DBJ Dscusson Paper Seres, No.20 A mcroeconomc foundaton for optmum currency areas: The case for perfect captal moblty and mmoble labor forces Masayuk Otak (Insttute of Socal Scence, Unversty of Tokyo) May 202 Dscusson Papers are a seres of prelmnary materals n ther draft form. No quotatons, reproductons or crculatons should be made wthout the wrtten consent of the authors n order to protect the tentatve characters of these papers. Any opnons, fndngs, conclusons or recommendatons expressed n these papers are those of the authors and do not reflect the vews of the Insttute.
A mcroeconomc foundaton for optmum currency areas: The case for perfect captal moblty and mmoble labor forces Masayuk Otak Insttute of Socal Scence, Unversty of Tokyo ohtak@ss.u-tokyo.ac.jp JEL classfcatons: E42, E44, F33, F5 Keywords: Optmum currency area, Captal wthout a natonalty, Non-cooperatve game between central banks, Dsparty n ncome dstrbuton, Ineffcency of resource allocaton Abstract Ths artcle provdes a mcroeconomc foundaton for Mundell s (96) optmum currency area theory. We consder twn countres where labor forces are fxed to each country although the real captal moves nternatonally. When the central bank n each country behaves non-cooperatvely, t wll rase the domestc nterest rate to attract more real captal and ncrease the rent of her resdences. However, the ferce competton between the central banks ultmately exacerbates the dsparty n ncome dstrbuton. Moreover, when the real captal does not have a natonalty, the worsened ncome dstrbuton also results n the neffcent resource allocaton. Thus, such twn countres should unfy ther central banks and coordnate ther monetary and nterest polces. In other words, these countres consttute an optmum currency area.
. Introducton Income dsparty s not lmted to developng countres. Advanced economes also face ths growng problem. Ths artcle consders why such an undesrable economc consequence s nvoked when constructng a mcroeconomc foundaton for the optmum currency area theory orgnatng from Mundell s (96) semnal work. As Mundell (96) emphaszes, the moblty of producton resources plays a crucal role when we consder whch economes should consttute an optmum currency area. We deal wth the case n whch both labor forces are mmoble and have a natonalty but real captal wth no natonalty can move nternatonally at the owners dscreton. Such a settng s plausble when we observe that foregn drect nvestment s generally preferable to certfyng work vsas. When small twn countres wth dentcal economc structure are n ths stuaton, ther central banks compete to nvte more real captals to enrch ther economy as long as the countres attan full employment. Nonetheless, such competton has the followng devastatng consequence. If one central bank pursues a hgh-nterest polcy to attract more real captal, the other central bank counteroffers wth a hgher nterest rate. Such a cumulatve process does not cease untl a surplus from workng that s the benefts of the hgh nterest polcy vansh entrely. Consequently, the non-cooperatve behavor of two central banks brngs about a serous ncome dsparty between captal and labor. Furthermore, snce captals are assumed to have no natonalty, the emergng dsparty also results n the large welfare losses for these two natons. The unfcaton of two central banks s desrable for overcomng such a dffculty. The small twn countres should be at least economcally ntegrated. Then, the same amount of money s suppled to ensure full employment and the nterest rates offered to non-natonalty real captals become dentcal. Accordngly, each country s suppled wth an amount of real captal, and thus, the ncome dsparty and neffcent resource allocaton wthn the natons are entrely resolved. That s, the small twn countres consttute an optmum currency area whenever the real captal moblty s complete. The remander of ths paper s structured as follows. In Secton 2, we construct a small twn country model based on Otak (2009). In Secton 3, we compare the non-cooperatve and cooperatve monetary polces, and prove the nevtablty of optmum currency areas. In Secton 4, we provde bref concludng remarks.
2. The Model 2. Structure of the Model We use a two-perod overlappng generaton model n a producton economy wth money. The world conssts of twn countres A and B, whose economc structures are dentcal, and the rest of the world. Each country has resdents who cannot move elsewhere and who lve n two perods wth the densty [0, ]. Each resdent specalzes n producng one dfferentated goods wth the help of real captals when he/she s young. Real captal, whose owners have no natonalty, exsts wth the densty [0,] [0,2]. Hence, each resdent can potentally deploy the real captals wth the densty [0,]. Captal ncome s also earned when the owner s young, and then captal tself s passed to a descendant. Once an owner determnes the locaton of hs/her captal, he/she lves n that country even after hs retrement. The mnmum rate of return from the rest of the world ( r ) s guaranteed to all captal owners. Furthermore, for smplcty, a unt real captal combned wth a resdent s busness sklls produces a unt good. The ncome dstrbuton between resdents and captal owners s determned by a negotaton. The negotaton process, whch was developed by Otak (2009), s assumed to be the followng two-stage game. Frst, a resdent determnes how much captal to deploy n order to maxmze hs/her ncome from busness sklls. Second, gven the volume of captal deployed and goods produced, the resdents and owners mutually determne the ncome dstrbuton n accordance wth the asymmetrc Nash barganng soluton. In addton, there s a central bank n each country, whch pursues the socal welfare of her resdents. Each central bank s polcy varables are the nomnal money supply and the real nterest rate (.e., the rate of return for captal). We assume that a central bank manpulates the real nterest rate by ntervenng n the negotaton process between her resdents and the non-natonalty captal owners. That s, a central bank can control the barganng power of her resdents through moral suason. 2.2 Constructon of the Model 2.2. Indvdual s Utlty and Consumpton Functons We assume that all ndvduals (ncludng captal owners) have the same concave and lnear homogenous lfetme functon:, 2 0 U u( c c ), c [ [ c ( z)] dz], where c ( z) s the consumpton of good z durng the -th stage of lfe. We can derve 2
the followng correspondng ndrect utlty functon IU : Yt yt Yt pt IU, yt,, () ( pt, pt ) (, ) pt pt where Yt s the nomnal ncome and pt s the prce ndex defned by 0 t p [ [ p ( z)] dz]. t Furthermore, the consumpton functon of the younger generaton C s C c( ) y t. (2) Fnally, the demand functon for good z s pz ( ) d Dz ( ) [ ] y, (3) p where y d s the aggregate demand. 2.2.2 The Producton Process by the Two-Stage Game To develop the aforementoned producton process, we consder the followng two-stage game. I. Each resdent maxmzes hs/her ncome from busness sklls by deployng non-natonalty captals. II. The resdent and captal owners negotate ther ncome dstrbuton n accordance wth the asymmetrc Nash barganng soluton the threaten pont of whch s [0, r ]. We must solve ths problem by backward nducton. In the second stage game, the correspondng generalzed Nash product Gz (, ) s GP(,) z [ p() z r ] [ r r ], (4) where s the genune barganng power of a resdent and s the modfed (actual) power of country s ' central bank; that s, denotes the power of moral suason of country s central bank. r s the domestc rate of return for a unt captal. The shape of the product s derved from two propertes of the model. Frst, the objectve functon s lnear on the real ncome as ndcated by (). Second, the producton (or demand) volume s already determned by the frst stage of the game. Maxmzng (4) wth respect to r, we obtan the equlbrum domestc rate of return r : r [ ] p( z) [ ] r. (5) Takng (5) nto consderaton, the maxmzaton problem of the frst stage can be expressed as 3
() z max [ pz () r ] Dz () [ ]max [ pz () rdz ] (). (6). p( z) p( z) The soluton to (6) s r ( ),. p z z Hence, the prce level p s constant over tme and the equlbrum nflaton rate s. Substtutng (3) and (7) nto (6), we obtan (7) ( z) d [ ] y, (8) p where d y denotes the real GDP of country. From (), t s clear that (8) corresponds to the socal welfare of resdents n country. 2.2.3 The Market Equlbrum In both countres, money s suppled through the unexpected transfer to the older generaton; thus, takng (2) nto consderaton, the equlbrum condton for the domestc aggregate goods market becomes d d d m y c() y m y, (9) c() where m denotes the real money supply wthn country. The second term of (9) corresponds to the aggregate expendture of the older generaton. Owng to the perfect moblty, the real captal market acheves equlbrum when k j 2, f > =, f = j 0, f < j (0) where k s the amount of captal that has been nvested n country. The model contans fve types of endogenous varable ( r, p,, y d, k ), two types of exogenous varable (, m ), and fve structural equatons (5), (7), (8), (9), and (0). Thus, the model s closed. 2.3 The Non-Cooperatve Game between Central Banks and the Dsparty n Income Dstrbuton Because of the nternatonal moblty of real captals and the representaton of socal welfare (8), each central bank s eager to attract more captal and enrch ts country. Such competton s descrbed by the followng two-stage game. In the frst stage, 4
A B central banks determne (, ). Next, they decde how much money they supply (.e., A B ( m, m )). To solve the equlbrum of ths game, we must begn wth the second stage. Snce the outcomes of the frst stage are summarzed by (0), takng the socal welfare (8) and the equlbrum condton for each aggregated goods market (9) nto consderaton, the best response of each central bank s to mantan the full-employment equlbrum, whch s defned as the amount of real captal that s assocated wth ts country. Hence, the followng domnant strategy n ths game corresponds to the result of the frst-stage game. That s, j 2, f > j m =, f =. j 0, f < () Snce full-employment s assured n the second stage, central banks strve to nvte as much real captal as possble. The followng theorem holds concernng the unqueness of the Nash equlbrum: Theorem. The unque Nash equlbrum s characterzed by p ( m,, ) (,,0). (2) Proof. <Suffcency> If (2) s satsfed, there s no actve ncentve to dverge the strateges because no addtonal gan s obtaned by lesser ( m, ). Hence, (2) s a Nash equlbrum. <Necessty> Suppose that s strctly postve n some Nash equlbrum. Then, p By selectng a much as p p j slghtly larger than j j. [ j ] 0., country j mproves her socal welfare as Thus, there s an ncentve to dverge from the equlbrum. Ths s a contradcton. The economc mplcaton of Theorem s qute serous. As long as two central banks extend the non-cooperatve game to attract more captal, the ncome dsparty deepens 5
aganst ther ntentons. Owng to the competton, resdents earnngs from busness sklls are utterly absorbed by the captal ncome. Snce, as seen n (8), the socal welfare of resdents s proportonal to ther ncome, the deepenng ncome dsparty also results n a less effcent economy. Such a phenomenon s promnent n East Asa, for example. In ths area, foregn drect nvestments flow manly from Japan and Chna to other countres. Although the captal accumulaton suffcently advances and a lmted number of captalsts surprsngly become rch, the labor ncome of most resdents ncludng those n Japan and Chna stagnates. Income dsparty s one of the most urgent problems n East Asa. 3. The Optmum Currency Area as the Unfcaton of Central Banks In the prevous secton, we showed that the non-cooperatve actons of central banks have qute harmful effects on both countres. In ths sense, we propose the unfcaton of central banks. Accordng to Mundell (96), a currency area s defned as follows: ``A sngle currency mples a sngle central bank (wth note-ssung power) and therefore a potentally elastc supply of nterregonal means of payment. (p. 568) We adopt ths defnton of a currency area. When central banks are unfed and a currency area s formed, the twn countres, A and B, can be treated as a sngle country, and hence, monetary coordnaton becomes possble. Because of the symmetry of the countres, the optmal coordnaton polcy s also symmetrc. Hence the two-step game extended n 2.3 requres equal allocaton of A B A B real captal. Thus, we obtan m m and 0. It s evdent that the socal welfare of each country (8) becomes. Ths s the maxmal value that each country attans. In ths sense, these twn countres together consttute an optmum currency area. 4. Concludng Remarks We reconsder the theory of optmum currency area from the perspectve of resource allocaton and ncome dstrbuton. The obtaned results are as follows. We concentrate on the case of twn countres under perfect captal moblty and mmoble labor forces. Ths assumpton seems natural f we consder the sgnfcance of the exstence of naton states. When each central bank pursues ts natonal nterests, that s, the socal welfare of ts mmoble labor force (.e., the resdents), dre economc consequences emerge. Each 6
central bank s led to adopt an artfcal hgh nterest polcy because more captal nduced by a rate slghtly hgher than the rates of ts rval central bank brngs about hgher ncomes for the busness sklls possessed by the resdents of that central bank s naton. However, such compettve and escalatng nterest-rasng s devastatng and cumulatve, and t does not end untl all the resdents ncome s absorbed by the real captal wthout natonalty. Thus, a serous ncome dsparty and a large decrease n socal welfare occur. It s clear that the naton s not an optmum currency area. When the two central banks are unfed and the monetary coordnaton becomes possble, such catastrophc competton ceases. Real captal s allocated equally by abolshng the compettve and artfcal hgh-nterest polcy. Just enough external money s suppled to ensure the full-employment equlbrum n each country. Thus, the socal welfare acheves ts maxmum. In other words, our twn countres under perfect captal moblty consttute an optmum currency area. It s also noteworthy that our approach s based on a rgorous dynamc mcroeconomc foundaton. In ths sense, we succeed n updatng and extendng the theory of Mundell (96). Fnally, we must note some lmtaton of our work. The frst s the dffculty of central bank unfcaton. Bureaucrats who operate the unfed central bank may be of dfferent natonaltes. Dfferences due to culture, ethncty, tradton, and etc., are not as easy to overcome as our theory assumes. It takes more tme than we expect to ensure fare polcy coordnaton. The second concerns the glut of foregn drect nvestment. In realty, the volume and range of moble real captal s large and wde. It s feared that the world as a whole may become a unque optmum currency area. Nevertheless, t s certan that such a tremendous and enlarged organzaton would never work well. It may be more practcal to place a levy on nternatonal captal movement, lke the Tobn tax. References [] Mundell, R. A. (96), ``A theory of optmum currency area, Amercan Economc Revew 5, pp. 657-665. [2] Otak, M. (2009). ``A welfare economcs foundaton for the full-employment polcy, Economcs Letters 02, pp. -3. 7