The Simple Analytics of Price Determination

Transcription

1 Econ. 511b Spring 1997 C. Sims The Simple Analyics of rice Deerminaion The logic of price deerminaion hrough fiscal policy may be bes appreciaed in an exremely lean model. We include no sochasic elemens, we assume here is no money, and we assume ha here is no capial accumulaion. There is, however, nominal governmen deb, which individuals perceive as giving hem an opion of shifing consumpion hrough ime. We assume a represenaive agen who maximizes wih respec o C and subjec o z 1 1 γ C e β d γ C + & r + τ = + Y (2) (1). (3) Equaion (2) is he usual budge consrain, equaing consumpion, asse accumulaion, and axes o he yield on wealh and exogenous non-asse income Y. Noe ha i is no a jump equaion: he ime-derivaive appearing in i is boh a lef and righ derivaive. can change only over ime, via a gap beween income and expendiure. The inequaliy (3) requires ha individuals no borrow from he governmen. Resricions weaker han (3) would also work, bu some such condiion prevening individuals from financing arbirarily large C by rolling over deb (negaive ) forever is required. The governmen s insananeous budge consrain is & = r τ. (4) The governmen can be hough of as choosing r, and τ subjec o (4), wih aken as given, or i can be hough of as choosing all variables in he sysem subjec o (4), (2) and privae opimizing behavior. To close he model we need wo more equaions characerizing governmen policy. For example, one of hese equaions can be a ax-seing, or fiscal policy, equaion, while anoher is an ineres-rae-seing, or moneary policy equaion. Equaions (2) and (4) imply he social resource consrain, which is simply We define he real ineres rae as C = Y. (5) & ρ = r, (6) where he derivaive in (6) is a righ-derivaive, referring o expeced inflaion from now on.

2 I. The rivae Agen s roblem FOC s for he privae agen are C: C γ = λ. (7) λ& λ & : + + β λ = λ r. (8) Subsiuing and manipulaing he resul, we obain γ C& & β C r =. (9) Equaion (9), having been derived from Euler equaions, is a jump equaion. The derivaives in i are forward-looking, righ-derivaives. In he firs examples below we will mainain he assumpion ha Y, and hence in equilibrium (via (5)) C, is consan, which implies via (9) and (6) ha ρ = β. II. egging he Real rimary Surplus and he Nominal Ineres Rae Suppose policy fixes he primary surplus as a consanτ = τ and he nominal ineres rae as a consan r = r. ecause in equilibrium under our assumpions ρ = β, we can wrie This means ha r = β + &. (1) = e ( r β ) for some iniial. We can rewrie he governmen budge consrain (4) as & ( β ), (11) = r τ e r. (12) This is an unsable linear difference equaion, whose general soluion is b r β τ e = + κe β where κ is some consan. Dividing hrough by, we ge τ = + β g κ β e r, (13). (14) Thus he only value for a = ha is consisen wih real governmen deb no exploding exponenially a he rae β is he ha saisfies (14) a = wih κ =, i.e. = τ β. (15)

3 In words, he iniial price level mus adjus o make he real value of he ousanding nominal deb equal o he discouned presen value of fuure ne surpluses. u how do we know ha could no explode exponenially in equilibrium? We can rule ou κ < by he fac ha, from (14), we can see ha his implies ha would evenually become negaive, violaing he no-borrowing consrain on individual behavior. Individuals would herefore see hemselves as no having enough resources o finance he C = Y consumpion level and also pay heir axes. This would generae an aemp o save, driving down he price level. If κ >, real deb grows wihou bound. An individual who is accumulaing wealh a his seady rae β will see him or herself as able o increase consumpion a all daes by, say, βκ 2. The individual, using his or her budge consrain o calculae he effecs of his increase in C, would conclude ha deb would no longer follow (12), which is an implicaion of he individual s budge consrain under C = Y, bu insead would follow b g e j F HG & ( r β ) r β βκ = r τ e + Y C = r e τ 2 I KJ b g. (16) This is anoher unsable linear difference equaion. Is general soluion is, analogous (14), o ξ β τ κ e = + +. (17) β 2 ecause his equaion, for a poenial ime pah of an individual s deb perceived by he individual as possible, mus saisfy he same iniial condiion as he equilibrium for he economy, i mus imply he same. Equaing (14) and (17) a = implies ξ = κ 2 >, i.e. ha wealh in he form of governmen deb sill grows exponenially forever, despie he higher consumpion. I is easy o see ha his would be rue so long as he amoun of he increase in he consan level of consumpion remains no higher han κ. We can conclude, herefore, ha only he soluion in which he iniial price level saisfies (15) is an equilibrium: If is lower han ha, so iniial wealh is higher, he FOC s and consrains imply ha wealh will accumulae forever, so rapidly ha individuals would perceive he possibiliy of financing a permanen increase in C ou of heir wealh; and if is higher han ha, so iniial wealh is lower, he FOC s and consrains imply ha individual wealh will become negaive in finie ime, violaing he solvency consrain on individuals (3). Noe ha we did no arrive a he unique equilibrium here simply by ruling ou explosive soluions. There is an explosive componen o he correc soluion. rices rise or shrink (depending on wheher r > β or r < β ) exponenially in equilibrium, according o (11), and nominal deb mus rise or shrink in proporion o. Noe also ha, hough he iniial deb level can be any posiive number, i canno be zero. This is because he budge consrain (12) hen implies iniial & <, which canno occur when individuals have he solvency consrain (3).

4 Equilibrium in his economy displays a kind of quaniy heory of deb deerminaion of he price level. The higher he iniial level of, he higher he iniial, and and move in proporion o one anoher. In fac, if we se r =, we ge an ordinary quaniy heory of money. Wih zero nominal ineres rae, he deb in his model becomes in effec non-ineresbearing money. Despie he absence of a ransacions moive for holding money, people hold i because here is seady deflaion a he rae r β = β, making money an asse wih an aracive reurn. The governmen mainains he deflaion by axing away a fixed proporion of he nominal sock of money per uni ime. This causes prices o drop in parallel wih he decline in (which in his case we may as well refer o as M), leaving real money balances M consan. III. Indeerminacy from assive Fiscal olicy Economiss someimes have wrien as if i were clear ha when governmens issue new deb hey are auomaically commiing hemselves o he fuure axes ha will be needed o pay off he deb and ineres on i. This is no rue governmens can issue deb wihou any commimen o increased fuure axes, and he resul will be price inflaion. u suppose a governmen is persuaded ha i should commi o fuure axes when i increases deb. One policy ha has his effec ses he real primary surplus τ o respond posiively o he level of ousanding real deb, for example τ = φ + φ1. (18) In order ha (18) represen o paying he ineres on he real deb, we require φ1 > β, i.e. he primary surplus increases by more han he increase in real ineres rae paymens as increases. I is easy o see ha subsiuing his ino he governmen budge consrain (4) produces & = ( r φ1) φ. (19) If we assume as before ha ineres rae policy simply pegs r = r a all imes, we have he same class of soluions (11) for and (19) becomes a consan-coefficien linear differenial equaion, wih general soluion = e φ + e φ β κ ( r β ) r φ1 1 = φ + φ β κ β φ 1 e 1 b b g, or (2) g.. (21) Noe ha, under our assumpion ha φ1 > β, and assuming also φ <, (21) describes a sable ime pah for real deb, regardless of he iniial price level. No iniial price level can be ruled ou as oo low, as his condiion does no imply ever-growing wealh for individuals, and none as oo high because his condiion does no imply ha deb mus become negaive. The iniial price level is indeerminae. In Leeper s local linear analysis of an economy wih money, his kind of fiscal policy is called passive, and he concludes ha o guaranee uniqueness, i mus be paired wih an acive ineres

5 rae policy, ha is, one ha aggressively increases he ineres rae in response o inflaion. A simple example of an acive ineres rae policy is one ha ses r = θ + θ1 p, (22) where p = log. Regardless of policy, he governmen budge consrain can be rewrien in real erms o produce &b = ρb τ, (23) where b =. Noe ha if we replace (4) wih (23) we lose some informaion, formally. Equaion (4) is a non-jump equaion, while (23), because i uses he forward-looking relaion beween he real and nominal ineres rae, is a jump equaion. However, (23) is a valid equaion, and we can use i provided we keep in mind ha i is, no b, ha is given by hisory. Subsiuing (18) ino (23) produces b 1g. (24) &b = ρ φ b φ Wih φ1 > β = ρ, (24) is a sable differenial equaion. Now using he ineres policy equaion (22) and he real ineres rae definiion (6) o eliminae r, we arrive a θ + θ p = β + p&. (25) 1 This is an unsable equaion in p. I has he unique sable soluion p = ( β θ ) θ1. u he unsable pahs for p generae no violaions of equilibrium condiions. The real rae of reurn remains fixed a ρ = β, even as he nominal rae diverges o ±. The real value of he deb converges exponenially oward is posiive seady sae value of φ bφ1 βg, regardless of where i sars from, even hough nominal deb explodes a ever-increasing exponenial raes. Thus an acive ineres rae policy succeeds only in making i very likely ha he price level will explode. I does no eliminae he indeerminacy ha arises because (24) is a sable equaion. The reader may wish o confirm ha wih he τ = τ policy of he previous secion, an acive ineres rae policy like (22) leaves he iniial price level deerminae, while implying in general ha he price level and nominal deb follow (more han exponenially) explosive pahs. IV. A alanced udge Amendmen In he US, he possibiliy of a policy which makes he convenional defici consan is being discussed. In our noaion his would be a policy of & =. This implies in urn r = τ. (26) Usually his policy is aken o be a prescripion for axes and expendiures, no a prescripion ha nominal ineres raes on deb be kep low enough o mainain a zero defici. So suppose we have an acive ineres rae policy like (22) o go wih he balanced budge. As we have already observed, (22) implies ha here is a unique iniial price level consisen wih non-explosiveness of he price level. This will imply a unique iniial r from (22), and hen from (26) and he given iniial value of a unique iniial ax level τ. Explosively decreasing p implies, because is fixed, explosively increasing. As we have discussed before, his is inconsisen wih individual

6 opimizaion. u explosively increasing p implies only explosively decreasing oward zero. (The fac ha i is log p ha decreases exponenially, becoming negaive, means ha iself only shrinks oward zero, remaining posiive.) This does no violae individual opimaliy condiions and i leads o no violaion of solvency consrains. Thus a convenional balanced budge policy, even coupled wih acive ineres rae policy, leads o indeerminacy of he price level. I rules ou low prices followed by explosive deflaion, bu no high iniial prices followed by explosive inflaion. We could, however, couple (26) wih an acive fiscal policy like τ = τ. This convers (26) o a prescripion for seing ineres raes o balance he budge. Using he definiion of he real rae and is consancy in equilibrium, we find ha (26) becomes &p + β = τ. (27) While no a linear differenial equaion, his is an unsable equaion in wih a unique consan soluion: = β. We migh be emped o conclude ha (27) will once again generae upwardly τ explosive pahs when iniial is oo high ha are noneheless consisen wih equilibrium. However, his nonlinear equaion can be solved analyically o imply some ineresing behavior. Rewriing i, we can find & τ β F I HG K J = 1. (28) We can expand he lef-hand side in parial fracions and inegrae, obaining (when τ > β ) F I HG K J = τ log + log β κ, (29) β β where κ is a consan of inegraion. Exponeniaing boh sides, we arrive a τ β = e ( κ + ) β. (3) Since is lef-hand side is bounded above by τ β for posiive, (3) implies ha from any iniial value of, we reach infinie in finie ime. Tha is, when he iniial is above is seady-sae value, inflaion mus be so explosive ha he price level goes o infiniy in finie ime. How do we inerpre his resul? The problem is ha o characerize behavior properly we need o consider policy behavior afer he dae a which becomes infinie. u (26) becomes a nonsense equaion wih infinie. If we inerpre = as implying ha here is no deb on which o pay ineres, hen he commimen o τ = τ is incompaible wih a balanced budge afer becomes infinie. One has o complee he model wih a specificaion for policy in his poshyperinflaion period. If we simply say ha τ drops back o zero when reaches infiniy, hen here is indeerminacy essenially he iniial real value of he deb can be larger or smaller depending on how long here is before reaches infiniy, and hence how long here is for he

7 τ = τ policy o be susained. If we suppose insead ha he governmen will drop he balanced budge policy afer reaches infiniy, mainaining a convenional surplus and τ = τ even afer he real deb has evaporaed, privae agens will see hemselves as needing o borrow a his poin o pay axes. Anicipaing his, hey will feel insolven earlier, save more, and hereby reduce he price level o ha consisen wih non-explosive equilibrium. V. The Ill-Faed Exercise Now we have he foundaion o answer he exercise you were asked o complee las week. We have in fac already answered, or shown o be unanswerable, some of he quesions. Quesion 3 on he exercise concerned he balanced-budge model of he las secion, wih fixed primary surplus. As we have seen, he quesion can be answered wihou considering how o respecify policy afer he price level reaches infiniy. However, if you relied on he plausible (bu no jusifiable from wha you were given in he problem alone) assumpion ha soluions wih explosive real deb were no equilibria, you would have concluded ha here is a unique iniial price level. The same analysis applies o quesion 4. Though we do no discuss i in secion IV, he passive fiscal policy of quesion 4 leads o no fundamenal change in he analysis if φ >. In ha case here is sill he problem ha he price level goes o infiniy in finie ime if i sars oo high, making he commimen o posiive primary surplus and zero convenional defici conradicory. If he consan erm φ in he fiscal rule is made negaive, however, and φ1 > β, he analogue of (27) becomes a sable equaion in, so here is non-uniqueness. Quesion 5 couples an acive ineres rae policy wih an acive fiscal policy and asks ha you prove non-exisence. u as we saw in secion II, acive fiscal policy does generally deliver a unique iniial price level in his model, wih ineres rae policy simply deermining he ime pah of prices afer he iniial dae. However, he exercise did no use hese noes version of an acive ineres rae policy, equaion (22), bu insead an equaion ha replaced p in (22) wih he price level. Wih such an ineres rae policy equaion, is implied o reach infiniy in finie ime, again producing an ambiguiy abou wha he policy equaion for ineres raes means when i implies infinie ineres raes. Quesion 1 was sraighforward. Solving he governmen budge consrain forward implied. 1 b = e τ. 1+ β. (31) We know we can do his because (23) implies ha for any oher soluion pah for deb, eiher real deb explodes, violaing individual opimizaion, or evenually becomes negaive, violaing solvency. Since is given, we can use (31) a = o deermine a unique. We also know from he privae secor FOC s ha and herefore ha &p + β = r (32) e r β = b g. (33)

8 Since we know he unique iniial, (33) gives us he full ime pah of, and hen (31) implies he full ime pah of and &. Using he numerical values supplied in he quesion, I concluded = 3e. 5, e = 2. 5, &.. = 1e , b = e. 3 Quesion 2, as already poined ou in class, was inernally inconsisen. ecause of he declining Y and hence C, he real ineres rae is implied o be negaive. Consumers mus hold non-negaive governmen deb, and hey have a sream of income ha is dwindling exponenially o zero. The governmen deb hey hold pays a negaive reurn. Thus hey do no have enough resources o pay a fixed real governmen ax forever. The fixed-τ policy proposed in he quesion is unsusainable. Quesion 6 is a combinaion of passive fiscal policy wih acive ineres rae policy, wih he level raher han log of price enering he ineres rae rule. This form of ineres rae rule leads o &p + β = θ + θ1. (34) This equaion has he same form as (27) and leads o he same conclusion, ha high iniial prices lead o infinie prices in finie ime. However here, because of he passive fiscal rule, he real value of deb follows a sable ime pah regardless of wha happens o he price level. Wha happens he insan afer he nominal deb and he nominal price level have boh reached infiniy is even more of a puzzle han in he model wih fixed (since wih fixed real deb vanishes as he price level reaches infiniy). Also, here low iniial values of, hough hey lead o rapidly shrinking, do no lead o explosion of and hus do no violae individual opimizaion. So he problem, in assuming here was a unique soluion, was misaken. However, one could have gone ahead and assumed we are ineresed in he unique sable soluion, which exiss. I has fixed a bβ θ g θ 1 and b following whaever ime pah is implied by he iniial and he equaion b& = ( β φ1) b φ. (35) If we began in seady sae, we would have b = φ ( β φ1), and hus = φ bβ θ g cθ 1bβ φ1gh. You were asked, hough, o suppose ha a = θ jumps from.3 o.5, while β was assumed o be.5. Wih his choice of θ, here is no sable price level. The price level reaches infiniy from any finie posiive saring value in finie ime. So here is no good answer o quesion 6.