Nontradable Goods and the Real Exchange Rate

Transcription

1 Nonradable Goods and e Real Excange Rae au Rabanal Inernaional Moneary Fund Vicene Tuesa CENTRUM Caólica Tis version: Augus 3, 22 Absrac Tis online appendix provides e equilibrium condiions of e model and e loglinear approximaion.

2 A Appendix: Te Baseline Model In is appendix, we presen e full version of a model wi radable and nonradable nal consumpion goods, in e spiri of Sockman and Tesar (995) and Dosey and Duare (28). We inroduce sicky prices in bo secors o be able o sudy in aion dynamics and eir role in a ecing e real excange rae. A. Houseolds A.. references Represenaive ouseolds in e ome counry are assumed o maximize e following uiliy funcion: U = E ( X = subjec o e following budge consrain: B H R + S B F R SB F Y " #) log C bc L +' ; () + ' B H + S B F + W L C + (2) E denoes e condiional expecaion on informaion available a dae = ; is e ineremporal discoun facor, wi < < : C denoes e level of consumpion in period, L denoes labor supply. Te uiliy funcion displays exernal abi formaion wi respec o e abi sock, wic is las period s aggregae consumpion of e economy C. b 2 [; ] denoes e imporance of e abi sock. ' > is inverse elasiciy of labor supply wi respec o e real wage. is a preference sock a follows an AR() process in logs log = log + " (3) We de ne e consumpion index as C =" c C T " " + ( c ) =" C N " " " " ; were " is elasiciy of subsiuion beween e nal radable (C T ) and nal nonradable (C N ) goods, and c is e sare of nal radable goods in e consumpion baske a ome. In is conex, e consumer price index a corresponds o e previous speci caion is given by c T " + ( c ) N " i " ; 2

3 were all prices are for goods sold in e ome counry, in ome currency and a consumer level, for bo radable and nonradable goods. Demands for e nal radable and nonradable goods are given by: C T T " = c C ; C N N " = ( c ) C : A..2 Incomplee Asse Markes For modelling simpliciy, we coose o model incomplee markes wi wo risk-free one-period nominal bonds denominaed in domesic and foreign currency, and a cos of bond oldings is inroduced o acieve saionariy. Ten, e budge consrain of e domesic ouseolds in real unis of ome currency is given by: B H R + S B F R SB F Y B H + S B F + W L C + (4) were W is e nominal wage, and are real pro s for e ome consumer. B H is e olding of e risk free domesic nominal bond and B F is e olding of e foreign risk-free nominal bond expressed in foreign counry currency. S is e nominal excange rae, expressed in unis of ome counry currency per uni of foreign counry. Te funcion (:) depends on e ne liabiliy posiion (i.e. e negaive ne foreign asse posiion) of e ome counry, B F, in percen of GD in e enire economy, and is aken as given by e domesic ouseold. (:) inroduces a convex cos a allows o obain a well-de ned seady sae, and capures e coss of underaking posiions in e inernaional asse marke. 2 A.2 roducion Secor Te producion of is economy is underaken by ree secors. Firs, ere is a nal goods secor, a uses inermediae radable inpus from bo counries and operaes under perfec compeiion, o produce e nal radable goods. Tis same secor also aggregaes varieies of e nonradable goods o produce a nal nonradable good a is sold o ouseolds. Te second As Benigno,.(2) poins i ou, some resricions on (:) are necessary: () = ; assumes e value only if B F; = ; di ereniable; and decreasing in e neigborood of zero. 2 Anoer way o describe is cos is o assume e exisence of inermediaries in e foreign asse marke (wic are owned by e foreign ouseolds) wo can borrow and lend o ouseolds of counry F a a rae ( + r ), bu can borrow from and lend o ouseolds of counry H a a rae ( + r ) (:) : 3

4 secor produces inermediae radable goods, wic are used as an inpu for e producion of nal goods bo in e ome and in e foreign counry. Te ird secor produces nonradable goods, a are used as inpus in e producion of e nal nonradable good. A.2. Final Goods Secor Te nal radable good is consumed by domesic ouseolds. Tis good is produced by a coninuum of rms, eac producing e same variey, labelled by, using inermediae ome X and foreign X f goods wi e following ecnology: = = x X + ( x ) = X f were is e elasiciy of subsiuion beween ome-produced and foreign-produced impored inermediae goods, and x is e sare of ome goods in e producion funcion. We furer assume symmeric ome-bias in e composie of inermediae radable goods. Te corresponding composie of ome and foreign inermediae radable goods abroad is given by = ( x ) = X + = x X f X and X f, a denoe e amoun of ome and foreign inermediae radable inpus o produce e nal radable good a ome, are also Dixi-Sigliz aggregaes of all ypes of ome and foreign nal goods, wi elasiciy of subsiuion : Z and X X f Z X () d X f (f) df were X () and X f (f) denoe individual quaniies from inermediae radable goods producers a ome and foreign. Te equivalen quaniies for foreign nal radable goods producers are X () and X f (f). Opimizing condiions by nal radable goods producers deliver e following demand funcions: X () = x () X f (f) = ( x) f (f) f T ; X () = ( x ) f T ; X f 4 () f (f) (f) = x f T f T

5 were and Z () T = Z d ; f f (f) x + ( x ) f df : We assume a e law of one price olds for inermediae inpus, suc a () = ()S, and f (f) = f (f)s, were S is e nominal excange rae. Te producion of e nal nonradable good is given by: Z X N (n) dn were we assume e same elasiciy > an in e case of nal radable goods produced wiin counry H. Te price level for nonradables is N Z p N (n) dn A.2.2 Inermediae Non-Tradable Goods Secor Te inermediae nonradable secor produces di ereniaed goods a are aggregaed by - nal good producing rms, and ulimaely used for nal consumpion by domesic ouseolds only. Eac rm produces inermediae nonradable goods according o e following producion funcion (n) = A Z N L N (n) (5) were A is a labor augmening aggregae world ecnology sock wic as a uni roo wi drif, as in Galí and Rabanal (25): log A = g + log A + " a (6) Tis sock also a ecs e inermediae radable secor producion funcion. Hence, real variables in bo counries grow a a rae g. Z N is e counry-speci c produciviy sock o e nonradable secor a ime wic evolves according o an AR() process in logs log Z N = ( N ) log( Z N ) + Z;N log Z N + " Z;N (7) Firms in e nonradable secor face a Calvo loery wen seing eir prices. Eac period, wi probabiliy N, rms receive a socasic signal a allows em o rese prices opimally. 5

6 We assume a ere is parial indexaion wi a coe cien ' N o las period s secorial in aion rae for ose rms a do no ge o rese prices. As a resul, rms maximize e following pro s funcion: Max N (n) E k= 82 X >< k N ;+k 6 4 >: N (n) N +k N +k 'N MC N +k ;d +k (n) 9 >= >; (8) subjec o ;d +k (n) = " N +k N (n) N +k N 'N # (9) were ;d (n) is oal individual demand for a given ype of nonradable good n, and Y N is aggregae demand for nonradable goods, as de ned above. ;+k = k +k is e socasic discoun facor, were = C bc is e marginal uiliy of consumpion. MC N corresponds o e real marginal cos in e nonradable secor. From cos minimizaion: MC N = W Z N A A.2.3 Inermediae Tradable Goods Secor Te inermediae radable secor produces di ereniaed goods a are sold o e nal secor goods producers in e ome and foreign counries. Mos funcional forms are similar o ose presened for e nonradable secor. Eac rm produces radable inermediae goods according o e following producion funcion Y () = A Z L () () were Z is e counry-speci c produciviy sock o e inermediae goods radable secor a ime wic evolves according o an AR() process in logs log Z = ( ) log( Z ) + Z; log Z + " Z; () Firms in e inermediae radable secor face e same Calvo loery as rms in e inermediae nonradable secor, wi relevan parameers and ' : 82 X >< Max () E k +k () 6 ;+k 4 k= >: +k ' MC +k Y ;d +k () 9 >= >; (2) 6

7 subjec o Y ;d +k () = X +k () + X " () = +k +k () +k ' # X (3) were Y ;d () is oal individual demand for a given ype of radable inermediae good, and X is aggregae demand for inermediae good, consising of ome demand, and foreign demand: MC X = " x T + ( x ) T corresponds o e real marginal cos in e nonradable secor. From cos minimizaion: MC = W Z A # A.2.4 Marke Clearing We assume a e demand sock is allocaed beween radable and nonradable goods in e same way a privae consumpion is. Hence e marke clearing condiions for bo ypes of nal goods, consising of privae consumpion and e demand sock in e radadable secor, are: were G N ; G T = C T + G T = C N + G N follow AR() processes in logs. Te bond marke clearing condiions are B H B F + B H = (4) + B F = (5) For e nonradable inermediae goods, e marke clearing condiion is: wile for e inermediae radable goods secor i is: For e labor marke: (n) = X N ; for all n 2 [; ] (6) Y () = X () + X (); for all 2 [; ] (7) L = L + L N = (8) = Z L ()d + Z L N (n)dn 7

8 A.3 Opimizing, Marke Clearing Condiions, and Moneary olicy In is subsecion we presen e full se of equaions caracerizing e symmeric equilibrium. Since all agens in eac economy are equal, en e per capia and aggregae consumpion levels are equal (C = C ), as well as e ne foreign asses levels (B F = B F ). A.3. Houseolds Te Euler equaions for ome and foreign ouseolds, and e opimal condiion of oldings by ome ouseold of e foreign bond are: = E R = E R S B F = Y were is e marginal uiliy of consumpion: E R Q + + Q = U C (C ) = C bc Te labor supply decisions in eac counry are: = U C (C ) = C b C W = L ' were: W = (L ) ' L = L + L N and L = L + L N 8

9 Houseold demand for nal radable and nonradable goods are given by: C T T " = c C ; C N N " = ( c ) C : C T = c and e CI s in eac counry are given by: T " C ; C N = ( N " c) C c c T T " + ( c ) N " + ( c ) " i " ; N " " Te real excange rae is Q = S A.3.2 Final Goods roducers Te producion of nal radable goods in bo counries is given by: = = x X + ( x ) = X f and = ( x ) = X + = x X f Demand for inermediae radable goods is: X = x T ; X = ( x ) T were X f = ( x ) f T Z () ; X f f (f) = x T Z d ; f f (f) df : 9

10 Te price of nal radable goods is: T = x + ( x ) f and T = x + ( x ) f Since we assumed a e law of one price olds for inermediae goods, i also olds in e aggregae, suc a = S, and f = f S, were S is e nominal excange rae. A.3.3 Nonradable Goods roducers Te price seing equaions are given by e following opimal expressions: ^p N N = ( ) E 8 >< >: X k k N +k k= X k k N +k k= ky s= ky s= ( N +s ) ' N N +s ( N +s ) ' N N +s MC N +k +k N +k +k +k 9 >= >; were MC N = W Z NA ; = C N + G N Te evoluion of e price level of nonradables is N N N N 'N + ( N ) ^p N were N = N. N 2 Te producion funcion is: = A Z N L N : In e foreign counry ese expressions are: ^p N N = ( ) E 8 >< >: X k k N +k k= X k k N +k k= ky s= ky s= 'N N +s MC N N +s ( N +s ) ' N N +s +k +k N +k +k +k 9 >= >;

11 were MC N = = C N Te evoluion of e price level of nonradables is N W Z N A ; + G N N N N 'N + ( N ) ^p N were N = N N 2 Te producion funcion is:. = A Z N L N : A.3.4 Inermediae Traded Goods roducers Te price seing equaions are given by e following opimal expressions: 8 X ky ^p >< k k ( +s ) ' +k MC +k Y +s = ( ) E k= s= X ky >: k k ( +s ) ' +k +k +k Y+k were k= s= MC = W Z A ; Y = X + X : Te evoluion of e price level of nal radables is ' + ( ) ^p +s +k 9 >= >; were =. 2 Te producion funcion is: Y In e foreign counry, is expressions are: 8 X ^p f >< k k f +k f = ( ) E k= X >: k k f +k k= = A Z L ky s= ky s= f 'f +s MC f f +k Y f +k +s f 'f +s f +s f +k +k Y f +k 9 >= >;

12 were MC f = Y f W Z f A ; = X f + Xf : Te evoluion of e price level of nal radables is f f f f 'f + ( f ) ^p f were f = f f 2 Te producion funcion is:. Y f = A Z f L f A.3.5 Moneary olicy Moneary policy in bo counries is conduced wi a Taylor rule a arges CI in aion and oupu grow deviaion from seady-sae values: R = R ( r ) R r = ( r ) Y =Y ( + g r ) y exp(" r ) R = R ( r ) R r = ( r ) Y =Y + g ( r ) y exp(" r ) A.3.6 Demand Socks G T = ( G T ) ( G T ) (G T ) G T exp(" GT ) G N = ( G N ) ( G N ) (G N ) G N exp(" GN ) = ( G T ) ( G T ) (G T ) G T exp(" GT G T G N ) = ( G N ) ( G N ) (G N ) G N exp(" GN ) A.3.7 Trade Balance and Ne Foreign Asse Dynamics We presen e evoluion of e rade balance and ne foreign asses of e ome counry, since e de niion ose in e foreign counry will mirror ose in e ome counry. Holdings of foreign bonds depend on e rade balance (NX ) as follows S B F = R SB F Y S B F + NX 2

13 Since inernaional rade only occurs a e inermediae goods level, ne expors equal expors minus impors of inermediae goods: NX = X f Xf Finally, we de ne nominal GD o be equal o aggregae nominal privae and public consumpion, ence Y = T (C T + G T ) + N (C N + G N ). 3

14 B Appendix: Log-Linear version of e Model Euler equaions bc = ( + g b) (r E 4p + ) + ( + g) E c + + ( + g b) b (9) b c = ( + g b ) r E 4p + + ( + g) E c + + ( + g b ) b (2) Risk saring E (q + q ) = ( + g) E c + ( + g b) bc ( + g) E c + b c ( + g b ) (2) + b b + b were SB () Y; b = F Y : Te labor supply scedules are given by: + g ~ = 'l + + g b ~ = 'l + g + + g b ec ec b ( + g b) ec b + ( + g b) "a (22) b b ( + g b ) ec + ( + g b ) "a (23) Tecnology ey = l + z " a (24) ey = l + z " a (25) ey N = l N + z N " a (26) ey N = l N + z N " a (27) Consumer price in aion 4p = c 4p T + ( c ) 4p N (28) 4p = c 4p T + ( c ) 4p N (29) Tradable in aion 4p T = x 4p + ( x ) (4p f + s ) (3) 4p T = x (4p s ) + ( x ) 4p f (3) rice seing in e nonradable secor 4p N ' N 4p N = E 4p N + ' N 4p N + N ew z N N (32) 4

15 4p N ' N 4p N = E 4p N + ' N4p N + N ew z N N (33) were N = ( N ) ( N ) = N ; N = ( N ) ( N ) = N ; N = p N p ; and N = p N p. rice seing in e inermediae radable good secor 4p ' 4p = E 4p + ' 4p + ew z T 4p f ' f 4p f = E 4p f + ' f 4pf + f ew z f were = ( ) ( ) = ; f = ( f ) ( f ) = f ; = p p T ; f T = p T p ; and T = p T p. Final consumpion demand Inermediae radable and nonradable demand Relaive rice Index f T = p f (34) (35) p T ; ec T = " T + ec (36) ec T = " T + ec (37) ec N = " N + ec (38) ec N = " N + ec (39) ex = + ey T (4) ex = + ey T (4) ex f = f + eyt (42) ex f = f + ey T (43) Le s de ne = pf p, since e law of one price olds = = p p f following relaive prices as a funcion of. ; en we can wrie e = ( x ) (44) = x (45) f = x (46) f = ( x ) (47) Relaive prices = + 4s + 4p f 4p (48) 5

16 T = T + 4p T 4p (49) T N = c c T (5) = T + 4p T 4p (5) N = c c T (52) q = q + 4s + 4p 4p (53) Taylor rules r = r r + ( r ) 4p + y 4y + " r (54) Ne foreign asses and ne expors r = rr + ( r ) 4p + y4y + " r (55) e b + g e b = fnx (56) fnx = Xf Y ex ex f were e B b = F S is e deb o GD raio, and fnx Y = NX Y, and were we ave assumed balanced rade in e seady sae. To solve for e seady-sae raios, we ave a: (57) Y = GT G = CT C = c Y = GN G X = x; Terefore radable GD over oal GD is X Y = CN C = c X f = x = X Y = x c : Hence Marke clearing: X f Y = Xf X T X T Y = ( x) c : ~y T = ( )~c T + ~g T (58) 6

17 were equals e fracion of governmen spending over oal oupu. For e nonradable good, e marke clearing condiion is: ~y N = ( )~c N + ~g N (59) Finally, for e inermediae radable goods secor: ey = x ex + ( x ) ex (6) Toal real GD oal labor For e foreign counry: ey = c ( T + ey T ) + ( c ) ( N + ey N ) (6) l = X X + l + X + ln (62) ~y T = ( )~c T + ~g T (63) ~y N = ( )~c N + ~g N (64) ey f = xex f + ( x) ex f (65) ey = c( T l = Mapping variables in e model wi observable variables. Socks + ey T ) + ( c) ( N + ey N ) (66) Y f Y + lf + Y f + l N (67) ec ec = 4c " a (68) ec ec = 4c " a (69) ey ey = 4y " a (7) ey ey = 4y " a (7) = + " = + " z = Z; z z f + " Z; = Z;f z f + "Z;f 7

18 z N = Z;N z N + " Z;N z N = Z;N z N + " Z;N g T = G;T g T + " G;T g N = G;N g N + " G;N g T g N = G;T g T + " G;T = G;N g N + " G;N and " a ; " r ; " r are iid socks. 8