Opimal Moneary Policy and Equilibrium Deerminacy wih Liquidiy Consrained Households and Sicky Wages Guido Ascari Universiy of Pavia and Kiel IfW Lorenza Rossi Universiy of Pavia Ocober 9, VERY PRELIMINARY Andrea Colciago Universiy of Milano-Bicocca Absrac We sudy Ramsey policies and opimal moneary policy rules in a model wih sicky wages and prices, where nancial marke paricipaion is limied, i.e. where a fracion of consumer are liquidiy consrained. The ineracion beween liquidiy coinsrained agens and wage sickiness resuls in a welfare-based loss funcion which depends on real wage gap beside depending on he oupu gap, wage in aion and price in aion as in previous sudies. Opimal simple rules are characerized by an in aion coe cien larger han one, no maer he degree of nancial marke parecipaion. We argue ha once wage sickiness, an unconroversial empirical fac, is considered, he degree of nancial marke parcipaion jus marginally a ecs he design of opimal moneary policy rules. JEL Classi caion Numbers: E, E4, E5, E6. Keywords: opimal moneary policy, sicky wages, liquidiy consrained household, deerminacy, opimal simple rules.
Inroducion [TO BE DONE] The Model In wha follows we follow Bilbiie (8). However, a major di erence is in he modelling of he labor marke. Whereas he has a perfecly compeiive marke, we asssume a monopolisically compeiive marke characerized by nominal wage sickiness, which we describe below.. Households The period uiliy funcion is common across households and i has he following separable form U = u [C (i)] v [L (i)] () where C (i) is agen i s consumpion, L (i) are labor hours and is a preference shock which has he following law of moion log = log + " ;. We assume a coninuum of di ereniaed labor inpus indexed by j [; ]. As in Schmi- Grohé and Uribe (5), agen i supplies all labor inpus. Wage-seing decisions are aken by labor ype-speci c unions indexed by j [; ]. Given he wage W j xed by union j, agens sand ready o supply as many hours on labor marke j, L j, as required by rms, ha is L j = W j W! w L d () where w > is he elasiciy of subsiuion beween labor inpus. Here L d is aggregae labor demand and W is an index of he wages prevailing in he economy a ime. Formal de niions of labor demand and of he wage index can be found in he secion devoed o rms. Agens are disribued uniformly across unions, hence aggregae demand of labor ype j is spreaded uniformly beween all households. I follows ha he individual quaniy of hours worked, L (i), is common across households and we will denoe i wih L. This mus saisfy he ime resource consrain L = R Lj dj. Combining he laer wih () we obain L = L d Z W j W! w dj (3) The labor marke srucure rules ou di erences in labor income beween households wihou he need o resor o coningen markes for hours. The common labor income is given by L d R W j W j w W dj. 3 The funcion u is incresing and concave while he funcion v is increasing and convex. Thus a share of he associaes of he unions are non ricardian consumers, while he remaining share is composed by non ricardian agens. 3 Erceg e al (), assume, as in mos of he lieraure on sicky wages, ha each agen is he monopolisic
.. Ricardian Households Ricardian households face he following ineremporal budge consrain E ;+ X + + S;+ V X + L d Z W j W j W! w dj + S; (V + P D ) P C S;: (4) We disinguish shares from he oher asses explicily since heir disribuion plays a crucial role in he res ofhe analysis. In each ime period, ricardian agens can purchase any desired sae-coningen nominal paymen X + in period + a he dollar cos E ;+ X +. The variable ;+ denoes he sochasic discoun facor beween period + and. The expression L d R W j W j w W dj represens labor income. V is average marke value a ime shares in inermediae good rms, D are real dividend payo s of hese shares and s; are share holdings. FOCS wih respec o C S; ; S; and X + are respecively u c (C S; ) = P ; (5) V = E + (V + + P + D + ) ; (6) ;+ = + ; (7) where is he lagrange muliplier on he ow budge consrain. The riskless nominal ineres rae is a soluion o E ;+ = ( + R ) : (8) Combining he previous condiions deliver he FOC for uiliy maximizaion of ricardian agens ( + R ) = E + u c (C S;+ ) P : (9) u c (C S; ) P +.. Non Ricardian Households. Non ricardian agens do no hold physical capial, do no enjoy rms pro s in he form of dividend income and are no able o rade in he nancial markes. The nominal budge consrain of a ypical non ricardian household is given by P C S; = L d Z W j W j W! w dj: () Agens belonging o his class are forced o consume disposable income in each period and delegae wage decisions o unions. For hese reasons here are no rs order condiions wih respec o consumpion and labor supply. supplier of a single labor inpu. In his case, assuming ha agens are spreaded uniformly across unions allows o rule ou di erences in income beween households providing he same labor inpu (no maer wheher hey are ricardian or no), bu i does no allow o rule ou di erence in labor income beween non ricardian agens ha provide di eren labor inpus. This would amoun o have an economy populaed by an in niy of di eren individuals, since non ricardian agens canno share he risk associaed o labor income ucuaions. Alhough his framework would be of ineres, i would imply a racabiliy problem. 3
. Wage Seing Nominal wage rigidiies are modeled according o he Calvo (983) mechanism. In each period a union faces a consan probabiliy w of being able o reopimize he nominal wage. We exend he analysis in GVL (7) and assume ha he nominal wage newly rese a, W f, is chosen o maximize a weighed average of agens lifeime uiliies. The weighs aached o he uiliies of ricardian and non ricardian agens are ( ) and, respecively. The union problem is max fw E j= X ( w ) j f +j [( ) u (C S;+j ) + u (C S;+j )] v (L +j )g () subjec o (3), (4) and (). 4 The FOC wih respec o f W is ( X E ( w ) +j ;+j + ( ) MRS H;+j j= MRS S;+j ) fw ( + w ) = () P +s where ;+j = v L (L +j ) L d +j W w +j and w = ( w ) is he, consan, ne wage mark-up in he case of wage exibiliy. The variables MRS H; and MRS S; denoe he marginal raes of subsiuion beween labor and consumpion of non ricardian and ricardian agens respecively..3 Firms In each period, a nal good Y is produced by a perfecly compeiive rm combining a coninuum of inermediae inpus Y (z) according o he following sandard CES producion funcion: Z Y = p p p Y (z) p dz wih p > (3) The producer of he nal good akes prices as given and chooses he quaniies of inermediae goods by maximizing is pro s. This leads o he demand of inermediae good z and o he price of he nal good which are respecively Y (z) = p h P(z) R i P Y ; P = P (z) p p dz Inermediae inpus are produced by a coninuum of monopolisic rms indexed by z [; ] using as inpus capial services, K (z), and labor services, L (z). The producion echnology is given by: Y (z) = A L (z) ; (4) R where A represens TFP. The labor inpu is de ned as L (z) = L j w w (z) w w dj. Firm s z demand for labor ype j and he aggregae wage index are respecively 4 Many reasons have been provided o jusify he presence of non ricardian consumers. A few of hem are miopia, fear of saving and ransacion coss on nancial markes. None of hese is, however, in conras wih rule of humb consumers delegaing wage decision o a forward looking agency, in his case a rade union. 4
L j (z) = W j w W L (z) ; W = R W j w =( w) dj : The nominal marginal cos, common across producers, is given by MC = W ; (5) A while rm z s real pro s are given by D (z) = h P(z) P MC P i Y (z). Price Seing Inermediae producers se prices according o he same mechanism assumed for wage seing. Firms in each period have a xed chance p o reopimize heir price. A price seer akes ino accoun ha he choice of is ime nominal price, e P, migh a ec no only curren bu also fuure pro s. The FOC for price seing is: E X s= h s p +s P p +s Y ep +s i ( + p ) MC +s = ; (6) which can be given he usual inerpreaion. 5 Noice ha p = ( p ) represens he ne markup over he price which would prevail in he absence of nominal rigidiies..4 Aggregaion Aggregae consumpion and aggregae pro s are de ned as: C = C H; + ( ) C S; ; (7) = ( ) S; (8) Since each agens holds he same quaniy of shares and he sum of shares mus equal, i has o be he case ha S; = S = : (9).4. Marke Clearing The clearing of good and labor markes requires 8 p >< Y (z) = P(z) Y d 8z Y d = Y ; >: P L j = W j W w L d 8j L = R Lj dj ; () where Y d = C represens aggregae demand, L j = R Lj (z) dz is he demand of labor inpu j and L d = R L (z) dz denoes rms aggregae demand of he composie labor inpu. 5 Recall ha is he value of an addiional dollar for a ricardian household. I is he lagrange muliplier on ricardian househols nominal ow budge consrain. 5
3 The model approximaion In his secion we log-linearize he model around he e cien seady sae, i.e. he seady sae obained by solving he Social Planner problem. Indeed, in order o compare our resuls on he opimal moneary policy wih he ones ge by our closed relaed papers, i.e. Anderson e al () and Bilbiie (8), we need o derive he cenral bank loss funcion by aking a second order approximaion of he households uiliy funcions around he e cien seady sae. Moreover, since we are going o sudy opimal moneary policy, we wan he moneary auhoriy o arge he welfare relevan oupu gap, i.e. he gap beween he acual and he e cien equilibrium oupu. The laer corresponds o he equilibrium oupu of he Social Planner rs bes allocaion. Appendix A shows how o derive he e cien equilibrium oupu. 3. The E cien Seady Sae Noe ha because of he presence of price and wage markups he seady sae of our economy is no e cien. Since we will approximae he dynamics of our economy around an e cien seady sae we assume ha he Governmen axes/subsidies rms hrough an employmen subsidy/ax a a consan rae ;and hen give/ake he money back o rms in a lump-sum way T: This means ha he rm s pro funcion becomes: D (i) = P (i) Y (i) P ( ) W (i) P N (i) T ; () where o balance he governmen budge we assume ha W(i) P N (i) = T :A he e cien seady sae pro s mus be zero, which implies ha C S = C N = C and hus ha agens have a common marginal rae of subsiuion beween labor and consumpion M RS. In his case he seady sae labor marke equilibrium would imply ha: w = Labor marke equilibrium in he e cien seady implies ha + p ( ) MP L = ( + w ) MRS: () MP L = MRS = ; (3) indeed, he aggregae producion funcion in he seady sae implies producion funcion M P L = A = Y N = : Then, () requires o be such ha: + p ( ) ( + w = : (4) ) Solving for we ge = : As argued above he implied value of leads o zero (+ p )(+ w ) seady sae pro s; o see his noice ha D = Y ( ) W P N T = Y Y + p ( + w ) + p ( + w )! Y = : (5) 6
3. The Log-linearized Model In he remainder we adop he following funcional forms for he uiliy of consumpion and labor hours: C u (C ) = + ; v (L ) = L+ (6) + wih represening he inverse ineremporal elasiciy of subsiuion in consumpion and measuring he elasiciy of he marginal disuiliy of labor and represening an AR() preference shock. In wha follows lower case leers denoe log-deviaion from he e cien seady sae. The Euler equaion for Ricardian households can be log-linearized as c S; = E c s;+ E (r + ) E + ; (7) where r = log +R, log P + P = + and + = log +. Consumpion of non ricardian agens reads as c H; = c +! ; (8) while he assumpion ha consumpion level are equal a he seady sae implies ha aggregae consumpion is c = ( ) c S; + c H; : (9) Log-linearizaion of he aggregae resource consrain around he seady sae yields y = c : (3) A log-linear approximaion o he aggregae producion funcion is given by y = l + a : (3) The New Keynesian Phillips Curve (NKPC) is obained hrough log-linearizaion of condiion (6) and reads as = E + + p mc ; (3) where p = ( p)( p ). The log-linear version of he rs order condiion for wage seing is p The wage in aion curve is X E ( w ) +s ^! +s mrs A +s = : s= w = E w + w w ; (33) where w = ( w )( w ) and w w =! (c + l ) is he log-deviaions of he wage markup ha unions impose over he average marginal rae of subsiuion. 6 Noice ha since unions 6 As poined ou by Schmi-Grohe and Uribe (5), he coe cien w is di eren form ha in Erceg e al (), which is he sandard reference for he analysis of nominal wage sickiness. The reason is ha we have assumed ha agens provide all labor inpus. In he more sandard case in which each individual is he monopolisic supplier of a given labor inpu, w would be equal o ( w )( w ) w (+ w) hence lower han in he case we consider. 7
maximize a weighed average of agens uiliies, he wage in aion curve akes a sandard form. Equaion (33) leads o a second order expecaional di erence equaion for he log-deviaion of he ime real wage:! = [! + (E! + + E + ) ] + w (l + c ) ; (34) where = w deermines boh he degree of forward and backward lookingness. (+ w). The parameer 3.3 Log-deviaions from he E cien Equilibrium Given our assumpions concerning seady sae axaion he naural and he e cien level of oupu coincide. In wha follows we will approximae he model around he e cien equilibrium. Logdeviaions of e cien oupu (from he e cien seady sae) are given by (see Appendix A for a deailed derivaion): y Eff = + + a + ( + ) ; (35) i.e. he e cien level of oupu is a funcion of exogenous produciviy and preference shocks, and herefore as sandard i is independen of policy. Also noice ha he e cien real wage level is! Eff = a ; (36) which again is known once he process for a is posied. Nex we move o obain deviaion from he e cien equilibrium. Recall ha he NKPC reads as = E + + p mc ; (37) where mc represen he log deviaion of he real marginal cos. The laer can be equivalenly wrien as mc =! y + l =! a ; (38) where we subsiue he log-linear version of he economy producion funcion, i.e., y = a + n : Thus, real marginal coss can be rewrien in erms of he real wage gap, mc = ~! ; (39) where we de ne ~! =! as he gap beween he curren and he e cien equilibrium real wage. In his case he NKPC (37) can be rewries as! Eff Nex consider he wage in aion curve = E + + p ~! : (4) w = E w + w w : (4) Recall ha w represens he log-deviaion of he wage markup from he (null) e cien level. Imposing marke clearing and using he producion funcion w =! ( + ) y + a + ; (4) 8
hus where x = y wrie y Eff ~ w = ~! ( + ) x ; (43) denoes he gap beween acual oupu and he e cien oupu. We can hen w = E w + + w ( + ) x w ~! : (44) The IS curve can be wrien as (see Appendix A for a deailed derivaion) x = E x + E r p + r Eff E w + E p + (45) where r Eff is he e cien rae of ineres de ned as: r Eff = y Eff + + E a + + : (46) Using he wage in aion curve and he price in aion curve E w + E p + = [(w E p ) + ( w p ) ~! w ( + ) x ] subsiuing ino he IS curve leads o w ( + ) ( ) x = E x + E r p + r Eff Furher noice ha by de niion! =!! = w p ( ) [(w p ) + ( w p ) ~! ] which implies ~!! + a = w p hus or w ( + ) x = E x + ( ) E r p + r Eff ( sw ) x = E x + E r p + r Eff ( ) [a! + ( + w p ) ~! ] ( ) [a! + ( + w p ) ~! ] where sw ( ) w(+) = ( ). Noice ha if > + w (+) = ( ) sw hen he ineres rae elasiciy of aggregae demand urns posiive. 9
3.4 A special case: sicky prices and exible wages When wages are exible a log-linear approximaion o he wage seing rule delivers! = c + l : (47) Furher, and independenly of he naure of price and wage seing, i holds ha mc =! (y l ): (48) As sandard we can rewrie real marginal coss in erms of oupu gap, as follows:pu gap in his case he NKPC ca be wriens mc = ( + ) x ; (49) = E + + p ( + ) x ; (5) which is he sandard NKPC considered also in Bilbiie (8). Recall ha wih liquidiy consrained households he IS curve reads as y = E y + + E a + E! + E r p + +; (5) hen, subsiuing for! +, using he equaion of he economy producion funcion and rearranging, we ge, y = E y + fw E r p + + fw ( + ) E fw a + + ; (5) where fw = ( + ) : Noice ha when > +(+) = ( ) fw he ineres rae elasiciy of aggregae demand urns posiive. Taking di erences wih respec o he e cien equilibrium, he IS can be rewrien as follows, x = E x + E r p + r eff ; (53) where, in he case of exible wages he e cien rae of ineres is de ned as: r Eff = ( + ) + ( + ) + + ( a ) a : (54) Noice ha ( ) sw > ( ) fw whenever + w (+) > +(+) i.e when w < or ( w ) ( w ) < w ( w ) ( w ) w w ( + ) + <
or + w = + p 4 + p () + w. Assuming =.99 hen he consiion ranslaes ino :38 w which implies ha ( ) sw > ( ) fw when wages have an average duraion longer han :6 quarers. Assuming an average duraion of 3 quarers as suggesed by empirical evidence ogeher wih = and = delivers ( ) sw = :79 Thus we need abou 8 percen of liquidiy consrained agens for he ineres rae elasicy of aggregae demand o urn posiive. 4 Deerminacy In order o close he model we need o specify an equaion for he nominal ineres rae. consider he following Taylor-ype ineres rae rule: We r = r r + ( r ) +i + y y +i : (55) When i =, (55) reduces o a backward looking rule, when i = i corresponds o a conemporaneous rule and when i = i becomes a forward looking rule. The rule is simple in he sense ha i depends on observable variables. 5 Opimal Moneary Policy [TO BE COMPLETED] In he Appendix A we derive a second order approximaion o he average welfare losses experienced by households in economies characerized by sicky wages and prices, where a fracion of consumers are liquidiy consrained, resuling from ucuaions around an e cien seady sae. Those welfare losses are given by: L = X ( ) ~! + ( + ) x + w ( w ) + p ( p w ) : (56) p = Noice ha he welfare loss above ness ha derived by Anderson e al () and ha in Bilbiie (8). Indeed, for = ; he erm in ~! in equaion (56) vanishes and he welfare funcion collapses o he one found by Anderson e al (). 7 In he case of exible wages, and wih a walrasian labor marke, i.e. in he absense of rade-unions, insead, we ge he welfare funcion 7 A MENO DI UN COEFFICIENTE MI SEMBRA. DOVREMMO CITARE IL PAPER DI SGU QUI IN NOTA
found in Bilbiie (8). Assuming ha wages are exible and ha, also in he case of exible wages, unions se households wage, we ge he following welfare funcion: 8 L = X = ( + ) + ( ) x + p ( p; ) ; (57) p which collapses o he sandard ex-book welfare-loss for = : In wha follows we will use (57) as a benchmark o compare he resuls obained under sicky wages. Noice also ha in he case in which he ineremporal elasiciy of subsiuion in consumpion, ; equals, he real wage gap does no a ec sociey s welfare loss. 6 Full Commimen [TO BE COMPLETED] We now wan o characerize he opimal moneary policy in he economy in which boh prices and wages are sicky and a fracion of consumers are liquidiy consrained. In paricular we will sudy he opimal moneary policy in response o a posiive produciviy shock, a ; and o a preference shock. As in Galì (8) we resric ourselves o he case of full commimen. This means ha he Cenral Bank maximizes he welfare funcion (56) subjec o he following consrains: 8 >< >: p = E p + + p~! + u w = E w + w ~! + w ( + ) x ~! = ~! + w p +!Eff ~! given. : (58) De ning, and 3 he lagrange mulipliers (LMs) on he consrains above, FOCs are: x : ( + ) x + w ( + ) = ; (59) p : w : ~! : p p p + 3 = ; (6) w w + + 3 = ; (6) w ( ) ~! + p w 3 + E 3+ ; (6) fully opimal policy requires seing = =, imeless policy insead ses = and = i.e.ses lagged LMs a heir seady sae value. In he remainder we characerize numerically he fully-opimal policy. Before urning o he opimal policy wih he baseline calibraion, noice ha, as discussed above when = he objecive funcion is independen of he share of non ricardian agens, hus he planner can implemen he same equilibrium pah for x,! and as in he full paricipaion economy. However he equilibrium pah of he ineres rae will depend on 8 For a deailed derivaion of he welfare funcion under exible wages see Appendix A.
he share of non ricardian consumers. In paricular, in response o a posiive produciviy shock, he planner engineers a sronger decrease in he ineres rae as he share of liquidiy consrained agens ges larger. This can be show by looking a he derivaive of he nominal ineres rae in he IS curve wih respec o ; which is: [TO BE COMPLETED] 6.. Opimal Responses o Shocks @r @ = ( ) E (! + + a + ) > : (63) In wha follows we will show ha opimal impulse response funcions (OIRFs) of he main economic variables following a echnology shock and a preference shock and a cos-push shock. The model is calibraed as follows. Preferences. Time is measured in quarers. The discoun facor is se o :99; so ha he annual ineres rae is equal o 4 percen. The parameer on consumpion in he uiliy funcion is se equal o and he parameers on labour disuiliy, ; is se equal o. Producion. Following Basu and Fernald (997), he value added mark-up of prices over marginal cos is se o :: This means ha he inermediae goods price elasiciy, p is se equal o 6: The Calvo (983) probabiliy ha rms do no rese prices, p ; is se equal o =3: Labour markes. The he elasiciy of subsiuion beween labor inpus, w is se equal o 6: The Calvo probabiliy ha unions do no rese wages, w ; is se equal o 3=4: Exogenous shocks. The process for he aggregae produciviy shock, a ; follows an AR() and is calibraed so ha is sandard deviaions is se o % and is persisence o.9. Similarly, he process for he aggregae produciviy shock, ; follows an AR() and is calibraed so ha is sandard deviaions is se o % and is persisence o.9. Finally, also he process for he cospush shock u follows an AR() and is calibraed so ha is sandard deviaions is se o % and is persisence o.9. Figure shows impulse response funcions o a one percen posiive produciviy shock for oupu, hours, Ricardian and non-ricardian households consumpion, price in aion, wage in aion, real wage and real ineres rae. Due o he increase of produciviy oupu, wage in aion, real wage and consumpion of boh ype of households increase. Di erenly, price in aion, hours and real ineres rae decrease. The moneary policy in his environmen faces wo disorions, sicky prices and and sicky wages. The rs disorion calls for zero in aion policy, i.e., o close he gap wih he exible price allocaions, while he second disorion calls for an acive moneary policy. As he moneary auhoriy is endowed wih a single insrumen, i mus rade-o s beween he wo compeing disorions. As a resul opimal policy deviaes from full price sabiliy, indeed he opimal IRF does no show an in aion rae always equal o zero, as in he case of exible wages. Speci cally, he moneary auhoriy wans o ake full advanage of he produciviy increase, herefore i reduces in aion o suppor higher demand. Noice, ha in aion shows a signi can overshoo afer a few periods. This capures he value of commimen as he moneary policy ries o in uence fuure expecaion o obain faser convergence oward he seady sae. A he same ime he moneary auhoriy 3
(a) OUTPUT (b) HOURS.5. 5 5 quarers (c) CONSUMPTION NON RICARDIAN.5.4 5 5 quarers (d) CONSUMPTION RICARDIAN.5 5 5 quarers (e) PRICE INFLATION.5 5 5 quarers (f) WAGE INFLATION..5 5 5 (g) REAL WAGE. 5 5 quarers (h) REAL INTEREST RATE.5.5 5 5 quarers λ= λ=.3 λ=.5 5 5 quarers Figure : Response o a echnology shock in he case of full commimen. Alernaive values of he share of liquidiy consrained agens. allows wage in aion and real wages o increase, so ha consumpion of liquidiy consrained household increases. Similarly, by allowing he real ineres rae o decrease he cenral bank is able o increase ricardian household consumpion, via he sandard consumpion smoohing mechanism. Noice ha, also in he case of sicky wages, he planner engineers a sronger decrease in he real ineres rae as he share of liquidiy consrained agens ges larger. In fac, has shown in gure, he dynamics of real wages does no change ha much as increases, so ha also he dynamics of consumpion of liquidiy consrained households remains almos una eced. This meas ha, he moneary auhoriy, in order o obain he same reducion of aggregae consumpion has o srongly reduce consumpion of Ricardian household. A policy which implies a sronger reducion of he real ineres rae. Figure shows impulse response funcions o a one percen posiive preference shock for oupu, hours, Ricardian and non-ricardian households consumpion, price in aion, wage in aion, real wage and real ineres rae. In response o a preference shock opimal moneary policy implies an increase in oupu, hours and consumpion. The real ineres rae increases so ha he moneary auhoriy is able o reach boh price and wage in aion sabiliy. Overall, he deviaions of he price level from he full price sabiliy case are raher small. This is so since he shock does no a ec direcly labour produciviy and herefore i does no generae an endogenous rade-o. Figure 3 shows impulse response funcions o a one percen posiive cos-push shock for oupu, hours, Ricardian and non-ricardian households consumpion, price in aion, wage in aion, real 4
.4 (a) OUTPUT.4 (b) HOURS.. 5 5 quarers (c) CONSUMPTION NON RICARDIAN.4 5 5 quarers (d) CONSUMPTION RICARDIAN.4.. 5 5 quarers x (e) PRICE INFLATION 6 5 5 x 6 (g) REAL WAGE 5 5 quarers x (f) WAGE INFLATION 7.5.5 5 5 quarers (h) REAL INTEREST RATE.4. 5 5 quarers λ= λ=.3 λ=.5 5 5 quarers Figure : Response o a preference shock in he case of full commimen. Alernaive values of he share of liquidiy consrained agens..4 (a) OUTPUT.4 (b) HOURS.. 5 5 quarers (c) CONSUMPTION NON RICARDIAN.4 5 5 quarers (d) CONSUMPTION RICARDIAN.4.. 5 5 quarers x 7 (e) PRICE INFLATION 5 5 quarers 5 x 7 (f) WAGE INFLATION 5 5 x 6 (g) REAL WAGE 5 5 5 quarers (h) REAL INTEREST RATE.4. 5 5 quarers 5 5 quarers λ= λ=.3 λ=.5 Figure 3: Response o a preference shock in he case of full commimen. Alernaive values of he share of liquidiy consrained agens. 5
(a) OUTPUT (b) HOURS 5 5 quarers (c) CONSUMPTION NON RICARDIAN 4 5 5 quarers (e) PRICE INFLATION 5 5 quarers (d) CONSUMPTION RICARDIAN 5 5 5 5 quarers (f) WAGE INFLATION.5 5 5 (g) REAL WAGE.5 5 5 quarers (h) REAL INTEREST RATE 4 5 5 quarers λ= λ=.3 λ=.5 5 5 quarers Figure 4: Response o a cos push sock under commimen. Alernaive values for he share of liquidiy consrained agens. wage and real ineres rae. In response o a cos-push shock opimal moneary policy implies [TO BE COMPLETED]. 6. Simple Rules In he reminder we assume ha he moneary policy auhoriy can credibly commi o a simple insrumenal rule of he form of he Taylor rule (55). For he momen we se r = ; so ha he planner chooses he values of he coe ciens and y ha minimize he expeced, as of ime zero, discouned sum of fuure welfare losses. We resric our search o policy coe ciens in he inerval [-5,5]. However, we will menion when our resuls are a eced by he size of he inerval. Table repors he opimal y and under he variuos speci caion of he insrumenal rule considered, ogeher wih he condiional welfare loss associaed o each of hem in he case of a echnology shock. We assume ha he sysem is iniially a he seady sae. Consider he case of a conemporaneous rule, i.e. i =. In ha case he oupu response is mued no maer he share of non ricardian agens, his is in line wih he resuls in SGU (5). The opimal in aion coe cien is always below he upper limi of he speci ed inerval. Remarkably, 6
Table = = :3 = :5 Policy Rule Sicky Wages ; y ; loss ; y ; loss ; y ; loss i = i = 3:; ; : 3:7; ; :9 4:5; ; :85 i = Flex Wages ; y ; loss ; y ; loss ; y ; loss i = i = 5; :3; 5; ; 5; :8; i = Table : Technology Shock. Opimal Simple Rules he opimal in aion coe cien response ges larger as he share of non ricardian agens increases. This mimics resuls obained under commimen, where he planner engineered a deeper reducion in he rae of ineres as he share of liquidiy consrained agen increased. Noice ha wage sickiness plays a relevan role in he design of opimal policy. Indeed, as shown in able, when wages are exible he opimal policy response o in aion his he lower bound of he speci ed inerval, more precisely i ges negaive. The opimal rule is also characerized by negaive values of coe cien y. Table shows he opimal y and under he variuos speci caion of he insrumenal rule (55) considered, ogeher wih he welfare loss associaed o each of hem in he case of a cos-push shock. Noice ha in he case of wage sickiness he opimal in aion coe cien is always larger han one, i.e i sais es he aylor Principle no maer ha share of non ricardian agens. This is no he case under exible wages. Di erenly form he case of a echnology shock, he cos push shock generaes apolicy rade-o even under exible wages. For his reason he opinal in aion coe cien says wihin he bundaries of he inerval over whcih we search. However as in he previous case i becomes negaive once ges above a cerrain hreshold. 6. Opimal Rules Vs Commimen [TO BE COMPLETED] 7 Conclusions [TO BE COMPLETED] 7
Table = = :3 = :5 Policy Rule Sicky Wages ; y ; loss ; y ; loss ; y ; loss i = i = :; ; 3:73 :; ; :9 :6; :; :65 i = Flex Wages ; y ; loss ; y ; loss ; y ; loss i = i = 4:; :3; :8 :4; ; : 4:8; :4; :3 i = 8 Technical Appendix Table : Cos Push Shock. Opimal Simple Rules 8. Derivaion of he e cien equilibrium oupu In order o derive he e cien equilibrium oupu, we need o rs calculae he soluion of he Social Planner problem (SPP) and o derive rs bes allocaion. The equilibrium oupu which solve he (SPP) corresponds o e cien equilibrium oupu. noe ha max fc H; ;C S; ;N g s: CH; + ( ) C S; L+ H; + ( ) L+ H; + C = Y = A N = C H; + ( ) C S; = A (L H; + ( ) L S; ) since we assume ha U CH = C H U CH" = C H = ; hen he Lagrangean L is = C H H; max L = C fc H; ;C S; ;N g + ( ) C S; rs order condiions imply which imply ha L+ H; + ( ) L+ H; + [C H; + ( ) C S; A (L H; + ( ) L S; )] @L = : C H; @C = H; @L = : ( ) C S; @C = ( ) S; C H; = C S; = C 8
he marginal uiliy of consumpion he wo consumer are idenical, hen given ha he consumer have idenical preferences in consumpion, i implies ha C H; = C S; = C wih respec o he labor supply again i implies @L @L H; = : L H; = A @L @L S; = : ( ) L S; = A ( ) L H; = L S; = L he marginal uiliy of labor of he wo consumer are idenical, hen given ha he consumer have idenical preferences in labor, i implies ha N H; = N S; = N Now i can combine he wo generic equaions hen C = L = A C L = A which is he sandard equaion wih one represenaive household Log-linearizing and considering ha c = y and y = a + l ; we ge he e cien equilibrium oupu y Eff = + + a + + : 8. Derivaion of he Welfare-based Loss Funcion In order o derive a second-order approximaion of he household uiliy funcion, we assume ha he seady sae of our economy is e cien. Under his assumpion, we have ha in he seady sae: V N;H = V N;S = W U C;H U C;S P = Y N = (64) where N H = N S = N = Y and C H = C S = C = Y: The las equaliy in (64) holds since he economy producion funcion is: Y = N A ; where A = in seady sae. As shown in secion once we ge he e cien seady rms pro s are zero in he seady sae and he wo households budge consrain is idenical, so ha C S = C N = C: In order o derive a second order approximaion of he households uiliy funcion, as in Bilbiie (8) we assume ha he Cenral Bank maximizes a convex combinaion of he uiliies of wo ypes of households, weighed by he mass of agens of each ype, i.e.: W = [U (C H; ) V (N H; )] + ( ) [U (C S; ) V (N S; )] (65) 9
we know ha in our model N H; = N S; = N for each, his means ha (65) can be rewrien as W = U (C H; ) + ( ) U (C S; ) V (N ) (66) A second order approximaion of U (C H; ) delivers U (C H; ) ' [U (C H ) + U CH (C H; C H ) + U ( )] + + hu CH C H (C H; C H ) + U CH (C H; C H ) ( ) + U ( ) i or U (C H; ) ' U (C H ) + U CH C H c h; + c h; + U + + + UCH C H C Hc h; + U C H C h c h; + U or U (C H; ) U (C H ) ' U CH C H c h; + c h; + U C H C H CHc h; + +U CH C h c h; + U + U + {z } ip U (C H; ) U (C H ) ' U CH C H c h; + c h; + U C H C H CHc h; + U C H C H c h; + ip U (C H; ) U (C H ) ' U CH C H c h; + c h; + U C H C H CHc h; + U C H C H c h; + ip noice ha U C H C H C Hc h; + U C H C H c h; = U CH C H U CH C H C H c h; U + U C H CH U CH c h; since U C H C H U CH C H = U C H C H CHc h; + U C H C H c h; = U CH C H c h; + U C H U CH c h; hus U (C H; ) U (C h ) ' U CH C H c h; + ( Given he assumed funcional form U CH U CH = ) c h; + U C H U CH c h; + ip
U (C H; ) U (C H ) ' U CH C H c h; + ( ) c h; + c h; + ip Similarly a second order approximaion o he uiliy of ricardian agens ( ) U (C s; ) U (C s ) ' ( ) U Cs C s c s; + ( ) c s; + c s; + ip Also a second order approximaion o V (N ) yields: V (N ) V (N) ' V N N ^n + + ^n Summing all he erms or W W = U CH C H c h; + ( ) c h; + c h; + + ( ) U Cs C s c s; + ( ) c s; + c s; W W = U CH C H c h; + ( ) c h; + c h; + + ( ) U Cs C s c s; + ( ) c s; + c s; V N N ^n + + ^n V N N ^n + + ^n Given our assumpions, seady sae consumpion levels are idenical as well as hours worked, in his case W W = U C C c h; + ( ) ch; + c h; + + ( ) U C C c s; + ( ) cs; + c s; V N N ^n + + ^n or W W = U C C c h; + ( ) c h; + U C Cc + ( ) U C C c s; + ( ) c s; V N N ^n + + ^n + ip From he economy producion funcion we know ha (67) ^n = y + d w; + d p; a where d w; = log R W j w W dj is he log of he wage dispersion and dp; = log R P i p P di is he log of he price dispersion. Boh erms are of second order and herefore hay canno be negleced in a second order approximaion. Noice ha ^n = (^y + d w; + d p; a ) = y + a y a
hus or or W W = U C C c h; + ( ) c h; + U C Cc + ( ) U C C c s; + ( ) c s; V N N y + d w; + d p; a + + y + a or, since U C C = V N N y a + ip W W = U C C c h; + ( ) c h; + U C Cc + ( ) U C C c s; + ( ) c s; V N N y + d w; + d p; a + + y ( + ) y a + ip W W = U C C c h; + ( ) c h; + U C Cc h; + ( ) U C C c s; + ( ) c s; V N N y + d w; + d p; a + + y ( + ) y a + ip W W U C C = c + ( ) c h; y + d w; + d p; hen using equilibrium condiion c = y W Nex noice ha hen W U C C + ( ) ( ) = y + c h; + ( y + d w; + d p; ^c H; = w + n c H; = w + n + w n ( ) c s; + U C Cc + a + + y ( + ) y a + ip ) c s; + c + a + + y ( + ) y a + ip = w + y + a y a + w y w a = (y a ) + w + w y w a
and hus ^c S; = = = ( ) ^c + ( ) ^c + ( ) ^y + ^c S; = ^c ( ) ^y ^w + ^y y a ^c H; ^c H; hen ^c H; + ( ) ^c S; = y + a y a + ^w + ^w ^y w a + ( ) ^y + ( ) collecing erms simplifying = ^c H; + ( ^w + ^n + ^w ^n ^c ^c H; ( ) ^c ( ^w + n ) ^w + ^y + a ^y a + ^w ^y ^w a ^w + ^y + a ^y a + ^w ^y ^w a ( ) ^c H; + ( ) ^c S; + w + + ( ) + ( ) {z } + + ( ) {z } + y + ( ) {z } a + y a + ( ) ( ) {z } + + w y ( ) ( ) {z } ) ^c S; = ( ) w + y + ( {z ) } ( ) a w a ( ) ^y ^w + ^y w a Using his resuls and considering ha a is independen of policy he welfare funcion can be rewrien as W W = ( ) U C C ( ) w ( + ) y ( ) ( ) w a + y + ( + ) y a (d w; + d p; ) + ip 3 y a
Nex we have o rewrie some erms. Recall ha ( + ) y Eff = ( + ) a + hus Also ( + ) y ( + ) y y Eff y eff = ( + ) = ( + ) = ( + ) y a + y y + y + y eff y eff y y eff ( + ) y y eff subsiuing for he previous resul ( + ) y y eff = ( + ) y + y eff ( + ) y a y which implies ha ( + ) y a + y = ( + ) y + y eff ( + ) y y eff In his case W W U C C = ( ) ( ) where x = y y Eff w w a and given ha y Eff ( + ) x (d w; + d p; ) + ip is independen of policy. Also noice ha w eff = a which is a erm independen of policy. Muliplying w Eff by w we ge: w w eff = w a Nex combining which implies w w w w a = w eff = w + w eff w eff = w w a + w w eff w eff w w eff w eff = ~! w eff 4
Subsiuing he laer ino he welfare loss funsion and considering ha w eff is a erm independen of policy, we ge W U C C = ( W ) ( ) ~! ( + ) x (d w; + d p; ) + ip Using Woodford Lemma and Lemma, we can nally wrie he presen discouned value of he Cenral Bank loss funcion as L = X ( ) ( ) ~! + ( + ) x + w ( w ) + p ( p w ) + ip p = Noice ha if < deviaion of he real wage from is e cien level leads o a lower sociey s loss. How can his be he case? Recal ha in a sandard model changes in he real wage have no wealh e ec on labor supply, since hey are exacly o se by changes in pro s for given hours. This is no he case here. However I don wheher his is he reason. Derivaion of he welfare funcion under exible wages: Remember ha in he case in which wages are fully exible, he labor supply is: w = c + n = y + y a d p; = ( + ) y a d p; hence, subracing he e cien equilibrium o he LHS and he RHS of he previous equaion! = ( + ) x d p; where we use he fac ha d p; d Eff p; = d p; (given ha d Eff p; = ). Moreover, we know a = a Eff and ha = Eff and erms muliplied by d p; are erms higher han second order. Then! = ( + ) x his meas ha he welfare-loss can be rerwrien as follows: L = X ( + ) x ( + ) ( ) + x + p ( p; ) p = Nowisanding wage exibiliy here is and addiional erm wih respec o a fully ricardian framework, given by (+)( ) x. Once again his is due o he presence of ro agens and similarly i disappears when =. Also, when <, he ideni ed addiional erm leads o a reducion in sociey s welfare loss. The presence of a union implies ha workers supply labor on demand. To see his and neglecing exogenous shocks, he producion funcion implies ^n = ^y 5
where recall ha n = n h; = n s;. Thus an increase in oupu leads o an increase in hours worked and viceversa for any given level of he real wage. Suppose now ha labor markes were compeiive. In his case liquidiy consrained agens would be characerized by he following FOC for labor supply bn h; + bc h; = bw (68) In can be shown ha under compeiive labor markes he dynamic of he aggregae wage would be idenical o ha obained in he case in which unions se wages, i.e. w = c + n. Subsiuing c h; = w + n h; and he aggregae wage ino (68) leads o n h; = ( ) y which shows ha in he case in which > hours of liquidiy consrained agens would move in he opposie direcion wih respec o oupu. Given he union forces hem, no maer he value of ; o increase hours worked as y icreases hey su er a loss which shows up in he welfare loss funcion. When < insead hours of liquidiy consrained agens would move in he same direcion of oupu, he gap wih respec o he decenralized equilibrium ges narrower and he loss reduces. When =, insead, liquidiy consrained agens would mainain heri labor supply consan, he loss in his case is measured uniquely by he disorions due o oupu changes as in he sandard framework. 8.3 Derivaion of he IS curve We know ha, by log-lienarizing he Ricardian Euler equaion c s; = E c s;+ E (r + ) + while from he consumpion funcion of rule-of-umb consumer we ge: c H; = l +! while aggregae consumpion is c = ( ) c s; + c H; hen solving he laer equaion for c s; subsiuing in he euler equaion we ge c s; = c c H; or c c H; = E c + c = E (c + c H;+ ) c H;+ E (r + ) ( ) E (r + ) ( ) + + 6
or subsiuing for c = y and for E c H;+ = E (l + +! + ) ( ) y = E y + E l + E! + E (r + ) given he aggregae producion funcion ( ) + y = l + a hen we subsiue l for l = y a ; hen we ge ( ) y = E y + E y + + E a + E! + E (r + ) ( ) + or solving for y y = E y + + E a + E! + E r p + + (69) rewriing equaion (69) in erms of oupu gap from he e cien equilibrium oupu we ge: y y Eff = E y + we de ne x = y y Eff + +y Eff + + E a + E! + E r p + + (7) y Eff ; hen x = E x + + y Eff + + E a + E! + E r p + + (7) remember ha E! + = E w + E p + ; hen equaion (7) becomes x = E x + + y Eff + + E a + E w + + E p + E r p + + (7) We now wan o rewrie he IS curve, in erms of he e cien rae of ineres. We know ha under he e cien equilibrium x = E x + = ; and E w + = E p + = ; hen given ha: E y Eff ineres as follows: r eff + = + r eff = y Eff + + E a + + (73) + E a + + + E +, hen we can rewrie he e cien rae of + = + + E a + + + E + hen, we can nally wrie he IS as x = E x + E r p + r Eff E w + E p + 7