Lecture 26. Lucas and Stokey: Optimal Monetary and Fiscal Policy in an Economy without Capital (JME 1983) t t

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1 Lecure 6. Luca and Sokey: Opimal Moneary and Fical Policy in an Economy wihou Capial (JME 983. A argued in Kydland and Preco (JPE 977, Opimal governmen policy i likely o be ime inconien. Fiher (JEDC 98 and Barro and Gordon (JME 983 gave uch cae in fical policy and moneary policy, repecively. Barro and Gordon eablihed an equilibrium uppored by repuaion. The weakne in ha paper i ha he privae agen are aumed o ue a one-period-penaly rigger raegy, in which cae, he fir be can no be achieved. If on he oher hand, an infinie-period-penaly rigger raegy i ued, he fir be can be aained (See Ireland commen on Sokey: Rule raher han Dicreion: 5 year laer, hp://fmwww.bc.edu/ec-p/wp53.pdf,.. In hi paper of Luca and Sokey, hey aked he following queion: in he cae in which he ax policy i known o be ime inconien, i here a deb porfolio ha he curren governmen can leave o i ucceor o ha he policy announced will be kep? 3. The anwer i poiive if he deb mauriy rucure i rich enough (namely, he governmen ha he opion of iuing ae-coningen deb. Given an opimal ax policy, here exi a unique deb policy ha make i ime-conien. 4. When he barer economy i exended o include money, ime-inconiency reappear unle we aume ha moneary policy i pree o mainain a pecified pah of nominal price. 5. We will focu on he barer economy cae. 6. General Seup: one good. No capial. The good i produced by labor. Governmen expendiure could be ochaic, which i he only poible ource of uncerainy in hi model. 7. Noaion:,..., g g g. f denoe he deniy funcion. The profile of governmen expendiure from ime o i denoed by g g,..., g. F denoe marginal diribuion of he hiory ( ( F g The condiional diribuion i denoed by: ( x x Leiure profile ( good. 8. Reource conrain: c. c. Conumpion profile (. Endowmen of labor i one uniy, which produce one uniy of c + x + g (. 9. Preference of he repreenaive agen: E β U( c, x = β U( c( g, x( g df ( g (. = =. Fir-Be Allocaion (Pareo-Opimal require marginal rae of ubiuion equaion marginal rae of ranformaion: Ux( c, x = (.3 Uc( c, x. If lump-um ax i available, he fir be can obviouly be achieved. Suppoe ha i i no available and we aume ha he only ax available i a fla rae ax τ levied again labor income ( x.. Suppoe he oher ource of finance i governmen deb: b b = = { },,,,... =

2 where g, g, and i he claim held by he individual a he beginning of period o conumpion good in period, coningen on he even g. b depend on ( 3. Individual Problem: A ime, ake coningen ax rae, { } price, p = { p}, bond holding, b and curren expendiure, max τ = τ, and coningen g, a given. E β U( c, x = β U( c( g, x( g df ( g = = p c ( τ ( x b ubjec o: [ ] [ τ ] p c ( ( x b dg (.4 = + 4. Fir order condiion: Ux( c, x = τ, =,,,..., for any g (.5 Uc( c, x Uc( c, x p( g β f ( g g =, =,,..., for any g (.6 Uc( c, x p 5. Thee condiion ogeher wih he budge conrain lead o an opimal coningen plan (, cx given τ, p, bg,. 6. Wha ranacion are needed a ime o ha he opimal coningen plan will be carried ou indefiniely? 7. When he marke mee in period, wih g known, he conumer purchae hi curren allocaion ( (, ( c g x g. Bu for bond holding + b, here are infinie number of choice a long a he following rebalancing equaion i aified: p b + p b dg = p c ( τ ( x [ ] =+ p[ c τ x ] dg+ g+ g =+ + ( ( for all, given (.7 8. How do we know ha any bond holding + b aifying (.7 i affordable? 9. Le u prove hi by inducion.. When =, any b aifying (.7 implie ha for any g, he following i rue: + = = p[ c τ x ] dg g = [ τ ] p b ( g p b dg p c ( ( x + ( ( for all given which i idenical o (.4. Suppoe a, any b aifying (.7 i affordable. Le u wrie ou (.7 for b :

3 + + = [ ( τ ( ] =+ + p[ c ( τ ( x ] dg+ for all g, g given =+ p b p b dg p c x To check wheher + b i affordable, we inegrae (.7 wih repec o g + and obain: p b dg + p b dg = p c ( τ ( x dg [ ] =+ + p[ c ( τ ( x ] dg+ for all g given =+ Subracing he wo equaion above, [ τ ] [ + ] + =+ p c ( ( x b + p b b dg = which i affordable. Hence we proved ha any + b aifying (.7 i affordable.. Le u now find he compeiive equilibrium. Namely olve for he coningen price p and he coningen conumpion c and leiure x, for any given τ, b, and F.. Noe ha afer ubiuing (.5 and (.6 ino (.4, we end up wih he following equaion: ( c b U ( c, x ( x U ( c, x c x β [ c b Uc c x x Ux c x ] df g g = + ( (, ( (, ( = (.8 3. From he governmen poin of view in period, given curren governmen conumpion, g, given he condiional diribuion of fuure governmen conumpion, F, and given he exiing (coningen governmen obligaion, b, any allocaion ( cx, mu aify he reource conrain (. and equilibrium condiion (.8. Converely, any allocaion ( cx, ha aify (. and (.8 can be implemened by eing he axe according o (.5. Equilibrium price, given hoe ax rae, are decribed by (.6, and he required deb rerucuring { b} = according o ( Therefore, o earch for an opimal governmen ax profile i equivalen of maximizing (. ubjec o (. and (.8. Subiue (. ino he objecive funcion and le λ be he muliplier aociaed wih he conrain (.8. The FOC become: ( U U λ c b ( U U x ( U U ( + λ c x + ( cc cx + ( xx cx = (.9 where he derivaive are evaluaed a ( c, x for =,,,... and for all g. (., (.8 and (.9 characerize he opimal ax plan from he poin of view of he governmen a ime.

4 5. The queion i wheher he opimal plan coninue o be opimal when he governmen re-opimize a ime =. 6. I can be hown ha he governmen in ime can manipulae he mauriy rucure of he deb, leaving he governmen a ime a deb profile, b, uch ha here exi a λ making he following equaion ( + λ ( Uc Ux + λ ( c b ( Ucc Ucx + ( x ( Uxx Ucx = coninue o hold a he original ( c, x for =,,,... and for all g. Hence he governmen a ime will coninue o adop he opimal policie announced by governmen. 7. Le u look a one deailed example. Ucx (, = c+ x.5( c+ x Uc = c, Ux = x, Ucc = Uxx =, Ucx = Aume ha here are hree period only,,,. Le β =. And uppoe for impliciy ha g = g = g = and b = b = = /6. Equaion (. and (.9 imply ha: ( + λ( c λ c = for all. 6 Therefore c = c = c. Pu hee ino (.8 yield: c ( c c = 6 Hence c = / 3, x = / 3, τ = /, p = for all =,,. 8. In hi imple example, axing a = generae exacly enough revenue o redeem he currenly mauring deb. And he opimal deb o leave for he ucceor i b = = /6. Suppoe he governmen a = made a miake and lef he governmen a = wih b = and b = / 3 (a rerucuring uing p= p= 9. Then a ime =, he opimaliy condiion for he governmen become: ( + λ( c λc = ( + λ( c λ c = 3 Clearly, c c. In fac, he oluion i: c =.3, c =.38. And conequenly he opimal axe become τ =.53, τ =.39. The equilibrium relaive price c p = = (here wa a ypo in he original paper. / p.9. c 3. By raiing ax rae a = and reduce he ax rae a =, he governmen can reduce curren conumpion and raie fuure conumpion, making he relaive price p / p below. A lower price in period reduce he deb burden of he

5 governmen a he deb = /3 i due a =. In oher word, he governmen ha he moivaion o devalue he deb. The original opimal plan i no long opimal. 3. In a model wihou capial and wihou money, Luca and Sokey how ha opimal governmen policy can be made ime conien if he deb mauriy i rich and he curren governmen leave he correc profile of deb o i ucceor. When he model i exended o include money, he governmen doe no have ufficienly many inrumen o reore ime coniency. 3. If you find he paper oo difficul o dige, hen focu on underanding he numerical example. Pay aenion o iem 7-3. Then ry he following homework: In he numerical example preened above, if he deb profile lef o governmen zero i b = b = = /5, wha deb profile he governmen a ime zero hould leave o he nex governmen in order o make ure i ax policy i unchanged when he nex governmen reconider he opimizaion problem. Wha would be he correponding opimal ax rae a ime zero? Noe ha when you find wo oluion, explain why you pick one and no he oher.