ADDITIONAL MATERIAL FOR. Sticky Prices and Volatile Output

Transcription

1 ADDITIONAL ATERIAL FOR Sicy rices and Volaile Oupu Journal of oneary Economics Vol. 46 No. 3 December 000 arin Ellison Andrew Sco

2 ONTENTS. The flexible price model. The inflexible price model 3. alibraion 4. oefficiens of he parameerised polynomials and accuracy es resuls 5. Resuls for a model wih inermediae nominal price rigidiy (ω Gauss programs

3 . The flexible price model model srucure The economy consiss of infiniely lived households and firms. There is a coninuum of differeniaed goods ha are produced by monopolisically compeiive firms using boh capial and labour inpus. These differeniaed goods are aggregaed ino a single composie good which can be used for eiher consumpion or invesmen. consumers ouseholds are assumed o be idenical and have preferences given by he uiliy funcion [ ( ] E0 0 (. where is consumpion of he composie good is hours wored and β is he discoun rae. measures he weigh of uiliy from consumpion relaive o uiliy from leisure ime and τ is he coefficien of consan relaive ris aversion. ouseholds are endowed wih one uni of ime each period and supply labour o firms who produce inermediae goods. ouseholds are also engaged in accumulaing capial which hey ren o firms. I is assumed ha households ener period wih nominal money balances equal o m carried over from he previous period. In addiion hese balances are augmened wih a lump sum ransfer equal o (g where is he aggregae money supply in period which because of our represenaive agen model equals m. The money soc hen follows he law of moion: g (. I assume ha he money supply growh erm g evolves according o an auoregressive process: log( g ρlog( g ξ (.3 In he above equaion ξ is a iid random variable wih mean log(g(ρ and variance σ ξ where log(g is he uncondiional mean growh rae. I is assumed ha g is revealed o all agens a he beginning of period. ouseholds are required o use heir previously chosen money balances o purchase he nonsorable consumpion good he socalled "cashinadvance" consrain: m ( g (.4 where is he price level. In his example as in ooley and ansen (989 he consrain always holds wih equaliy. A sufficien condiion for his consrain o be binding is ha he money supply growh erm g always exceeds he discoun facor β. The represenaive consumer mus choose consumpion invesmen X and nominal money holdings subjec o he budge consrain:

4 X m m h w r Π ( g (.5 where w and r are respecively he compeiively deermined wage rae and renal rae of capial. Π h is he exogenously given share of profi received by he household. Fixed coss of producion are calibraed so ha firms mae zero profis in he long run. This has he added advanage ha he share of profis received by households will be zero on average and so can effecively be ignored. The capial soc obeys he law of moion µ X (.6 wih µ he depreciaion facor. The household s problem is o maximise (. subjec o (.6 (.8(. where (.4 and (.5 hold wih equaliy. firms The demand funcion faced by each firm is derived by specifying an aggregaor for he differeniaed goods. The specificaion used here is due o Dixi and Sigliz (977: Di i D (.7 where D is he demand for he composie good during period and Di is he demand for good i when firm i ses a price of i. ε is he elasiciy of demand for good i. The firm produces oupu of good i Yi according o he producion echnology: α α Yi θ i i Fi (.8 apial leers denoe nominal per capia variables. Fi is a fixed cos of producion. The variable θ is an exogenous shoc o echnology ha follows he law of moion z z ln γ ε θ exp( z (.9 where ε is an iid random variable wih mean zero and variance σ ε. I assume z is revealed o all agens a he beginning of he period. In his model he value of γ is se o exceed one and hence he produciviy shoc is nonsaionary. Firms minimise coss which equal Ri Wi. The cos minimisaion condiions for he firm are he same in aggregae because he ne producion funcion is homogeneous of degree one in capial and labour. α w ( α mc θ α α r αmc θ α (.0 (. 3

5 Lower case leers denoe real per capia variables. mc is he real marginal cos of producion. The insananeous real profi of a firm is L i i mc D Fi (. The firm s problem in he flexible price model is o maximise (. subjec o (.7(.. onsumer uiliy maximisaion Le λ and η be he mulipliers on he resource and cashinadvance consrains. The consumer s firs order condiions are: ( mc E (A. E (A. (A.3 ( F (A.4 m m Fi (g (A.5 m (A.6 Firm profi maximisaion The firs order condiion for profi maximisaion gives mc which implies ha if he elasiciy of demand for firm i s produc is consan hen he marginal cos will also be consan. I can herefore drop he subscrips on mc. Soluion of he model for he special case when τ Sar by aing he parameerised expecaion of equaion (A.. h ( z g ; 4

6 5 Rearranging equaions (A. (A.3 and (A.6 enables me o convenienly wrie log(g( ORJJ exp g E m m E (A.7 This nea way of analyically calculaing an expecaion arises because g has a lognormal disribuion. Derending occurs by dividing and by ] [ and by muliplying λ by he same facor. I use he symbol o denoe a derended variable. Afer derending and aing a loglinear approximaion equaion (A.4 can be solved for. ] ORJ ORJ log(mc log( log( log( (log [log( Now derend and rearrange he resource consrain (A.5. The money erms will cancel in equilibrium. Fi Finally derend equaion (A.7 and ae logarihms. ( log( log(g( ORJJ log( log log( Seadysae calculaion In he long run he derended version of equaion (A. can be solved o give he seadysae capiallabour raio. mc Similarly equaion (A.5 gives percapia consumpion in seady sae. Fi

7 Subsiuing he seadysae derended version of (A.7 ino he corresponding version of (A.4 yields an expression ha can be solved o give seadysae hours wored. ( F exp ORJJ log(g(. The inflexible price model The inflexible price model follows he flexible price model oulined above excep for pricing behaviour which means ha firms no longer maximise insananeous profis (.. pricing rice adjusmens are assumed o follow a alvoype saggering process. In any one period a proporion (ω of firms are allowed o se a new independen price. The remaining firms ω se a price equal o heir previous period s price level. When firms change heir price hey se i equal o he curren opimal level of prices for each firm see (. below. The process is memoryless in he sense ha he probabiliy of being able o change price is independen of he ime elapsed since he las independen price change. Therefore every period all firms have he same ex ane probabiliy of changing prices. A firm will hen se is opimal price by maximising he presen discouned value of expeced fuure profis 0 ( E mc D where ( E 3 (. wih respec o. In his equaion β is he normal discoun facor and ω eners because here is a probabiliy of also being loced ino he price in he nex period. Λ appears because nominal profis have o be weighed by he expeced value of nex period s marginal uiliy of income. The firm s problem in he inflexible price model is o maximise (. subjec o (.7(.. onsumer uiliy maximisaion Le λ and η be he muliplier on he resource and cashinadvance consrains. The consumer s firs order condiions are: ( mc E E 6

8 ( F F m Firm profi maximisaion The firs order condiion for profi maximisaion is ( ( E [ D mc ] 0 ( E [ D ] 0 Aggregaion The aggregae price index is defined as i 0 di so he alvo price adjusmen rule will be ( 3 3 The aggregae resource consrain we wan has aggregae oupu as a funcion of aggregae inpus. To ge his we have o define new measures of aggregae oupu and he price level. i 0 di Y i Yi di 0 0 Ydi Y ( 3 3 7

9 8 The aggregae resource consrain is herefore F Solving he derended problem when τ Derend he variables using a facor of ] [ Θ. and are divided by Θ. λ and η are muliplied by Θ. All prices are derended by dividing by / Θ. Variables mc and r are no rending. ( (B. ( ( 0 0 E ( mc (... (B. ( (B.3 mc E (B.4 E (B.5 (B.6

10 9 F ( (B.7 F (B.8 (B.9 Sar by aing he parameerised expecaion of equaion (B.4 o give λ. ; g z h Afer rearranging equaions (B.5 (B.6 and (B.9 he ey equaion is log(g( ORJJ exp( g E m m E c (B.0 ( log( log(g( ORJJ log( log c log( So i is sraighforward o solve for consumpion given λ and he money soc. rice immediaely follows from equaion (B.9. Equaion (B. he implies he new price se by hose firms allowed o change heir price in period. Equaion (B.3 gives he value of he alernaive price index. (

11 0 Equaions (B. and (B.7 need o be solved simulaneously for and mc. Sar by wriing (B. as follows. 3 mc (B. where E ( mc ( ( 0 3 E ( Subsiuing equaion (B.7 in (B. gives an equaion ha can be solved for. 3 ( ( To parameerise hese expecaions wrie ; g z h E ; g z h 3 3 Rearranging (B.7 gives he marginal cos. ( ( mc

12 Seadysae calculaion Seadysae price raios can be derived from equaions (B. and (B.3. op J op J op op Seady sae marginal cos is from equaion (B.. op J J mc Seadysae capiallabour raio is given by (B.4. F ercapia consumpion is derived from (B.8. F Subsiuing (B.0 in (B.7 yields an expression ha can be solved for seadysae hours wored. F ( log(g exp( rices fall ou easily from (B.9.

13 3. alibraion Table 3.: Values calibraed for U daa arameer Value Descripion γ seadysae growh of produciviy σ ε sandard deviaion of produciviy shocs α seadysae capial share of income g 0.0 seadysae growh of money supply σ ξ 0.05 sandard deviaion of money supply shocs ρ auocorrelaion of money supply process ε.064 elasiciy of demand for firm i s produc ω 0.5 degree of price inflexibiliy µ rae of depreciaion of capial soc β 0.99 discoun rae h /3 seadysae labour supply τ ris aversion parameer 0.39 weigh of consumpion vs. leisure in uiliy The model is calibraed o correspond o he U economy by uilising daa from a variey of macroeconomic and microeconomic sources. The parameers o be calibraed fall ino four main groups o be described in deail below. The firs wo of hese groups concern he wo sochasic variables in my models namely he produciviy erm and he money supply process. arameers from hese groups can be pinned down from widely available macroeconomic daa. The hird group describes he degree of compeiiveness and price inflexibiliy in he economy. The parameers here have o be derived from microeconomic daa. The final group consiss of fairly sandard parameers. produciviy erm The seadysae capial share of income α is calibraed a following olland and Sco (995 who use whole economy measures of oupu and capial o give a capial share value more consisen wih simple sochasic growh models. The produciviy erm is measured as he Solow residual i.e. he variaion in oupu ha is no accouned for by changes in capial and labour inpus. Esimaing for he U over he period 963q 994q4 gives he following: z z ε σε money supply process The cashinadvance model implicily focuses on he ransacions demand for money so he money supply process is calibraed using a narrow measure of he U money supply 0. An auoregression for 969q4 994q in he form necessiaed by he model gives he following esimaes:

14 log( g log( g ξ σξ compeiiveness and price inflexibiliy The degree of compeiiveness in he economy will direcly influence he seadysae price o marginal cos raio so calibraion can be carried ou from eiher esimaes of price elasiciy of demand or from esimaes of he price o marginal cos raio. The precise relaionship beween hese wo variables has already been derived in he economy s seadysae calculaions in secions and. icroeconomic evidence from he U economy examined by asel e al. (995 shows he average price o marginal cos raio o be.94. From he seadysae calculaions his implies ha he elasiciy of demand for firm i s produc ε is.064. Alhough his ey parameer is calibraed solely on he basis of a single microeconomic sudy furher suppor for his paricular calibraion is given by US daa. A survey paper by Tellis (988 concludes ha he median measured price elasiciy is jus under. Obaining an esimae of he degree of price inflexibiliy is he mos difficul and unreliable par of he calibraion exercise. Indeed Yun (996 maes no aemp a his calibraion and simply esimaes he model for differen degrees of inflexibiliy rying ω as and A sudy of pricing policy in U companies underaen by all Walsh and Yaes (996 suggesed ha approximaely 6% of companies do no change heir price over a year implying ω and a price inflexibiliy parameer ω of approximaely 0.5. owever I also esimae he model wih ω 0.5 and give he resuls in secion 5 for comparison. Table 3.: Fixed cos and ime horizon calibraions for differing price inflexibiliy rice inflexibiliy ω Fixed coss Fi Time horizon forward The amoun of fixed coss los in producion Fi are calibraed so ha seadysae profis in he economy are equal o zero. Because seadysae profis depend on he price inflexibiliy parameer i is necessary o calibrae Fi separaely for each level of inflexibiliy. The calibraions are shown in able 3.. and are calculaed using he seadysae calculaions of secions and. In he inflexible price models he alvo adjusmen rule means ha here is always a posiive probabiliy of oday s price sill being valid a any poin in he fuure. A profi maximising firm will herefore need o loo infiniely ino he fuure when deciding on wha opimal price o se. This is clearly a compuaional difficuly; o solve he model i is necessary o decide on a ime horizon ha firms consider when seing a new price. To calibrae his parameer forward i is assumed ha firms se heir ime horizon such ha profis afer he end of he ime horizon only have a negligible effec. Negligible is defined here o be so small ha he effecive discoun facor for hese fuure profis is less han 0 9 so ha he ime horizon is defined by: forward forward I s.. ( βω < 0 9 3

15 ence he values of forward also depend on he price inflexibiliy parameer. This is naural since if prices are more inflexible a firm will be more liely o have o eep oday s price in he fuure. Table 3.. also shows he forward calibraions. oher parameers Evidence from Aanasio and Weber (993 suggess calibraing he coefficien of ris aversion τ a on a microeconomic level. This also has he advanage of enabling he uiliy funcion in equaion. o be wrien in logarihmic form: U log( ( log( The relaive weigh of consumpion vs. leisure in he uiliy funcion can be obained from he sandard calibraion ha seadysae hours wored h are one hird of available hours and by using he seadysae calculaions in secions and. The value of is approximaely There is a sligh variaion wih he price inflexibiliy parameer bu i was decided o eep his parameer consan because i is quie "deep". The consequen variaion in seadysae hours wored across he flexible and inflexible price models remains small. The sandard US seings calibrae he rae of depreciaion of physical capial µ a.5% per quarer and he discoun rae β a % per quarer. 4

16 4. oefficiens of he parameerised polynomials and accuracy es resuls Table 4.: oefficiens of he parameerised polynomials Flexible price model onsan ln z g [ln ] ln ( z [ z ] [ z ] 3 Inflexible price model ω 0.5 h h h h3 onsan ln ( /0 z g [ln ( /0] ln ( /0 z ln ( / [ z ] [g ] ln ( / g Inflexible price model ω 0. 5 h h h3 onsan ln ( /0 z g [ln ( /0] ln ( /0 z ln ( / [ z ] [g ] ln ( / g

17 Table 4.: Accuracy es resuls Flexible price model lower ail upper ail Inflexible price model ω 0.5 h 5.6 % 3. % h h h3 lower ail upper ail 3.8 % 7.6 % 4.0 % 5.6 % 5.4 % 3.4 % Inflexible price model ω 0. 5 h h h3 lower ail upper ail 3.8 % 7.0 %.8 % 6.6 % 4. % 6.0 % Noe: Accuracy es based on 3000 where 500 draws are used. 6

18 5. Resuls for a model wih inermediae nominal price rigidiy (ω 0.5 Table 5.: yclical behaviour of he inflexible price model economy (ω0.5 Variables SD rho 3 3 Oupu onsumpion Invesmen ours rices Inflaion oney Velociy Nominal Ineres raes Noes: Deviaions from odricresco rend. All he saisics are averages of 50 draws each of 30 observaions. SD denoes he sandard deviaion of a variable and rho is correlaion wih is lagged value. 7

19 6. Gauss programs The Gauss programs o solve es and simulae he model are shown in able 6.. The specral decomposiion programs are adaped from he maser program for easures of Fi for alibraed odels Journal of oliical Economy Vol. 0 No indly provided by ar Wason. Table 6.: Gauss programs Flexible price model Sicy price model Descripion flexi.pea sicy.pea Solves he model using he arameerised Expecaions Algorihm (EA of den aan and arce Journal of Business and Economic Saisics 990 flexi.acc sicy.acc hecs he accuracy of he parameerised expecaions using he es of den aan and arce Review of Economic Sudies 994 flexi.ab sicy.ab alculaes he sylised facs for able. flexi.spc sicy.spc alculaes he specral decomposiion of he model and compares i o U daa. flexi.log sicy.log Loglinearisaion of model for use in specral decomposiion program above y.asc c.asc h.asc i.asc p.asc y.asc c.asc h.asc i.asc p.asc U oupu daa U consumpion daa U hours wored daa U invesmen daa U price index daa 8