Monetary and Fiscal Policy for Oil Exporting Economies: A DSGE Model Approach

Transcription

1 Moneary and Fiscal Policy for Oil Exporing Economies: A DSGE Model Approach Jean-Pierre Allegre, Mohamed Tahar Benkhodja and Tovonony Raza ndrabe EconomiX Absrac This paper proposes a DSGE framework inegraing moneary and scal policy in order o analyze he shor-run and long-run e ecs of oil price shocks on oil exporing economies. We focus our analysis on resource-rich developing counries. Indeed, our aim purpose is o see o wha exen such counries can face changes in oil prices -he sabilizaion arge- and, a he same ime, o promoe policies favoring economic diversi caion -he inergeneraional arge. EconomiX, Universié Paris Oues Nanerre la Défense.

2 In line wih he balanced-growh pah heory, we assume ha real variables will share he same evoluion as he labor-augmening echnology process. Therefore, o render he model saionary, we scale real variables by and nominal variables by he consumer price level P. Tranformed variables are represened by lower-case leers which in he lieraure is known as "inensive form" represenaion. For insance, c = C = and p ; = P ; =P represen respecively he saionary level of consumpion and relaive price of domesic goods. owever, here are some noeworhy excepion when scaling he level of capial and wage. Given he predeermined naure of he capial sock and he convenion ha K represens he sock of capial in he beginning of period, he saionary level of capial sock is de ned as k + = K + =. Moreover, given he assumpion ha nominal wage evolves in line wih labor-augmening produciviy growh, i is necessary o scale i boh wih and P. Tha is, saionary level of wage is de ned as w = W = P. I is imporan o noe ha he level of hours worked is already saionary and no furher ransformaion is needed. Finally, he growh rae of he labor-augmening echnology process g ; = = is assumed o evolve according o: ln(g ; ) = ( g ) ln(g ) + g ln(g ; ) + g where g iid N ; 2 g and g is he seady-sae value of g ;. ouseholds The populaion size in he oil-exporing counry is normalized o uniy, h = [; ]. Represenaive household wihin he oil-exporing (ome) counry maximizes a sring of discouned fuure value of uiliies given by: X E k U +k (c +k (h); ha +k (h); +k (h)) () k= where he period uiliy funcion of he household is de ned as: U ( ) = B ln c hg ; c ( ) + + We assume perfec insurance markes wihin home counry and ha households share he same preference echnology. Thus, c represens he represenaive household s composie consumpion index, B and are respecively he preference and he labor supply shocks, ha is an exernal habi ha is de ned as ha = hg ; c, and represens represenaive household s di ereniaed labor supply (number of hours worked). Finally, parameer represens I implies ha household s individual variable X (h) for X = fc; ; K; W; B; B ; DIV g will be equal o he corresponding aggregae variable X. Formally, we allow individual household o receive ne cash in ow from paricipaing in a sae-coningen securiies ha insures idenical wage income and, hence, opimal allocaion in equilibrium across households.! (2) 2

3 he inverse of Frisch elasiciy of labor supply. Moreover, we assume ha he aggregaion of individual labor supply across oil, expor and domesic secors is represened by he following Cobb-Douglas aggregaor: 2 = O O; + X X; + ; where is he housdehold s labor supply elasiciy of subsiuion beween di eren secors of producion. The parameer i, for i = fo; X; g and where P i i =, represens labor supply share of household o secor i. Opimal allocaion of labor supply beween di eren secors is herefore given by: i; = i wi; w The overall wage index evolves according o: (3) for i = fo; X; g (4) w = O w O; + X w X; + w ; (5) Finally, combining he las wo equaions yields he following nominal aggregae labor income: w = w O; O; + w X; X; + w ; ; (6). Consumpion, price and demand We assume ha he consumpion baske of a represenaive household is composed of non-oil goods c NO; and exclusively impored re ned-oil c RO;. Thus, oal consumpion is represened by he following CES funcion: c = ( C ) C (c NO; ) C C + C C C (c RO;) C C C where C is he elasiciy of subsiuion beween oil and non-oil goods, and C represens he share of re ned-oil energy in he represenaive household s consumpion baske. Given his consumpion funcion index, he consumpion-based price index (CPI), which we de ne henceforh he "headline-cpi ", is de ned as: p = 2 For =, he labor supply aggregaor is given by: h i ( C ) (p NO; ) C + C (p RO; ) C C (7) = Y i=;x; (i;) i i i 3

4 where p NO; is he core-consumpion price index, henceforh "core-cpi ", which is de ned in equaion (49). In urn, opimal allocaion of expendiure beween non-oil goods and re ned-oil energy is given by: c NO; = ( C ) (p NO; ) C c (8a) c RO; = C (p RO; ) C c (8b) Therefore, and as is sandard in he lieraure, combining he expression of headline-cpi and ha of opimal allocaion of expendiure yields he following de niion of oal nominal expendiure: c = p NO; c NO; + p RO; c RO; (9).2 ousehold s opimizaion problem ouseholds have access o boh domesic and privae foreign bonds markes which we denoe respecively b d = B d = P and b = B = P. We assume ha inernaional marke is incomplee and domesic households rade only risk-free asses. owever, he nominal ineres rae paid or received by households when selling or buying foreign bonds depends on nancial inermediaion premium hrough a risk-premium charges on op of nominal world ineres rae R. This is done o ensure saionariy of equilibrium following Schmi-Grohé and Uribe (23). Moreover, we assume as in Sähler and Thomas (22) ha he risk premium is an increasing funcion of he home counry (ne) deb posiion. Tha is, { = exp nfa y nfa y wih > and where nfa = NF A = P and y are respecively he period ne foreign asse posiion and gross domesic producs. Moreover, we assume ha home-counry ne foreign asses are composed of privae ne foreign asses b and public ne foreign asses f, he sovereign funds, ha come from oil expors. Moreover, we assume ha foreign asses are denominaed in US Dollar. Tha is, nfa = Z P B P P + F P nfa = z ~ (b + f ) () where z = Z P =P denoes US-bilaeral real exchange rae. Nominal exchange raes is expressed as he home-currency price of one uni of foreign currency. Tha is, a decrease in Z is inerpreed as a home-currency appreciaion. Finally, ~ = = represens produciviy di erenial beween home and foreign counries. Di eren labor-augmening echnology process are assumed o share he same sochasic rend as in Chriso el e al. (28). This assumpion permis o rea produciviy di erenial as saionnary variable. 4

5 Each period, individual represenaive household faces he following budge consrain: ( + c )c + p I; i p b + d + z ~ b = inc + Q R (g ; ) b d () +z ~ g; { R b where Q is he risk premium shock ha arises from he presence of domesic nancial inermediaion. As is argued by Chriso el e al. (28), he use of curren real e ecive exchange rae z seems from he fac ha ne foreign asse posiion is a predeermined variable. ousehold s oal income inc is composed of dividends derived from impor, expor and domesic non-radable inermediae rms (div = P i=m;x; p i; wih p i; = p i; = P ), reurn on e ecive capial sock minus he cos associaed wih variaions in he degree of capial uilisaion u, labor income and lump-sum ax or ransfer = T = P. Tha is, inc = div + r k u (u ) p I; g ; kp + ( w )w where r k = R k =P is he real reurn on e ecive capial and (u ) is he cos associaed in changing he degree of capial uilisaion u wih () =. Moreover, we assume ha households accumulae unis of privae capial used in oil, expor and domesic non-radable rms. We follow Chrisiano e al. (25) and Sähler and Thomas (22), and assume ha privae capial evolves according o he following law of moion: k p + = ( )g; kp + i p I S g ; i p (2) i p where I is de ned o be he (privae) invesmen shock and S(:) = I i 2 g p ; i p g 2 is a posiive cos funcion for changing he level of invesmen which has he following properies: S() =, S () = and S () >. Moreover, and given he main role of resource movemens e ec in our analysis (encore à deailler), we assume as for labor supply ha he aggregaion of individual capial across oil, expor and domesic non-radable secors is represened by he following Cobb-Douglas aggregaor: k p = O kp O; + X kp X; + kp ; where is he elasiciy of subsiuion of capial beween di eren secors of producion and is assumed o be he same as for ha of labor supply (encore à deailler si nécessaire). Parameer i, for i = fo; X; g and where P i i =, represens he share of secor i in he represenaive household s capial baske. I is herefore sraighforward o obain he following overall real reurn on capial sock: r k = O ro; k + X rx; k + r; k 5

6 Opimal allocaion of capial supply beween di eren secors is herefore given by: k p i; = i r k i; r k! k p for i = fo; X; g (3) Therefore, combining he las wo equaions yields he following nominal aggregae capial income and e ecive capial income: r k k p = r k O;k p O; + rk X;k p X; + rk ;k p ; r k ~ k p = r k O; ~ k p O; + rk X; ~ k p X; + rk ; ~ k p ; Finally, households choose c +k ; b d +k ; b +k ; kp +k+ ; ip +k ; u +k o maximize he discouned fuur value of heir uiliies () subjec o heir budge consrains () and he law k= of moion for capial (2). Solving his maximizaion problem yields he following sandard rs order condiions: B c hg ; c = + c (4a) + = E (g ;+ + ) Q R (4b) + = E g ;+ z + + { R (4c) z + h i Q = E g ;+ r k +u + (u + ) p I;+ + ( )Q + (4d) p I; = Q I i p S g ; i p S i p i p g ; i p g ; i p + (4e) E (Q + I + + S i p + i p 2 ) g ;+ i p + g ;+ i p r k =.3 Wage seing (u ) p I; (4f) ouseholds supply monopolisically a disincive variey of labor and se nominal wages in saggered conracs fashion à la Calvo (983). Each period, individual household is allowed o se is nominal wage only afer receiving a random signal wih consan probabiliy ( W ), so ha w (h) = ~w o (h). owever, whenever household is no allowed o adjus is conracs, 6

7 wage is indexed o las period CPI in aion 3 rae according o he following indexaion rule: w (h) = W w (h) (5) If W = here is no indexaion and if W = here is a perfec indexaion of wage o pas in aion. When seing nominal wage, household chooses W (h) o maximize is lifeime uiliy () subjec o budge consrains () and labor demand: + W; w (h) W; (h) = (6) w The rs order condiion o his opimizaion problem yields he following expression of wage seing equaion: where ~w o (h) = f W; f 2 W; fw; = W; B ({ w ) + ( + W; ) + W E f W;+ fw; 2 = W; { w B c hg w ; c W + c + W E fw;+ 2 + and where we have assumed ha here is no wage dispersion across households, ha is R (h)dh = { w = wih { w =. I is worh noing ha when W =, here is a perfec wage exibiliy, ~w o (h) = ~w o = w, and wage seing equaion becomes: ( w ) ( + c ) w = ( + W; ) ( ) c hg ; c (7) Therefore, under perfec wage exibiliy, real wage is a ime varying mark-up over marginal rae of subsiuion beween consumpion and labor. Moreover, given ha all individual households ha adjus in period choose he same wage ~w o and hose which do no simply se wage according o he indexaion rule in (5), one can derive he following expression of wage index: 2! 3 W; W W; w = 4 W w + ( W ) ( ~w o ) W; 5 (8) 3 We follow Erceg e al. (999), Smes and Wouers (23), and Adolfson e al. (27) when aking CPI in aion as wage indexaion o pas in aion. Some open DSGE model such as he SIGMA model by Erceg e al. (26) and ha of Jacquino e al. (26) insead use wage in aion o index wage o pas in aion. 7

8 2 Governmen Each period, he governmen is subjec o he following budge consrain: b d + w w + c c + = (g ; ) Q R b d + p ; g where g is he public purchases or he governmen spending commonly de ned in he lieraure. As in Bouakez e al. (28), re ned-oil consumed domesically is produced abroad wih price assumed o evolve according o: p RO; = ( RO ) g ; p RO; + RO z p O; (9) ou p RO; = g ; p RO RO; z p RO O; where p O; = P O; is he inernaional oil price and RO 2 [; ] indicaes he degree of domesic re ned-oil price subsidy from he governmen. Moreover, we assume ha he governemen uses he surplus of revenues earned from oilexporing secor o sabilize oil-revenues agains inernaional oil-price ucuaions and o suppor lagging secor (expor) and domesic producers, especially if oil-resources is inended o be depleed as we formally invesigae in his paper. Namely, i is done by leing sovereign funds F evolves according o: =P us z ~ f +p I; i g + p O; z p O; o = z ~ g ; { R f + pro; z p O; cro; + O; where i g is he public invesmen which is assumed o be produciviy-enhancing for he lagging secor and domesic producers (à déailler). We assume ha he law of moion for public capial sock evolves according o: k g + = ( )g ; kg + ig (2) We assume ha oil price p O; is se in he inernaional marke and is labelled in US Dollar. Thus, i is assumed o be exogenous for oil rm and evolves according o: ln p O; = p ln (p O O) + p O ln p O; + p O where p O iid N ; 2 p O where p O is he seady sae value of crude-oil price and p O he crude-oil price shock. The governmen subsidizes revenues of oil rms using sovereign funds p O; z p O; o avoid damaging consequences of oil-price swings in he economy. Moreover, we assume ha demand for crude-oil is exogenous and evolves according o: ln o = o;d ln (o) + o;d ln o + o;d where o;d iid N ; 2 o;s ou o = o;d o p o O; ~ y where ln o;d = o;d ln o;d + o;s, o;s iid N ; 2 o;s where o is he seady sae value of crude-oil supply and o;d he oil-demand shock. For insance, a posiive shock may be inerpreed as an exogenous increase in he demand of crude-oil. 8

9 3 Firms 3. inermediae good rms In he domesic marke, here exiss hree ypes of inermediae good rms ha behave as monopolisic suppliers of heir di ereniaed inermediae goods: a coninuum of domesic inermediae good rms h (f) indexed by f 2 [; ] which produce a di ereniaed inermediae goods ha are sold domesically, a coninuum of expor inermediae good rms x (f) which produce a di ereniaed inermediae goods ha are sold exclusively o domesic exporing rms, and a coninuum of impor inermediae good rms m (f) which impor a di ereniaed inermediae goods ha are produced abroad and sell hem wihou any ransformaion o domesic nal-good rms. 3.. Domesic inermediae good rms Domesic inermediae good rms use boh labor ; (f) and e ecive capial sock k ~ p ; (f) = u g ; kp ; (f) o produce oupu according o he following consan reurns o scale echnology: h i h (f) = a (k g ) k ~ p ; (f) [; (f)] (2) where a is he aggregae produciviy shock, k g he public capial sock ha is assumed as in Sähler and Thomas (22) o be produciviy-enhancing, 2 [; ] he parameer ha measures he degree of public invesmen ino privae producion, and a xed cos. Cos minimizaion problem Given he real wage w ; =p ; and he renal cos of capial r; k, domesic inermediae good rm f chooses he opimal level of labor ;(f) and capial ~k p ; (f) ha minimizes is oal inpu cos w ; p ; ; (f) + r k ; ~ k p ; (f) subjec o he producion echnology (2). Firs order condiions of he rm s cos minimizaion problem wih respec o labor and e ecive capial sock are given by: w ; = MC ; (f) ( ) (h (f) + ) p ; ; (f) (22a) r k ; = MC ; (f) (h (f) + ) ~k p ; (f) (22b) where MC ; (f) denoes he agrange muliplier associaed wih he consrain and inerpreed as he marginal cos of producing domesic inermediae oupu. Therefore, capiallabor raio and marginal cos will be idenical across inermediae good producers, and are 9

10 given respecively by: ~k p ; (f) ; (f) = w ; =p ; r k ; = ~k p ; ; (23) MC ; (f) =! w; =p ; r k ; a = MC ; (24) (kg ) Price seing Domesic inermediae good rms behave as a monopolisic supplier of heir goods. They o er heir goods in he quaniy demanded a he curren price p ; (f) which is assumed o be sicky and se in saggered fashion à la Calvo (983). Tha is, a fracion ( ) of randomly seleced rms is able o se new prices ~p o ; (f) each period, whereas a fracion of rms keeps heir prices unchanged and simply follow he following indexaion rule: p ; (f) = ; p ; (f) where he parameer measures he degree of indexaion o pas in aion. Given he expression of real marginal cos, rms ha are allowed o se prices will maximize he discouned sum of heir expeced pro s subjec o demand from nal good rms (24). Solving his maximizaion R problem and assuming here is no price dispersion across home rms, ha is h (f)df = { h = h wih { =, yield he following rs order condiion ha characerizes domesic inermediae good rm s opimal pricing decision: ~p o ; p ; = f ; f 2 ; (25) where f; = { h p ; + p ; MC ; + E f ;+ n o f; 2 = { h p ; + E ; ( ;+ ) f;+ 2 The erm + p ; denoes ime-varying mark-up of prices over marginal coss a domesic inermediae good rms level and is assumed o evolve according o he following rule: ln + p ; = ln + p + p ; where p ; is a domesic good markup shock or domesic rm cos-push shock ha evolves according o: p ;,! i.i.d N ; 2 p

11 Finally, given ha all rms ha adjus in period choose he same opimal price ~p o ; whereas hose ha do no simply index prices o pas in aion, he aggregae domesic price index p ; evolves according o: 2 p ; = 4! p ; ; p ; + ( ) ~p o ; p ; 3 5 p ; (26) 3..2 Expor inermediae good rms Expor inermediae good rms use boh labor X; (f) and e ecive capial sock k ~ p X (f) = u g ; kp X; (f) o produce oupu ha will be used by expor reail rms according o he following consan reurns o scale echnology: x (f) = a (k g ) h ~ k p X; (f) i X [X; (f)] X (27) As for domesic inermediae good rms, di ereniaed expor inermediae goods are supplied in a monopolisically compeiive markes. Cos minimizaion problem of expor inermediae good rms herefore yields he following sandard expression of capial-labor raio: and real marginal cos: ~k p X; (f) X; (f) = X X w X; =p X; r k X; = ~k p X; X; (28) MC X; (f) =! wx; =p X X; r k X X; a = MC X; (29) (kg ) X X We assume ha prices are sicky and se in saggered fashion à la Calvo (983). Therefore, opimal expor inermediae good rm s opimal pricing decision is given by: ~p o X; p X; = f X; f 2 X; where fx; = { X x p X; + p X; MC X; + X E f X;+ n o fx; 2 = { X x p X; + X E X X; ( X;+ ) fx;+ 2 where + p X; denoes ime-varying mark-up of prices over marginal coss a expor inermediae good rms level and is assumed o evolve according o he following rule: ln + p X; = ln + p X + p X;

12 where p X; is an expor good markup shock or an expor rm cos-push shock ha evolves according o: p X;,! i.i.d N ; 2 p X Finally, given ha rms ha are able o adjus choose he same prices ~p o X;, he aggregae expor price index p X; evolves according o: 2 p X; = 4 X! X p X; X; p X; + ( X ) ~p o X; p X; 3 5 p X; (3) 3..3 Impor inermediae good rms There exiss a coninuum of domesic reailer rms which impor goods in inernaional rade marke where he law of one price holds "a he dock". In order o generae incomplee exchange rae pass-hrough ino impor prices, we follow Monacelli (23) and assume ha inermediae imporing rms behave as a monopolisic rm when seing home currency price of impored goods. Therefore, deviaions from he law of price assumpion, hence incomplee exchange rae pass-hrough, occur due o he opimal mark-up problem ha imporing rms have o face when seing prices. We assume ha prices are sicky and se in saggered fashion à la Calvo (983). A fracion ( M ) of randomly seleced imporing rms is able o se new prices ~p o M; (f) each period, whereas a fracion M of imporing rms keeps heir prices unchanged and simply follow he following indexaion rule: p M; (f) = M M; p M; (f) where he parameer M measures he degree of indexaion o pas in aion. Firms ha are allowed o se prices will maximize he following discouned sum of heir expeced pro s subjec o he demand consrain from nal good rms. Solving his maximizaion problem gives he following rs order condiion ha characerizes impor inermediae good rm s opimal pricing decision: ~p o M; = f M; p M; fm; 2 (3) where fm; = { M m p M; + p M; MC M; + M E f M;+ n o fm; 2 = { M m p M; + M E M M; ( M;+ ) fm;+ 2 wih MC M; = Z P X; P M; 2 = z p X; p M; (32)

13 and p X; = P X; =P. Finally, given ha all imporing rms ha adjus in period choose he same opimal price ~p o M;, whereas hose ha do no simply index prices o pas in aion, he aggregae impor price index p M; evolves according o: 2 p M; = 4 M! M p M; M; p M; + ( M ) ~p o M; p M; 3 5 p M; (33) 3..4 Employmen Finally, we assume ha labor supply is no perfecly subsiuable and is given by he following Dixi-Sigliz ype aggregaor funcion: Z (f) = (h; f) +W; + W; dh where he elasiciy of subsiuion among di eren ypes of labor supplied by households is ime-varying. I is assumed o be random and given by: ln ( + W; ) = ln ( + W ) + W; where W; is he wage markup shock ha evolves according o: W;,! i.i.d N ; 2 W Firms choose he bes combinaion of di eren ypes of labor ha minimizes heir labor coss. The soluion of he cos minimizaion problem yields he following labor demand of individual household h: (h; f) = (h) = w (h) w w (h) w + W; W; (f) + W; W; (34) from which we can derive an expression of nominal wage index of each secor i. Tha is, Z w = w (h) W; W; dh 3

14 3.2 Oil rms When reaing individually an Oil-exporing counry, an imporan quesion arise: should we ake ino accoun or no he phenomenon of peak oil producion? 4 Oil nowadays plays an imporan role in he modern sociey and has a grea in uence economically and poliically. Therefore, i is di cul o accep his noion of oil depleion even if one has no doub ha by he naural order, oil resource is limied. This common belief is re eced in he lieraure (à faire, cier les aricles qui ne raie pas de cee "depleion") where he noion of depleion is ignored or supposed o arrive in he far fuure and is no reaed e ecively (one usually rea Oil supply o follow an auoregressive process). If one considers world oil producion, one can jusify his assumpion given possible fuure new discoveries of oil deposis around he world, change in consumpion habi, change in marke rules and advance in echonology o explore already available sies. owever, when aking individually an oil-exporing counry, should we coninue o ignore oil depleion? The answer is cerainly no. When suying opimal economic policy for hose counries, i is necessary o give a rigourous formulaion of oil resources depleion in he analysis. The seminal paper of ubber (956) reaed his phenomenon quanialively. Ye he model su ers from some limiaions, as deailed in Cavallo (24), and is considered as an economeric model raher han a heorical one, i has succeed in predicing US oil producion peak in 97. Indeed, a beer undersanding of he resources depleion will lead o opimal conduc of economic policy analysis and, hence, an opimal allocaion of he surplus issued from naural resource discorvery. Ignoring his naural facs is a source of he well known resources curses (à faire, une revue). In his paper, we consider a model of depleable resources following olland (28) ha generaes opimally a peak oil producion hrough an opimizaion framework. The model is in line wih oelling heoreical framework bu where peak in producion is inroduced hrough he exisence of wo opposing forces which are an increase and a decrease in producion. On he one hand, oil producion rises following an increase in demand, improvemens in echnology and addiional reserves exploraion. One he oher hand, oil producion decreases due o scarciy and oil reserves depleion. I is worh noing ha he objecive of his paper is o inroduce in he analysis he phenomenon of resources depleion hough a rigourous modelizaion of peak oil producion bu no o deemine he exac dae of peak oil (à faire, faire un revue de l éa de peak oil dans le monde). e us denoe o he producion of crude oil. Producing o incurs exracion coss which we denoe C (eo ; rs ) ha increase wih he quaniy of crude oil exraced eo due o scarciy and decrease wih he quaniy of reserves rs available a he end of period. 5 Tha is, o = eo C (eo ; rs ) (35) 4 The phenomenon of peak oil refers o fuure decline in oil producion due o resources depleion. 5 As argued by Pindyck (978), exracion cos depends negaively o quaniy of reserve available. Tha @rs <. 4

15 wih C (eo ; rs ) = (eo ) 2 rs Change in reserves in urn is assumed o evolves according o: rs = g ; rs + f O; ; k ~ p O; ; s eo (36) where rs = and f O; ; k ~ h i p O; ; s = (s ) O s ~k p O; [O; ] O is he addiional reserve explored and developped o decrease marginal cos of exracion. 6 Cumulaive discoveries s in urn is de ned as: s = g ; s + f O; ; k ~ p O; ; s (37) where s =. Therefore, pro s of Oil rms is given by: O; = o w O; p O; O; r k O; ~ k p O; Oil rms will maximize he discouned value of heir expeced pro s: X E k O;+k k= subjec o law of moion of reserves (36) and cumulaive discoveries (37). Firs order condiion of his maximizaion problem is given by: = Ceo (eo ; rs ) + (38a) w O; =p O; = ( O ) ( + ) f O; ; k ~ p O; ; s = O; (38b) ro; k = O ( + ) f O; ; k ~ p O; ; s = k ~ p O; (38c) = E + Crs (eo ; rs ) (38d) n = E + s E f O;+ ; k ~ o p O;+ ; s =s (38e) The rs order condiion (38a) indicaes ha marginal revenue from oil producion is equal o marginal cos of exracion C eo (eo ; rs ) plus marginal cos of scarciy. I is worh noing ha marginal cos of exracion is increasing wih quaniy exraced and ha marginal cos 6 I is assumed ha f () depends posiively on labor and k ~ p > ), and negaively on O reserves already available < ). The laer assumpion inroduces resources depleion in he 5

16 of scarciy is negaively correlaed wih he size of reserve available. The rs order condiions (38b) and (38c) indicaes ha marginal cos of labor and capial (w O; and ro; k ) are equal o marginal bene issued from new resources exploraion and discovery ( f ()) ne of marginal scarciy cos of new addiional reserves ( f ()). Finally, we assume ha he crude oil price is se in inernaional marke. owever, in order o miigae he damaging e ec oil price swings on domesic economy, he governmen is allowed o smooh inernaional oil price variaion hrough he managemen of he sovereign funds. Therefore, he law of moion of domesic crude oil price wries: 3.3 Final-good rms p O; = ( O ) g ; p O; + O z p O; (39) ou p O; = g ; p O O; z p O O; 3.3. Final privae consumpion-good rms Non-oil nal privae consumpion-good rms produce homogeneous goods q C NO using a bundle of domesic h C NO and impored m C NO inermediae goods. The producion funcion ha ransforms inermediae goods ino nal consumpion oupu is given by: q C NO = " C;NO C;NO h C NO C;NO C;NO + ( C;NO ) C;NO m C NO # C;NO C;NO C;NO C;NO where C;NO is he elasiciy of subsiuion beween home and foreign non-oil bundles of goods, and C;NO measures he degree of home producion bias. The adjused level of impored inpu m C NO is de ned as: m C NO = m C NO m C NO Tha is, we assume ha adjusing he bundle of impored goods in producing non-oil nal consumpion-goods incurs cos, represened by mc NO, for rms. Adjusmen cos is inroduced in he model o insure smooh response of impor goods share in he producion process following a change in he relaive price of impored goods. owever, he level of impored goods is allowed o jump following a change in he oal demand. Thus, we follow Chriso el e al. (28) and assume ha he adjusmen cos funcion is de ned as: m C NO = m C NO 2 " M m C NO where M C NO > and M refers o impor demand shock. m C NO =q C NO m C NO =qc NO # 2 (4) 6

17 The bundle of domesic and impored inpus are respecively an aggregaion of a coninuum of di ereniaed home-produced and impored inermediae goods. The aggregaion echnology is assumed o be a CES funcion given by: h C NO = m C NO = Z Z + p h C NO + (f) p ; ; df + p m C NO + (f) p M; M; df Therefore, aken as given nominal price of domesic p ; (f) and impored p M; (f) inermediae goods, rms opimal allocaion of di ereniaed inpus ha minimizes heir oal expendiures yields respecively he following demand for individual domesic and impored inermediae goods: (4) (42) h C NO (f) = p; (f) p ; +p ; p ; h C NO (43) m C NO (f) = pm; (f) p M; +p M; p M; m C NO (44) Combining hese demand funcions wih CES aggregaion funcions (4) and (42) yields he following aggregae price indices of he bundle of, respecively, domesic and impored inermediae goods: Z p p ; = (p ; (f)) p ; ; df (45) p M; = Z (p M; (f)) p p M; M; df Finally, given he aggregae domesic and impor price indices p ; and p M;, opimal allocaion of expendiure beween he bundle of home-produced h C NO and impored m C NO goods o produce non-oil nal privae consumpion-good q C NO is given by: C;NO h C NO p; = C;NO q C NO (47) p NO; m C NO = ( C;NO ) pm C NO ; p NO; where he adjused aggregae impor price index p M C NO ; is de ned as: p M C NO ; = " m C NO m C NO 7 m C NO (46) C;NO q C NO (48) =q C NO m C NO =qc NO # p M;

18 whereas he aggregae price index of non-oil nal privae consumpion-good p NO;, which we denoe "core-cpi ", is given by: C;NO p NO; = ( C;NO ) p M C NO ; + C;NO (p ; ) C;NO C;NO (49) Final invesmen-good rms Final invesmen-good rms have he same srucure as non-oil nal privae consumpion-good rms. They produce homogeneous nal invesmen-goods q I using a bundle of domesic h I and impored m I inermediae goods. The producion funcion ha ransforms inermediae goods ino nal invesmen oupus is given by: q I = I h I I I + ( I ) I m I I I I I (5) I where I is he elasiciy of subsiuion beween home and foreign non-oil bundles of goods, and I measures he degree of home producion bias in producing nal invesmen-goods. The adjused level of impored inpu M I is given by: m I m = I m I where he adjusmen cos funcion is de ned: " m I = m I 2 M m I m I =q I m I =qi wih M I > and M refers o impor demand shock. The bundle of domesic and impored inpus are respecively an aggregaion of a coninuum of di ereniaed home-produced and impored goods. The aggregaion echnology is assumed o be a CES funcion given by: h I = m I = Z Z h I (f) + p ; df + p ; # 2 m I (f) + p + p M; M; df Final invesmen-good rms opimal allocaion of di ereniaed inpus ha minimizes heir expendiure yields respecively he following demand of individual domesic and impored inermediae goods: (5) (52) h I (f) = m I (f) = p; (f) p ; pm; (f) p M; 8 +p ; p ; +p M; p M; h I (53) m I (54)

19 Finally, opimal allocaion of expendiure beween he bundles of home-produced h I and impored m I goods o produce nal privae invesmen good is given by: I h I p; = I q I (55) p I; I m I pm = ( I ) I ; q I (56) where he adjused aggregae impor price index p M I ; is given by: p M I ; = " m I m I p I; m I =q I m I =qi # p M; whereas he aggregae price index of nal invesmen good p I; is given by: p I; = h i I ( I ) p M ; I + I (p ; ) I I (57) I is worh noing ha he di erence beween non-oil privae consumpion and invesmen producion is due o he di erence in he values of elasiciy of subsiuion and home producion bias parameers. Whenever C;NO = I and C;NO = I, core-cpi and invesmen price index will be idenical, p NO; = p I;. (on peu assumer que C;NO > I c es à dire que les invesissemens nécessien plus de biens d imporaion) Public nal consumpion-good rms In conras o nal privae consumpion and invesmen goods, nal public consumpiongoods are produced using only a bundle of domesic inermediae goods. Tha is, here exiss a full home bias producion for he public consumpion-goods and he producion echnology is given by he following CES aggregaion funcion: Z q G = h G (f) + p + p ; ; df Therefore, given nominal price of domesic inermediae goods p ; (f) and aggregae home price index p ;, rms opimal allocaion of di ereniaed inpus ha minimizes heir expendiures yields he following demand for he individual domesic inermediae goods: h G p; (f) (f) = p ; +p ; p ; (58) q G (59) Finally, given he assumpion of full home bias producion for nal public consumpiongoods, aggregae public consumpion price index is equal o home price index. Tha is, p G; = p ;. 9

20 3.3.4 Expor nal-good rms As for nal public consumpion-goods, exporing rms produce homegeneous radeable goods x using expor inermediae goods x (f). The producion funcion ha ransforms expor inermediae goods ino nal expor-goods is given by: Z x = (x (f)) + p X; df + p X; Therefore, given nominal price of expor inermediae goods p X; (f) and aggregae expor price index p X;, exporing rms opimal allocaion of di ereniaed inpus ha minimizes heir expendiures yields he following demand for he individual expor inermediae goods: px; (f) x (f) = p X; +p X; p X; (6) x (6) Combining his demand funcion wih producion funcion (6) yields he following expor price index: Z p p X; = (p X; (f)) p X; X; df (62) We assume as for impors ha he law of one price holds "a he dock" o have symmery in he invoicing sraegy of domesic and foreign radeable rms. Therefore, exporing rms will follow he producer currency pricing (PCP) sraegy and se he price of heir goods in domesic currency. We assume ha he srucure of demand in foreign counry for domesic expored goods follows he same srucure as he demand of foreign goods in (47) and (55). Tha is, x = px; z ~ y (63) where as in Chriso el e al. (28), is he ime-varying expor share of domesic exporing rms which is supposed o capure non-price relaed preference of foreign households of domesic expored goods, y represens foreign oupu. Moreover, x and p X; are respecively de ned as: x x = x " # x p X; = x g ; x =y g; x =y p X; where he foreign adjusmen cos funcion for domesic expored goods is given by: " # 2 x = x X g x ; x =y 2 g; x =y and X refers o expor demand shock. 2

21 3.3.5 Aggregae demand of inermediae goods rms Aggregae demand of individual domesic and impored inermediae goods are obained by aggregaing individual demand across nal good rms. Using (43), (53) and (59) aggregae demand for individual domesic inermediae goods is given by: p; (f) h (f) = p ; +p ; p ; h (64) where h (f) = h C NO (f) + h I (f) + h G (f) and h = h C NO + h I + q G. In urn, using (44) and (54), aggregae demand for individual impor inermediae goods is given by: pm; (f) m (f) = p M; +p M; p M; m (65) where m (f) = m I (f) + m C NO (f) and m = m I + m C NO. 4 Cenral bank To close he model, i assumed ha he moneary auhoriy adjuss he shor erm nominal ineres rae R in order o persue a chosen moneary policy rule. The ineres rae insrumen rule is formally de ned as: R R rno R = R NO; r rz Z R NO Z R exp R (66) where parameer R indicaes he degree of ineres-rae smoohing. Parameers r no, r,r z in urn are policy coe ciens ha measure he degree of cenral bank response o, respecively, core-cpi, headline-cpi and US-Dollar nominal exchange rae changes, respecively. Variables NO, and Z = = are seady-sae values of NO;, and Z, respecively. Finally R iid N ; 2 R is an exogenous moneary policy shock. In he analysis, we consider di eren ypes of moneary policy rule and allow he cenral bank eiher o arge in aion rae or exchange rae level. 4. In aion argeing Whenever he cenral bank chooses in aion argeing rule, i may furher eiher conrol he headline-cpi or he core-cpi NO;. In eiher cases, nominal exchange rae is fully exible. If he cenral bank chooses o conrol he headline-cpi, we se r no = r z =, leading o wha we denoe IT rule. In urn, if i chooses o conrol he core-cpi, we se r = r z =, leading o wha we denoe CIT rule. 2

22 4.2 Exchange rae argeing As for in aion argeing, whenever he cenral bank chooses exchange rae argeing rule, we allow i o x nominal US-Dollar exchange rae Z in which oil-revenues are denominaed. Therefore, if he cenral bank chooses o arge a given level of he nominal e ecive exchange rae, we se r no = r =, leading o wha we denoe ET rule. 5 Marke Clearing 5. abor marke clearing abor marke clears when labor demand of di eren secors i = fo; X; g mach excacly labor supply of households a he wage level se monopolisically. Tha is, Z Z +W; dh i; (f)df = i; = + i; (h) W; (67) Aggregaing over households h he labor demand in (34) yields: ha is, Z (h)dh = Z Z w (h) w + W; W; dh (h)dh = { w; (68) where { w; measures he degree of wage dispersion across households and evolves according o: ~w o + W; { w; = ( W ) W;! + W; W + w W; W { w; (69) w w I is worh noing ha wage dispersion equals o one in he seady sae and vanishes in he rs-order approximaion around seady sae. Aggregae oal wage paid by he rms in urn is given by: 5.2 Capial marke clearing Z w (h) (h)dh = w Capial marke clears when he e ecive use of capial services equals demand from rms in di eren secors i = fo; X; g. Tha is, u Z g ; kp i; (h)dh = u g ; kp i; = ~ k p i; = Z ~k p i; (f)df (7) 22

23 5.3 Inermediae goods marke clearing Marke clears for each inermediae-good rms when he supply of heir producs equals domesic or foreign oal demands in (64), (6) and (65). Aggregaing over rms f yields: Z Z Z h (f)df = { ; h x (f)df = { X; x m (f)df = { M; m where { i;, for i = f; X; Mg, are price dispersion across rms ha evolves according o: ~p o i; { i; = ( i ) p i; +p i; p i;! +p i; i p i; i; + i { i; (7) i; and h (f) = h C NO (f) + h I (f) + h G (f) h = h C NO + h I + q G m (f) = m I (f) + m C NO (f) m = m I + m C NO Therefore, aggregae real ressource of he oil exporing economy consiss of non-radable goods h, radable manufacured goods x and crude oil o. Tha is, In nominal erms, domesic aggregae ressource is given by: p Y; y = Z y = { ; h + { X; x + o (72) p ; (f)h (f)df + Z p X; (f)x (f)df + p O; o p Y; y = p ; h + p X; x + p O; o (73) Tha is, he GDP de aor can be obained as: p Y; = p ; h y + p X; x y + p O; o y 23

24 5.4 Final goods marke clearing Final goods marke clears when supply of nal goods equals demand. Tha is, q C NO = c NO; q I = i p + (u ) g ; kp + ig = i (74) q G = g from which we obain he following expression of nominal aggregae ressource using opimal allocaion of expendiure beween di eren domesic and impored bundles of di ereniaed goods and he expression of nominal oal consumpion expendiure. Tha is, where p Y; y = p c + p I; i + p ; g + (p X; x + p O; o ) (p M; ~m + p RO; c RO; ) (75) i = i p + (u ) g ; kp + ig ~m = mc NO m C NO + mi m I and mc NO = mi = m C NO m I m C NO m C NO m I m I m I =qi m I =qi C m NO =q C NO m C NO =qc NO 5.5 Bonds marke clearing In equilibrium, holdings of governmen domesic bonds equals o zero each period given he assumpion ha governmen budge consrain is closed each period by lump-sum axes. Tha is, Z b d (h)dh = b d = Morever, marke clears in he inernaional marke of foreign privae bonds when supply mach exacly holdings of foreign bonds by domesic households. Tha is, Z b (h)dh = b 24

25 Given he de niion of domesic ne foreign asse in (), i evolves according o: Z P (B + F ) P P = Z P P { R B + F P P P + T B P z ~ (b + f ) = z ~ g ; { R b + f + b wih rade balance b de ned as: b = + p X; x + z p O;o z p X;m + p O;c RO; where is a fricion ha arise due o he presence of impor adjusmen cos in nal-good rms. I is de ned as = p M; mc NO m C NO + mi m I and is equal o zero in he seady sae, =. 25

26 References [] M. Adolfson, S. aseen, J. inde, and M. Villani. Bayesian esimaion of an open economy dsge model wih incomplee pass-hrough. Journal of Inernaional Economics 72(2), 48 5 (July 27). [2] G. A. Calvo. Saggered prices in a uiliy-maximizing framework. Journal of Moneary Economics 2(3), (Sepember 983). [3] A. J. Cavallo. ubberšs peroleum producion model: An evaluaion and implicaions for world oil producion forecass. Naural Resources Research 3(Issue 4), 2 22 (december 24). [4]. J. Chrisiano, M. Eichenbaum, and C.. Evans. Nominal rigidiies and he dynamic e ecs of a shock o moneary policy. Journal of Poliical Economy 3(), 45 (February 25). [5] K. Chrisoffel, G. Coenen, and A. Warne. The new area-wide model of he euroarea: A micro-founded open-economy model for forecasing and policy analysis. Working Paper Series 944, European Cenral Bank (Ocober 28). [6] C. J. Erceg,. Guerrieri, and C. Gus. Sigma: A new open economy model for policy analysis. Inernaional Journal of Cenral Banking 2() (March 26). [7] C. J. Erceg, D. W. enderson, and A. T. evin. Opimal moneary policy wih saggered wage and price conracs. Journal of Moneary Economics 46(2), (Ocober 2). [8] S. P. olland. Modeling peak oil. The Energy Journal 29(Number 2), 6 8 (28). [9] M. K. ubber. Nuclear energy and he fossil fuel. Drilling and Producion Pracice. American Peroleum Insiue. (June 956). [] T. Monacelli. Moneary policy in a low pass-hrough environmen. Working Paper Series 227, European Cenral Bank (Apr. 23). [] R. M. P. Jacquino and M. Spizer. An open-economy dsge model of he euro area. Technical Repor, European Cenral Bank (26). [2] R. S. Pindyck. The opimal exploraion and producion of nonrenewable resources. Journal of Poliical Economy 86(5), 84 6 (Ocober 978). [3] S. Schmi-Grohe and M. Uribe. Closing small open economy models. Journal of Inernaional Economics 6(), (Ocober 23). 26

27 [4] F. Smes and R. Wouers. An esimaed dynamic sochasic general equilibrium model of he euro area. Journal of he European Economic Associaion (5), (9 23). [5] N. Sähler and C. Thomas. Fimod - a dsge model for scal policy simulaions. Economic Modelling 29(2), (22). 27

28 A Seady-Sae A. ouseholds The seady sae of he rs order condiion characerizing he households opimal purchase of consumpion good is obained from (4a). I is given by: = ( + c ) hg c (76) From he ucas and foreign bond pricing (4b) and (4c), one can derive he seady sae value of he discoun facor. Tha is, 7 R = R = g Z (77) Similarly, from he capial bond pricing (4d), one an derive he following seady sae expression linking he "marginal Q" of Tobin and real ineres rae of capial: Q = r k g + where we have used he seady-sae properies of he degree of he capial uilizaion u = and he degree of capial uilizaion cos (u) =. Using he invesmen equaion (4e), and noing ha i S(g ; ) = I i 2 g ; g = a s.s. i 2 i S i i (g ; ) = I g ; g = a s.s. i i (78) one obains: Q = p I (79) Combining (78) and (79) yields he seady sae value of real ineres rae of capial r k = p I g + (8) Using he sr-order condiion wih respec o capial uilizaion r k = (u ) p I; and he expression of he cos associaed in changing he degree of capial uilisaion u yields: r k = u; p I 7 For non-zero seady sae in aion, one have: R = g Z = 28

29 Tha is, u; = g + (8) Finally, using he law of moion for capial (2), one obains he following expression of he seady sae value of invesmen: { p = ( )g k p (82) A.2 abor Marke Equilibrium On he labor supply side, he seady sae of rs order condiion characerizing he wage seing decision of households (7) wries: w = + ( + c ) W ( w ) hg c (83) Moreover, we assume ha he level of wage is equal across secor a seady-sae. Tha is, w O = w X = w = w (84) Therefore, using he las expression and ha of he opimal allocaion of labor supply beween di eren secors (4), households seady-sae labor supply o di eren inermediae rms are de ned as: i = i for i = fo; X; g (85) The same assumpion is also assumed o hold for he level of real ineres rae of capial. Tha is, r k O = r k X = r k = r k (86) and k p i = i k p for i = fo; X; g (87) where he las equaion is obained using he opimal allocaion of capial supply beween di eren secors (3). In urn, on he labor demand side, using he expression of capial-labor raio of domesic non-oil inermediae rms (23) and capial marke clearing (7) yields: w ; p ; r k ; w p r k = u g ; kp ; ; = g k p Using he expression of w and r k in (83) and (8) yields: k p + g k p = c = + W ( + c ) ( w ) hg c p p I g + g + W ( + c ) ( w ) hg p p I g + 29

30 Therefore, k p + c = g + W ( + c ) ( w ) hg p p I g (88) + Moreover, from wage seing of household, one have: fw; = W; B ({ w ) + ( + W; ) + W E f W;+ f W = + + W W ( W ) = W ( W ) + W (89) and f 2 W; = W; { w ( W w ) + W E fw;+ 2 + f W 2 = W ( w ) + W f 2 W f W 2 = W ( W ) ( w ) (9) A.3 Capial marke equilibrium A.3. Domesic inermediae rms The rs-order condiion of non-oil inermediae goods cos minimizaion problem wih respec o capial is given by: r k ; = MC ; a (k g ) h i ~k p ; [; ] ~k p ; = MC ; a (k g ) ; ~k p ;! r k ; = MC ; a (k g ) ; u g ; kp ;! where he las expression is obained using capial marke clearing (7). From he price seing condiion (25) of non-oil inermediae rms, one obain under exible price seing ( = ) he following relaion ha links marginal cos and gross mark-up: ~p o ; = + p ; MC ; = (9) p ; 3

31 Therefore, a seady sae: k p r k = + p k g g k p g = + p k g k p = g r k + p k g! Using he expression of r k in (8) and he assumpion (86), one have: A.3.2 k p = g Expor inermediae rms " k g p I g + + p # (92) By he same analogy, he seady-sae capial-labor raio for expor inermediae rms is de ned as: " k p X X k g # = g X p I g + X + p (93) X A.3.3 Oil rms For oil rms, he capial labor raio is obained from he expression of he marginal cos of labor and capial in (38b) and (38c). Tha is, ~k p O; O; = u g ; kp O; = O w O; =p O; O; O From he las equaion, we derive he following seady sae capial labor raio for oil rms. r k O; k p O; O; = g O O w O =p O r k O (94) From he rs order condiion (38a) and (38d), on have eo rs = 2 eo rs + = 2 3

32 and Therefore, one obains: ( ) = eo 2 rs ( ) 2 4 = ( ) = 4 ( ) 2 [2 + 4 ( )] + = from which he seady sae value of he marginal cos of scarciy is de ned as: [2 + 4 ( )] = [2 + 4 ( )] Therefore, eo rs = 2 From he oil producion funcion (35), he seady sae crude oil producion is given by: o = eo (eo) 2 rs 2 = eo eo rs (95) Tha is, eo = o 2 = In urn, he seady sae value of reserves is de ned as: 2o + rs = 2 eo = 4o ( ) ( + ) (96) (97) Using he expression of cumulaive discoveries (37), one have: s = g s + (s) s g k p O O O g s = (s) s g k p O O O O g s + s = g k p O O O O O Tha is, s = " g k p O O O g 32 O # + s (98)

33 Finally, using (38e), one obains: ( ) = s ( + ) f O ; g k p O ; s =s = s ( + ) (s) ( s +) g k p O O O O = s (s) ( s +) g k p O O O O s (s) ( s +) g k p O O O Tha is, = A.4 Goods-marke equilibrium A.4. Domesic inermediae rms s (s) ( s +) g k p O O O O + s (s) ( s +) g k p O O O < (99) O Using capial marke clearing (7), seady-sae domesic inermediae rms producion funcion is given by: h = k g g k p = k g k p g Using households seady-sae labor supply o di eren inermediae rms in (85), one have: h = k g k p g () The seady-sae expression of non-oil domesic inermediae goods demand in (47), (55) and (59) is de ned as: C;NO C;NO h C p NO = C;NO q C p NO = C;NO cno p NO p NO C;NO p = C;NO ( C ) (pno) C c p NO I h I p = I q I p = I p I p I h G = q G = g I I ({ p + { g p ) = I { p I where he las equaliy comes from he aggregae nal goods marke clearing (74), he assumpion ha () = a seady-sae and he opimal allocaion of expendiure o non-oil goods (8a). Therefore, he seady-sae value of domesic inermediae goods oal demand is given by: h = C;NO ( C ) p p NO C;NO I (pno) p C c + I { + g () p I 33 O

34 Finally, combining domesic goods supply () and demand () yields: k g g k p C;NO I p = C;NO ( C ) (pno) p C c+ I {+g p NO p I (2) Moreover, real pro s of non-oil inermediae rms is de ned as: ; = h r; k k ~ p ; + w ; ; p ; = h MC ; (h + ) where he las equaliy is obained from he opimizaion problem of domesic rms in (22). From zero-pro condiion a he equilibirum and dividing by h, one have: MC + h = Therefore, using (9), he expression of he seady sae xed cos wries: = p h (3) Concerning opimal pricing of home inermadiae goods, one have: f = f 2 = h p ( ) + p MC (4a) h p ( ) (wih = = ) (4b) A.4.2 Expor inermediae rms Similarly, he seady-sae value of he aggregae supply of non-oil expor inermediae goods wrie: x = X k g k p X g X X (5) X In urn, he seady-sae value of he aggregae demand of expor inermediae goods can be derived from (63). Tha is, x = p X z where we use he fac ha x = and assume ha he seady sae value of he produciviy di erenial beween home and foreign counries is equal o one, ~ =. Finally, by he same analogy wih domesic inermadiae rms, he expression of he seady sae xed cos wries: = p X x (6) 34 y

35 Concerning opimal pricing of home inermadiae goods, one have: f X = f 2 X = x p X ( X) + p X MCX (7a) x p X ( X) (wih X = = ) (7b) A.4.3 Impor inermediae rms The seady-sae expression of impored inermediae goods demand in (48) and (56) wries: m C pm NO = ( C;NO ) m I pm = ( I ) p I p NO C;NO q C pm NO = ( C;NO)( C ) I q I pm = ( I ) p I p NO I ({ p + { g pm ) = ( I ) p I C;NO (pno) C c I { where we use he fac ha he seady-sae value of adjusmen cos is equal o zero. Tha is, m C NO = m I = m C NO = m I = (8) Therefore, he seady sae consumpion and invesmen impor share are de ned as: m C NO C;NO pm = ( C;NO )( C ) (pno) C = m c (9a) c p NO m I I pm = ( I ) = mi (9b) { p I Concerning opimal pricing of oil impored inermadiae goods, one have: f M = f 2 M = m p M ( M) + p M MCM m p M ( M) (a) (b) A.5 Relaive prices By de niion, he seady sae value of relaive consumpion prices is equal o uniy. Tha is, p = () In urn, we assume ha he seady-sae level of real oil price is equal o (à jusifer e à compleer) p O =?? 35

36 From he law of moion of re ned and domesic crude oil price (9) and (39), one have p RO = p O = p O Using he de nion of he CPI index (7), one have: p NO = " C (p RO ) C C # C From he de niion of he impor rms marginal cos (32). One obains: MC M = p X p M where we assume ha he PPP holds a seady saes. Tha is, z = (2) I is worh noing ha from opimizaion problem of imporing rms ((a) and (b)) MC M = + p (3) M Threfore, he seady sae expression of impor price is given by p M = + p M p X where inernaional price of non-oil goods p X are aken as given by imporing rms. Using he de niion of he core-cpi index (49), he seady sae value of domesically produced gods is equal o " # (p NO ) C;NO ( C;NO ) (p M ) C;NO C;NO p = C;NO We assume ha he seady-sae price of domesically produced expor (p X ) and public (p G ) goods is given by: p X = p G = p Using he de niion of he invesmen price index (57), one have: p I = h i ( I ) (p M ) I + I (p ) I I Finally, one can derive he seady-sae value of gdp-de aor using he aggregae resource consrain (73). Tha is, h p Y = p y + p x X y + p o O y 36

37 where h y x y o y = h y = = ( x y o y ) = x y (calibraed) = o y (calibraed) wih erms h= and y= de ned in (5), and (9), respecively. Therefore, knowing ha p X = p. A.6 Seady sae labor p Y = p ( x y o y ) + p X x y + p O o y p Y = p ( o y ) + p O o y (4) Using he producion funcion () and seady sae value of xed cos (3), one obains: h = k g k p ( + p (5) )g From he aggregae real ressource consrain (72) and assuming ha price dispersion vanishes in seady sae, { i = for i = f; X; wg (6) one obains he following expression of seady-sae oupu per capia: y = h + x y y + o y y where x y = x=y and o y = o=y denoes he seady sae share of non-oil and oil expored goods, respecively (x y e o y doi êre calibrer. Cela perme de simpli er mais aussi de disinguer les di érens pays). For convenience purpose, he non-oil and oil expor per capia is herefore de ned as: x = x y y = x y y o = o y y = o y y Finally, using (5), he expression of seady-sae oupu per capia wries: (7) (8) y = k g ( x y o y ) ( + p )g 37 k p (9)