Dynamic General Equilibrium Model with Uncertainty: Uncertainty regarding the future path of the economy

Transcription

1 Dynamc General Equlbrum Model wh Uncerany: Uncerany regardng he fuure pah of he economy Absrac Ths paper oulnes he concepualsaon and prelmnary resuls of a smple Dynamc General Equlbrum model wh uncerany. In forward-lookng dynamc Compuable General Equlbrum (CGE) models economc agens are endowed wh perfec foresgh, so boh consumers and frms ancpae any exogenous shocks and adjus her maxmsng behavour from he frs me perod onwards. Perfec foresgh hen would appear o negae any unceran response o a shock. Takng a smple model wh Ramsey economc growh dynamcs, hs paper llusraes a frame work ha ncorporaes uncerany by allowng alernave fuure me pahs resulng n uncerany n he model. The base lne growh pah wll be conrased wh smulaons where he probably of he shock vares boh n magnude and over me. 1. Inroducon The concep of rsk has been examned from many dfferen dscplnes: from an economcs perspecve, from a socologcal perspecve, from a fnancal perspecve. Rsk s a complex consruc. Rsk has been defned n many dfferen ways. One frequenly ced defnon of rsk s Kngh s defnon (1921). He defnes rsk as measurable uncerany. Rsk s a pervasve par of all acons. As MacCrmmon and Wehrung pon ou (1986), along wh deah and axes, rsk s one of he ceranes of lfe. Some rsks are dramac such as naural dsasers or crossng a busy sree, whle ohers are much more nsdous such as borrowng money or workng n a pollued envronmen. Some rsk mgh affec many people, whle oher rsk may be solaed o one person. Whn he realm of economcs, rsk has played a role n varous heores from consumer heory o he producer heory and nvesmen heory. In expeced uly heory, ndvduals make choces among rsks, each represened by a probably dsrbuon. Indvduals dffer n her wllngness o bear rsk. Some are rsk averse, some are rsk lovng, and some are rsk neural. An ndvdual who prefers a ceran gven oucome o a rsk scenaro wh he same expeced oucome s descrbed as beng rsk averse. The ndvdual s rsk neural f ndfference always holds and rsk lovng f he preference s reversed. The curvaure of he uly funcon reflecs her rsk aude. The nex secon (Secon 2) oulnes he naure of rsk and uncerany conaned n CGE models and draws ou he mplc assumpons regardng rsk n CGE models, whch are radonally deermnsc n naure. Secon 3 assesses

2 prevous research ha has aemped o ncorporae rsk and uncerany no CGE models. Secon 4 n hs paper descrbes he explc reamen of rsk n a CGE model nvolvng he creaon of mulple fuure pahs for he model, where agens are able o predc each pah and make decsons, gven an elemen of rsk averson, n he presence of hs uncerany. Secon 5 concludes. 2. Compuable General Equlbrum Models and Implc Rsk 2.1 CGE Modellng The secon oulnes he basc characerscs of he dfferen ypes of CGE models: he sac model, he dynamc recursve model, he sngle secor dynamc forward-lookng model and he mul-secor dynamc forward-lookng model. For each ype of model, he mplc assumpons of rsk are oulned The Whn Perod CGE Model (Sac / Inra-emporal Model) The sac (whn perod) CGE model follows he neracons and relaonshps of a marke economy and solves for a se of prces ncludng producon prces, facor prces and exchange rae and levels of producon ha clear all markes. The sac model recreaes an Arrow-Debreu (1954) general economc equlbrum model. The model conans a represenave consumer. Each consumer has an nal endowmen of he N commodes and a se of preferences resulng n demand funcons for each commody. Marke demands are he sum of all consumers' demands. Commody marke demands depend on all prces and sasfy Walras' law. The oal value of consumer expendures equals consumer ncomes, a any se of prces. Technology s descrbed by consan reurns o scale producon funcons. Producers maxmze profs. The zero homogeney of demand funcons and he lnear homogeney of profs n prces (.e. doublng all prces double money profs) mply ha only relave prces are of any sgnfcance n such a model. The absolue prce level has no mpac on he equlbrum oucome (Ruherford and Palsev, 1999). Equlbrum n hs model s characerzed by a se of prces and levels of producon n each ndusry such ha he marke demand equals supply for all commodes. Snce producers are assumed o maxmze profs, and producon exhbs consan reurns o scale, hs mples ha no acvy (or cos-mnmzng echnque for producon funcons) does any beer han break even a he equlbrum prces. Mahesen (1985) has shown ha an Arrow-Debreu model can be formulaed and solved as a complmenary problem. Demand

3 for and supply of goods and servces readjus unl all excess demands and excess supples are elmnaed hrough changes n prces. The producon funcon s specfed no erms of labour and capal and he amoun of each ype of hese npus employed by a producer n a parcular secor s based on he secor specfc producon echnology and npu prces. The mos common funconal form ends o be a Consan Elascy of Subsuon (CES) funcon o express hs relaonshp. Perfecly compeve markes operae o deermne hese equlbrum prces. Addonally, n equlbrum, no secor earns above-normal profs, markes clear for all facors and producs, and, n an open economy, he value of mpors for nermedae use and fnal demand equals he value of expor earnngs. In erms of mplc rsk, he reurn on capal capures all he nheren rsk assocaed wh he nvesmen and owners of capal are pad an approprae reurn, gven he level of rsk. Elasces capure he rade-off beween he choce of varous producs and of he nheren rsk assocaed wh he curvaure of he uly funcons. In such models, here s no obvous place for rsk or uncerany o be facored no he model Dynamc Recursve Model A dynamc recursve model nvolves solvng a model for perod (he nra-emporal model) and hen solves he model for +1 and so on. Wha lnks he models ogeher s he nvesmen rule. However, he nvesmen rule hen deermnes o a large exen, he resuls of he model. The ner-emporal (beween-perod) model lnks he sac (nra-emporal or whn-perod) models by updang he varables ha are exogenous n he base year from one perod o he nex. Ths process can be seen n he fgure below. Fgure 1 Dynamc Recursve Model Invesmen Rule Invesmen Rule Perod =1 =2 =3.. Exogenous Varables updaed The problem wh dynamc recursve models s ha expecaons are no aken no accoun. For he mos par, economc agens behave as n a one-sho example many mes. In recursve models, agens lack a coheren represenaon of expecaons. In realy, alhough economc agens do no have full nformaon regardng he fuure, hey behave as

4 f he fuure maers. Recursve models end o be characersed by "overshoo and collapse" behavour. Gven he fac ha hese models are essenally one-sho model advanced one me perod and updaed wh he values of he exogenous varables for he nex me perod, he assumpons regardng rsk are smlar o he nra-emporal model, ha s, he neres rae (reurn o capal) reflecs all he nheren rsk assocaed wh savng and nvesng Sngle-Secor Dynamc Forward-Lookng Model The dynamc forward-lookng compuable general equlbrum model depcs he crcular flow of oupu, ncome and expendure n he goods and facor markes accounng for prce-based backward and forward lnkages across varous producon secors for he economy over he enre model horzon. In each me perod, a represenave household, who s endowed wh labour and capal, supply hese facors of producon o frms. In urn, he frms use hese npus o produce goods and servces. As owners of he facors of producon, households are compensaed accordng o her margnal conrbuon for each facor hey provde o producon. Income earned from work and/or from supplyng capal s hen eher spen on curren consumpon of domesc or foregn producs, or saved for fuure consumpon. Frms, hen, use hose savngs o purchase a compose nvesmen good, whch replace depleed capal and add o her capal sock. Ex pos oal nvesmen equals he ex-ane amoun of savngs n he economy. Economc agens follow maxmsng behavour, ha s, households and frms make opmal choces gven her ner-emporal budge consrans. Households maxmses he presen value of her lfeme uly, frms maxmse he value of her profs. In every perod, prces adjus o guaranee equlbrum n he model so ha demand equals supply. Labour s perfecly moble across secors n he model. Labour wll flow o an ndusry wh a hgher margnal revenue produc from one wh a lower margnal revenue produc unl he demand and supply adjus from prce changes, n hs case, he prce beng he prevalng wage rae. Demand for and supply of goods and servces re-adjus unl all excess demands and excess supples are elmnaed hrough changes n prces. Perfecly compeve markes operae o deermne hese equlbrum prces. Addonally, n equlbrum, no secor earns abovenormal profs, markes clear for all facors and producs, and he value of mpors for nermedae use and fnal demand equals he value of expor earnngs. Includng a publc

5 secor, any changes n axes wll change economc behavour and evenually marke prces va he model s equlbrum condons. Fgure 2 Dynamc Forward Lookng Model: Infne Horzon Capal Accumulaon Capal Accumulaon Perod =1 =2 =3.. Perfec Foresgh Expecaons In conras o he dynamc recursve model, he dynamc forward lookng model does no jus have a rule ha lnks one me perod o he nex bu capal s accumulaed n each fuure me perod. Furher, frms maxmse he ne presen value of her profs and consumers maxmse her ne presen value of her uly. They have raonal expecaons abou fuure me perods and can see he fuure. Decsons made n perod =1 (and subsequen me perods) ake no consderaon evens ha occur n fuure me perods. Economc agens can adjus o shocks before hey occur. A represenave consumer maxmses he presen value of her lfeme uly: 1 max U( c ) = 1 1+ρ Where denoes he me perod ρ denoes he dscoun facor or ndvdual me-preference parameer (1) U denoes he uly funcon c denoes consumpon n perod Termnal Condon The mplemenaon of a dynamc general equlbrum nvolves hree seps. These seps are calbraon of model parameers, replcaon of he benchmark economy, and compuaon of ransonal dynamcs n relaon o exernal shocks or polcy changes n he model economy (he couner-facual). Numercal models can only be solved for a fne number of perods. An adjusmen needs o be made o produce a model whch approxmaes over he

6 nfne horzon. If here was no adjusmen process hen all capal would be consumed n he las perod and nohng would be nvesed. Lau e al. (2002) propose a mehod for approxmang he nfne horzon equlbra wh endogenous capal accumulaon. Ths mehod has become he sandard way of approxmang nfne horzon general equlbrum models. Ther cenral dea s o spl he neremporal uly funcon no wo pars. Ths can be wren as: U = T β U( C ) + β ( C ) (2) = 1 = T+ 1 The second erm n hs uly funcon collapses no a consan erm. The dynamc forward lookng model can be seen graphcally n he followng way: Fgure 3 Dynamc Forward Lookng Model: Fne Horzon Capal Accumulaon Capal Accumulaon Termnal Condon Perod =1 =2 =3.. =n Perfec Foresgh Expecaons Dfferng from he dynamc forward lookng nfne horzon model, he fne horzon model s solved for a ceran number of me perods afer whch he ermnal condon hen s operaonalsed. Ths mehod gves he model racably. In a sngle secor model, he wo sub-problems are lnked hrough he capal sock n perod T+1. Havng decomposed he model, a good ermnal approxmaon occurs where he capal sock n perod T+1 s close o he opmal value n he nfne-horzon par of he model. If he rue value of he capal sock n he pos-ermnal perod s known hen he rue consumpon and savng pahs can be calculaed n he nerm ransonal perod. Afer an exernal shock, however, he rue value of he capal sock n he pos-ermnal perod may no be known. In hs case, wha Lau e al. (2002) recommend s, raher han o mpose he long-run seady-sae value of capal sock (where he model horzon may be exraordnarly long for o converge o he seady sae), he sae varable, K T+ 1, can be deermned as par of he equlbrum calculaon by argeng he assocaed conrol

7 varable, I T. Ths s done by mposng a consran ha defnes how ermnal nvesmen grows. Gross nvesmen n he ermnal perod s herefore deermned by he sze of he capal sock n he ermnal perod, he seady sae growh rae, and he rae of capal deprecaon. Ruherford (2004) has suggesed he followng: Termnal nvesmen growh se equal o he long-run seady-sae growh rae: I I T T 1 = 1+γ (3) Termnal nvesmen growh rae se equal o he growh rae of consumpon: I I T C = C T T 1 T 1 (4) Termnal nvesmen growh rae se equal o he growh rae of aggregae oupu: I I T Y = Y T T 1 T 1 (5) Lau e al. (2002) argue ha hs sae-varable argeng mehod s a superor mehod o he one oulned by Barr and Manne (1967). The laer mehod nvolves an ncreased wegh on uly of consumpon n he ermnal perod, and a consran on nvesmen n he ermnal perod. However he sae varable argeng mehod has wo advanages over he echnques based on opmsaon mehods. Sae varable argeng provdes a more precse approxmaon of he nfne horzon equlbra. In oher words, he model s more effcen (akes fewer perods o approxmae he nfne horzon saddle pah) when he sae varable argeng mehod s used) Furher, sae varable argeng does no requre ex ane specfcaon of he growh rae of he ermnal perod or mpose a specfc capal sock n he pos-ermnal perod. Ths wll be he mehod hs research follows because s herefore suable for models wh endogenous growh where he ermnal sock s no deermned ex ane. For boh he sngle-secor dynamc forward-lookng model wh he nfne and fne horzon versons, rsk s mplc as s wh he sac and dynamc recursve models Mul-Secor Equvalen of Dynamc Iner-emporal Model Ths secon explans he dfference beween he sngle-secor and mulple-secor dynamc neremporal model. The change from homogenous capal o heerogeneous capal has

8 more mplcaons han jus addng a secor-specfc subscrp. In hs model, heerogeneous capal can nroduce exogenous rsk prema. Frms nves, usng savngs. The marke renal rae of capal s deermned by marke forces, he supply of and demand for capal. Toal nvesmen demand equals he use of nvesmen goods from domesc and mpored sources. Economy-wde, a compose nvesmen good s derved from he fnal nvesmen demand column from he IO able. The compose nvesmen good s allocaed o secor-specfc nvesmen so ha he margnal producvy of capal s equal across secors. Invesmen opporunes are arbraged when he ne rae of reurn from each secorally dfferenaed nvesmen does no exceed he rae of neres. When nvesmen s underaken n ha secor he ne rae of reurn n ha secor wll equal he rae of neres. These relaonshps can be expressed n he followng equaons: R δ r, (6) I, 0 (7) I R δ r ) 0 (8), (, = Where R, = Gross of deprecaon rae of reurn n secor a me δ = Secor-specfc deprecaon rae r = rae of neres a me Ths arbragng condon means ha secors wh hgh gross reurn and lower deprecaon rae generae more gross nvesmen demand. In he seady-sae, nvesmen wll grow a he same rae n all secors, and he reurn o capal wll be equalsed across all secors. However, durng he ransonal phase, s possble for he ne reurn n one secor o fall below ha of anoher. As a resul, nvesmen can be shu down n he low reurn secor. Smlar o he sngle-secor model, asses deprecae. Gross secoral nvesmen ncreases he capal sock as well as replaced deprecaed capal. K, 1 = K, ( 1 ) + I, + δ (9) Dfferen form he oher models; he mul-secor dynamc forward lookng model can nroduce a rsk prema beween he dfferen raes of reurn on secor-specfc nvesmen. Examples of appled models where hs has been nroduced can bee found n Secon 3 of hs paper. Neverheless, hs neoclasscal rsk s exogenous. Hence CGE models are deermnsc n naure. Appled general equlbrum models gnore he sochasc elemens

9 ha may affec boh he producon and consumpon sdes of he economy. Wh forwardlookng models, perfec foresgh negaes he possbly of endogenous rsk as producers and consumers would foresee he rsk n he fuure and change her behavour accordngly. Based on hese lmaons mgh be argued ha here s no room for rsk o be ncorporaed no a CGE model. The CGE mehodology does no allow. The nex secon revews a selecon of nsances where elemens of rsk have been ncorporaed no appled models. 3. Addressng he Issue of Rsk and Compuable General Equlbrum Models: A Revew of he Leraure 3.1 Modeller Uncerany There have been several ways o he ssue of rsk and uncerany have been addressed concernng CGE modellng. Uncerany can orgnae whn he economc sysem or uncerany can occur as par of he modellng process, ha s, modeller uncerany. Modeller uncerany relaes o he rsk of reporng ncorrec resuls or he uncerany relang o exogenous parameers whch have been npued no he model. The modellng process may nvolve uncerany regardng he rue value of he model parameers, such as elasces. Elasces are somemes appled o CGE models from one regon or pons n me ha have been esmaed economercally from daases from dfferen regons or dfferen me perods, ha s, here s a msmach beween he daa sample and he source of varaon n he economercs and he polcy expermen explored n he CGE model (Herel e al., 2004). Sysemac sensvy analyss, va Mone Carlo analyss or Guassan Quadraure procedure (DeVuys and Preckel, 1997) s a way o accoun for hs ype of uncerany n CGE models. Ths echnque has been used by Blake (2005), for example. Modellers may also been unceran abou he resuls obaned n her CGE model smulaons. There s a rsk of reporng naccurae resuls. Because he source daa for CGE models are usually Inpu-Oupu ables and Socal Accounng Marces (SAMs) from a parcular benchmark year, he assumed producon funcons and consumer preference funcons are calculaed deermnscally by a process of calbraon raher han beng esmaed economercally. As such, -raos and confdence nervals ha can be used for sascal esng do no exs hence here s uncerany over he accuracy of resuls. Research exss ha aemps o valdae resuls usually hrough economerc echnques. Valenzuela e al. (2005), n an aemp o valdae resuls from he global CGE model,

10 GTAP, employ he mehod of sochasc smulaon o reflec random producon varably for he commody, whea. They model uncerany n whea oupu usng shocks derved from a mes-seres (ARMA) model of whea producon o measure he randomness nheren n annual oupu. Repeaedly solvng a CGE model, whle samplng from he resduals of he mes-seres model, creaes a dsrbuon ha maes he correspondng marke prce changes for whea, by regon. The sandard devaons based on hese model resuls are compared o he observed oucomes for annual whea prce changes n order o valdae he model. They fnd ha he smulaed oucomes for some regons are remarkably close o he observed oucomes bu for oher regons he model does no perform as well. Two oher peces of research o employ smlar mehods: Gehlhar (1997) and Lu e al. (2004), where he former uses a backcasng smulaon o evaluae he valdy of GTAP model resuls versus observed oucomes concernng Eas Asan economc growh n he 1980s. Gehlhar fnds ha he CGE model performs adequaely a he qualave level (drecon of change n rade share), bu s weak n s predcve power. Lu e al. buld on Gehlar s approach and develop and approxmae lkelhood funcon o assess he qualy of model performance over a 6 year perod, 1986 o Economc Rsk In economc heory, here has been an aemp o ncorporae rsk no he economc sysem n a CGE model n wo ways: hrough he nroducon of rsk prema and hrough assumpon regardng economc agens expecaons Rsk Prema McKbbn and Wlcoxon (1999), n her G-Cubed model, ncorporae exogenous rsk premums n her ner-emporal dynamc mul-secor mul-regon CGE model. They do hs hrough he full negraon of real and fnancal markes. Wh he assumpon of perfecly negraed asse markes across regons, he expeced reurns on loans (neres raes plus rsk premums) denomnaed n one regon (currency) s equal o he expeced reurns n anoher regon (currency) adjused by he exchange rae so here s no arbrage. Whn each economy, he expeced reurns o each ype of asse are equalzed by arbrage, adjusng for adjusmen coss of physcal capal sock and exogenous rsk premums. In long run equlbrum he reurn of capal across secors s he same, ye n he shor run, smulaons can allow for arbrage and hence rsk premums across dfferen capal asses. Counry rsk has also been modelled hrough exogenous rsk premums usng a dynamc CGE model (Malcolm, 1998). The sandard GTAP model assumes ha he global bank

11 equalzes rsk-adjused raes of reurn so ha he rsk-adjused raes for all regons are equal o a weghed average of reurns around he world. Malcolm (1998) explcly defnes hs rsk prema and hence examnes he effecs of changes n hese rsk prema. In hese mulsecor models, he rsk premums are exogenous. One mehod of endogensng rsk no a CGE model has been developed by Arnd and Tarp (2000). They employ a CGE model o analyse he neracons beween agrculural echnology mprovemen, rsk, and gender roles n agrculural producon n Mozambque. They nroduce a parcular ype of echnology rsk no he model, assumng ha a safey frs sraegy s pursued, ha s, hey assume ha households am o produce a ceran exogenous amoun of cassava (he crop of neres n he sudy) for rsk reducon purposes only. Once resources have been allocaed o produce a mnmum amoun of cassava, resources are hen dsrbued o oher agrculural and non-agrculural acves accordng o he marke. The safey frs rsk-averson sraegy s appled by addng an endogenous varable whch serves as a rsk premum. Ths rsk varable eners n wo funcons n he model: he facor demand funcon and he facor ncome equaon wh he rsk premum n he numeraor of he facor demand funcon and he rsk premum s n he denomnaor of he facor ncome equaon. Ths means ha f a rsk premum exss (>1), facor demand for he commody wll be hgher han n he rsk-less pure prof maxmzng poson and facor ncome wll be lower han n he rsk-less scenaro. Arnd and Tarp conclude here are consderable dfferences n producon and prce movemens for cassava beween he rsk and no rsk scenaros Incomplee Informaon Uncerany can be vewed n he conex of ncomplee nformaon a marke neffcency. The lack of nformaon regardng he fuure may gve producers wll an ncenve o supply oo much of some producs and oo lle of ohers. Alernavely, consumers may no purchase a produc even hough hey would benef. One mehod of smulang ncomplee nformaon has been o conras sac expecaon (ncomplee nformaon), where consumer and producers have full nformaon n each curren me perod bu know nohng of he fuure, o raonal expecaons (perfec nformaon) where consumers and producers have perfec knowledge of boh curren and fuure marke condons. The ssue of uncerany wh regards o fuure nformaon s explored by Arnd and Bacou (2000). Takng a relavely sandard CGE model of Mozambque, hese auhors explore he value of clmae forecas nformaon ha operaes and neracs a a farm level, a a

12 markeng sysem level and a a full economy level. Under he premse ha predcable droughs are less damagng han randomly occurrng droughs, hree smulaons are modelled. Ths frs smulaon nvolves an unancpaed drough (droughs beng smulaed by a Hcks-neural echnology declne), where agrculural acvy s fxed and can no be adjused based on realzed clmae oucomes. The second and hrd smulaon allow farmers and boh farmers and markeng agens, respecvely, o reac o a receved perfec clmae forecas. Resuls show ha reducon n rsk assocaed wh a perfec forecas resuls n lower decreases n real GDP and welfare as resources are reallocaed from drough noleran o more drough oleran acves. Followng along he same lnes bu n a dfferen conex Adams e al. (2001) explore he ssue of mng and announcemen of polcy changes whn a dynamc CGE model. The wo scenaros modelled nvolve, frs, he nroducon of a once-off quoa whou any prevous announcemen on he producon of pgs n he Dansh economy and he second scenaro nvolves an announced gradual phased n producon quoa. No surprsngly, he adjusmen pah of he economy s smooher when he polcy s announced compared o he surprse polcy mplemenaon. The key facor s he relaonshp beween nvesmen and expecaons. In he announced scenaro nvesors correcly ancpae fuure adjusmens n prces and renal raes on capal when makng nvesmen decsons and he capal sock sars o adjus from he sar of he smulaon. In he surprse scenaro, expecaons are sac and nvesors adjus fully only when he quoa s mplemened (raher han announced). Wh he announced polcy, scenaros ha conrac he economy occur earler and more gradually so wheher he announcemen or surprse mplemenaon s preferable depends on agen s aude o rsk and her mplc dscoun rae. The more rsk averse hey are or he lower hey dscoun fuure consumpon, he more lkely s hey prefer an announced polcy. Focusng on nernaonal capal mobly, Ianchovchna e al. (1999) develop a dsequlbrum approach for a dynamc mul-secor mul-regon general equlbrum model. As well as modellng exogenous rsk prema as McKbbn and Wlcoxon do, he key feaure of hs model s ha here are errors n nvesors assessmen of poenal reurns o capal. They argue ha nvesors expecaons are scky and ha when he observed raes of reurn change, nvesors are unceran wheher hs change s emporary or permanen. I s only wh a lag ha hey adjus her expecaons. Inally, nvesors make small adjusmen, and f he change n he rae of reurn persss, hey make furher changes

13 n her expecaons unl he expeced rae equals he observed rae of reurn. Ths was he explanaon for he Asan fnancal crss of The auhors argue ha he developmens n Eas Asa reflec he fac ha nvesors have no foreseen correcly reurns o capal. They argue ha hs can be represened hrough a smple recursve soluon procedure o mmc nvesmen heory of adapve expecaons. They argue ha he lmaon of forward-lookng ner-emporal models s he assumpon of perfec foresgh of reurns o capal. In hs case, he fnancal crss would mply nvesors dd no have perfec foresgh. Boussard e al. (2002), as well as allowng for mperfec expecaons, examne he ssue of agrculural rade lberalzaon, addng nsably n he model by endogensng rsk hrough lags n delvery, and rsk averson. Uncerany s nroduced no he model hrough a producon lag n he agrculural secor. Pckng up he work done by Ezekel (1938), who developed cobweb heorem, he researchers specfy a lag beween he producon and consumpon decson for he agrculural secor. The marke equlbrum s beween he prevous year s producon and curren consumpon. Producon decsons are aken on he bass of expeced prces, raher han equlbrum prces. Equlbrum prces are used as npus, expecaons are mporan, n hs model, only for nex year s producon. In urn, ncome n he curren year depends heavly on expecaons for he fuure year; mplyng frms can gan or lose. As such, frms bear rsks. In sum, Boussard e al. nroduce rsk, mperfec nformaon and producon lag n he agrculural secor n a sandard CGE model o model uncerany. They fnd, n conras o he classcal perfec foresgh model where global gans are assocaed wh rade lberalzaon, he model wh rsk averson, mperfec nformaon and a producon lag n he agrculural secor shows negave changes n real ncome. Imperfec nformaon consrans he economy from reachng s opmal. In a laer pece of work, Boussard e al. (2004) explan how he endogenous rsk dffers from exogenous rsk. Neoclasscal rsk s exogenous, s delvered form above, ousde he model. The behavour of agens has no relaonshp o he level of rsk nvolved and canno nfluence he degree of rsk. Endogenous rsk s a consequence of expecaon errors. These errors are nconssen wh he raonal expecaons hypohess. In he CGE rsk leraure a dynamc recursve model s mplemened o replcae ncomplee nformaon abou he fuure. However, one weakness wh hs research s ha he dynamc recursve model has some nheren problems. Whle he researchers ced above argue ha a dynamc forward-lookng model does no allow he exsence of

14 mperfec nformaon or errors n expecaons, o argue economc agens do no make any decsons based on wha hey know abou he fuure and / or aemp o, say smooh consumpon or producon s o err n he oher drecon. In he nex secon, an alernave mehod s oulned ha reas rsk explcly hrough creang mulple fuure pahs for he model, wh agens able o predc each pah and make decsons n he presence of hs uncerany. As CGE models have become more sophscaed, here have been effors o model elemens of rsk and uncerany. There have been wo man caegores: modeller uncerany, where here s uncerany regardng exogenous parameers and resuls n general and aemps o allow for uncerany n hese cases; and uncerany regardng fuure expecaons where agens behavour under sac expecaons s conrased wh expecaon regardng perfec foresgh expecaons. The nex secon descrbes a dfferen way o ncorporae uncerany no a CGE model. 4. Uncerany Regardng Fuure Pahs 4.1 Concepualsaon In hs dynamc forward-lookng model, uncerany wll ener he model hrough he uncerany n an exogenous shock. An exogenous varable, for example an expors, can be smulaed o vary sochascally (wh a desgnaed probably) abou he base year mean. The sandard assumpon of raonal expecaons wll hold so ha he represenave consumer, frms, and governmen are endowed wh perfec foresgh, and so ancpae any exogenous shocks and adjus her maxmsng behavour from he second me perod (he perod n whch he uncerany, bu no necessarly he shock wll occur). The uncerany n he model nvolves he choce of fuure pahs by consumers and frms. Gven hs uncerany, how do consumers and frms change her behavour? For example, n he paral equlbrum leraure, frms wll under-nves n secors where here s uncerany. Do hese resuls hold n he general equlbrum conex? I may even be ha he uncerany may be so hgh as o sfle nvesmen such ha, may be unprofable o nves n he curren me perod. The economc mpacs of he mng (when he shock occurs and how long he uncerany remans) as well as he magnude of he exogenous shocks wll also be analysed.

15 A graphcally represenaon of he concepual model s shown n Fgure 4. Several dynamc forward lookng models are possble. One scenaro may be ha baselne growh pah s conrased wh a posve shock and a negave slump. These mpacs are hen followed hrough o he nh me perod. Expecaons are conssen hroughou so ha n economc agens have hgher expecaons under he posve shock scenaro from he frs me perod, even hough he posve shock does no ake place unl perod =2. Uncerany s represened by he lkelhood of a posve shock or a negave shock o occur a any gven me perod. Fgure 4 Dynamc Forward Lookng Model: Wh Uncerany Posve Shock Uncerany Probably of oucomes =2 =3.. =n Termnal Condon Hgher Expecaons Perod =1 =2 =3.. =n Cenral Case =2 =3.. =n Negave Shock Lower Expecaons 4.2 Applcaon Ths secon of he paper oulnes how he dynamc forward-lookng model wh uncerany s operaonalsed. A smple model feaurng hree secors (Agrculure, Manufacurng and Servce), 2 facors of producon (Capal and Labour) n an open economy s used o llusrae how uncerany wll work. The SAM of hs fcous economy s dsplayed n Table 1 below.

16 Table 1. SAM INTERMEDIATE DEMAND FINAL DEMAND Agrculure Manfacurng Servce Consumpon Invesmen Expors Agrculure Manfacurng Servce Impors Capal Labour The model can be hough of as conssng of wo pars. The frs me perod can be vewed akn o a sac model where here s only one possble sae of he economy. The second par of he model s he dynamc par where here are dfferen pahs ha he economy can ake. The model s nended o be as sandard as possble so as o hghlgh he uncerany feaure. The represenave consumer s endowed wh prmary facors (labour, capal). The value of expors equal he value of mpors so here s a rade balance. Afer he represenave consumer sells prmary facors, and purchases back capal n he ermnal perod, he remanng ncome s devoed o prvae consumpon Producon Frms are assumed o be prce akers who choose varable npus and s level of nvesmen n order o maxmse profs. Each ndusry s modelled usng he consan elascy of subsuon (CES) famly of funcons, whch ncludes Leonef, Cobb-Douglas and Consan Elascy of Transformaon (CET) funcons. Each producon secor produces wo ypes of commodes: domesc goods D, and goods for expor Q, X,. These goods are assumed o be mperfec subsues, and hey have a consan elascy of ransformaon. For producon, each secor uses capal, labour, and nermedae goods. As such, he secor's producon funcon s Q = g(d,x,,, = ) f(k,, L,,A, j, ) where g s oupu ransformaon funcon, and f s npu ransformaon funcon. Oupu ransformaon s assumed o be he consan elascy of ransformaon (CET): Q η 1 η 1 e η e η, = Θ δ D, + (1 δ ) X, where Q, = oupu η 1 η

17 X, = Expors D, = Domesc producon η = he elascy of ransformaon n oal supply e δ = he calbraed share of expors Θ = he calbraed shf parameer n he ransformaon funcon Overall marke clearng n he produc marke means he Armngon aggregae s used for prvae consumpon, nvesmen, and as an nermedae npu for producon. A, j, n + = CC, + I, IO, j, j where: A,j, = he Armngon CES aggregae CES aggregae of domesc producon and mpors CC, = compose consumpon I, = compose nvesmen n j IO,, = Demand for nermedae npus j For npu combnaon, we have a Leonef aggregaon of facors of producon. Capal and labour eners as a Cobb-Douglas value-added aggregae. Inermedae npus from dfferen secors ener as a Leonef aggregae no a secor's producon funcon: f(k,, L,, A, j, ) = α mn B L K (1 α ) A,mn a,1,1 A, a,2,2 A,..., a, j, j An nermedae npu o a secor from a secor j s an Armngon aggregae of domesc oupu and mpor. Users regard hese goods as mperfec subsues, and hese goods are assumed o have a consan elascy of subsuon (CES). Q γ 1 γ 1 m γ m γ, = Ω δ D, + (1 δ ) M, where γ 1 γ Q, = oupu M, = Impors D, = Domesc producon

18 γ = he elascy of subsuon beween domesc goods and servces and mpored goods and servces m δ = he calbraed share of mpors Ω = he calbraed shf parameer n he subsuon funcon Each frm s producon echnology s represened by a consan elascy of subsuon (CES) funcon. VA, = A o 1 o o σ o ( 1 δ )( K, ) + δ ( L, ) σ o 1 σ o σ o σ o 1 Where VA, s he gross value added of secor, o A reflecs he level of echnology n he producon funcon, K, and L, are he amouns of capal and labour used n secor, o o δ s he share parameer of labour n producon, and σ s he elascy of subsuon and he subscrp ndcaes ha he parameers apply o oupu. Ths s a consan reurns o scale producon funcon. The nra-emporal (sac) producon can be depced as n Fgure 5. Fgure 5. Producon Srucure Inermedae Inpus CES Impors Domesc Expors CD Oupu GL Inermedae Inpus Value Added A 1 GL.. A j Capal CD Labour CES = Consan Elascy of Subsuon CD = Cobb-Douglas GF = Generalsed Leonef

19 4.2.2 Consumpon The represenave consumer has an endowmen of prmary facors of producon: capal and labour. They demands nvesmen and consumpon goods. The nvesmen secor oupu are exogenous whle consumpon demand s deermned by uly maxmzng behavour. Consumer uly consss of a Cobb-Douglas uly ndex defned over Armngon aggregaon of domesc and mpored commodes n C= Π c =1 α where c γ 1 γ 1 m γ m γ, = Χ δ CD, + (1 δ ) CM, where γ 1 γ c, = oupu CM, = Impored producon of consumpon good CD, = Domesc producon of consumpon good γ = he elascy of subsuon beween domesc goods and servces and mpored goods and servces m δ = he calbraed share of mpors Χ = he calbraed shf parameer n he subsuon funcon Inal Values for Capal and Invesmen To calbrae he model he benchmark level of capal ( K 0 ) and he real rae of reurn ( R 0 ) need o be calculaed, gven he benchmark capal sock ( KS 0 ) and benchmark level of nvesmen ( I 0 ) such ha: K 0 (1 SAME * DEP) = I 0 * where DEP s he deprecaon rae, GROW s he consan ( GROW + DEP) growh rae of he economy n he seady sae and SAME he same-perod proporon of new capal ha comes on lne n he same perod. R 0 KS0 = ( K 0 + SAME * I 0 ) Growh Raes for Quanes and Prces

20 Assumng he base perod of an economy s on a seady-sae growh pah, hen all quanes ncludng capal, labour, consumpon and oupu grow a he same consan rae, GROW. For example, for labour: L = L (1 GROW ) 0 + Lkewse, gven a consan neres rae and consan deprecaon rae, all fuure prces are desgned n erms of presen value: The presen value of he wage W s: W ( 1 R0 ) = where DEP s he deprecaon rae and (1 DEP) Ro s he real rae of reurn Capal and Invesmen The consumer faces wo consrans. The frs consran s ha oal oupu produced n he economy can eher be consumed or nvesed, I (saved). The second consran s ha capal deprecaes a he rae,δ. Hence, c and = F L, K ) I (, (10) K + + 1= K (1 δ ) I (11) where K denoes capal and F( ) s he producon funcon. Capal can eher be bough or rened. Ths s refleced n he producon of capal so he curren perod capal earns he reurn R and nex perods capal earns PK + 1. Lkewse for nvesmen, curren perod nvesmen earns PK and nvesmen n he nex me perod earns PK + 1 whch hen dscouned and deprecaes Addng Uncerany Takng hs dynamc forward-lookng CGE model, n perod = 2, here s uncerany on whch pah he economy mgh follow, gven an exogenous shock. The nex sep s o nroduce a number of dfferen pahs ha he economy mgh ake. Clearly, s no possble o model all possble fuure me pahs. The number of possble pahs chosen mgh ha span a feasble soluon. Le p be he number of possble pah, hen Φ( p) s he probably ha a specfc pah s aken. I s a necessary condon ha Φ ( p), ha s, ha he probably of he sum of he pahs sum o 1. Ths occurs afer perod = 1, ha s, when he exogenous shock occurs. From he second me perod onwards, all he secors P 1

21 and quanes need o be mulpled by he probably ha hs parcular pah s chosen. The relaonshps descrbed above (Secon 4.1) exs for he frs me perod and n each pah Φ ( p), where * K ndcaes he frs me perod and for pah Φ. For example he capal accumulaon equaon now becomes: K ( + I *, = 2 = K 1 ) * K Φ, ndcaes capal n perod Φ δ (12) K Φ, + 1 KΦ, ( 1 ) + IΦ, = δ (13) where * n Equaon 12 denoes frs-year values and Equaon 13 represens he capal accumulaon equaon from he second perod onwards. On he consumer sde, uly gven by Equaon 1 s now dependen on he me pah chosen s represened by: 1 U Φ = CΦ, = 1 1+ ρ * U = C + U * Φ (15) σ 1 P σ σ 1 * σ U = Φ( p) UΦ (16) Φ * Where U s frs perod uly * C s frs perod consumpon ρ s he dscoun facor or ndvdual me-preference parameer UΦ s dscouned uly, and σ s he rsk averson parameer 5. Concluson CGE models have radonally ended o be deermnsc n naure. Recenly, here have been aemps o add sochascy no he model n varous ways. Undersandably, here are lmaons o modellng uncerany n hs way. I s no possble o model all fuure possble fuure me pahs. One area o develop he research furher s o deermne whch fuure me pahs are more mporan o model; neverheless, he model explcly ncludes rsk and agens reacon o rsk. Ths paper oulnes one way o add uncerany no a dynamc forward-lookng model. More forward, he model needs furher esng on he daase of a real economy. (14)

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