha i can be approximaed by comparing he marke value of he firm s ousanding sock and deb wih he esimaed replacemen cos of is capial sock. The former is easy, he laer somewha more difficul, bu if you measure q in his way you ge a q based on average revenue and cos raher han he more useful marginal q. Because of hese measuremen problems, mos empirical work using q has been based on average raher han marginal q

19 Dynamic analysis On pages 415 hrough 419, Romer develops a phase-diagram analysis of he join evoluion of he capial sock and q over ime. The mehod should be familiar from our analysis of he Ramsey growh model. There are wo variables, K and q, boh of which have changes (ime derivaives) ha depend on he level of one or boh variables. One variable (K) is a rue sae variable in ha i canno jump insananeously. The oher (q) is a conrol variable ha can change insananeously. The equilibrium is a saddle poin. The conrol variable q jumps a any insan o he value given by he saddle pah a he curren value of K. The economy hen converges down or up along he saddle pah o a seady sae in which neiher q nor K is changing. Suppose ha everyhing else in he economy remains consan, bu ha he capial sock is for some reason below is opimal level. This means ha in he iniial equilibrium q > 1, and he capial sock will expand along he saddle pah. As he capial sock expands, q rereas oward one, evenually converging o a seady-sae equilibrium wih he capial sock a is opimal level and q = 1. Afer developing he basic convergence properies of he model, Romer hen analyzes hree examples. Firs he shows ha an increase in oupu demand will cause he q = 0 curve o shif o he righ. The value of q iniially jumps upward o he saddle pah, simulaing posiive invesmen and an increase in he capial sock as he economy converges down he saddle pah. In he second example, an increase in he ineres rae lowers q and reduces he equilibrium capial sock in he long run. The final example shows ha an invesmen ax credi increases he profiabiliy of capial and raises he long-run capial sock. Noe ha all of hese effecs will be differen depending on wheher he exogenous change is permanen or emporary. A permanen change resuls in an immediae leap o he new saddle pah followed by convergence along he pah. A emporary change will no pu he economy on he saddle pah because he exogenous variables are known o be changing in he fuure. (The saddle pah describes he convergence of he sysem when he fuure values of exogenous variables are sable.) Insead, he economy jumps o an unsable saddle pah ha leads back o he original pah a he momen ha he policy revers o is original sae. Once he emporary change is reversed, he economy mus be on he saddle pah associaed wih he original (and ulimae) equilibrium and converges, since no fuure changes in exogenous variables are expeced

20 E. Empirical Sudies of Invesmen The empirical analysis of aggregae invesmen spending has been one of he grea frusraions of poswar macroeconomics. Unlike mos of he oher behavioral funcions of he basic Keynesian sysem, he invesmen funcion has no fi he aggregae daa for mos counries well. 20 All heories based on profi maximizaion predic ha he flow of invesmen expendiures should be sensiive o he cos of capial. Someimes his sensiiviy is modeled hrough q, someimes hrough he user cos of capial, bu in eiher case, ineres raes and ax raes should affec invesmen. However, he correlaion beween aggregae invesmen and real ineres raes is exremely low in mos samples and is someimes posiive raher han negaive. Despie he creaive use of lagged effecs and refined measures of q and he user cos of capial, no consisenly reasonable aggregae empirical specificaion has emerged. This lack of empirical suppor is all he more frusraing because of he overwhelming consensus ha he basic heoreical approach based on q or on he user cos of capial is correc. Of course failing o find a significan relaionship in aggregae daa does no necessarily imply ha he heory is wrong. Measuremen error, biases associaed wih reverse causaliy, and he effecs of omied variables hose ubiquious nemeses of economeric analysis may preven our economeric ess from finding he rue underlying relaionship among he variables. One possible explanaion for he apparen failure of aggregae invesmen demand equaions is ha we do no have good measures of he variables ha shif he demand curve. The fundamenal facor hrough which coss affec invesmen demand is he expeced profiabiliy of incremens o he capial sock, which depends mainly on he marginal produc of capial and he firm s expecaion of he fuure demand for is produc. Only he crudes proxies for hese laer variables can be observed, so he effecs of demand shifs are usually no well represened in he economeric specificaion. If, as is plausible, he larges changes in invesmen resul from shifs in hese unobserved variables raher han from changes in he cos of capial, hen i is no surprising ha he observed empirical relaionship beween invesmen and he cos of capial is weak. In conras, he empirical link beween aggregae invesmen and corporae cashflow measures seems o be quie srong. Firms inves more when hey are earning los of money, almos regardless of he opporuniy cos. Is his because cash flow 20 This secion of he chaper relies heavily on work done joinly wih Malcolm Spiler under he sponsorship of a Goldhammer Summer Collaboraive Research Gran during he summer of