Dynamic Stochastic General Equilibrium Models: A Tool for Monetary Policy in Light of Lucas Critique

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1 Mddle Easern Fnance and Economcs ISSN: Issue 4 (20) EuroJournals Publshng, Inc. 20 hp:// Dynamc Sochasc General Equlbrum Models: A Tool for Moneary Polcy n Lgh of Lucas Crque Syed Kashf Saeed COMSATS Insue of Informaon Technology, Islamabad, Paksan Shahd Mehmood Sargana COMSATS Insue of Informaon Technology, Islamabad, Paksan Usman Ayub COMSATS Insue of Informaon Technology, Islamabad, Paksan Fasal Nawaz COMSATS Insue of Informaon Technology, Islamabad, Paksan Absrac Dynamc sochasc general equlbrum (DSGE) models are he macro models based on Mcro foundaons. These are playng an mporan role n conducng moneary polces a varous cenral banks lke Norges Bank, bank of England ec. The prmary emphasze of hese models s on expeced fuure oucomes whle deermnng curren oucome. Due o ncluson of Mcro foundaons and expecaons hese are n lne wh Lucas Crque. Alhough very popular n academa and polcy makers sll hese models are n ransonal phase o prove her usefulness n forecasng and polcy makng. Keywords: Dynamc sochasc general equlbrum, Mcro foundaons, Euler Equaon, oupu gap.. Inroducon The las wo decades can be called era of scenfc revoluon whch has resuled n he gganc developmen of mahemacal and sascal ools resuled n he enormous change n he dscplne of macro economcs. The sad developmen has changed he approach of economss o look a problems and also provde he ools for solvng heorecal ssues whch were, prevously, ou of quesons due o lack of echnology requred. Besdes he remendous mprovemen n research ools, pas wo decades have also full of economc crses, sarng of Tequla Crss n 995, Asan Fnancal crss n 997 and laes Global Fnancal Crss n 2008 ec. Two mporan quesons have been rased afer heses crses. Frs, are our economc polces (.e. Moneary and fscal) suffcen o handle hese crses and second are our economc models capable n forecasng and mgang he adverse effec of hese crses. Economc leraure places moneary polcy a mporan level whle dscussng boh of hese quesons. A lo of research has been conduced for desgnng opmal moneary polcy [Taylor, 993; Woodford, 2003; and Clarda e al 999, 2000; Monacell, 2005]

2 Mddle Easern Fnance and Economcs - Issue 4 (20) 7 Lucas (976) argues ha whle preparng he macroeconomc polcy and predcng s effec enrely on he grounds of hsorc daa can lead o dsasrous or a leas neffecve macroeconomc polces. Ths argumen s known as Lucas crque n economcs leraure. Hs own words are as follows: Gven ha he srucure of an economerc model consss of opmal decson rules vary sysemacally wh changes n he srucure of seres relevan o he decson maker, follows ha any change n polcy wll sysemacally aler he srucure of economerc models (Lucas, 976, p.4) Lucas explans hs vew and poses a queson also on he effecveness of very large counry wde macroeconomc models. He argues ha f we are neresed n esmang he mpac of polces hen our parameer should be Deep parameer,.e. hese should be polcy nvaran. He calls hese as Srucural parameers whch should be based on such varable whch relaes o ndvdual preference, resource consrans, echnology mprovemen ec. Afer hs, polcy makers wll be able o predc ha how ndvdual wll behave, frms wll reac and aggregaon of ndvdual and frm decson wll lead us o esmae he macroeconomc mpac of any suggesed economc polcy. I s he grea conrbuon of Lucas o movae macroeconomss owards mcro foundaons for her models. Kydland and Presco (977) also sressed on mporance of mcro foundaons especally ndvdual (Raonal) expecaons. They argue n her own words as: Even f here s an agreed-upon, fxed socal objecve funcon and polcymakers know he mng and magnude of he effecs of her acons, dscreonary polcy, namely, he selecon of ha decson whch s bes, gven he curren suaon and a correc evaluaon of he end- of-perod poson, does no resul n he socal objecve funcon beng maxmzed. The reason for hs apparen paradox s ha economc plannng s no a game agans naure bu, raher, a game agans raonal economc agens. here s no way conrol heory can be made applcable o economc plannng when expecaons are raonal. (Kydland and Presco,977) The need for opmal moneary polcy desgn n he lgh of Lucas crque, avalably of mahemacal and sascal ools wh gganc compung power, combnely, resuled n evoluon of Dynamc Sochasc general Equlbrum (DSGE) models. Ths shor paper nends o provde very bref orenaon owards Mcro foundaon based Macroeconomc models. The paper run as follows. Secon 2 dscuss abou basc conceps of Dynamc Sochasc General Equlbrum (DSGE) model. Secon 3 provdes mcro foundaon for DSGE model whereas secon 4 dscusses varous avalable mehodologes for smulang, calbrang and esmang DSGE model and secon 5 concludes. 2. Wha s DSGE Model? DSGE s he abbrevaon of Dynamc Sochasc General Equlbrum model conanng hree mporan jargons requrng need o explan. Frs s he Dynamc, second s Sochasc and las s General Equlbrum.

3 8 Mddle Easern Fnance and Economcs - Issue 4 (20) Source: Sbordone e al (2008), FRBNY Economc Polcy Revew Dynamc analyss relaes o decson made a he respecve me perod. Decsons made a me for + can be on he bass of Raonal Expecaon or Adapave Expecaons, as shown n Fgure. Fgure : Dynamc Analyss Expecaons (Raonal/Adapave) Therefore Dynamc aspecs of he model sresses ha oday s oucome are deermned on he bass of fuure expecaons. DSGE models hghlgh he mporance of expecaons n managng economy. I can be sad ha o manage radeoff beween nflaon and oupu wha Cenral bank need o do s o manage expecaons. Sochasc Analyss relaes o random shocks whch affec any economy and hen her mpac s propagaed hroughou some fuure me perod. Wh he help of hese random shocks unceranly are njeced no economc sysem oherwse economy would be runnng as predced whou any uncerany or flucuaons. These random shocks can be supply shocks, demand shock ec Fgure 2: Sochasc Analyss Impulse Propagaon Flucuaon General equlbrum analyss relaes o undersandng model dynamcs afer akng no he consderaon of hree relevan secors. These hree relevan blocks/secors for DSGE models are Supply sde of economy, Demand sde of economy and he Moneary Polcy. Models regardng each block are based on explc assumpons of major economc agens n each block.e. household, frms and moneary auhores.

4 Mddle Easern Fnance and Economcs - Issue 4 (20) 9 Fgure 3: General Equlbrum Analyss Thus DSGE model are dynamc model ncorporang sochasc componens nvolvng all relevan secors of economy. 3. Mcro Foundaon of DSGE Model As explaned already ha n las weny fve years, macroeconomc modelng has made enormous developmen n all aspecs (boh emprcs and heory). Wh he help of Mcro foundaon, DSGE models have made be possble o escape or a leas mnmze he Lucas (976) crque. The dsncve feaure of DSGE models s ha decson rules of economc agens are derved, from assumpons abou preferences and echnologes, by solvng neremporal opmzaon problems. The uncerany presen n model s generaed by exogenous sochasc processes or random shocks ha dsurb demand sde and/or supply sde of he economy and resulng devaons from neres-rae feedback rule of cenral bank. DSGE model, generally, are composed of hree secor ha s Household, Frms and moneary auhores. The dscusson of how mcroeconomcs of each secor conrbues owards baselne model s gven below. Basc Noaon of model has been adaped from Walsh (2003) and Gal and Moncell (2005). a) Households Decson and Demand Sde of Economy An nverse relaonshp beween oupu and real neres rae s an negral componen of demand sde of economy. Ths nverse relaonshp s resul of paern n household consumpon decsons. A raonal represenave household, of an economy always nends o maxmze hs uly by radng off hs consumpon ( C ) and lesure ( N ). The preferences for such a household can be descrbed by an ner-emporal Consan Relave Rsk Averson uly funcon (CRRA) as E β U ( C, N ) = 0 σ + ω C N + + U = E β = 0 σ + ω Where, β Iner-emporal dscoun facor, descrbes he rae of me preferences, σ represens he nverse of elascy of Iner-emporal subsuon n consumpon, also gves he degree of relave rsk averson and ω s he nverse of wage elascy of labor supply. σ > 0, ω > 0 and β (0,). C, he compose consumpon ndex of foregn and domescally produces goods s defned as ()

5 20 Mddle Easern Fnance and Economcs - Issue 4 (20) η η η η C = ( α ) η ( C η H, ) α ( C η F, ) (2) Where η s he prce elascy of subsuon beween home and foregn goods s, α (0,) s he rade share, also measures he degree of openness. C H, s an ndex of consumpon of domesc goods and C F, s an ndex of consumpon of mpored (foregn) goods. Raonal households under resrcon of lmed budge ry o mnmze he cos of consumpon. PC + E Q D D + W N + T (3) (, + + ) W N s he Nomnal wage mes he number of hours worked, D s he payoff from secures held and T are he ne of axes and ransfer paymens. So Lef hand sde s provdng us he expendure whereas Rgh Hand sde s sources of Income. Applyng Lagrangan opmzaon funcon o maxmze eq () subjec o eq (3) wll resul n σ σ C + C = β ( + ) P E P + Above equaon s also called Euler Consumpon Equaon for he opmal ner-emporal allocaon. So as per Euler equaon, when real neres rae ncreases or fuure consumpon s expeced o decrease or dscoun rae ncreases, curren consumpon s expeced o decrease. Log-lnearzng equaon above equaon and applyng he conceps ha margnal rae of subsuon beween lesure and consumpon s equal o real wage, resuls n cˆ = E ˆ c + ( ˆ Eπ + ) σ In above equaon, î s he shor erm nomnal rae and π s he consumer prce nflaon rae. Assumng he closed economy model, one can safely assumes ha y = c + g, and also denong ^ oupu gap as x = y y, hen above equaon can be shown as follows; x = φ [ Eπ + ] + E x + + e (4) Where φ =, g σ s he dsurbance erm whch s AR() process e = µ e + η. Equaon (4) s smlar o radonal IS equaon and herefore s also called Demand sde equaon n DSGE model. b) Frms Decson Supply sde of he economy n DSGE model ncorporaes he behavor of he frms n seng up her prces keepng n vew her demand as well. Raonal enrepreneurs a varous frms, n he same way, nend o maxmze he profs of frms. Convenonally he prof maxmzaon frms used o consder he resrcon from demand and producon sde. The concep of prce rgdes provded anoher dynamcs o frm s opmzaon decson. Now s assumed ha a ceran percenage of busness don change he prces whereas remanng percenage does change prces. To ncorporae he prce rgdy n producon funcon, Calvo prces (983) are nroduced. For smplcy, s assumed ha labor s he sngle facor of producon. Thus Y = Z N η Ths phenomenon has been dscussed a lengh, please see Bls and Klenow (2004); Nakmura and Sensson (2008)

6 Mddle Easern Fnance and Economcs - Issue 4 (20) 2 Z s he counry specfc aggregae producvy dsurbance whch s assumed o be sochasc wh consan reurns o scale, ha s, E( Z ) =. Ths can also be called as Producvy shcok. Aggregae oupu can be defned as ( ρ ) ( ρ ) Y = Y ( ) 0 j dj ( ρ ) ( ρ ) = 0 = + ln Y ln Y ( j) dj ln( Z ) ln( N ) y = z + n Frm j mnmze cos subjec o producng he frm specfc good Y W L = N + ϕ ( Y Z N ) P D, Frs order condon gves W P D, MC = ϕ = Z mc = w p z D, So we can see he relaonshp among real margnal cos, labor facor producvy and real wages. Assumng Calvo prce assumpon, only ( θ ) frms revsng prces hus, θ measures prce rgdy. Domesc prce ndex s defned as θ ( ) P P j P H, H, = H, P H, 2 H Frms maxmze her profs afer seng he new prce max p p a me as: E ( θ ), + Y + ϕ + (5) P = o + ε p j (6) s.. Y j, = C P I s nuve for frms o consder elascy of demand whle prces are beng se. Subsung he demand funcon [equaon 6] n maxmzaon funcon [equaon 5] resuls n ε ε p p p max E ( θ ), + C + ϕ + = o P + P + P + Takng Frs order condon and applyng algebra wll resul n

7 22 Mddle Easern Fnance and Economcs - Issue 4 (20) P P σ ( β θ ) ( ) ϕ ( ) E C P ε ε E C P k = 0 H, = ε j, P σ ( β θ ) ( ) ( ) + + = 0 ε σ P + ( βθ ) ( C + ) ϕ + = 0 P ε σ P + E ( βθ ) ( C + ) = 0 P E ε = ε The above equaon descrbe rule for opmal prce; and conssng of new prce and Calvo prcng can be shown as ε ε ε ( θ ) P = p + θ P ε (7) ε s he markup. Prce ndex ε In above equaon, we have wren prce ndex as weghed ndex of newly se prce prce ndex from prevous perod P. Where ε ε P = P dj 0 Takng log of he above equaon and ransformng no seady sae resuls n ( ) p and p = θ p j, + θ p (8) The above equaon shows he general prce level n seady sae whch s weghed average of θ and θ frms do no adjus her prce. he frms whch adjus her prces each perod, ( ) ε ε P θ P p = ( θ ) + (9) P Now Equaon (6) abou frms opmal prce seng rule can be wren as η η k σ P k k σ P + + k E ( βθ ) ( C+ k ) F = µ E ( βθ ) ( C+ k ) k = 0 P k = 0 P (0) Applyng Taylor seres approxmaon and resrcng o wo perods he above equaon can be wren as fˆ = ( θ β ) ˆ ϕ + θ β ( E ˆ ϕ + E π ) () + + θ θ ( ) ˆ π = θβ ϕ + θβ[( ) E π Eπ ] θ θ θ π ( ) ˆ = θβ ϕ + θβ[( ) E π ] θ + θ π ˆ = λϕ + β Eπ + (4) Where ˆ ϕ s real margnal cos and Eπ + s he expeced nflaon. Above equaon s he forward lookng Phlp curve whch also descrbes he supply sde of he economy. Ths forward lookng Phlp curve s hghlghng he relaonshp beween nflaon, (2) (3)

8 Mddle Easern Fnance and Economcs - Issue 4 (20) 23 economc acvy and expecaon abou nflaon. So f economc acvy wll be hgher, hgher wages wll be pad o labor resulng ncrease n margnal cos whch wll ulmaely resul n rse n nflaon. Now can be easly undersood ha how macro model lke DSGE are based on mcro foundaons provded by household behavor and frm behavor. These models are based on coeffcens provded hrough mcroeconomcs varables lke Coeffcen of relave rsk averson, subjecve dscoun facor, Frsch elascy of labor supply, fracon of frms ha canno adjus prces, elascy of subsuon beween dfferen goods ec. Afer ncorporang hese mcro foundaons, now macro model s expeced o be less prone o Lucas crque. c) Moneary Auhores Afer dervng demand sde and supply sde equaon whch represens households secor and frm/producon secor, now s mporan o nclude moneary auhores no model so ha equlbrum can be aaned n oupu-nflaon space. Generally, s assumed ha cenral banks should follow some rule for seng neres rae hus seng preference on rule raher han dscreon. The man objecve of moneary polcy s o sablze boh oupu and nflaon, so hey need o fnd a radeoff beween wo. Wh hs objecve hey are supposed o follow some rule lke Taylor Rule (993). The arge neres rae for each perod s a funcon of he gaps beween expeced nflaon and oupu wh her respecve arge levels. Therefore Clarda e al (2000) posulae he lnear equaon. r s desred nomnal rae when boh nflaon and oupu gap are a her arge level. { π Ω } π ) + y E{ X Ω } = r + βx(e,k x,q r For ncorporang Ineres Rae Smoohng, he arge rae funcon can be shown as: r = ρl(r ) + ( ρ)r Where βπ and γ x are he coeffcens for expeced nflaon rae and oupu gap; and ρ s he neres rae smoohng facor. Afer ceran algebrac manpulaon, he Moneary Reacon funcon can be shown as r = ρ ( L) r + ( ρ )[ α + ( β )[ E{ π Ω } π ] + γ ( x )] + ε (5), k, q Now our smple and small moneary polcy model s complee conssng of Demand sde equaon, supply sde equaon and moneary reacon funcon and whole model s able o, a leas parally, answer Lucas Crque. The model s equlbrum condons can be summarzed by he followng equaons: x = φ [ Eπ + ] + E x + + e = ˆ + E π λ ϕ β π + r = ρ ( L) r + ( ρ )[ α + ( β )[ E{ π Ω } π ] + γ ( x )] + ε, k, q Followng s he canoncal represenaon of he closed economy model. As we can see ha ne expors and exchange rae are no here n he model. Generally here are wo varaons o above model closed economy model. Frs s he open economy model wh he assumpon when purchasng power pary holds and second s he varaon of he above model when purchasng power pary does no hold. 4. Usng DSGE Model for Polcy Makng Afer dervng DSGE model, nex problemac sep s how o use hese models for polcy makng. The ssues nvolve are more compung power and unconvenonal mehods. Evaluaon of DSGE models can be done n prmarly hree ways, Frs Calbraon, second Smulaon and hrd Esmaon. In he nal perod of DSGE models, calbraon was he mos useful way bu now s use has been grealy declned manly due o avalably of compuaonal power and

9 24 Mddle Easern Fnance and Economcs - Issue 4 (20) sophscaed economerc mehods. Smulaon and esmaon are grealy n used for all sze of models, from small and medum sze models used by manly academa and for large sze model manly used by cenral banks. Canova (2007) has exensvely dscussed dfferen mehodologes avalable for DSGE model. Followng mehods are beng grealy used for esmaon of DSGE models; Generalzed mehod of momens (GMM) Smulaed mehod of momens (SMM) Srucural VAR based on Mnmum Dsance Maxmum lkelhood mehod (ML) Bayesan Mehods I s mporan o noe ha GMM mehods, workng as sngle equaon, only use par of he nformaon mpled by he model, herefore mosly models use esmaon mehods oher han GMM. Maxmum lkelhood (ML) mehod, alhough generally used, bu quesoned a lo due o some heorecal ssues. For example sochasc sngulary problem, he assumpon ha he model correcly represens he process generang he daa up o a number of parameers, s over-sensvy o model msspecfcaon, denfcaon ssues ec. Canova (2007) and An and Schorfhede (2007) has provded enormous dscusson on he ssue. In comparson wh ML mehods, Bayesan mehods are supposed o handle hese esmaon ssues for DSGE n beer way because n hese mehods lkelhood are augmened wh prors abou parameer dsrbuons. Due o hs augmenaon, he poseror dsrbuons provde robus nferences abou he parameers. We sll canno say ha Bayesan mehods are free from flaw bu sll s able o answer some quesons. Tovar (2008) argues one mporan pon by sayng ha he mehods for solvng and esmang DSGE models are no he sandard ones found n he older leraure. A good example of hs s he economerc mehods employed. For nsance, Bayesan echnques are no ye a sandard par of economerc courses n PhD programmes n economcs. 5. Concluson Lucas Crque rghly quesons he very bascs of every macroeconomc model and encourages macro economs o hnk abou mos mporan quesons ha models should no be Polcy varan and me nconssen. Kydland and Presco (982) and ohers ook some seps and now we are wnessng DSGE models based on Mcro foundaons. DSGE models are undoubedly based on powerful mehods and ools based on varous secors of economy, provde coheren overvew and are able o answer changes n srucural parameers. Bu hese are no beyond crcsm (Sms, 2006). Sll s mcro foundaons are quesoned and very mporanly s communcaon power s quesoned. Bu need o be n mnd ha DSGE models are n her nal sage and hese problem wll be dscussed n due me. Alhough academa seems o lke DSGE model as s research publcaon are on ncrease bu only me wll ell wheher problems relaed o DSGE ouperform s benefs or no. References [] Bls, M., and P. J. Klenow Some Evdence on he Imporance of Scky Prces. Journal of Polcal Economy 2, no. 5 (Ocober): [2] Calvo, G. (983). Saggered Prces n a Uly-Maxmzng Framework. Journal of Moneary Economcs 2 (3), [3] Canova, F (2007): Mehods for appled macroeconomc research. Prnceon Unversy Press. [4] Clarda, R., Galí, J. and Gerler, M. (999). The Scence of Moneary Polcy: A New Keynesan Perspecve, Journal of Economc Leraure 37:

10 Mddle Easern Fnance and Economcs - Issue 4 (20) 25 [5] Clarda R., Gal J., Gerler M., [2000], Moneary Polcy Rules and Macroeconomc Sably: Evdence and Some Theory, Quarerly Journal of Economcs, Vol. CXV, ssue, pp [6] Clarda, R., Galí, J. and Gerler, M. (200). Opmal moneary polcy n open versus closed economes: An negraed approach, The Amercan Economc Revew Papers and Proceedngs 9(2): [7] Clarda, Rchard, Jord Gal, and Mark Gerler (2002), A Smple Framework for Inernaonal Polcy Analyss, Journal of Moneary Economcs, Vol. 49, pp [8] Del Negro, M and F Schorfhede (2007): Moneary polcy analyss wh poenally msspecfed models, NBER Workng Paper Seres 3099, May. [9] Del Negro, M and F Schorfhede (2004): Prors from equlbrum models for VARs, Inernaonal Economc Revew, Vol. 45, pp [0] Del Negro, M and F Schorfhede (2003): Take your model bowlng: forecasng wh general equlbrum models, Economc Revew, Federal Reserve Bank of Alana, Fourh Quarer. [] Gal, J and M Gerler (2007): "Macroeconomc modelng for moneary polcy evaluaon", Journal of Economc Perspecves, Vol, 2, No.4, Fall. [2] Galí, J, and Monacell, T (2005):.Moneary Polcy and Exchange Rae Volaly n a Small Open Economy, Revew of Economc Sudes, vol. 72, [3] Lucas, Rober Jr. (976), Economerc Polcy Evaluaon: A Crque, n: K. Brunner and A. Melzer (eds.), The Phllps Curve and Labor Markes, Carnege-Rocheser Conference Seres on Publc Polcy, Volume, pages [4] Monacell, Tommaso (2005), Moneary Polcy n a Low Pass-Through Envronmen, Journal of Money, Cred, and Bankng, Vol. 37, No. 6, December, pp [5] Nakamura, E., and J. Sensson Fve Facs abou Prces: A Reevaluaon of Menu Cos Models. Quarerly Journal of Economcs 23, no. 4 (November): [6] Kydland, F and Presco, E. (977), Rules Raher Than Dscreon: The Inconssency of Opmal Plans, Journal of Polcal Economy 85, [7] Kydland, F and Presco, E. (982), "Tme o buld and aggregae flucuaons", Economerca, Vol. 45, pp [8] Kydland, F and Presco, E. (996), "The compuaonal expermen: an economerc ool", Journal of Economc Perspecves, Vol. 0, no., wner 996, pp [9] Sbordone, A. Tambalo, A. Rao, K and Walsh, K (2008), Polcy Analyss Usng DSGE Models: An Inroducon, FRBNY Economc Polcy Revew. [20] Sms, C (2006): Commen on Del Negro, Schorfhede, Smes and Wouers. Avalable a: hp://sms.prnceon.edu/yfp/dssw806/dssealecommen.pdf [2] Taylor, John B. (993), Dscreon versus Polcy Rules n Pracce," Carnege-Rocheser Conference Seres on Publc Polcy, 39, December, [22] Tovar, C (2008): DSGE Model and Cenral Banks, BIS Workng Paper No Sepember. [23] Uhlg, H. (995). A Toolk for Analyzng Nonlnear Dynamc Sochasc Models Easly. Tlburg Unversy - Cener for Economc Research - Dscusson Paper [24] Walsh, C. E. (2003), Moneary Theory and Polcy (2 nd Ed.), MIT Press. [25] Woodford, M (2003): Ineres and prces. Prnceon Unversy Press.