E β t log (C t ) + M t M t 1. = Y t + B t 1 P t. B t 0 (3) v t = P tc t M t Question 1. Find the FOC s for an optimum in the agent s problem.

Transcription

1 Noes, M. Krause.. Problem Se 9: Exercise on FTPL Same model as in paper and lecure, only ha one-period govenmen bonds are replaced by consols, which are bonds ha pay one dollar forever. I has curren marke value /r, where r is an infiniely long ineres rae. The represenaive agen maximizes wih respec o C, B, and he objecive funcion E β log C subjec o he consrains, for = 0,...,, C + + B B + + v r =0 = + B τ 2 B 0 3 v = C 4.. Quesion. Find he FOC s for an opimum in he agen s problem..2. Answer. The FOC s are C: B: M: = λ + + π 5 C + v λ λ+ + r + + µ 6 r + r + λ λ+ + + π C M 2 + θ 7 v: π = λ C γ + v 2 8 Subsiuing π in he C and M-FOC s, and defining z as yields z z v v + v 2 v 2 γ + v 2 z+ + 0

2 Now se r r, his simplifies he B-FOC. Dividing he M-FOC by he B-FOC and assuming ha boh asses are sricly posiive, so ha he mulipliers on he nonnegaiviy consrains are zero, yields. v 2 γ + v = 2 + r This ies down v = v. Also, i follows ha z = z..3. Quesion 2. Verify ha, when iniial B > 0, he condiions under which here is a unique equilibrium price level under he policy combinaion r r, τ τ are he same as hose under which here is a unique equilibrium price level under R R, τ τ in he one-period bond model of he lecures and he Simple Model paper..4. Answer. The remaining FOC is v 2 M γ = βe + v 2 + Wih he money demand curve above, his is equivalen o + rβ = E M + Noe ha wih one-period bonds we would arrive insead a βr = E M, 4 + in which he one-period gross ineres rae R simply plays he role of + r. Equaion 3 migh seem consisen wih a sunspo equilibrium, because as long as he expecaion is saisfied, here is no violaion of he FOC. However, consider he GBC B B + r Muliply wih / and ake expecaions o ge r B r Rearranging, using he above condiions r = B τ 5 + M = B M = B r β + + rβ τv + + v Y P τ This is an unsable difference equaion unless B /r is consan a is seady sae value B Mr = + rβ + τv + + v Y. 7 β 6

3 I is no hard o verify ha, since v is increasing in r from, he righ-hand side of his expression is increasing in r. This reflecs he fac ha higher nominal ineres raes increase he opporuniy cos of holding money and correspond o higher seadysae inflaion, which increases seignorage revenue and hereby backs higher levels of real deb. If we make an analogous sequence of subsiuions and expecaion-akings for he model wih one-period bonds, we arrive a, insead of 6, = B β + Rβ τv + + v Y 3, 8 where in he one-period model B is he number of one-period bonds issued a. In 8 we have R playing he role ha + r plays in 6, and B, he number of oneperiod bonds which for one-period bonds is of course he same hing as heir marke value held by agens a, playing he role ha B /r, he nominal marke value of consols held by agens a, plays in 6. Thus he unique consan equilibrium value of B/Mr ha we display in 7 for he consol model is also he unique consan equilibrium value of B/M in he one-period bond model, so long as we keep R = + r. Now here are wo aspecs of his resul o check: Firs of all, is his soluion for B/Mr unique? If yes, wha is he implied behavior of he price level? Suppose B / increases wihou bound, i.e., sars above is seady sae value. From he definiion of velociy, we have ha = C v = v v This implies ha real money balances are bounded because is assumed o be bounded. For B / o increase wihou bound, i mus be ha B / mus grow ad infinium. Bu ha canno be an equilibrium by he now familiar argumen ha agens would violae ransversaliy by holding infinie amouns of an asse. They could increase uiliy arbirarily by selling off some par of i. Suppose B / falls. I canno fall below zero as boh componens are consrained o be posiive. However, i could be he case ha i falls unil B = 0. From he firs order condiion for consols, one sees ha µ > 0 implies ha v > v. This in urn implies ha z < z. For he res of he argumen, see Sims Simple Model. Knowing ha B / = B/M, one can reurn o he governmen budge consrain, before aking expecaions: B = + r B + M + +v τv. 20 r r

4 The corresponding equaion for he one-period bond model is B B = R + M + +v τv. 2 Y Rearranging 20 and 20 and using he fac ha a bu possibly no, in he firs period, he r = r or R = R policy is permanenly in place, we arrive a + + r B = + B + r r + τv +v 22 and B + R = + B + +v + τv, 23 respecively, as in he paper. Given, and he realizaion of, his implies a pah for and hence a pah for, by v + +v = 24 The argumens for exisence and uniqueness are hus he same as in he one period bond case..5. Quesion 3. Show ha if we sar in an equilibrium wih consan ineres rae and axes, chosen so ha here is no rend in prices, an unanicipaed change o a new policy wih same ax level, bu a lower nominal ineres rae, produces differen ime pahs for prices depending on wheher one is in he consol or one-period bond case, even if he new ineres rae is such ha long run inflaion is he same for boh ypes of economies..6. Answer. We can refer o 3 and 4. We have already argued ha, so long as he new lower ineres rae is he same for boh shor-bond and consol models, he righ-hand sides of hese wo equaions are he same. However he coefficien in parenheses ha muliplies he money growh rae on he lef-hand side is differen in he wo models in he firs period afer he swich o he lower ineres rae. This already answers he quesion as posed. However i is ineresing o go furher and observe firs ha afer he iniial period, he wo equaions are he same, so ha he money growh rae, and hence he rae of inflaion, will be he same in boh models. In paricular, he expeced money growh rae is lower wih lower r. We can see his direcly from 22 and 23. To see he iniial effec on prices of he moneary expansion he r or R decrease we muliply 3 and 4 by v /v, o conver / o C / C. The resul 4

5 is C + + r B = v C r v + B v + v r + v +v τ 25 C B + R = v C v + B v + +v + v τ. 26 v The righ-hand side in hese wo equaions has he same value by our previous argumens. The firs wo erms on he righ-hand side are increased relaive o he equilibrium values before he moneary expansion. The las erm on he righ will be slighly decreased, bu so long as ransacions coss are a small par of Y, he influence of his las erm will be small and he overall effec of he expansion will be o raise he righ-hand side. In 26, he shor-bonds case, his can be seen unambiguously o imply ha in he firs period of he new policy will be lower han i would have been in he original equilibrium. I.e., he expansionary moneary policy no only produces lower expeced money growh and expeced inflaion, i makes he iniial price level jump downward. Or o pu he maer anoher way, he only way o produce a decline in he ineres rae is o underake a deflaionary policy ha conracs M. The siuaion is differen wih 25, he consol-deb case. There, because he coefficien in parenheses on he lef increases as r increases, he required decline in iniial P will cerainly be smaller han in he case of shor deb, and i can easily be, when ineres-bearing deb is large in value relaive o non-ineres-bearing deb, ha he decline in r produces an iniial rise in P. This occurs because he decline in r produces a capial gain for bond-holders a he iniial price level; even hough he higher anicipaed fuure seignorage revenue increases he equilibrium real value of he deb, he drop in r is likely o produce an increase in he value of he deb ha exceeds he equilibrium increase, so prices mus rise o compensae. This resul is of some ineres, because i suggess ha in he presence of a fiscal policy ha is unresponsive o he level of he deb, ineres rae policy has he usual inflaionary or deflaionary impacs only if here is subsanial long-erm deb ousanding. 5