Optimal Monetary Policy When LumpSum Taxes Are Unavailable: A Reconsideration of the Outcomes Under Commitment and Discretion*


 Gwendoline Hood
 3 years ago
 Views:
Transcription
1 Opimal Moneary Policy When LumpSum Taxes Are Unavailable: A Reconsideraion of he Oucomes Under Commimen and Discreion* Marin Ellison Dep of Economics Universiy of Warwick Covenry CV4 7AL UK Neil Rankin Dep of Economics Universiy of Warwick Covenry CV4 7AL UK Augus 2005 * We acknowledge helpful conversaions wih many people: Berhold Herrendorf, Henrik Jensen, Jonahan Thomas and Ted To, o name jus a few. Thanks also o wo anonymous referees, and o paricipans in numerous seminars. All errors and opinions are of course our own. Marin Ellison acknowledges suppor from an ESRC Research Fellowship, Improving Moneary Policy for Macroeconomic Sabiliy in he 2s Cenury (RES ).
2 2 Absrac We reexamine opimal moneary policy when lumpsum axes are unavailable. Under commimen, we show ha, wih alernaive uiliy funcions o ha considered in Nicolini s relaed analysis, he direcion of he incenive o chea may depend on he iniial level of governmen deb, wih low deb creaing an incenive owards surprise deflaion, bu high deb he reverse. Under discreion, we show ha he economy will no necessarily end o he Friedman Rule, as Obsfeld found. Insead i may end o he criical deb level a which here is no cheaing incenive under commimen, and inflaion and could well be posiive here. Keywords ime consisency, opimal inflaionax smoohing, discreion, commimen, Friedman Rule JEL Classificaion E52, E6
3 . Inroducion As is well known, opimal, welfarebased moneary policy, even in a flexprice, efficienly funcioning economy, is subjec o a ime consisency problem if he governmen does no have lumpsum axes or ransfers a is disposal. This is because surprise inflaion may be a subsiue source of nondisoring revenue, alleviaing he problem of being unable o reach he firsbes, Friedman Rule oucome of a zero nominal ineres rae. In his world, moneary policy mus be implemened by openmarke swaps of money and governmen deb, and he opimal secondbes policy is o use deb o smooh ineremporally he disorions caused by inflaion. Surprise inflaion can help reduce hese disorions. Alhough his problem is familiar, significan puzzles abou i remain. Firs is ha he opimal moneary policy is likely o be degenerae, in ha he bes surprise rae of inflaion o pick is infiniy, because his maximises he lump sum of revenue appropriaed by he governmen. Aenion was drawn o his feaure by Lucas and Sokey (983). However, a welldefined opimum can be obained if money eners he economy in such a way ha here is a welfare cos of curren inflaion, since his cos mus hen be balanced agains he benefis. This feaure was inroduced in an ineresing conribuion by Nicolini (998). The presence of he cos, on he oher hand, leads o a second puzzle: i may now be he case ha he ime consisency problem akes he form of an incenive o creae surprise deflaion (i.e. inflaion lower han expeced). This invers all our usual ideas abou ime inconsisency in moneary policy. A hird puzzle arises if we assume policy is conduced under discreion, raher han (as implici in he discussion so far) under commimen. In his case lack of rus in he governmen s projeced policy migh be conjecured o lead o higher expeced and acual inflaion; bu in fac i has been argued ha discreion will lead in he long run o lower inflaion in paricular o convergence o he Friedman Rule where inflaion is negaive. Such an argumen is presened in wo papers by Obsfeld (99, 997). Here we refer o he case of Obsfeld s analysis where he auhoriies objecive funcion is as close as possible o privae uiliy funcions.
4 2 In his paper we reconsider his opimal moneary policy problem. We do so using Nicolini s model of a simple cashinadvance economy, hereby avoiding he firs of he abovemenioned puzzles. We exend his analysis of he case of commimen a lile, bu our main conribuion is o he case of discreion, which he did no sudy. The model enables us o conduc a pure welfarebased analysis of opimal policy under discreion, avoiding Obsfeld s need o posulae an ad hoc objecive funcion for he policymaker. Our analysis shows ha he second and hird of he abovemenioned puzzles are relaed. Specifically, we find ha under discreion i is no necessarily rue ha in he long run he economy will converge on he Friedman Rule. Depending on consumer preferences, i may converge on a differen seady sae where inflaion is above he FriedmanRule level, and quie possibly posiive. We call his he imeconsisen seady sae. The force which makes such a seady sae possible urns ou o be he incenive which exiss owards surprise deflaion under commimen. This laer couneracs he more familiar incenive owards surprise inflaion, and makes i possible ha, under commimen, a criical level of inheried governmen deb exiss such ha he wo incenives exacly cancel ou, leading o no empaion o behave in a imeinconsisen manner. I ranspires ha his criical deb level is he same as he one associaed wih he ime consisen seady sae o which he economy may converge under discreion. The paper is organised as follows. Secion 2 describes he srucure of he economy. We examine opimal moneary policy under commimen in Secion 3. In Secion 4 we sudy opimal moneary policy under discreion, providing boh analyical resuls and numerical compuaions. Secion 5 concludes. 2. The Srucure of he Economy The economy consiss of many idenical households who consume he single ype of oupu good, supply a single ype of labour, and hold money  moivaed by a cashinadvance consrain  and bonds. Markes are perfecly compeiive and all prices are flexible. The governmen issues money and bonds. Bonds are aken o be real, or indexed, as in he
5 3 papers already cied. 2 A key consrain on he governmen is ha i does no have access o lumpsum axes and ransfers: he money supply mus be changed hrough open marke operaions, i.e. purchases or sales of bonds in exchange for money. Since our focus is purely on moneary policy, we shall ignore convenional disoring axes, and rea governmen spending on goods and services as exogenous. Technology akes he form y = n, where y is oupu and n is labour inpu. Hence compeiive firms will se a price equal o he wage, and make zero profis. For he represenaive household, he opimisaion problem is: maximise β [ Uc αn ] Σ= 0 ( ) (U > 0, U < 0) s.. pc M, M + bp( + R) + pn = pc + M + b p, for = 0,...,, wih M 0, b 0 given. () M,b are, respecively, he quaniies of money and real bonds held a he sar of period ; R is he nominal ineres rae beween  and ; c is consumpion; n is labour supply; and p is he price level. The key feaure of his problem is ha M 0 and b 0 are given. In his way he model incorporaes a welfare cos of curren inflaion : a rise in p 0 (and equal proporional rise in all p, > 0) depresses he real value of he household s iniial cash holdings, so ha (provided he cashinadvance consrain is binding) curren consumpion is squeezed. This liquidiy squeeze occurs because he household is assumed o be unable o visi he asse marke a he sar of he period: in any period, goods markes open before asse markes. Such a iming assumpion was inroduced by Svensson (985), and is he reverse of he more common iming assumpion in cashinadvance models associaed wih Lucas (e.g. Lucas 2 Alhough i would be more realisic o assume nominal bonds, his would obviously make i easier o generae a surprise inflaion resul, so o assume real bonds canno be said o faciliae our main conclusion. An ineresing relaed analysis which does assume nominal bonds is by DiazGimenez e al. (2002).
6 4 (982)). Is value in he presen conex is ha i capures he idea ha curren inflaion has a cos because agens canno insananeously adjus heir porfolios. 3 From he firsorder condiions for he household s problem we can readily derive: U'( c ) = α[ + R ], =,...,, (2) + p [ + R + ] =, = 0,...,, (3) p β (in which i has been assumed ha R > 0, so ha he cashinadvance consrain binds). (3) indicaes ha he real ineres rae is a consan equal o he inverse subjecive discoun rae. The nominal ineres rae hus moves oneoone wih he expeced inflaion rae. (2) indicaes ha consumpion is uniquely and negaively relaed (since U < 0) o he nominal ineres rae, or equivalenly o he expeced inflaion rae. This reflecs he disoring effec of he inflaion ax: he earnings from an increase in curren labour supply canno be spen immediaely, owing o he cashinadvance consrain, and heir fuure value is hence eroded by expeced inflaion, which herefore provides a disincenive o labour supply and so o consumpion. Wriing m M /p, we noe ha m = c provided ha he cashinadvance consrain binds, which reminds us ha expeced inflaion equivalenly reduces he demand for real balances. Equilibrium in he privae secor of he economy may now be deermined. A his poin, suppose an arbirary moneary policy defined by a sequence of moneary growh raes, { µ } 0 (where µ [M + M ]/M ). Then equilibrium consumpion mus saisfy: β c = U ( c ) c + + α + µ, (4) which has been obained from (2), (3) and p + /p = (+µ )c /c + (he laer being implied by m = c ). Under perfec foresigh (4) deermines c in a pure forwardlooking manner, as a 3 As Nicolini (998) shows, i can be inerpreed as a more compac version of he limied paricipaion model of Grossman and Weiss (983) and Roemberg (984). There, agens are divided ino wo groups who are only allowed o visi he bank in alernae periods.
7 5 funcion of curren and fuure moneary growh raes { µ + s } s = 0. Wih consumpion ied down by (4), oupu is deermined via he goods marke clearing condiion: y = c + g (5) where g is he level of governmen spending. A key role is played in wha follows by he governmen s budge consrain. The singleperiod consrain may be wrien: bp( + R) + pg = M M + b p. (6) Noe he absence of lumpsum axes or ransfers. Under our iming assumpions, (b,m ) are predeermined in period, and he only policy acion open o he governmen (since we rea g as exogenous) is an openmarke sale or purchase of bonds, which raises or lowers b +, lowering or raising M +, respecively. We may inegrae (6) forwards, incorporaing (3) (and appealing o a No Ponzi Game condiion), o obain: b0( + R0) = Σ= 0βµ m g. (7) β This is he governmen s ineremporal budge consrain expressed in erms of cashflow seigniorage, i.e. µ m. We may also rewrie i as: b0( + R0) = m0 + Σ= β Rm g. (8) β (8) gives he same consrain in erms of opporuniycos seigniorage, i.e. R m. 4 Noice ha m 0 eners (8) differenly from m,m 2, ec. Algebraically speaking, his is he source of ime inconsisency, as we discuss furher below. 4 See Herrendorf (997) for a useful discussion of hese wo definiions. To obain (8) from (7), noe ha [M +  M ]/p can be rewrien as m + β[+r + ]  m.
8 6 3. Opimal Moneary Policy Under Commimen In his secion we assume ha here is some commimen mechanism which obliges he governmen o adhere o he moneary policy which i chooses in period 0, { µ } 0. Is opimisaion problem is hen: maximise w.r.. { µ } 0 Σ 0 β [ Uc ( ) αc αg] = ( ) ( ) β α s.. + R0 b0 + g = c0 +Σ= β U c c, β c = U ( c ) c + + α + µ for = 0,...,, wih (+R 0 )b 0, g given. (9) We have subsiued ou R using (2), m using m = c, and n using he producion funcion and (5). I is clear ha, raher han rea he µ s as he conrol variables, we can equivalenly rea he c s as he conrol variables, leaving he µ s o be deermined residually by he difference equaion consrains. This reduces he problem o: maximise w.r.. { c } 0 Σ= 0 β [ Uc ( ) αc αg] ( + R ) b g = c Σ β U ( c ) c β α. (0) s = We easily see ha his problem has an unconsrained maximum where: U ( c ) = α for all. () This is he firsbes, Friedman Rule soluion, where R = 0. The governmen s ineremporal budge consrain in general prevens he aainmen of his oucome unless iniial governmen deb is sufficienly negaive. Thus, in a secondbes siuaion, c will be less han is FriedmanRule level, and an opimal policy mus rade off he deviaion of c from his wih he deviaion of any oher c s.
9 7 I is sraighforward o derive he following se of firsorder condiions for he problem (0): U ( c ) α = 0 U ( c ) α + [ σ ( c ) ] U ( c )/ α for =,...,. (2) Here, σ(c ) is he relaive risk aversion measure of curvaure of U(c ), i.e. cu ( c ) / U ( c ). I is immediae from his ha c is he same for all =,...,. I is furher apparen ha his common c ( c, say) is in general differen from c 0. Hence an opimal ime pah of c ypically akes he form of eiher a sep up or a sep down (see Figure ). Inspecion of (2) furher shows ha if σ < hen we obain c 0 < c (a sep up ), and if σ > hen we obain c 0 > c (a sep down ). In he special case where σ =, he opimal ime pah is fla. 5 The above resuls are as in Nicolini (998). Nicolini however resrics aenion o uiliy funcions of he consan relaive risk aversion (CRRA) class, so ha σ is an exogenous parameer. Here we will explore he consequences of more general uiliy funcions. A visual aid o doing so is provided by Figure 2. The diagram explois he fac ha, under an opimal policy, c = c for =,...,, in order o represen he problem in reduced form. The indifference curves over (c 0,c) are given by he lifeime uiliy funcion wrien as Uc c g+ Uc c g, and he governmen s ineremporal budge ( 0) α 0 α β[ β] [ ( ) α α ] consrain conracs o: c β 0 () ( 0) 0 U c c c = + β α R b β g. (3) Figure depics his as a backwardbending curve. This reflecs he seigniorage Laffer curve : as fuure inflaion raes (and hus nominal ineres raes) are raised, c falls, bu neverheless fuure seigniorage revenue a firs rises, permiing less revenue o be raised hrough curren seigniorage, and hus permiing higher c 0. However (depending on he shape of U(c)) he Laffer curve may have a peak, such ha revenue sars o fall as fuure inflaion coninues o be increased and c coninues o be reduced. Iniial governmen deb, (+R 0 )b 0, 5 Corresponding o hese ime pahs of consumpion, i is easy o show ha he ime pahs of deb are a sep down, a sep up, and fla, respecively.
10 8 acs as a parameer which fixes he posiion of he budge consrain: higher iniial deb shifs i o he lef. By varying iniial deb we hus race ou a locus of poins of angency wih he indifference curves  he expansion pah. The equaion of he expansion pah is jus he firsorder condiion (2), wih c subsiued by c. Consider now he shape of he expansion pah. Firs, i clearly passes hrough he poin ( c, c ), where c is he Friedman Rule consumpion level given by (). Second, from he resuls presened earlier, he expansion pah mus lie above (below) he 45 o line a levels of c such ha σ(c) < (>). I follows ha if U(c) is such ha here exiss a value c c a which σ(c) =, hen he expansion pah mus cross he 45 o line a his criical value of c. (For his o be of ineres, we obviously need c c < c ). Third, we would inuiively expec he expansion pah o be upwardsloping (i.e. under an opimal policy lower iniial governmen deb would be used o raise boh c 0 and c) 6. Hence, fourh, if c c exiss hen he quesion arises as o wheher he inersecion wih he 45 o line occurs from above or from below. Differeniaing (2) and evaluaing where c 0 = c = c c, we obain: dc α =. (4) dc0 c α + cσ [ U α] c This is clearly less han or greaer han one as σ (c c ) is (respecively) posiive or negaive. Tha is, he expansion pah cus he 45 o line from above if he relaive risk aversion parameer is increasing in consumpion a he inersecion poin, and from below if i is decreasing. 7 Figure 2 illusraes he case where an inersecion of he expansion pah and he 45 o line exiss, and where i occurs from above. In his case here mus also exis a criical value of (+R 0 )b 0 such ha, if iniial deb happened o ake his value, he opimal policy would be o choose c 0 = c = c c. The associaed budge consrain is he one which passes hrough he poin C. If iniial deb were slighly higher han his level, he budge consrain would lie slighly farher o he lef, and he opimum would be a a poin like A, where c 0 < c. Conversely if 6 The condiion for his is ha he RHS of (2) be decreasing in c. I urns ou ha he same condiion is necessary for he opimisaion problem s secondorder condiions o be saisfied, so we need his condiion o hold. I holds provided ha σ(c) is increasing, consan, or no oo srongly decreasing, in c. 7 (4) is no he correc expression for he slope a he FriedmanRule poin. The slope here is insead given simply by σ, and so is greaer or less han one as σ is greaer or less han one.
11 9 iniial deb were slighly lower, he opimum would be a a poin like B, where c 0 > c. In he case in Figure 2, hen, he shape of he opimal ime pah of consumpion depends on he iniial level of deb: high deb gives rise o a sep up shape, and low deb o a sep down shape. Wih he CRRA uiliy funcion used by Nicolini (998), on he oher hand, he expansion pah lies eiher always above, or always below, he 45 o line, and he magniude of iniial governmen deb is herefore irrelevan, qualiaively speaking, o he shape of he opimum ime pah of consumpion. In order o assess wheher here is a poenial ime consisency problem wih he opimal policy, i mus be compared wih he opimal policy if he governmen were permied o reopimise in period. This reopimisaion is only hypoheical, since we have assumed ha here is a commimen mechanism o preven he governmen from acually deviaing from is period0 plan. Le us denoe he opimal choice of c from he perspecive of period s (s ) as c s. We are paricularly ineresed in wheher c is smaller or greaer han c 0. Since c = M /p and M is predeermined, c < c 0 indicaes ha he governmen in period would wish o se p higher han was planned in period 0, i.e. ha here is an incenive o surprise inflaion. Conversely, if c > c 0, he incenive is o surprise deflaion. Following Nicolini s mehod we may readily show ha, if he period0 opimal ime pah akes he form of a sep up, hen in period here is an incenive o surprise inflaion; while if i akes he form of a sep down, hen in period here is an incenive o surprise deflaion. A summary skech of he wo possible relaionships beween he period0 and period opimal ime pahs for consumpion is hus as in Figure. Togeher wih he earlier resuls, his hen implies ha if here exiss a criical consumpion level (sricly less han he Friedman Rule level) a which he coefficien of relaive risk aversion in consumpion equals one, hen associaed wih his is a criical level of governmen deb such ha, if iniial deb happens o ake his value, here is no ime inconsisency. Moreover, if relaive risk aversion is increasing (decreasing) in consumpion a he criical level, hen iniial deb above he criical deb level will be associaed wih a empaion o creae surprise inflaion (deflaion); and iniial deb below, wih a empaion o creae surprise deflaion (inflaion). This is an exension of Nicolini s (998) main finding. I
12 0 says ha if relaive risk aversion is no a consan, hen he direcion of ime inconsisency may depend on he level of iniial governmen deb. An inuiively plausible oucome, we would argue, arises in he case of increasing relaive risk aversion. Here, while he raher unorhodox incenive o creae surprise deflaion dominaes for low levels of deb, he more convenional incenive o creae surprise inflaion dominaes for high levels. This oucome occurs if he expansion pah has he shape shown in Figure 2. 8 A naural followup quesion concerns how likely i is o find a uiliy funcion possessing he properies jus highlighed. Alhough he CRRA funcion is a common uiliy funcion wih some convenien feaures, here is of course no shorage of alernaives. Consider, for example, quadraic uiliy, Uc () = cc [ ˆ c/2]. (Wih his, ĉ α is needed o ensure a nonnegaive FriedmanRule consumpion level.) σ(c) = c/[ cˆ c] for his funcion, so a value of c such ha σ = clearly exiss, namely a c = ĉ /2. Moreover, σ is clearly everywhere increasing in c. The associaed expansion pah hen looks exacly like ha in Figure 2, provided ha ĉ > 2α. 9 Anoher common funcional form, he consan absolue risk aversion (CARA) funcion, can similarly yield an expansion pah like ha in Figure 2. We hence conclude ha he preferences required in order for he above resul o apply are no especially unusual. I is rue ha CRRA preferences are widely used in macroeconomics, parly because hey have he convenien propery of being consisen wih balanced growh. However, quadraic preferences are also widely used in consequence of some oher convenien properies: for example, in he presence of uncerainy hey generae he exac consumpionasarandomwalk resul, and hey underpin meanvariance porfolio analysis. In he presen paper our aim is o explore he implicaions of some familiar uiliy funcions as a heoreical exercise, raher han o claim any one funcion fis he facs well. Clearly 8 A referee poins ou ha, in an exension o his main analysis, Nicolini (998) also noes ha here may be a criical level of iniial deb such ha here is no empaion o ime inconsisency. This is where σ > and deb is nominal. Nominal deb generaes an incenive owards surprise inflaion which could exacly offse he incenive owards surprise deflaion. However Nicolini does no develop he analysis of his case. 9 This condiion ensures ha ĉ /2 is less han he FriedmanRule consumpion level, ĉ α. If i is violaed, he expansion pah lies everywhere above he 45 o line. Even hough quadraic uiliy by iself does no herefore guaranee an expansion pah like ha in Figure 2, we can sill say ha, wih quadraic uiliy, if an incenive o surprise deflaion is ever o exis (i.e. if par of he expansion pah is ever o lie below he 45 o line), hen a criical consumpion level as defined in he main ex mus also exis.
13 much more elaboraion of our barebones model would be needed before i could be confroned wih he daa. A furher naural quesion concerns why he direcion of ime inconsisency should depend on he size of σ. As noed, in a CIA model wih Lucas s iming assumpion, he empaion o chea in moneary policy is always owards surprise inflaion. In he presen framework, he incenive o raise all curren and fuure prices equiproporionally in order o appropriae revenue in a nondisoring way is counerbalanced by he incenive no o generae a large welfare cos of curren inflaion hrough he liquidiysqueeze mechanism explained earlier. The direcion of he empaion o chea depends on how hese incenives change, relaive o he incenives for he seing of subsequen periods inflaion, as ime advances. Firs, observe ha a empaion o reduce curren inflaion relaive o wha was planned would arise if an increase in curren inflaion (increase in p 0 /p , from he perspecive of period 0) were o provide a smaller gain in he PV of seigniorage and a larger loss of lifeime uiliy 0 han a projeced increase in fuure inflaion (increase in p /p 0, from he perspecive of period 0). This is because, when he fuure arrives, he seigniorage benefi would be smaller han i was from he vanage poin of he period before and he uiliy cos would be larger, so he governmen would perceive an incenive o deviae downwards from wha i had planned. Such changes in incenive occur when σ > because fuure consumpion demand is hen inelasic wih respec o fuure inflaion (or, equivalenly, +R ), as can be seen from (2); while curren consumpion demand is always unielasic wih respec o curren inflaion, as follows from he cashinadvance consrain. Hence a uni increase in p 0 /p  causes a larger loss of lifeime uiliy han a (/β)uni increase in p /p 0. I also causes a smaller gain in he PV of seigniorage, because he inelasic fuure consumpion demand means a weaker dampening effec on revenue of he shrinkage in he ax base as he inflaion rae is raised, so ha seigniorage increases more srongly wih fuure han wih curren inflaion. 0 Through is direc effec on uiliy, ignoring he indirec effec via is budgeary impac
14 2 4. Opimal Moneary Policy Under Discreion µ s s 0 If he governmen is unable o commi o a given policy plan made in period, { + }, hen i is no raional for households forecass of fuure µ = +s s, as of ime (denoe e hese by µ + s ), o equal hose in he plan. Insead, forecass should be based only on variables observable a ime. The governmen s inheried sock of deb is one obvious observable on which o condiion forecass, since i is clear from he previous secion ha iniial deb is a key deerminan of he governmen s opimal policy choices. Oher observables could also be used, such as curren and pas values of µ. However, his would inroduce an elemen of repuaionbuilding behaviour ino policy. Since we wish o focus on purely discreionary behaviour, we use only deb as he basis for privae forecass. Formally, he concep of equilibrium employed will be ha of Markovperfec equilibrium. Firs, i is useful, as Obsfeld does, o define he concep of governmen commimens k b( + R ) + Σ β g. (5) s s= 0 + s Commimens is jus he spending, in presenvalue erms, over which he governmen does no have discreion. Formally, we will use k raher han deb as he sae variable in wha follows, alhough, given our assumpion ha g is imeinvarian, k is simply deb plus he consan, g/(β). Noe ha he governmen s budge consrain in erms of k is (cf. (6)): k = µ m + β k +. (6) We now suppose ha, in period, households forecas (µ +,k +2 ) using he rules: µ = ˆ( φ k ), (7) e + + k = ψˆ ( k ). (8) e This is he same basic idea as in Obsfeld (99, 997). His 99 paper sudies a small open economy version of he opimal inflaion ax smoohing problem, while his 997 paper does he same for a closed economy. The key difference from he presen analysis, as noed in he Inroducion, is ha Obsfeld s model does no include a welfare cos of curren inflaion. This obliges him o use a governmen objecive funcion which differs from he privae uiliy funcion, being obained by adding on an ad hoc cos of curren inflaion erm o he laer.
15 3 (Noe ha k + is observable by households in period  since hey can observe µ m and hus use (6)  so ha i is appropriae o base heir forecass upon i.) ˆ(.) φ and ψ ˆ (.) are for he momen reaed as arbirary funcions, bu hey will laer be deermined by imposing a raionaliy requiremen as par of he condiions of Markovperfec equilibrium. To generae an speriod ahead forecas, (8) may be used repeaedly in (7): µ = ˆ φ( ψˆ ( k )), (9) e s + s + n where ψˆ (.) ψˆ( ψˆ(... ψˆ(.)...)) denoes he nh ierae of he funcion ψ ˆ (.). These forecasing rules can nex be used o deermine equilibrium consumpion in period. Recall ha he equilibrium value of c is given by he privaesecor law of moion, (4), solved in a forwardlooking manner. The relevan µ s o use in his equaion are now he expeced values as given by (9), raher han he values from he governmen s policy plan, since wha couns for deermining he acual c are households expecaions. I is helpful o consider he deerminaion of c in wo sages. Firs, given k + and hus a sequence of expeced values { e µ + s } s = generaed by (9), we use (4) for periods + onwards o solve for c +. This yields households forecas, as of period, of he equilibrium value of c +. Since i e is coningen on he observed k +, we may wrie i as c ˆ( ) + = θ k+. Second, using ˆ( θ k + ) in (4) for period, we obain he equilibrium value of curren c : β c = U ( ˆ θ( k )) ˆ θ( k ) + + α + µ ek ( + ). (20) + µ Curren c hus depends on wo variables: he currenly observed µ, and he currenly observed k +, whose effec operaes via influencing expecaions abou fuure µ +s s. The funcion e(.) capures his expecaions effec. The form of e(.) derives from he form of ˆ(.) θ, which is in urn derived from he forms of ˆ(.) φ and ψ ˆ(.). The governmen now reas (20) as par of he economy s srucure, and hus as a given. Is opimisaion problem under discreion is herefore o choose { µ } 0 o maximise lifeime
16 4 uiliy of he ypical agen subjec o (20) and (6) for = 0,...,, and o given k 0. By subsiuing (20) ino (6) and ino he maximand, we can express his as: maximise w.r.. { } 0 ek ( ) ek ( ) Σ β U α αg µ µ µ + + = ek ( + ) s.. k = µ + βk + µ +, for = 0,...,, wih k 0 given. (2) (2) reveals ha he governmen s dynamic opimisaion problem under discreion  unlike ha under commimen  has a sandard recursive form. Tha is, every period, he new value of he sae (k + ) depends only on he curren value of he sae (k ) and on he curren value of he conrol (µ ), while he flow maximand also depends only on hese same wo variables (alhough i is k + which appears in he flow maximand, k + is an implici funcion of (k,µ ) via he consrain). This srucure means ha he problem can in principle be solved by dynamic programming. In urn, dynamic programming ensures ha he soluion is ime consisen. 2 The dynamic programming perspecive also makes clear ha he soluion o our problem can be expressed as a pair of feedback rules on he sae; for example µ = φ( k ), (22) k+ = ψ ( k). (23) (22)(23) define he governmen s opimal moneary policy having aken as our saring poin he public s arbirary forecasing rules, (7)(8). We noice ha (22)(23) relae he same variables as (7)(8); hence, for (7)(8) o provide raional forecass by he public, we need he funcions ˆ(.) φ and φ (.), and also ψ ˆ (.) and ψ (.), o be he same. In his case households will forecas correcly no maer wha he value of k. The discreionary, or 2 Given ha he funcion e(.) has as ye unknown properies, here is a quesion as o wheher he opimisaion problem (2) is well defined. Here we proceed as if his is he case, bu we reurn o he quesion below.
17 5 Markovperfec, equilibrium is hus a pair of forecasing rules which have he propery ha hey reproduce hemselves in he guise of opimal governmen policy rules. I is convenien, as earlier, o rewrie he opimisaion problem in order o rea he c s raher han µ s as he conrols. To do his we use (20) o subsiue ou he µ s from (2), so ha he problem is ransformed o: maximise w.r.. { c } 0 Σ 0 β [ U ( c ) αc αg] = s.. k = e( k+ ) c + β k+, for = 0,...,, wih k 0 given. (24) The firsorder condiions for his are hen easily derived: U ( c+ ) α = [ U ( c) α][ + e ( k+ )] for = 0,...,. (25) β Repeaing here he consrain from he problem (24), k = e( k ) c + βk, (26) + + we see ha (25)(26) consiue a firsorder dynamical sysem in (k,c ), which deermines he evoluion of he economy under he opimal discreionary policy. Given ha k is a predeermined variable whereas c is no, c 0 will generally be ied down in relaion o k 0 by he ransversaliy condiion. This hence deermines a paricular soluion of he sysem (25) (26) which consiues he soluion of he opimisaion problem. We denoe his soluion as: c = θ ( k ). (27) Equivalenly, (27) is he opimal feedback rule of he conrol upon he sae variable. If he opimum is such ha he economy converges on a seady sae, hen (27) is also he saddlepah soluion of (25)(26). Alhough θ (.), like e(.), is sill a presen an unknown funcion, once i has been deermined we can subsiue i back ino (26) o ge:
18 6 ek ( ) + βk = k+ θ ( k). (28) + + This implicily deermines he opimal feedback rule for k +, (23). The opimal feedback rule for µ, (22), is similarly recoverable by subsiuing (27) and (23) ino (20). The funcions e(.) and θ (.) hus play a cenral role in he Markovperfec equilibrium, since, if hey can be deermined, he oher unknown funcions φ(.) and ψ(.) follow. I is also worh noing ha in equilibrium he funcion θ (.) mus urn ou o be he same as he funcion ˆ(.) θ. This is because if households forecasing rules are always o yield correc predicions, heir forecas of c + coningen on k + mus coincide wih he governmen s opimal, and hus acual, choice of c + coningen on k +. We now aim o sudy he properies of he discreionary equilibrium. We sar wih he seady saes. Inspecion of (25) suggess wo ways in which a saionary soluion of he dynamical sysem (25)(26) may occur. Firs, (25) is clearly saisfied a he Friedman Rule, where U  α = 0 for all. We shall refer o his as he Friedman Rule seady sae (FRSS), I is inuiively clear ha if iniial governmen deb is sufficienly negaive ha he underlying secondbes problem is absen, hen ime inconsisency is removed, and a governmen which sared in his forunae posiion would have no incenive o move away from i. Obsfeld (99, 997) similarly idenifies he Friedman Rule allocaion as a seady sae of he discreionary equilibrium in his analysis. However, whereas for Obsfeld he Friedman Rule allocaion is he only seady sae, his is no necessarily rue here. A second way in which a saionary soluion of (25) could occur is if here exiss a k + a which e (k + ) = 0. More specifically, we migh conjecure ha his would be rue a he value of k + corresponding o he criical deb level as we defined i for moneary policy under commimen. The moivaion for such a conjecure is ha we know from Secion 3 ha here is no ime inconsisency if iniial deb happens o equal he criical value, and ha, under commimen, if he governmen sared wih his amoun of deb i would choose o say here. Thus i migh be hypohesised ha, wih his criical amoun of iniial deb, he opimal policy under discreion would be he same as under commimen. This second ype of seady sae we shall refer o as he ime consisen seady sae (TCSS).
19 7 To prove ha a TCSS exiss under discreion if, in he problem under commimen, here exis criical consumpion and deb levels as defined in Secion 3, consider again he definiion of he funcion e(.) (given in (20)). Differeniaing his funcion wih respec o k, we obain: β e ( k) = [ σ( c)] U ( c) θ ( k) (29) α (in which we have equaed ˆ(.) θ wih θ (.), for he reason explained). Alhough θ ( k) is unknown, we do know from Secion 3 ha σ(c) = a he criical consumpion level c c. This is herefore sufficien o prove ha e = 0 a he corresponding criical commimens level (call his k c ). Hence a saionary soluion of (25) does indeed arise a k c. The level of inflaion a he TCSS is higher han he negaive inflaion rae (β) which prevails a he Friedman Rule. Is value depends on he uiliy funcion: from (2) and (3), he general expression for he inflaion rae is (β/α)u (c) , and a c c his could be posiive or negaive. Hence, if we can show ha here are condiions under which he discreionary equilibrium converges on he TCSS (for he general case in which he iniial k 0 k c ) hen i follows ha he longrun desinaion of he economy under discreion is no necessarily a siuaion of deflaion. We may also noe ha he level of deb a he TCSS is higher han he negaive level required o susain he Friedman Rule, and he same is rue of he level of commimens. Wheher hey are negaive or posiive in he absolue again depends on he uiliy funcion, and also on g. 3 The evoluion of he economy under discreion is governed by (23), which we saw o ake is form implicily from (28). Differeniaing (28), and evaluaing a a generic seady sae k, we have: dk + θ ( k) = ψ ( k) =. (30) dk β + e ( k) + Local convergence o eiher he FRSS or he TCSS clearly requires ha  < ψ <. Hence we shall proceed by aemping o solve for θ ( k) and e (k) a each ype of seady sae, in 3 Specifically, seing c 0 = c = c c in (3), we have k c = ( β)[(β/α)u (c c )]c c, and b c (+R c ) = k c  g/(β).
20 8 order o deermine wheher his condiion can be saisfied. An expression for e has already been obained in (29). Differeniaing (29) for a second ime yields he following expression for e, which will be useful below: 2 2 e = ( β / α)[ σ U ( θ ) + ( σ) U ( θ ) + ( σ) U θ θ ]. (3) In order o find an expression for θ, recall ha θ ( k ) is he saddlepah soluion of he dynamical sysem (25)(26). By aking a linear approximaion o his sysem abou a seady sae, we may find an expression for θ. Such a calculaion yields: + θ U α = e + ( β + e ) + β ± e + ( β + e ) + β 4 β( β + e ) 2β U U 2 2 U α 2 2. (32) We now observe ha (29), (3) and (32) consiue a sysem of hree simulaneous equaions in he four unknowns ( e, e, θ, θ ). Alhough we canno solve i as i sands, if we proceed o evaluae i a eiher he FRSS or TCSS, i urns ou ha we obain addiional resricions which are sufficien o make soluion possible. Consider firs he local dynamics of he FRSS. Seing U = α, noice ha e drops ou of (32). (29) and (32) hen consiue a sysem of wo equaions in jus wo unknowns, e and θ, from which we may hope o solve for e and θ. Appendix A presens he relevan calculaions. We show ha he sysem can be reduced o a quadraic equaion in θ as he single unknown. Once θ has been deermined, e follows from (29) and ψ from (30). Since he quadraic implies wo possible soluions for θ, here are also wo possible soluions for e and ψ. The poin of paricular ineres is wheher eiher of he soluions for ψ have absolue value less han one. The key resul demonsraed in Appendix A is ha if σ <, here is exacly one soluion for ψ wih absolue value less han one; bu if σ > here are no soluions wih absolue value less han one. Combining his wih our earlier finding, we hus conclude ha:
21 9 Proposiion The FriedmanRule consumpion level and associaed negaive deb level consiue a seady sae of he discreionary equilibrium. Moreover, if he coefficien of relaive risk aversion in consumpion is less (greaer) han one a his seady sae, hen, wihin a neighbourhood of i, a discreionary equilibrium which converges on i exiss (does no exis). In he case σ < we also find, more specifically, ha ψ lies in (0,) (ensuring ha convergence is monoonic), and ha θ and e are negaive. Thus, as he Friedman Rule is approached, deb seadily falls (or equivalenly  once deb becomes negaive  governmen asses seadily rise) and consumpion seadily rises. Inflaion also seadily falls (becomes more negaive). This is he oucome obained by Obsfeld (99, 997). Such an oucome differs from wha happens under commimen, where, if σ <, consumpion also rises along an opimal ime pah, bu only beween periods 0 and l. Deb correspondingly falls beween periods 0 and, bu remains permanenly above is Friedman Rule level. The clue as o why a governmen acing under discreion goes farher in reducing is deb over ime han a governmen acing under commimen, lies in he negaive e, as Obsfeld poined ou. e < 0 means ha lower deb (lower k + ) induces higher curren consumpion hrough affecing privae agens expecaions (recall (20)). From he governmen s perspecive, e < 0 hus increases he reurn o a marginal reducion in he deb: no only does lower deb nex period mean he fuure inflaionax revenue needed is lower, bu also, by lowering privae expecaions of fuure inflaion 4, i raises curren consumpion. By conras, under commimen, privae agens do no use he level of governmen deb as he basis for heir forecass  insead, hey rus he governmen o carry ou oday s opimal plan. Hence a change in he deb level per se does no have his added Obsfeld effec. 5 A numerical illusraion of a se of discreionary equilibria (one for each iniial value of k ) converging on he FRSS is given in Figure 3. This is calculaed for he CRRA uiliy 4 To see his, noe ha since θ is negaive, and recalling ha θ = ˆ θ, a reducion in k + raises privae agens forecass of c +. c + is negaively relaed o expeced inflaion p + /p by (2)(3). 5 Alhough, under his oucome, uiliy is higher in he long run under discreion han under commimen, lifeime uiliy from he perspecive of period 0 is lower, because consumpion in he shor run can be shown o be lower. This is as i should be, because inabiliy o commi acs as a consrain on he opimal policy.
22 20 funcion wih σ = 0.5, α = 5 and β = (We have also se g = 0 for all.) A descripion of he algorihm used for he compuaions is provided in Appendix C. The FriedmanRule consumpion level in his example is c = 0.04, and he corresponding level of commimens needed o susain his is k = Panel (a) shows he e(k ) funcion for his example. This confirms he negaive effec of he sock of deb (o which commimens are here equivalen), operaing via privaesecor expecaions, on c . Panel (b) shows he θ(k ) funcion (he line labelled c ), indicaing he negaive oal effec of he sock of deb on he governmen s opimal choice of c. Noe ha he range of c values considered exends from he maximum, FriedmanRule, value down o approximaely one quarer of ha value, so ha he picure is no jus concerned wih a small neighbourhood of he FRSS. The funcion is clearly quie close o being linear. Panel (c) plos he ψ(k ) funcion and also, for reference, he 45 o line. As Proposiion predics, he slope is less han one  in fac, i is abou which confirms ha he economy does indeed converge monoonically on he FRSS for any iniial level of deb above For comparison, panel (b) also plos he wo consumpion levels which resul from he problem under commimen. The line c 0 gives consumpion in he firs period of an opimal plan as a funcion of he iniial deb k 0 ; while he line c gives consumpion in he second and all laer periods, again as a funcion of k 0. We hus see ha for he same inheried level of deb, in his example consumpion in he shor run is chosen o be lower under discreion han under commimen and, correspondingly, inflaion is chosen o be higher. When σ >, however, Proposiion indicaes ha i is no possible for he oucome obained by Obsfeld o occur in our model. The force driving he economy away from he FriedmanRule oucome in such a case is discussed below in he conex of Proposiion 2. This suggess ha in his case he desinaion may insead be he imeconsisen seady sae, where he laer exiss. We now urn o he local dynamics of he TCSS. A he TCSS iself, σ = and e = 0, as shown previously. Hence θ drops ou of (3), and he simulaneous sysem of (29), (3) and (32) can be reduced o he sysem of jus (3) and (32), in he unknowns θ and e. From his we may hope o solve for θ and e. Appendix B presens he relevan calculaions. We show
23 2 ha he sysem can be reduced o a quadraic equaion in θ as he single unknown, analogous o, bu differen from, he quadraic applying a he FRSS. Once θ has been deermined, ψ follows from (30). Since he quadraic implies wo possible soluions for θ, here are also wo possible soluions for ψ. The poin of paricular ineres is wheher eiher of he soluions for ψ have absolue value less han one. The key resul demonsraed in Appendix D is ha if σ > 0, here is exacly one soluion for ψ wih absolue value less han one; bu if σ < 0 here are no soluions wih absolue value less han one. Combining his wih our earlier findings, we hus conclude ha: Proposiion 2 If he criical consumpion and deb levels as defined for he problem under commimen exis, hen a seady sae of he discreionary equilibrium also exiss a hese consumpion and deb levels. Inflaion could be posiive or negaive a his seady sae. Moreover if, under commimen, iniial deb above he criical level was associaed wih a empaion o creae surprise inflaion (deflaion), and below, wih a empaion o creae surprise deflaion (inflaion), hen, under discreion, wihin a neighbourhood of he seady sae, an equilibrium which converges on i exiss (does no exis). As Proposiion 2 emphasises, in he neighbourhood of he criical consumpion level here is a close relaionship beween he opimal policy under commimen and ha under discreion. Moreover, in he case where under commimen a empaion o surprise inflaion is associaed wih deb above he criical level and vice versa (i.e. in he case where σ > 0), we can show ha 0 < ψ <, hus ensuring ha convergence under discreion is monoonic; and we can also show ha θ and e are negaive. If k 0 > k c, consumpion hen seadily rises over ime unil i reaches he criical level. Along his pah, inflaion and deb are falling. If, on he oher hand, k 0 < k c, hen consumpion seadily falls over ime, wih accompanying rises in deb and inflaion. By comparison, under commimen, pahs wih he same k 0 would involve a rise or a fall in consumpion (respecively) beween periods 0 and, and a corresponding fall or rise in deb during period 0, bu hey would no coninue all he way o (c c,k c ). The reason why he evoluion is carried furher under discreion is ha e < 0 a k 0 > k c and e > 0 a k 0 < k c, as follows from he fac ha e < 0 a he TCSS. e < 0 means ha here
24 22 is an Obsfeld effec, as described above, so ha he governmen perceives addiional benefis of deb decumulaion when unable o commi. e > 0, by conras, implies ha here is a reverse Obsfeld effec : now a marginal increase in endofperiod deb raises curren consumpion hrough he effec on agens expecaions, giving he governmen an incenive o accumulae deb relaive o he case in which i can commi. Inuiively, e > 0 here because his is he region of k 0 in which, under commimen, he empaion is o creae surprise deflaion. In his region, for a governmen following he opimal commied policy, afer one period he higher deb which i faces would creae an incenive for i o choose consumpion o be greaer han in he plan: under discreion, his incenive is refleced in he way deb influences consumpion hrough expecaions. 6 Figure 4 provides a numerical example of a se of discreionary equilibria (one for each iniial value of k ) converging on a TCSS. This is calculaed for he quadraic uiliy funcion wih ĉ = 0.08, α = and β = (Again, g = 0 for all.) The algorihm used is again ha described in Appendix C. The implied criical value c c, i.e. where σ(c) =, is 0.04, and correspondingly k c = The e(k ) funcion for his example is depiced in panel (a). As prediced by he heory, i shows ha he gradien is zero a k c and he shape is concave: i.e. small increases in deb here affec consumpion via he public s expecaions in opposie ways, depending on wheher deb is high or low. Panel (b) plos he oal effec of deb on he governmen s opimal choice of c. The dynamics of deb are illusraed in panel (c). To make he picure clearer, we plo k + k on he verical axis, since k + as a funcion of k urns ou o be very close o he 45 o line. I can be seen ha he TCSS a k c is indeed sable, even if convergence is very slow. Panel (c) also shows he FRSS (a k = ) and so reveals how i is unsable. For comparison, in panel (b) we also plo he wo consumpion levels which resul from he problem under commimen (cf. Figure 3). We see ha while, for deb above 6 Having esablished ha e = 0, e < 0 in he neighbourhood of he TCSS, i is sraighforward o check and confirm ha he secondorder condiions for he governmen s opimisaion problem are saisfied when e(.) has hese properies. In he neighbourhood of he FRSS we canno deermine e analyically. However our numerical experimen confirms ha a maximum exiss for he chosen parameer values, and also suggess ha under CRRA uiliy e = 0 more generally may no be a bad approximaion. I is again sraighforward o check and confirm ha he secondorder condiions are saisfied when e(.) is linear.
25 23 k c, consumpion in he shor run is lower under discreion han commimen, for deb below k c he opposie is rue. Our resuls demonsrae he main claim made in he Inroducion, which is ha i is no ineviable ha under discreion he opimal policy will converge on he Friedman Rule. As we have jus argued, he force which may preven he aainmen of he Friedman Rule, or even drive he economy away from i, is he fac ha under commimen here can be an incenive o surprise deflaion raher han surprise inflaion. I may be remarked ha, alhough he condiions for local convergence o he FRSS and TCSS are no he same, i urns ou hey appear similar when viewed in erms of he expansion pah diagram. For local convergence o be possible, a eiher seady sae he expansion pah mus cu, or mee, he 45 o line from above. This is because he sabiliy condiions in Proposiions and 2 are he same as hose governing he slopes of he expansion pah a he relevan poins, as is clear from he resuls in Secion 3. Thus, if he expansion pah has he shape illusraed in Figure 2, hen, under discreionary policy, local convergence o he FRSS is no possible, bu local convergence o he TCSS is. If insead he expansion pah lies everywhere above he 45 o line  excep a he Friedman Rule poin where i mus mee i  hen he FRSS is he only seady sae under discreion, and local convergence o i is possible Conclusions Our main findings having been summarised in he Inroducion, we here commen on heir generaliy. Firs, i is no essenial o he resuls ha we have used a cashinadvance framework. A common alernaive is he moneyinheuiliy funcion approach. If his is adoped, hen, in order o capure a welfare cos of curren inflaion, an analogous modelling device o he use of Svensson s iming in cashinadvance is o use beginningofperiod nominal balances deflaed by he curren price as he real balances variable in he uiliy funcion (as is done, in a differen applicaion, by Neiss (999)). In early work we employed 7 These are he only wo possible shapes for he expansion pah under quadraic or CARA uiliy.
26 24 such an approach and obained resuls which exacly parallel hose presened here. In fac, i can be shown ha he wo approaches are mahemaically equivalen. However he inerpreaion required of he funcions is differen: wha maers in he moneyinheuiliy funcion approach is he shape of he uiliyofrealbalances funcion, raher han he shape of he uiliyofconsumpion funcion. Second, he precise condiions which we have obained in his paper for a imeconsisen seady sae of he discreionary equilibrium o exis, and for convergence o i o be possible, are ones which we would no expec o be robus o oher ways of modelling he welfare cos of curren inflaion. The way he laer is represened here is simple and convenien bu here are more sophisicaed ways in which i could be capured, such as in he limied paricipaion models referred o earlier. Neverheless we would sill expec ha a imeconsisen seady sae, as we have defined i, could occur in such models, albei wih modificaions o he exac condiions for is exisence. Hence we anicipae ha our conclusion ha opimal moneary policy under discreion does no necessarily converge on he Friedman Rule would sill apply in more general models.
27 c plan plan 0 c plan 0 plan Figure Opimum ime pahs of consumpion under commimen c c A C B expansion pah 45 o 0 c c 0 Figure 2 The governmen s opimisaion problem under commimen in reduced form
28 2 FIGURE 3 Discreionary equilibrium wih CRRA uiliy σ = 0.5, α = 5, β = 0.95; FRSS value of k = 0.04
29 3 FIGURE 4 Discreionary equilibrium wih quadraic uiliy ĉ =0.08, α = , β = 0.95; TCSS value of k = 0.04; FRSS value of k =
30 References DiazGimenez, J., Giovanei, G., Marimon, R. and Teles, P. (2002) Nominal Deb as a Burden on Moneary Policy, unpublished paper, Dep of Economics, Universidad Carlos III de Madrid Grossman, S. and Weiss, L. (983) A TransacionsBased Model of he Moneary Transmission Mechanism, American Economic Review 73, Herrendorf, B. (997) Time Consisen Collecion of Opimal Seigniorage: A Unifying Framework, Journal of Economic Surveys, 46 Lucas, R.E. (982) Ineres Raes and Currency Prices in a TwoCounry World, Journal of Moneary Economics 0, Lucas, R.E. and Sokey, N. (983) Opimal Fiscal and Moneary Policy in an Economy wihou Capial, Journal of Moneary Economics 2, Neiss, K.S. (999) Discreionary Inflaion in a General Equilibrium Model, Journal of Money, Credi and Banking 3, Nicolini, J.P. (998) More on he Time Consisency of Moneary Policy, Journal of Moneary Economics 4, Obsfeld, M. (99) A Model of Currency Depreciaion and he DebInflaion Spiral, Journal of Economic Dynamics and Conrol 5, 577 Obsfeld, M. (997) Dynamic Seigniorage Theory: An Exploraion, Macroeconomic Dynamics, Roemberg, J. (984) A Moneary Equilibrium Model wih Transacions Coss, Journal of Poliical Economy 92, Svensson, L.E.O. (985) Money and Asse Prices in a CashinAdvance Economy, Journal of Poliical Economy 93,
Economics Honors Exam 2008 Solutions Question 5
Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I
More informationA Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)
A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke
More information11/6/2013. Chapter 14: Dynamic ADAS. Introduction. Introduction. Keeping track of time. The model s elements
Inroducion Chaper 14: Dynamic DS dynamic model of aggregae and aggregae supply gives us more insigh ino how he economy works in he shor run. I is a simplified version of a DSGE model, used in cuingedge
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More information4. International Parity Conditions
4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency
More informationGraduate Macro Theory II: Notes on Neoclassical Growth Model
Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationImpact of Debt on Primary Deficit and GSDP Gap in Odisha: Empirical Evidences
S.R. No. 002 10/2015/CEFT Impac of Deb on Primary Defici and GSDP Gap in Odisha: Empirical Evidences 1. Inroducion The excessive pressure of public expendiure over is revenue receip is financed hrough
More informationUnderstanding Sequential Circuit Timing
ENGIN112: Inroducion o Elecrical and Compuer Engineering Fall 2003 Prof. Russell Tessier Undersanding Sequenial Circui Timing Perhaps he wo mos disinguishing characerisics of a compuer are is processor
More informationOptimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More informationChapter 6: Business Valuation (Income Approach)
Chaper 6: Business Valuaion (Income Approach) Cash flow deerminaion is one of he mos criical elemens o a business valuaion. Everyhing may be secondary. If cash flow is high, hen he value is high; if he
More informationRelative velocity in one dimension
Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies
More informationA OneSector Neoclassical Growth Model with Endogenous Retirement. By Kiminori Matsuyama. Final Manuscript. Abstract
A OneSecor Neoclassical Growh Model wih Endogenous Reiremen By Kiminori Masuyama Final Manuscrip Absrac This paper exends Diamond s OG model by allowing he agens o make he reiremen decision. Earning a
More informationII.1. Debt reduction and fiscal multipliers. dbt da dpbal da dg. bal
Quarerly Repor on he Euro Area 3/202 II.. Deb reducion and fiscal mulipliers The deerioraion of public finances in he firs years of he crisis has led mos Member Saes o adop sizeable consolidaion packages.
More informationThe Grantor Retained Annuity Trust (GRAT)
WEALTH ADVISORY Esae Planning Sraegies for closelyheld, family businesses The Granor Reained Annuiy Trus (GRAT) An efficien wealh ransfer sraegy, paricularly in a low ineres rae environmen Family business
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationRepresenting Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationBALANCE OF PAYMENTS. First quarter 2008. Balance of payments
BALANCE OF PAYMENTS DATE: 20080530 PUBLISHER: Balance of Paymens and Financial Markes (BFM) Lena Finn + 46 8 506 944 09, lena.finn@scb.se Camilla Bergeling +46 8 506 942 06, camilla.bergeling@scb.se
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More informationGraphing the Von Bertalanffy Growth Equation
file: d:\b1732013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and
More informationJournal Of Business & Economics Research September 2005 Volume 3, Number 9
Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy YiKang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo
More informationThe Greek financial crisis: growing imbalances and sovereign spreads. Heather D. Gibson, Stephan G. Hall and George S. Tavlas
The Greek financial crisis: growing imbalances and sovereign spreads Heaher D. Gibson, Sephan G. Hall and George S. Tavlas The enry The enry of Greece ino he Eurozone in 2001 produced a dividend in he
More informationChapter 8: Regression with Lagged Explanatory Variables
Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One
More informationPresent Value Methodology
Presen Value Mehodology Econ 422 Invesmen, Capial & Finance Universiy of Washingon Eric Zivo Las updaed: April 11, 2010 Presen Value Concep Wealh in Fisher Model: W = Y 0 + Y 1 /(1+r) The consumer/producer
More informationOption PutCall Parity Relations When the Underlying Security Pays Dividends
Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 22523 Opion Puall Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,
More informationAppendix D Flexibility Factor/Margin of Choice Desktop Research
Appendix D Flexibiliy Facor/Margin of Choice Deskop Research Cheshire Eas Council Cheshire Eas Employmen Land Review Conens D1 Flexibiliy Facor/Margin of Choice Deskop Research 2 Final Ocober 2012 \\GLOBAL.ARUP.COM\EUROPE\MANCHESTER\JOBS\200000\22348900\4
More informationWhy Did the Demand for Cash Decrease Recently in Korea?
Why Did he Demand for Cash Decrease Recenly in Korea? Byoung Hark Yoo Bank of Korea 26. 5 Absrac We explores why cash demand have decreased recenly in Korea. The raio of cash o consumpion fell o 4.7% in
More informationA Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
More informationMACROECONOMIC FORECASTS AT THE MOF A LOOK INTO THE REAR VIEW MIRROR
MACROECONOMIC FORECASTS AT THE MOF A LOOK INTO THE REAR VIEW MIRROR The firs experimenal publicaion, which summarised pas and expeced fuure developmen of basic economic indicaors, was published by he Minisry
More informationMorningstar Investor Return
Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion
More informationChapter 10 Social Security 1
Chaper 0 Social Securiy 0. Inroducion A ypical social securiy sysem provides income during periods of unemploymen, illhealh or disabiliy, and financial suppor, in he form of pensions, o he reired. Alhough
More informationDYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper
More informationWorking Paper No. 482. Net Intergenerational Transfers from an Increase in Social Security Benefits
Working Paper No. 482 Ne Inergeneraional Transfers from an Increase in Social Securiy Benefis By Li Gan Texas A&M and NBER Guan Gong Shanghai Universiy of Finance and Economics Michael Hurd RAND Corporaion
More informationHedging with Forwards and Futures
Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buyside of a forward/fuures
More informationState Machines: Brief Introduction to Sequencers Prof. Andrew J. Mason, Michigan State University
Inroducion ae Machines: Brief Inroducion o equencers Prof. Andrew J. Mason, Michigan ae Universiy A sae machine models behavior defined by a finie number of saes (unique configuraions), ransiions beween
More informationChapter 1.6 Financial Management
Chaper 1.6 Financial Managemen Par I: Objecive ype quesions and answers 1. Simple pay back period is equal o: a) Raio of Firs cos/ne yearly savings b) Raio of Annual gross cash flow/capial cos n c) = (1
More informationStochastic Optimal Control Problem for Life Insurance
Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian
More informationCHARGE AND DISCHARGE OF A CAPACITOR
REFERENCES RC Circuis: Elecrical Insrumens: Mos Inroducory Physics exs (e.g. A. Halliday and Resnick, Physics ; M. Sernheim and J. Kane, General Physics.) This Laboraory Manual: Commonly Used Insrumens:
More informationCRISES AND THE FLEXIBLE PRICE MONETARY MODEL. Sarantis Kalyvitis
CRISES AND THE FLEXIBLE PRICE MONETARY MODEL Saranis Kalyviis Currency Crises In fixed exchange rae regimes, counries rarely abandon he regime volunarily. In mos cases, raders (or speculaors) exchange
More informationThe naive method discussed in Lecture 1 uses the most recent observations to forecast future values. That is, Y ˆ t + 1
Business Condiions & Forecasing Exponenial Smoohing LECTURE 2 MOVING AVERAGES AND EXPONENTIAL SMOOTHING OVERVIEW This lecure inroduces imeseries smoohing forecasing mehods. Various models are discussed,
More informationMarkit Excess Return Credit Indices Guide for price based indices
Marki Excess Reurn Credi Indices Guide for price based indices Sepember 2011 Marki Excess Reurn Credi Indices Guide for price based indices Conens Inroducion...3 Index Calculaion Mehodology...4 Semiannual
More informationIndividual Health Insurance April 30, 2008 Pages 167170
Individual Healh Insurance April 30, 2008 Pages 167170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve
More informationEfficient Risk Sharing with Limited Commitment and Hidden Storage
Efficien Risk Sharing wih Limied Commimen and Hidden Sorage Árpád Ábrahám and Sarola Laczó March 30, 2012 Absrac We exend he model of risk sharing wih limied commimen e.g. Kocherlakoa, 1996) by inroducing
More informationLecture Note on the Real Exchange Rate
Lecure Noe on he Real Exchange Rae Barry W. Ickes Fall 2004 0.1 Inroducion The real exchange rae is he criical variable (along wih he rae of ineres) in deermining he capial accoun. As we shall see, his
More informationRandom Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More informationTHE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS
VII. THE FIRM'S INVESTMENT DECISION UNDER CERTAINTY: CAPITAL BUDGETING AND RANKING OF NEW INVESTMENT PROJECTS The mos imporan decisions for a firm's managemen are is invesmen decisions. While i is surely
More informationLEASING VERSUSBUYING
LEASNG VERSUSBUYNG Conribued by James D. Blum and LeRoy D. Brooks Assisan Professors of Business Adminisraion Deparmen of Business Adminisraion Universiy of Delaware Newark, Delaware The auhors discuss
More informationINSTRUMENTS OF MONETARY POLICY*
Aricles INSTRUMENTS OF MONETARY POLICY* Bernardino Adão** Isabel Correia** Pedro Teles**. INTRODUCTION A classic quesion in moneary economics is wheher he ineres rae or he money supply is he beer insrumen
More informationDOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR
Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 7 33 DOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR Ahanasios
More informationYTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.
. Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure
More information4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay
324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find
More informationInductance and Transient Circuits
Chaper H Inducance and Transien Circuis Blinn College  Physics 2426  Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual
More informationThe Real Business Cycle paradigm. The RBC model emphasizes supply (technology) disturbances as the main source of
Prof. Harris Dellas Advanced Macroeconomics Winer 2001/01 The Real Business Cycle paradigm The RBC model emphasizes supply (echnology) disurbances as he main source of macroeconomic flucuaions in a world
More informationChapter 9 Bond Prices and Yield
Chaper 9 Bond Prices and Yield Deb Classes: Paymen ype A securiy obligaing issuer o pay ineress and principal o he holder on specified daes, Coupon rae or ineres rae, e.g. 4%, 5 3/4%, ec. Face, par value
More informationPart 1: White Noise and Moving Average Models
Chaper 3: Forecasing From Time Series Models Par 1: Whie Noise and Moving Average Models Saionariy In his chaper, we sudy models for saionary ime series. A ime series is saionary if is underlying saisical
More informationPhillips Curve. Macroeconomics Cunningham
Phillips Curve Macroeconomics Cunningham Original Phillips Curve A. W. Phillips (1958), The Relaion Beween Unemploymen and he Rae of Change of Money Wage Raes in he Unied Kingdom, 18611957, Economica.
More informationCannibalization and Product Life Cycle Management
MiddleEas Journal of Scienific Research 19 (8): 10801084, 2014 ISSN 19909233 IDOSI Publicaions, 2014 DOI: 10.5829/idosi.mejsr.2014.19.8.11868 Cannibalizaion and Produc Life Cycle Managemen Ali Farrukh
More informationLongevity 11 Lyon 79 September 2015
Longeviy 11 Lyon 79 Sepember 2015 RISK SHARING IN LIFE INSURANCE AND PENSIONS wihin and across generaions Ragnar Norberg ISFA Universié Lyon 1/London School of Economics Email: ragnar.norberg@univlyon1.fr
More informationPrice elasticity of demand for crude oil: estimates for 23 countries
Price elasiciy of demand for crude oil: esimaes for 23 counries John C.B. Cooper Absrac This paper uses a muliple regression model derived from an adapaion of Nerlove s parial adjusmen model o esimae boh
More informationNetwork Effects, Pricing Strategies, and Optimal Upgrade Time in Software Provision.
Nework Effecs, Pricing Sraegies, and Opimal Upgrade Time in Sofware Provision. YiNung Yang* Deparmen of Economics Uah Sae Universiy Logan, UT 84322353 April 3, 995 (curren version Feb, 996) JEL codes:
More informationAnalysis of tax effects on consolidated household/government debts of a nation in a monetary union under classical dichotomy
MPRA Munich Personal RePEc Archive Analysis of ax effecs on consolidaed household/governmen debs of a naion in a moneary union under classical dichoomy Minseong Kim 8 April 016 Online a hps://mpra.ub.unimuenchen.de/71016/
More informationDebt Relief and Fiscal Sustainability for HIPCs *
Deb Relief and Fiscal Susainabiliy for HIPCs * Craig Burnside and Domenico Fanizza December 24 Absrac The enhanced HIPC iniiaive is disinguished from previous deb relief programs by is condiionaliy ha
More informationNiche Market or Mass Market?
Niche Marke or Mass Marke? Maxim Ivanov y McMaser Universiy July 2009 Absrac The de niion of a niche or a mass marke is based on he ranking of wo variables: he monopoly price and he produc mean value.
More informationDebt management and optimal fiscal policy with long bonds 1
Deb managemen and opimal fiscal policy wih long bonds Elisa Faraglia 2 Alber Marce 3 and Andrew Sco 4 Absrac We sudy Ramsey opimal fiscal policy under incomplee markes in he case where he governmen issues
More informationChabot College Physics Lab RC Circuits Scott Hildreth
Chabo College Physics Lab Circuis Sco Hildreh Goals: Coninue o advance your undersanding of circuis, measuring resisances, currens, and volages across muliple componens. Exend your skills in making breadboard
More informationRisk Modelling of Collateralised Lending
Risk Modelling of Collaeralised Lending Dae: 4112008 Number: 8/18 Inroducion This noe explains how i is possible o handle collaeralised lending wihin Risk Conroller. The approach draws on he faciliies
More information1. Explain why the theory of purchasing power parity is often referred to as the law of one price.
Chaper Review Quesions. xplain why he heory of purchasing power pariy is ofen referred o as he law of one price. urchasing ower ariy () is referred o as he law of one price because he deerminaion of he
More information1. The graph shows the variation with time t of the velocity v of an object.
1. he graph shows he variaion wih ime of he velociy v of an objec. v Which one of he following graphs bes represens he variaion wih ime of he acceleraion a of he objec? A. a B. a C. a D. a 2. A ball, iniially
More informationWorking Paper Monetary aggregates, financial intermediate and the business cycle
econsor www.econsor.eu Der OpenAccessPublikaionsserver der ZBW LeibnizInformaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Hong, Hao Working
More informationINVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS
INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,
More informationMeasuring macroeconomic volatility Applications to export revenue data, 19702005
FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a
More informationPerformance Center Overview. Performance Center Overview 1
Performance Cener Overview Performance Cener Overview 1 ODJFS Performance Cener ce Cener New Performance Cener Model Performance Cener Projec Meeings Performance Cener Execuive Meeings Performance Cener
More informationLecture III: Finish Discounted Value Formulation
Lecure III: Finish Discouned Value Formulaion I. Inernal Rae of Reurn A. Formally defined: Inernal Rae of Reurn is ha ineres rae which reduces he ne presen value of an invesmen o zero.. Finding he inernal
More informationWorking Paper Social security systems, human capital, and growth in a small open economy
econsor www.econsor.eu Der OpenAccessPublikaionsserver der ZBW LeibnizInformaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Kaganovich, Michael;
More informationChapter 2 Problems. 3600s = 25m / s d = s t = 25m / s 0.5s = 12.5m. Δx = x(4) x(0) =12m 0m =12m
Chaper 2 Problems 2.1 During a hard sneeze, your eyes migh shu for 0.5s. If you are driving a car a 90km/h during such a sneeze, how far does he car move during ha ime s = 90km 1000m h 1km 1h 3600s = 25m
More informationRelationships between Stock Prices and Accounting Information: A Review of the Residual Income and Ohlson Models. Scott Pirie* and Malcolm Smith**
Relaionships beween Sock Prices and Accouning Informaion: A Review of he Residual Income and Ohlson Models Sco Pirie* and Malcolm Smih** * Inernaional Graduae School of Managemen, Universiy of Souh Ausralia
More informationSPECIAL REPORT May 4, Shifting Drivers of Inflation Canada versus the U.S.
Paul Ferley Assisan Chief Economis 4169747231 paul.ferley@rbc.com Nahan Janzen Economis 4169740579 nahan.janzen@rbc.com SPECIAL REPORT May 4, 2010 Shifing Drivers of Inflaion Canada versus he U.S.
More informationChapter 4: Exponential and Logarithmic Functions
Chaper 4: Eponenial and Logarihmic Funcions Secion 4.1 Eponenial Funcions... 15 Secion 4. Graphs of Eponenial Funcions... 3 Secion 4.3 Logarihmic Funcions... 4 Secion 4.4 Logarihmic Properies... 53 Secion
More informationTable of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities
Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17
More informationFollow links Class Use and other Permissions. For more information, send to:
COPYRIGHT NOTICE: David A. Kendrick, P. Ruben Mercado, and Hans M. Amman: Compuaional Economics is published by Princeon Universiy Press and copyrighed, 2006, by Princeon Universiy Press. All righs reserved.
More informationCOMPLEMENTARY RELATIONSHIPS BETWEEN EDUCATION AND INNOVATION
Discussion Paper No. 731 COMPLEMENTARY RELATIONSHIPS BETWEEN EDUCATION AND INNOVATION Kasuhiko Hori and Kasunori Yamada February 2009 The Insiue of Social and Economic Research Osaka Universiy 61 Mihogaoka,
More informationForecasting and Information Sharing in Supply Chains Under QuasiARMA Demand
Forecasing and Informaion Sharing in Supply Chains Under QuasiARMA Demand Avi Giloni, Clifford Hurvich, Sridhar Seshadri July 9, 2009 Absrac In his paper, we revisi he problem of demand propagaion in
More informationFifth Quantitative Impact Study of Solvency II (QIS 5) National guidance on valuation of technical provisions for German SLT health insurance
Fifh Quaniaive Impac Sudy of Solvency II (QIS 5) Naional guidance on valuaion of echnical provisions for German SLT healh insurance Conens 1 Inroducion... 2 2 Calculaion of besesimae provisions... 3 2.1
More informationFourier series. Learning outcomes
Fourier series 23 Conens. Periodic funcions 2. Represening ic funcions by Fourier Series 3. Even and odd funcions 4. Convergence 5. Halfrange series 6. The complex form 7. Applicaion of Fourier series
More informationAnalysis of Pricing and Efficiency Control Strategy between Internet Retailer and Conventional Retailer
Recen Advances in Business Managemen and Markeing Analysis of Pricing and Efficiency Conrol Sraegy beween Inerne Reailer and Convenional Reailer HYUG RAE CHO 1, SUG MOO BAE and JOG HU PARK 3 Deparmen of
More informationTrading on ShortTerm Information
428 Trading on ShorTerm Informaion by ALEXANDER GÜMBEL This paper shows ha invesors may wan fund managers o acquire and rade on shorerm insead of more profiable longerm informaion. This improves learning
More informationOptimal Stock Selling/Buying Strategy with reference to the Ultimate Average
Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July
More informationStock Trading with Recurrent Reinforcement Learning (RRL) CS229 Application Project Gabriel Molina, SUID 5055783
Sock raing wih Recurren Reinforcemen Learning (RRL) CS9 Applicaion Projec Gabriel Molina, SUID 555783 I. INRODUCION One relaively new approach o financial raing is o use machine learning algorihms o preic
More informationPrincipal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya.
Principal componens of sock marke dynamics Mehodology and applicaions in brief o be updaed Andrei Bouzaev, bouzaev@ya.ru Why principal componens are needed Objecives undersand he evidence of more han one
More informationTSGRAN Working Group 1 (Radio Layer 1) meeting #3 Nynashamn, Sweden 22 nd 26 th March 1999
TSGRAN Working Group 1 (Radio Layer 1) meeing #3 Nynashamn, Sweden 22 nd 26 h March 1999 RAN TSGW1#3(99)196 Agenda Iem: 9.1 Source: Tile: Documen for: Moorola Macrodiversiy for he PRACH Discussion/Decision
More informationUsefulness of the Forward Curve in Forecasting Oil Prices
Usefulness of he Forward Curve in Forecasing Oil Prices Akira Yanagisawa Leader Energy Demand, Supply and Forecas Analysis Group The Energy Daa and Modelling Cener Summary When people analyse oil prices,
More informationMonetary and Fiscal Policy Interactions with Debt Dynamics
Moneary and Fiscal Policy Ineracions wih Deb Dynamics Sefano Gnocchi and Luisa Lamberini Preliminary and Incomplee November, 22 Absrac We analyze he ineracion beween commied moneary and discreionary fiscal
More informationVector Autoregressions (VARs): Operational Perspectives
Vecor Auoregressions (VARs): Operaional Perspecives Primary Source: Sock, James H., and Mark W. Wason, Vecor Auoregressions, Journal of Economic Perspecives, Vol. 15 No. 4 (Fall 2001), 101115. Macroeconomericians
More informationMotion Along a Straight Line
Moion Along a Sraigh Line On Sepember 6, 993, Dave Munday, a diesel mechanic by rade, wen over he Canadian edge of Niagara Falls for he second ime, freely falling 48 m o he waer (and rocks) below. On his
More informationModule 4. Singlephase AC circuits. Version 2 EE IIT, Kharagpur
Module 4 Singlephase A circuis ersion EE T, Kharagpur esson 5 Soluion of urren in A Series and Parallel ircuis ersion EE T, Kharagpur n he las lesson, wo poins were described:. How o solve for he impedance,
More informationOn the degrees of irreducible factors of higher order Bernoulli polynomials
ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on
More informationTwo Compartment Body Model and V d Terms by Jeff Stark
Two Comparmen Body Model and V d Terms by Jeff Sark In a onecomparmen model, we make wo imporan assumpions: (1) Linear pharmacokineics  By his, we mean ha eliminaion is firs order and ha pharmacokineic
More informationMultiprocessor SystemsonChips
Par of: Muliprocessor SysemsonChips Edied by: Ahmed Amine Jerraya and Wayne Wolf Morgan Kaufmann Publishers, 2005 2 Modeling Shared Resources Conex swiching implies overhead. On a processing elemen,
More information