Multi-National Public Goods Provision under Multilateral Income Transfers & Productivity Differences

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2 Mult-atonal Publc Goods Provson under Multlateral Income Transfers & Productvty Dfferences by Tatsuyosh Myakosh Laxun Zhao Hose Unversty Kobe Unversty Abstract Ths paper examnes multnatonal publc goods provson under multlateral ncome transfers and productvty dfferences across countres. We assume the exstence of a planner who uses lnear approxmaton for utlty maxmzaton for all countres. The man fndngs are: () A country s an ncome recever f t has an advantage n producng publc goods; () the planner country can determne the values of transfers for all countres wth an adjustment cost; () all countres obtan an dentcal level of utlty; (v) the country wth the lowest adjustment cost s the best canddate for the planner country. All results are derved based on well-known nformaton regardng the cost of producng the publc goods and on ncome levels. Keywords: nternatonal publc goods; multlateral ncome transfers; productvty dfference; planner; adjustment cost; welfare JEL classfcaton: H4, F Please address correspondence to: : Faculty of Scence & Engneerng, Hose Unversty, Tokyo, Japan; myakosh@hose.ac.jp. (Correspondng author): Research Insttute for Economcs & Busness, Kobe Unversty, Kobe , Japan; E-mal: zhao@reb.kobe-u.ac.jp; Fax:

3 . Introducton Internatonal publc goods (possbly ncludng laws & safety, polces, nfrastructure, organzatons and sometmes even personnel, etc.) are usually systematcally underprovded by prvate market forces, and such under-provson can have mportant cross-border externalty effects. For nstance, polluton orgnatng n some countres may affect health status n others, terrorsts may travel from one country to another, fnancal volatlty (such as the Federal Reserve Bank s recent QE and QE (quanttatve easng)) n one naton may generate follow-on fraglty elsewhere, and so on. Often, natonal polcymakers are less lkely to consder the well-beng of foregn ctzens n settng ther own polces regardng publc goods. Thus, the provson of nternatonal publc goods requres some form of multlateral coordnaton and even ncome transfers, especally when multple countres are nvolved. The present paper examnes the nternatonal provson of a publc good n a setup of multple countres, by ntroducng multlateral ncome transfers and adjustment costs under productvty dfferences across countres. We assume the exstence of a planner country who adopts lnear approxmaton for utlty maxmzaton and an adjustment cost for ncome transfers. The followng man results are obtaned: Frst, country becomes an ncome recever f t has an advantage n producng publc goods; second, the partcular sze of multlateral ncome transfer can be pnned down for each country; thrd, all countres obtan an dentcal level of utlty; fourth, a decrease n adjustment cost leads to a Pareto mprovement, and thus, the country wth the lowest adjustment cost s the best canddate for the planner country (e.g., a country wth effcent nsttutons or strong

4 poltcal powers). These results are obtaned under well-known nformaton on the cost of producng publc goods and on the ncome levels for all countres. A large body of lterature has evolved snce Warr s (983) semnal work that shows the real equlbrum s unaffected by ncome transfers when publc goods are prvately provded. Bergstrom et al. (986, 99) generalze and renforce ths result by gvng proofs for exstence and unqueness of equlbrum. Usng a two-country model, Ihor (996) nvestgates the welfare effects of publc goods provson wth productvty dfferences across countres, and shows that an ncome transfer from a low productvty (or hgh cost) country to a hgh productvty country brngs a Pareto mprovement. As an example, Murdoch and Sandler (986, p.84) study the provson of weapons wthn the orth Atlantc Treaty Organzaton (ATO). Subsequently, a few authors have examned blateral ncome transfers between two or more countres under productvty dfferences (e.g., Ihor (999), Caplan et al. (000), Km and Shm (006) and Cornes and Hartley (007)). Cornes and Itaya (00) examne the case of several publc goods and show that voluntary contrbuton typcally generates not only too low a level of publc good provson but also the wrong mx of publc goods. However, there s lttle lterature on multlateral ncome transfers across multple countres, especally wth regard to the followng ssues. Frst, how are the dfferent productvtes among the multple countres compared, and whch countres are ncome senders or recevers? Second, how much do ncome recevers receve? Thrd, whch country manages the ncome transfer? To our knowledge, the present paper s the frst to nvestgate the effects of multlateral ncome transfers across multple countres wth an explct planner rule, partcularly emphaszng the three questons rased above. In Ihor (996) and Cornes and Hartley (007) provde very nformatve surveys.

5 addton, the methods we adopt n the model, e.g., lnear approxmaton for utlty maxmzaton by the planner and the adjustment cost for ncome transfers, are convenent but stll not wdely used n the lterature. The rest of the paper s organzed as follows. Secton outlnes the basc model. Secton 3 bulds the benchmark case of blateral ncome transfers. Secton 4 deals n detal wth multlateral transfers among multple countres. Secton 5 llustrates our equlbrum wth the case of two countres. And fnally Secton 6 concludes.. The Basc Model Consder countres wth one publc good and one prvate good. Country (= ) consumes an amount x of the prvate good and contrbutes an amount g to the supply of the nternatonal publc good. The total supply of the nternatonal publc good, G, s just the sum of g provded by each country. Country s utlty s gven by U U( x, G), where U s strctly ncreasng and quas-concave, and x and G are normal goods for each country. Country s budget constrant s gven by x p g y, where y 0 s ts exogenously gven natonal ncome and p 0 s the relatve prce (cost of producton) of publc goods n terms of prvate consumpton n country. As n Ihor (996), a low (hgh) p means a hgh (low) productvty n producng the publc good. We also make the Cournot ash assumpton that each country beleves that the contrbutons of others are ndependent of ts own. Then, we can rewrte G g g j as the total publc j good, where s the sum of g j provded by countres j other than. g j j 3

6 Defnton. A Cournot ash equlbrum n ths model s such that for each ( = l,,...,), ( x, G ) solves: max x, G U ( x, G) st.. x pg y p g, j j x 0, G g 0, U j j,,.., () As s well known, n (), each country mplctly chooses not only ts own supply g of nternatonal publc goods contrbuton, but also n effect the equlbrum level of the total supply G, and each country consumes the same level of G. The exstence and unqueness of ths soluton are proved by Cornes and Hartley (007). The frst order condton for each s p U / x U / G ( =,,..,). Utlzng the mplct functon theorem, the equatons can be solved to gve x ( =,,,). ext, we use Defnton and the expendture functon to analyze the effects of ncome transfers, followng Ihor (996). Defne the expendture functon at the Cournot ash equlbrum as: E mn x pg st.. U( x, G) U () xg, where U U( x, G ) s the utlty level at the equlbrum for country. Then, E EU p, and at the equlbrum, t s equal to the ncome n (): (, ) EU (, p) yp g, (3) j j Warr (983) sums the budget constrants of the countres and then obtans G under p = for all. In contrast, p n our model, and the equlbrum G wll be derved soon as n (5a) and (5b). 4

7 where the ncome on the rght-hand sde contans actual ncome y and the externaltes from other countres provson of the publc good. By Shephard s Lemma, we can derve: EU p (, p) GU (, p), GU (, p) G, (4) where G U p s the compensated demand functon of country for the nternatonal (, ) publc good. From (3) and (4), the equlbrum can be summarzed by:, (5a) (, ) ( ) EU p y G GU p G, (5b) (, ), (,,..., ) where p, p and (5a) s obtaned by multplyng (3) by and j j summng up. These + equatons determne the utltes amount of the nternatonal publc good U of the countres and the G at the equlbrum. Ths formulaton s smlar to Ihor (999, p.48-49), extendng Ihor s (996) two-country model to countres. Let us now use the above framework to examne both blateral and multlateral ncome transfers. Takng the total dervatves of (5a) and (5b) consstng of + equatons gves:, (6a) ( ) EUdU dg dy Gu du dg 0, ( =,,..,), (6b) where E ( U, p )/ U E and U G( U, p )/ U G. Substtutng (6b) nto (6a), we U can decrease the number of equatons from + to. Moreover, by usng the relatonshp of the expendture functon () for country, 5

8 E U x U pg U, (7) we can rewrte (6a) and (6b) as follows:, (8a) ( x pg ) du E du U U ( ) G du dy U u G du G du 0, (,j =,,), (8b) u ju j whch can be arranged n vector and matrx format, ( x UpG U( )), EU,..., EU du dy G U, G U, 0,0...,0 du 0 0, G U, G3 U,0..., , (9a) , 0, 0,0... G ( ) U, GU du 0 where the determnant of the matrx on the left hand sde s: ( x pg ( )) G U U U ( x pg ( ))( G G ) G E ( G G G ), U U U U U U j U ju U (9b) and, are sub-determnants of. Usng (7), we have EU xu pgu, whch can be nserted nto (9b) to yeld: ( x pg ( ))( G G ) ( G G G ) U U U U U U U G x ( G G G ) U U j U ju U ( x pg )( G G ) G x ( G G G ). U U U U U U j U ju U (0) The partal dervatves of the compensated demand functon, x U and G U, are postve n terms of utlty because they are normal goods. Then, the determnant s postve. 6

9 3. Blateral Income Transfers Under blateral ncome transfers, ncome s transferred from country j to f p ; that s, n (9a): dy dy 0, dy 0 for all k, j. Then, j p j k dy ( j) dy () The welfare effects of ths transfer are derved usng Cramer s rule on (9a): ( )( ) du p p G dy p p j k ku j 0, (a) du ( p p )( ) G dy p p j j j k j ku j 0. (b) Condtons (a) and (b) are the same as n Ihor (996), Cornes and Hartley (007), etc. That s, a transfer from country j wth low productvty (hgh prce p j ) of provdng the publc good to country wth hgh productvty (low prce p ) mproves welfare n both countres. 4. Multlateral ncome transfers ow, we consder multlateral ncome transfers among many countres. We frst determne the planner, whch can be anyone but only one among the countres. The planner country h calculates the utlty ncrease du h from the present level by takng a frst order lnear approxmaton around the present utlty level, as n Myakosh et al. (00). Intutvely, the planner uses a gradent method to transfer the ncome n order to acheve the hghest utlty. Ths can be done by takng a lnear approxmaton startng at the present levels of all players, and then step by step, approachng a hgher level. Gven the above, the ncome transfers are 7

10 Uh( yy, yy,..., yy) Uh( y, y,..., y) ( U / ) h y y (3a) In dervatve format, du U ( y dy, y dy,..., y dy ) U ( y, y,..., y ) h h h ( U / ) h y dy (3b) where dy s ncome transfer to country, and U y s the partal dervatve of utlty h / n ncome (solved from (9a)).Then, the planner maxmzes the utlty transferrng ncome to each country. U h (or d U h ) by Wth multlateral ncome transfers, t s often the case that whle country A s a recever from country B, t may be also a sender to country C. Here we defne a recever (sender) as a net recever (sender) of ncome. The planner country collects from the net senders and redstrbutes to net recevers, for whch we assume a transfer fee s requred, that can be nterpreted as admnstratve costs for back and forth complcated transactons among multple countres. Observe that ths type of transfer fee to our knowledge has not been proposed yet. We assume that the fee s pad wth past donaton or endowment, whch sgnfcantly smplfes our analyss. 3 Then, the maxmzaton problem of the planner can be reformulated as: MaxU { y } a y h h.. 0,, st y y, (4a) 3 As wll be proven later, the planner can obtan Pareto-mprovement after ncome transfers. Thus, ths fee can be pad wth endowment frst and returned later wth a fracton of the ncome gans, as long as the fee s suffcently small. 8

11 where U du, y dy, a U / y, 0 (4b) h h h h The varables wth a ' tlde' are devatons from an ntal allocaton, as contaned n (4b). The frst constrant n (4a) means that the ncome transfers are mplemented wth fxed total ncome across the countres; the second constrant mples all ncome transfers are mplemented wth a gven total cost 0, where the unt cost s ~ y n any country (t s squared because for a postve transfer, ~ y 0 (ncome recever), and for a negatve transfer, ~ y 0 (ncome sender)). Techncally, snce we have adopted lnear approxmaton for utlty maxmzaton, n order to ensure the exstence of an nteror soluton, some form of nonlnear constrant s needed, and thus we assume the transfer fee takes a nonlnear form as n the second constrant. Fnally, a h s derved from (9a) as follows: a U 0, ( =,,.h,.,). (5) h h k hgku y Then, the Lagrange equaton and Kuhn-Tucker condton for (4a) are respectvely: and (,, ) (0 ) ( ) h L y a y y y (6a) (). a y 0, ( ). y 0, h ( ). y, ( ). ( ) 0, ( ). 0 v y v (6b) 9

12 The multpler s nonnegatve and depends on the prce parameters out of a h n (4a) and (5). In partcular, s zero when p s the same across countres, whle t s postve when p dffers. 4 can be solved by nsertng condton () nto condton () n (6b) f > 0 (.e., at least one prce s dfferent). Then, at least one a h for country s dfferent n (5) such that a 0. Hence > 0, and ~ y 0 n condton () n (6b). h To be more specfc, when > 0, condton (v) n (6b) can be rewrtten as y. Usng condton () n (6b), we get y~. a 0,, ( ). (7) h a h y ah a h On the other hand, by nsertng y~ n (7) nto condton (v) n (6b), we have: ( a a ) h h. (8) Also we obtan ~ y, ~ y,..., ~ y by substtutng (8) nto (7): ( ahj a ) h y, (j =,,.h,.,). (9) j ( a a ) h h Then, by nsertng (5) nto (9) and denotng b / p (country s productvty of producng the publc good), we can get the followng soluton { y~ j } ( j =,,.h,.,): 4 If all prces are the same, (5) gves ah condton () n (6b), we have a 0. Then condton h h, dentcal for all. In fact, aggregatng condton () and usng () n (6b) means ether = 0 (n ths case y s not determned) or y = 0 for all. 0

13 khgku j bj b y j (0) ( ) a h a h b b The denomnator s not zero, gven dfferent prces across countres. ow let us nterpret (0). The frst fndng s, when country j s productvty b j s hgher than the average ( b good, and thus t becomes an ncome recever ( ), then t has an advantage n producng the publc y~ j >0). Secondly, gven a partcular level of the adjustment cost ζ, the planner can determne the sze of ncome transfers for all countres, 5 by maxmzng the utlty for each country. Specfcally, for the planner country h, usng nformaton on xu for and { G, G,..., G }, the optmal utlty s obtaned by nsertng (0) nto (4a): U U U du h jkhgku j bj b b b. () And for any other country o, we can approxmate ts utlty by nsertng (0) nto (3b): du o jk ogku j bj b b b. () 5 Cornes and Hartley (007, Corollary 4.) mply a soluton for blateral ncome transfers where only the country wth the hghest productvty provdes the publc good. However, ther soluton does not show how much publc good there s, how much ncome s transferred, and whether ths country stll remans the most productve when t produces a large amount.

14 ote that f the planner s swtched from country h to country o, the optmal utlty level of country o s obtaned by replacng as n (), unsurprsngly. G wth kh ku G n (), whch s the same ko ku Thrdly, from the above, all countres obtan the same maxmum utlty level, d U. When the transfer fee s treated as sunk as n (0), then the optmal ncome transfers do not depend on the dentty of the planner,.e., any country could be the planner. Fnally, however, f the unt adjustment cost ( y as n (4a)) for ncome transfers s reduced, the utlty levels for all countres ncrease, yeldng a Pareto mprovement. Thus, the country wth the lowest adjustment cost s the approprate canddate for the planner country. In realty the level of adjustment cost may depend on country sze, nsttuton effcency, poltcal powers, and so on. ote especally that the above fndngs are all derved based on well-known nformaton on the cost p of producng publc goods, and on the ncome level y. 5. An llustraton of Multlateral Income Transfers We provde an llustraton usng the example of =. Due to (4a), the maxmzaton problem of a planner country (say country ) s as follows: ~ Max U ~ y, ~ y } { a ~ y a ~ y, (3a) s t. ~ y ~ y 0, (3b). ~ ~ y y. (3c) Moreover, usng (5) we have:

15 a G U / 0, a GU / 0 and p, p. (4) Then, the objectve functon n (3a) can be rewrtten as: U G p y p y / A p y p y : A G / 0 (5) U U Insert Fgure about here In Fgure, the term ~ U / P A dentfes the utlty level, constrant (3a) s the downward slopng sold lne and constrant (3c) s the regon wthn the crcle. Thus, wthout the transacton fee, constrant (3c) dsappears and there s no crcle. If P P, the objectve functon (5) concdes wth constrant (3b). Then the optmum transfers are on constrant (3b) and wthn (3c), but not unquely decded; and the optmum utlty level s zero. ext consder when P P (we take the case of P P ). The objectve functon (5) reaches the maxmum utlty for planner country at pont C. The optmum ncome transfer exsts at pont C wth ~ y 0 and ~ y 0 and the utlty level s A p ~ y p ~ y. Also note that f constrant (3c) does not exst, the optmal ncome transfer ~ y 0 endlessly decreases to obtan the maxmum utlty, by movng upwards on the downward slopng sold lne. Ths formulaton of maxmzaton wthout (3c) s used n prevous studes. Meanwhle the non-planner country receves ncome from planner country and obtans utlty level: p ~ y p ~ y B where B /. On the other hand, f country becomes the planner, ts objectve functon s B p ~ ~ y y p where B G U /. When P P, the optmal ncome transfer s gven G U 3

16 at pont C n Fgure and ts utlty level s obtaned as B p ~ y p ~ y, whch s the same as that of country n the prevous case when t s not a planner. The reason s as follows. The optmal transfers are decded only by the constrants (3b) and (3c). Moreover, the utlty ncrease (see (3a)) s approxmated by a lnear functon of the ncome transfers of all countres. The dfference between the utlty ncreases n countres and s only a constant coeffcent A or B, whch does not affect the optmal y or y. Thus whchever country becomes the planner, t necessarly obtans the ntersecton of the two constrants (3b) and (3c). Fnally, when the unt adjustment cost of ncome transfers ncreases, constrant (3c) becomes hgher and n turn the utlty for each country decreases. The same logc holds vald n a model of (>) countres. 6. Concludng Remarks When countres dffer n productvty, how should ncome be transferred among multlateral countres so that proper publc goods are provded? Ths paper has proposed an explct rule that s effectve: Assumng a planner among countres, that can maxmze ts utlty by lnear approxmaton and transferrng ncome to each country for a fxed adjustment cost. Ths adjustment cost s requred to form agreements among ncome-sendng and ncome-recevng countres. The planner country uses only wellknown nformaton on the cost of producng publc goods and on the ncome levels of all countres. More mportantly, all countres obtan the same level of utlty and the country wth the lowest adjustment cost s the best canddate for the planner country. A very nterestng extenson of our model s to ntroduce outsourcng of publc goods provson. Our mechansm s smlar to the market of emsson permts. To combat 4

17 global warmng, a country can buy emsson permts from other countres (see Bohrnger (005), Rosendahl (008) and Flachsland et al. (009)). Our mechansm allows countres to outsource the producton of the publc good, whch can serve as an alternatve to ncome transfers. Ths remans a frutful avenue for future research. Acknowledgement: Ths paper s a much mproved verson based on Myakosh (008). Earler versons were presented n semnars at the Helsnk School of Economcs and Osaka Unversty, the East Asan Economc Assocaton Conference n Manla (008), and the Conference on Dynamcs, Economc Growth, and Internatonal Trade n Manla (008). We are grateful to J. Välmäk, P. Ilmakunnas, P. Haaparanta, M. Pohjola, P.C. Cruz, B.S. Jensen, S. Ikeda, C.Y. Horoka, conference and semnar partcpants for ther helpful comments. Any shortcomngs reman our own. 5

18 References Bergstrom, T., Blume, L. and Varan, H. (99), Unqueness of ash equlbrum n prvate provson of publc goods: an mproved proof, Journal of Publc Economcs 49, Bergstrom, T., Blume, L. and Varan, H. (986), On the prvate provson of publc good, Journal of Publc Economcs 9, Bohrnger, C.and Lange, A.(005), On the desgn of optmal grandfatherng schemes for emsson allowance, European Economc Revew 49, Cornes, R.C. and Hartley, R. (007), Aggregatve publc good games, Journal of Publc Economc Theory 9, 0 9. Cornes, R.C. and Itaya, J. (00), On the prvate provson of two or more publc goods, Journal of Publc Economc Theory (), Caplan, A.J., Cornes, R.C. and Slva, E.C.D. (000), Pure publc goods and ncome redstrbuton n a federaton wth decentralzed leadershp and mperfect labor moblty, Journal of Publc Economcs 77, Flachsland,C., Marschnsk,R., and Edenhofer,O. (009), Global tradng versus lnkng: Archtectures for nternatonal emssons tradng, Energy Polcy 37, Hamlton, J. and S.M. Slutsky (007), Optmal nonlnear ncome taxaton wth a fnte populaton, Journal of Economc Theory 3 (), Ihor, T. (996), Internatonal publc goods and contrbuton productvty dfferentals, Journal of Publc Economcs 6, Ihor, T. (999), An economc analyss of publc transfers, Japanese Economc Revew 50, Km, J. and Shm, S. (006), Incentve mechansms for nternatonal publc goods under uncertanty of producton costs, Economcs Letters 9, Myakosh, T. (008), A planner of global ncome transfers: nternatonal publc goods and productvty dfferentals, Dscusson Paper 08-38, December. Myakosh, T., Tsukada Y., Kono, T., and Koyanag, M. (00), Economc growth and publc expendture composton: optmal adjustment usng the gradent method, Japanese Economc Revew 6, Murdoch, J.C. and Sandler, T.(986), Complementarty, free rdng, and the mltary expendtures of ATO alles, Journal of Publc Economcs 5, Rosendahl, K.E. (008), Incentves and prces n an emssons tradng scheme wth updatng, Journal of Envronmental Economcs and Management 56, Warr, P. (983), The prvate provson of a publc good s ndependent of the dstrbuton of ncome, Economcs Letters 3, 07 6

19 ~ y ~ y C( 0 ~ y, ~ y ) ~ y ~ y ~ y ~ y ~ C U / P A 45 P P ~ ~ y U / P A ~ y P / P Fgure : The case of = 7