Chaper 20 Opimal Fiscal and Moneary Policy We now proceed o sudy joinly opimal moneary and fiscal policy. The moivaion ehind his opic sems direcly from oservaions regarding he consolidaed governmen udge consrain. Specifically, a road lesson emerging from our sudy of fiscalmoneary ineracions is ha money creaion and hus inflaion poenially helps he fiscal auhoriy o pay for is governmen spending. Alernaively, a road inerpreaion we made when we sudied opimal moneary policy earlier was ha seady-sae inflaion (more precisely, any seady-sae deviaion from he Friedman Rule) aced as a ax on consumers. A ha sage, we did no noe ha a deviaion from he Friedman Rule, acing as a ax, poenially raised revenue for he governmen; now, wih our noion of a consolidaed udge consrain, we are in a posiion o undersand his laer idea. Here, he quesion ha we ake up is: if oh moneary and fiscal policy are conduced opimally, wha is he opimal seady-sae mix of laor axes and inflaion needed o finance some fixed amoun of governmen spending? Our approach o answering his quesion will hew very closely o he mehods of analysis we have already developed in our separae looks a opimal moneary policy (wihou regard for fiscal policy) and opimal fiscal policy (wihou regard for moneary policy). The model we use o answer his quesion mosly comines elemens we have already seen. To overview he key elemens of he model we will use o ry o hink aou our main quesion, our model will: - Feaure an infinie numer of periods - Model money using he money-in-he-uiliy funcion (MIU) approach - Feaure laor income axes as he only direc fiscal insrumen (i.e., no consumpion axes and no axes on savings) - Feaure a consolidaed governmen udge consrain - Feaure a simple linear-in-laor producion echnology - Focus on he seady sae Because y now mos of hese model elemens are familiar o us, we will no spend much ime developing he deails of he asic model; raher we will spend mos of our ime analyzing he opimal policy prolem and is soluion. Spring 204 Sanjay K. Chugh 289
Firms The way in which we model firms is as we have ofen done: he represenaive firm simply hires laor each period in perfecly-compeiive laor markes and sells is oupu. The producion echnology we assume here is also as simple as possile, linear in laor: y f ( n) n. Firms profis in period (in nominal erms) are hus simply Py Wn, where he noaion is sandard: P is he nominal price of goods, W is he nominal wage, and n is he quaniy of laor. When he firm is maximizing profis, we assume i akes as given oh he nominal price P and he nominal wage W. 62 Susiuing he linear producion echnology ino he profi funcion and opimizing wih respec o n (he only hing he firm decides here is how many unis of laor o hire on a period-y-period asis) yields he firm firs-order condiion P W 0. If we define, as usual, he real wage as w W / P, he resul of firm profi-maximizaion is w (.85) Condiion (.25) is one of he equilirium condiions of he simple model we are developing, and is he only one ha arises from he firm (supply) side of he model. Consumers As menioned aove, we will model consumers using our money-in-he-uiliy funcion (MIU) specificaion. The represenaive consumer egins period wih nominal money holdings M, nominal ond holdings B, and sock (a real asse) holdings a. The period- udge consrain of he consumer is Pc P B M Sa ( ) Wn M B ( S D) a, (.86) where he noaion again is as in he MIU model presened earlier: S is he nominal price of a uni of sock, D is he nominal dividend paid y each uni of sock, and P is he nominal price of a one-period nominal ond wih face-value $. Because we coninue o assume ha all onds are one-period onds and he face value of each ond is FV, we have ha P (which you should recall), where i is he ne nominal i ineres rae on a nominal ond held from period o period +. Noe he erm 62 Nohing more han our usual assumpion of price-aking ehavior; here, price-aking descries he firm s ehavior in oh oupu markes and inpu markes. Spring 204 Sanjay K. Chugh 290
( ) Wn in he udge consrain: i represens oal afer-ax laor income in period. The consumer akes oh he wage W and he ax rae as given. 63 Noe he asence in he consumer udge consrain of he lump-sum amoun of ransfer from he governmen which was presen in our earlier sudy of opimal moneary policy. This is one sule u crucial difference in he model we are using here o sudy joinly opimal fiscal and moneary policy. The presen value of lifeime uiliy of he consumer is, as expeced, given y M M 2 M uc,, 2 nuc,, n uc2,, n2..., (.87) P P P2 in which each period s uiliy depends on consumpion c, real money alances M / P, and leisure n and, also as is sandard y now, fuure uiliy is discouned y he facor. Seing up a sequenial Lagrangian (wih he muliplier on he consumer s ime- udge consrain), M M 2 M uc,, 2 nuc,, n uc2,, n2... P P P2 ( ) Wn M B( S D) a Pc P B M Sa (.88) W n M B S D a P c P B ( ) ( )... M S a In period, he consumer chooses ( c, n, M, B, a ). Proceeding mechanically, he firsorder-condiions wih respec o each of hese five choice variales, respecively, are: M u c,, np 0 P M u3 c,, n ( ) W 0 P (.89) (.90) 63 As efore, we, he modeler, know from he firm opimaliy condiion (.25) ha i will (in equilirium) e he case ha W P ; however, he consumer need no undersand his; all he consumer does is ake whaever W is as given. Spring 204 Sanjay K. Chugh 29
M u2 c,, n P 0 (.9) P (.92) P 0 S ( S D ) 0 (.93) Condiions (.29) hrough (.33) descrie how consumers make opimal choices; as such, hey represen equilirium condiions. As usual, hough, i is insrucive o no work wih hese raw firs-order condiions direcly, u insead comine hem ino inerpreale expressions of he form MRS equals a price raio which are he cornersone of consumer heory. From here on, o save on noaion, we will adop he following convenion regarding argumens of funcions. Raher han wrie, for example, M u c,, n o sand for he marginal uiliy of consumpion in period, we will P simply wrie u, and i will e undersood ha he second suscrip indicaes ha i is ime- argumens (specifically, c, M / P, and n ) ha are inside he funcion. Thus, u 2 sands for he marginal uiliy of real money alances in period, u 3 sands for he marginal uiliy of leisure in period, u sands for he marginal uiliy of consumpion in period +, u 2 sands for he marginal uiliy of real money alances in period +, and so on. u Wih his noaional convenion, condiion (.89) implies ha. Insering his in P condiion (.90) and rearranging, we have u ( ) w 3 u, (.94) where, as usual, w W / P sands for he real wage in period. We have seen condiion (.94) counless imes y now: i is simply he consumer s consumpion-leisure opimaliy condiion, saing ha he MRS eween consumpion and leisure (he lefhand-side) equals he afer-ax real wage. Condiion (.94) is an equilirium condiion of he model, and i descries how consumers make opimal consumpion-leisure radeoffs. Nex, condiion (.92) ells us P. Using his fac in condiion (.9), we can u2 wrie P 0, or, equivalenly. Nex, recall ha wih onds ha always have P Spring 204 Sanjay K. Chugh 292
a face value of one, u herefore wrie 2 P i P, meaning we can wrie he previous expression as i u, or, simplifying, 2 i u P i. Recalling ha, we can P u i, (.95) 2 u i which saes ha when consumers are making opimal choices, he MRS eween consumpion and money (he lef hand side) depends on he nominal ineres rae. 64 Condiion (.95) is he consumpion-money opimaliy condiion of his model, in analogy wih he consumpion-leisure opimaliy condiion, and is an equilirium condiion of he model. Finally, he firs-order condiion on sock, equaion (.93), can e manipulaed (along wih condiion (.89) and he ime-+ version of condiion (.89)) o yield a consumpion-savings opimaliy condiion, u u ( r ). (.96) To recall deails, refer ack o he analysis of he consumer s opimizaion prolem in Chaper 7. Resource Consrain As always, he resource consrain descries all of he differen uses of oal oupu (GDP) of he economy. In he model here, oupu is produced y he linear-in-laor producion echnology and, as in he model we used o sudy jus fiscal policy, here are wo uses for oupu: privae consumpion (y consumers) and pulic consumpion (i.e., governmen expendiures). Hence he resource consrain in any arirary period is 64 Don e misled y he noaion: here, u 2 sands for he marginal uiliy of real money alances ecause real money alances is he second argumen of he uiliy funcion. In much of wha we ve done efore, he second argumen of he uiliy funcion was leisure, meaning ha in previous models u 2 sood for he marginal of leisure; in he model we are sudying here, he marginal uiliy of leisure is u 3 ecause leisure is he hird argumen of he uiliy funcion. This is simply a noaional choice, however; we could have jus as readily chosen o make leisure he second argumen and real money alances he hird argumen. Spring 204 Sanjay K. Chugh 293
c gov n. (.97) Governmen The governmen is a consolidaed fiscal-moneary auhoriy, as in our sudy of fiscalmoneary ineracions. The period- udge consrain of he consolidaed governmen is W n P B M M Pgov B, (.98) which is an adapaion of he consolidaed period- governmen udge consrain we encounered in Chaper 5; he only difference is ha raher han regular ax revenue eing specified arirarily as T, here we have W n. The consolidaed governmen udge consrain (GBC) has he same inerpreaion as in Chaper 5: he GBC saes ha governmen spending on goods and services as well as repaymens of mauring governmen de (he righ-hand-side of expression (.98)) can e covered y hree sources (he lef-hand-side of expression (.98)): laor income ax revenue, issuance/sales of new governmen onds, and money creaion. Compare he GBC (.98) wih he governmen udge consrains ha underpinned our analysis of opimal moneary policy in Chaper 7 and opimal fiscal policy in Chaper 9: M M W n P gov, respecively. 65 and Equilirium and Seady-Sae Equilirium The nex sep, as usual, is o descrie he privae-secor equilirium. Because he general idea is he same as in our earlier (separae) sudies of opimal moneary policy and opimal fiscal policy, we do no discuss his in deail here. Raher, we simply proceed o lis he equilirium condiions and hen condense hings down o a small se of seadysae equilirium condiions. The firm opimaliy condiion, expression, (.85), is he only equilirium condiion arising from he supply side of he model. On he demand side of he model, expressions (.94), (.95), and (.96) descrie, respecively, he represenaive consumer s opimal consumpion-leisure radeoff, opimal consumpion-money radeoff, and opimal consumpion-savings radeoff. As such, all hree are also equilirium condiions of our model. In principle, he resource consrain is an equilirium condiion of he model, as well. Bu, as we were ale o do in our sudy of opimal fiscal policy, we can use he 65 Of course, in our previous sudy of jus opimal fiscal policy, we did no pu explici ime suscrips on hings nor did we formulae he analysis in nominal erms, u he wo modificaions are sraighforward. Spring 204 Sanjay K. Chugh 294
consumer s udge consrain, given y expression (.86), in place of he resource consrain. Hence, expression (.86) is he final condiion descriing he privae-secor equilirium. We are concerned wih seady-saes, so we mus impose seady-sae on all of he equilirium condiions. A his sage, imposing seady-sae should e a relaively sraighforward exercise. Le s analyze in some deail, hough, he seady-sae version of he consumer udge consrain. For reasons ha will ecome a i more clear when we formulae he opimal policy prolem elow, le s assume ha B 0 always. Also, i urns ou ha for our purpose (sudying opimal fiscal and moneary policy) he seady-sae quaniy of sock he consumer has is irrelevan, hus le s also assume (wihou furher proof of is irrelevance) ha a 0 always. 66 Wih hese simplifying assumpions, we can wrie (.86) in real erms (i.e., dividing hrough y P ) as c M wn M ( ), (.99) P P or, puing oh erms involving money on he same side of he equaion, M M c wn. (.00) ( ) P P Defining m M / P as real money alances, and using he manipulaion M M P m, we have P P P m c ( ) wn m. (.0) Imposing seady-sae, c ( ) wnm. (.02) 66 Noe ha we are making hese asserions afer we have already oained he consumer s FOCs. If we had made hese assumpions efore compuing FOCs, he srucure of he enire model would e drasically differen; as i sands, i is relaively innocuous, u for reasons ha we leave for a more advanced course in macroeconomic heory. Spring 204 Sanjay K. Chugh 295
Nex, we know ha in seady sae, he inflaion rae equals he money growh rae; if i did no, hen real money alances could no e consan in he seady sae. 67 Readoping our noaion from efore, le g e he seady-sae growh rae of he nominal money supply. Then, g c( ) wnm g ; (.03) noice he appearance of he minus sign on he righ-hand-side. Susiuing w, we have ha he consumer s choice of consumpion depends on his choice of laor supply and real money holdings, g c( ) nm g. (.04) As we did in our analysis in purely opimal fiscal policy, we nex susiue his expression for seady-sae equilirium consumpion ino he remaining privae-secor equilirium condiions, he (seady-sae version of he) consumpion-leisure opimaliy condiion (.94) and he (seady-sae version of he ) consumpion-money opimaliy condiion (.95). Noe ha we do no need o make his susiuion ino he (seady-sae version of he) consumpion-savings opimaliy condiion ecause if we impose seady sae on equaion (.96), we find, as always, ha r. Of course, y he exac Fisher equaion and he fac ha g in seady sae, his can in urn e expressed as i i, which reveals ha in seady-sae equilirium, g g i, (.05) which of course was also rue in our discussion of purely opimal moneary policy. Making he susiuion for c in he consumpion-leisure and consumpion-money opimaliy condiions hus give us 67 In oher words, having already assered ha real money alances ecome consan in he seady sae, i mus e, y he definiion of real money alances, ha M / P is consan. The only way for M / P o e consan is for he numeraor and he denominaor o oh e changing a he same exac rae. This is nohing more han our usual monearis/quaniy-heoreic noion ha in he long run (i.e., in seady sae), he money growh rae is equal o he inflaion rae. Spring 204 Sanjay K. Chugh 296
g u3 ( ) nm, m, n g g u ( ) nm, m, n g (.06) and g u2 ( ) nm, m, n g g. (.07) g g u ( ) nm, m, n g In wriing hese wo expressions, we have re-inroduced he argumens o he marginal uiliy funcions and also used he relaionship in (.05) o eliminae he nominal ineres rae. Condiions (.06) and (.07) condense he enire descripion of he privae-secor equilirium of he economy down o wo condiions. Joinly, hese wo condiions should e hough of as defining a pair of funcions ng (, ) and mg (, ). 68 Formulaion of Opimal Policy Prolem Our ojecive is o sudy joinly-opimal seady-sae fiscal and moneary policy. The policy prolem is o choose a and a g ha maximizes he represenaive consumer s uiliy aking ino accoun he funcion ng (, ), he funcion mg (, ), and he governmen udge consrain. Because we are only concerned wih seady-sae policy, o move owards his goal, le s firs rearrange he governmen udge consrain (.98) and urn i ino a seady-sae expression. Firs, recognize as usual ha P and divide i hrough y P o pu everyhing in real erms: 68 You should hink of his jus as he equilirium reacion funcion cg ( ) in our consideraion of purely opimal moneary policy and n () in our consideraion of purely opimal fiscal policy. The echnical difference here is ha he governmen has wo policy insrumens (he laor ax rae and he money growh rae) and here are wo seady-sae equilirium ojecs o e deermined. However, for a wide class of uiliy funcions used in quaniaive macroeconomic models, i can e shown (in a more advanced reamen of moneary heory) ha laor will depend only on he laor ax rae and money alances will depend only on he money growh rae. Thus, we simply asser in he res of wha we do ha his is rue. Spring 204 Sanjay K. Chugh 297
B M M B wn gov i P P P. (.08) On he righ-hand-side, noice, as is always he case in a consolidaed fiscal-moneary M M udge consrain, he appearance of seignorage revenues, sr. As aove and P in Chaper 5, define B / P as he real amoun of governmen de ousanding a he end of period. Also, reak up he seignorage revenue erm as M M M M M M P. Recalling ha m M / P is real money P P P P P P alances and recalling ha P / P /( ), we can rearrange (.08) furher o ge m B P wn m gov i P P, (.09) or, a lile more compacly, m w n m gov i (.0) B B P where, in he las sep, we used he manipulaion. Our nex sep P P P is o impose seady sae on expression (.0); doing so and comining erms, Because wn m gov i. (.) g in seady-sae, we can wrie he previous expression as wn m gov i g g. (.2) We can condense his expression even furher. The Fisher equaion ells us i( r)( ), which in urn can e expressed as i( r)( g). Nex, we know from our consumpion-savings opimaliy condiion (equaion (.96)) ha in seady g sae, r /. Thus, in seady-sae, i, which we saw in expression (.05). Insering all of his on he lef-hand-side of (.2), we have Spring 204 Sanjay K. Chugh 298
g wn m gov g g. (.3) Afer several manipulaions and rearrangemens, we have arrived a a very useful inermediae form of he seady-sae equilirium version of he governmen udge consrain. 69 Expression (.3) shows ha governmen spending mus e financed in he long run (i.e., in he seady sae) y a cominaion of laor income axes (he firs erm on he lef-hand-side), seignorage revenue (he hird erm on he lef-hand-side), and deflaion of governmen de (he second erm on he lef-hand-side). This las revenue source, deflaion of governmen de, can e hough of as a seadysae version of he ideas of he fiscal heory of he price level and he fiscal heory of inflaion ha we sudied earlier. In ha analysis, recall ha he wo ideas were disinc, and he disincion eween hem lay in when he inflaion wrough y an acive fiscal policy was going o occur: he fiscal heory of he price level saed ha i would occur now, while he fiscal heory of inflaion saed ha i would occur a some ime in he curren period or fuure periods, or perhaps spread ou over muliple periods. In seady sae, however, which is wha we are focused on here, he very noions of now and laer disappear: in seady-sae, ime disappears, hus now and laer are lurred. Hence, in seady-sae we canno disinguish eween he fiscal heory of he price level and he fiscal heory of inflaion; he wo roll ino wha we are here calling deflaion of governmen de. 70 I urns ou ha for he purpose a hand (sudying he opimal seady-sae mix of money growh/seignorage and laor axes) he deflaion of governmen de channel is no imporan. 7 Thus, from now on, we will assume 0 (i.e., he governmen has no de oligaions), which also jusifies why we assumed aove ha B 0 when we were descriing he privae-secor equilirium. The GBC can hus now e wrien as g wn m gov g. (.4) Recall our mode of analysis of opimal policy prolems: a he sage of deermining he opimal policy, he governmen (in his case, he consolidaed fiscal-moneary governmen) akes ino accoun all equilirium condiions, including funcions ha descrie how he privae secor responds o any arirary policy ha i ses. Thus, here 69 Noe ha in our sudy of he FTPL and he FTI, we were no focused on he seady-sae version of he governmen udge consrain; here, we were explicily concerned wih he dynamics (of inflaion and seignorage revenue) implied y he ineremporal governmen udge consrain. 70 In ye oher words, he fiscal heory of he price level and he fiscal heory of inflaion are inherenly dynamic conceps. 7 We leave he precise reasons ehind he seady-sae irrelevance of he de-deflaion mechanism for a more advanced course in moneary heory. Spring 204 Sanjay K. Chugh 299
are hree more hings o do wih (.4): inser he equilirium seady-sae real wage rae w (recall equilirium condiion (.85)), inser he funcion ng (, ), and inser he funcion mg (, ). Making hese inserions, g n(, g) m(, g) gov g. (.5) The governmen s policy prolem hus oils down o he governmen choosing and g o saisfy is udge consrain (.5). The reason ha he opimal policy prolem oils down o jus he governmen udge consrain is jus as i was in our sudy of opimal fiscal policy: he funcions n (, g ) and m (, g ) already capure how he privae secor responds o a given policy he governmen chooses. There are in principle an infinie numer of cominaions of (, g ) ha saisfy (.5). In Chaper 9, when we arrived a he analogous place in he analysis, wha we had was one equaion (he governmen udge consrain) in one unknown (he ax rae); here we have one equaion in wo unknowns. Clearly, if we knew eiher or g, hen we would know he oher as well ha is, if we somehow pick eiher or g, hen equaion (.5) would again reduce o one equaion in one unknown. In order o pin down one of he policies, le s proceeding o compue he firs-order condiions of (.5) wih respec o and g ; using he produc and quoien rules, hey are, respecively, n m g n (, g) 0 g (.6) and n m g m(, g) 0 2 g g g. (.7) ( g) Condiions (.6) and (.7) define eiher he opimal laor ax rae or he opimal growh rae of he money supply; hey do no define oh. We will make his poin more clear hrough an example in he nex secion, u when free o choose wo variales (here, policy variales) o saisfy one equaion, one is of course no really free o choose oh of hem. As was he case in Chaper 9, we canno make any more progress acually compuing he opimal values of and g wihou making some Spring 204 Sanjay K. Chugh 300
assumpions aou he uiliy funcion. 72 This is he ask we ake up in he nex secion. In he nex secion, we firs assume a convenional form for he uiliy funcion, make some progress owards analyzing he joinly-opimal policy, and draw on lesions we have learned previously o draw some general conclusions. 72 Noe ha in he analysis of only opimal moneary policy, we were ale o compleely solve for opimal moneary policy (in isolaion from fiscal policy) wihou making any assumpions aou he uiliy funcion. Things are differen in he analysis of only opimal fiscal policy and he join analysis ecause of he presence of he governmen udge consrain ha is, he presence of a financing concern (i.e., how should he governmen raise revenue?) makes hings much more complicaed, and he level of generaliy of proofs/resuls ha we can oain is no as high as i was in in he case of only opimal moneary policy. Spring 204 Sanjay K. Chugh 30
A Workhorse Uiliy Funcion A uiliy funcion ha is a saple in modern macroeconomic models and one ha we have had many occasions o work wih already is he addiively-separale funcion ha is log in consumpion and money alances and linear in leisure. For he res of our analysis, we hus assume ha he uiliy funcion is ucm (,, n) logclog mlog( n), (.8) which means he marginal uiliy funcions are u / c, u2 / m, and u 3 /( n). Before we can use equaions (.6) and (.7) o figure ou wha eiher he opimal ax rae or he opimal money growh raes is given his uiliy funcion, we mus firs deermine wha he funcions n (, g ) and m (, g ) are for his uiliy funcion (ecause we need hese funcions for use in expressions (.6) and (.7)). In order o deermine he funcions n (, g ) and m (, g ), recall ha we mus use condiions (.06) and (.07). Using he marginal uiliy funcions associaed wih our assumed uiliy funcion in hese wo condiions, respecively, we have and g ( n ) m g n (.9) g ( n ) m g g. (.20) m g The ask is o solve equaions (.9) and (.20) for n and m. There are oviously a numer of ways one can aack his prolem since all i requires is some rue-force (hough edious ) algera. Le s firs solve (.20) for m. Afer a couple of seps of algera and rearrangemen, we have ( n ) ( g) m. (.2) 2g Nex, ake his expression for m and inser i in equaion (.9); doing so, we have ( n ) g ( )( g) n n g 2g n. (.22) Spring 204 Sanjay K. Chugh 302
Canceling some erms gives us n n g n n 2g (.23) or even more compacly, n g n 2g. (.24) Solving his for n, we find 2g ng ( ), (.25) 2 3g 2 which shows ha n is a funcion of g u no a funcion of. This is no a general saemen, of course, u raher simply a propery of he uiliy funcion we are using here; noneheless, i is an ineresing propery o noe. 73 Nex, we need he funcion m (, g ). To compue i, inser (.25) ino (.2); doing so gives us ( )( g) m (, g). (.26) 2 3g 2 If neiher expression (.25) nor expression (.26) srikes you as paricularly informaive don worry, hey really are no. They are inermediaes, hough, required o ake our nex sep, which is o inser hese funcions and heir parial derivaives ino expressions (.6) and (.7). Omiing all he algeraic seps in compuing he appropriae parials and doing he necessary susiuions, ec, making hese inserions leaves us wih he following wo expressions ha deermine eiher he opimal ax rae or he opimal money growh rae: and g 0 23g 2 (.27) (2 )( ) (2( ) 3 g) 2 0. (.28) 73 Wih log uiliy, opimal laor supply is no a funcion of he laor ax rae. Spring 204 Sanjay K. Chugh 303
Assuming, equaion (.28) can only e saisfied if 2, i.e., a laor ax rae of 200 percen! Clearly, his makes no economic sense ecause, as we saw in Chaper 9, if he laor ax rae were even jus 00 percen, noody would ever work. This is he sense, hen in which we mean aove ha he FOCs of he opimal policy prolem here deermine eiher he opimal money growh rae or he opimal laor ax rae. 74 Equaion (.27), on he oher hand, immediaely ells us g is he opimal money growh rae. We ve of course seen his policy prescripion efore i s simply he Friedman Rule. Thus, in he conex of he join conduc of fiscal and moneary policy, he opimal moneary componen of policy is o implemen he Friedman Rule, which recall, means ha here should deflaion of prices and, equivalenly, a zero nominal ineres rae. The las sep is hen o solve for he laor ax rae. Insering he Friedman Rule g in he governmen udge consrain (.5) ells us ha he laor ax rae mus saisfy n(, g) m(, ) gov. (.29) 74 A useful analogy is o hink in erms of he soluion of a general quadraic equaion. The quadraic formula ypically reurns wo soluions, u in pracice, only one of hem makes sense (i.e., in applicaions in economics, physics, engineering, ec.) Spring 204 Sanjay K. Chugh 304