Internationally Mobile Factors of Production and Economic Policy in an Integrated Regional Union of States

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1 Dscusson Paper No Internatonally Moble Factors of Producton and Economc Polcy n an Integrated Regonal Unon of States Perry Shapro Jeffrey Petchey August 2001 Adelade Unversty Adelade 5005 Australa

2 CENTRE FOR INTERNATIONAL ECONOMIC STUDIES The Centre was establshed n 1989 by the Economcs Department of the Adelade Unversty to strengthen teachng and research n the feld of nternatonal economcs and closely related dscplnes. Its specfc objectves are: to promote ndvdual and group research by scholars wthn and outsde the Adelade Unversty to strengthen undergraduate and post-graduate educaton n ths feld to provde shorter tranng programs n Australa and elsewhere to conduct semnars, workshops and conferences for academcs and for the wder communty to publsh and promote research results to provde specalsed consultng servces to mprove publc understandng of nternatonal economc ssues, especally among polcy makers and shapers Both theoretcal and emprcal, polcy-orented studes are emphassed, wth a partcular focus on developments wthn, or of relevance to, the Asa-Pacfc regon. The Centre s Drector s Professor Kym Anderson (kym.anderson@adelade.edu.au) and Deputy Drector s Dr Randy Strnger (randy.strnger@adelade.edu.au) Further detals and a lst of publcatons are avalable from: Executve Assstant Centre for Internatonal Economc Studes Adelade Unversty SA 5005 AUSTRALIA Telephone: (+61 8) Facsmle: (+61 8) Emal: ces@adelade.edu.au Most publcatons can be downloaded from our Home page: ISSN seres, electronc publcaton

3 CIES DISCUSSION PAPER 0134 Internatonally Moble Factors of Producton and Economc Polcy n an Integrated Regonal Unon of States Perry Shapro Jeffrey Petchey Centre for Internatonal Economc Studes Unversty of Adelade ces@adelade.edu.au August 2001

4 Internatonally Moble Factors of Producton and Economc Polcy n an Integrated Regonal Unon of States Perry Shapro 1, Jeffrey Petchey 2 Abstract There s a general class of model used to examne the expendture and tax strateges adopted towards moble factors by states that are members of regonal unons (eg., federatons, confederatons or common markets). We develop a varant of ths model, extended to allow for mperfect factor moblty between the regon and the world, and characterze a game between states n whch tax and expendture polces are the strateges. It s shown that a Nash equlbrum to ths game exsts (though there may be multple equlbra). We also argue that, unlke prevous results, states have an ncentve to levy non-zero taxes, or subsdes, on the mgrant factor (wth the precse mx of strateges dependng on the degree of factor moblty). Generally, the strateges adopted lead to an neffcent supply of the factor to the regon, as well as an neffcent dstrbuton across member states. Keywords: regonal unons, tax competton, economc ntegraton, common factor market, labor and captal moblty, externaltes, taxes, subsdes, publc goods. JEL Classfcatons: H21, H41, F15, F20. 1 Department of Economcs, Unversty of Calforna, Santa Barbara, Calforna, and Department of Economcs, Unversty of Melbourne: Emal: pshapro@bgpond.com: Phone: (nternatonal) (Correspondng author): Federalsm Research Program, School of Economcs and Fnance, Curtn Unversty of Technology, GPO Box U 1987, Perth, Western Australa, 6845, Australa: Emal: petcheyj@cbs.curtn.edu.au: Phone (nternatonal)

5 1. Introducton Regonal unons of states nclude exstng federatons and emergng confederatons such as the European Unon. Both forms of unon are characterzed by rules that prohbt member states from mpedng the movement of factors of producton between member states. In other words, they have common nternal factor markets. Usually, member states have retaned suffcent soveregnty over publc expendture and tax nstruments to allow them to nfluence, through ther polcy choces, the dstrbuton of moble factors wthn the common market. The effcency of polcy choces made by states wthn these unons has been examned extensvely usng a fscal competton model that, whle varyng accordng to the partcular applcaton, has certan general features. The model supposes that there are two states, each endowed wth a concave producton technology wth at least two factors of producton, dentcal mmoble ctzens (workers) and a second factor (captal or labor) that s moble between states but whose supply to the regonal unon s fxed. In other words, the unon s assumed to have a closed common factor market. States levy a tax on the moble factor, and sometmes ctzens as well, to provde a publc good whch benefts the mmoble ctzens. The moble factor mgrates between states to satsfy an equal net return condton. Compettve factor markets are assumed so that each factor receves a monetary reward equal to the value of ts margnal product. The moble factor may, or may not, be owned by domestc ctzens (absentee ownershp can be allowed). States choose ther polces to maxmze a payoff functon (often the utlty functon of the mmoble ctzens) subject to budget constrants and the moble factor equal return condton. The structure mples nterdependence between the polcy choces of the states that s modeled by adoptng the Nash equlbrum concept. The polces chosen by states are commonly shown to be neffcent. Two papers that employ aspects of ths general structure are the ones by Wldasn (1991a) and Wellsch and Wldasn (1996). The frst of these supposes a decentralzed economy of two states each wth two types of labor; one rch mmoble group and a group of poor moble workers who mgrate between states to equate per capta utlty. Snce the

6 total supply of moble workers s fxed the model s one of a closed common labor market. The rch control the publc choce process so that the nterests of each jursdcton are those of the rch class. Further, the rch are altrustc towards the poor and hence each jursdcton chooses some level of ncome redstrbuton n favor of the moble poor. It s argued that f sub-natonal governments choose ths redstrbuton the resultng Nash equlbrum s neffcent: redstrbuton creates fscal externaltes that are gnored by competng jursdctons. Varous coordnaton optons and central nterventon are then examned and shown to be effcency enhancng. Wellsch and Wldasn (1996) extend ths framework n at least two mportant ways. Frst, they ncorporate nternatonally moble captal whch mgrates between states and the world to satsfy a condton that ts net rate of return should equal some gven world return. Second, the authors open the common labor market by allowng for the possblty of nternatonal mgraton of the moble group. Ths was a major nnovaton to the general model of common markets wthn regonal unons snce, as noted, untl then the effcency of nter-state competton had been examned wthn the context of models wth closed common factor markets. Wellsch and Wldasn agan characterze an neffcent Nash equlbrum n whch competng sub-natonal jursdctons choose ther polces for some exogenously determned and arbtrary number of nternatonal mgrants. It s shown that the optmal tax/subsdy on moble captal s zero. Ths mrrors the result n Oates and Schwab (1988). The authors also examne how the Nash polces are nfluenced by changes n the arbtrary levels of nternatonal mgraton. They fnd that more mgraton always rases socal welfare f mgrants are net taxpayers and lowers socal welfare f the mgrants are a net fscal burden. A game n whch states drectly choose the level of mmgraton s also consdered. Rather than model the game explctly, Wellsch and Wldasn draw conclusons about ts possble features by referrng to ther results on the Nash equlbrum obtaned usng the assumpton of some exogenously gven level of mgraton. They conclude that f mmgrants are net fscal burdens the Nash equlbrum s one n whch there s no nternatonal mgraton. If mgrants are net fscal contrbutors some 2

7 postve level of mmgraton s chosen by jursdctons. Fnally, a "corrected" Nash equlbrum s examned n whch a central government provdes matchng transfers to each jursdcton (the polcy nstrument s the matchng rate) that take account of the fscal externaltes and acheve an effcent outcome. The authors also note that there s no general exstence proof for ths class of model though one can construct examples, usng specfc functonal forms, where exstence and unqueness are assured 3. Brad (1996) consders a smlar model wth the addton of an ad valorem, rather than a lump-sum tax, on the moble factor. He examnes only the symmetrc Nash equlbrum, one n whch all jursdctons are the same. Hs analyss explots the smlarty n equlbrum polcy choces to examne the predcted outcome. He also assumes that the factor of producton s perfectly, and freely, moble nternatonally. It allows hm, as t has prevous analysts, to assume a constant return to the moble factor of producton. We develop a model of a regonal unon n the sprt of the general model dscussed above. The model s of a two-state regon wth free mgraton of a moble factor between states. An nnovaton of the work s to ncorporate a separate moble factor supply condton that also allows the total supply of the moble factor to the regon to vary dependng on the payoff t receves wthn the regon relatve to the world payoff. Unlke prevous studes, we allow ths mgraton to be mperfect due to mgraton frctons. Therefore, n equlbrum t s possble for the regonal payoff to dffer from the world payoff. The mgraton frcton s captured through a parameter, ε, that we take to be a measure of the level of ntegraton between the regon and the world. The parameter can vary from zero (a completely solated regon) to nfnty (a fully ntegrated regon). Usng ths basc structure, we are able to develop a game wth polces as strateges and characterze a Nash equlbrum n whch the total supply of the moble factor to the regon, and ts regonal dstrbuton, are determned by the compettve tax and expendture polces of the states. 3 Some examples are provded n Wldasn (1991b). 3

8 The frst contrbuton of the paper, though not a drect consequence of the ncluson of the moble factor supply functon, s to provde a general exstence proof for a Nash equlbrum. As noted above, no proof of global exstence has been developed for ths class of models. The exstence result s an example of the well-known Gale Mas- Colell Theorem (1975). It proceeds by transformng the game wth polces as strateges nto a game wth the desred moble factor supples as strateges. Exstence of equlbrum for ths dual game s assured by the applcaton of well-known fxed pont results. Every choce of moble factor supply s assocated wth a best feasble polcy. Thus, the exstence of an equlbrum n the dual game ensures the exstence of an equlbrum n the prmal game, the one n whch the strateges are the states tax and expendture polces. The second result s to show that states have an ncentve to choose a non-zero moble factor tax/subsdy n equlbrum. Ths s n contrast to the fndngs n Wellsch and Wldasn, and Oates and Schwab. The non-zero tax/subsdy s a consequence of two factors. The frst, that s ndependent of the degree of ntegraton, s an externalty generated by the mgratng factor. It tends to make the tax postve or negatve (a subsdy) dependng on the sgn of the externalty. The second, whch s dependent on the degree of ntegraton as captured by ε, we call a hrng cartel effect. If ε s less than nfnte (less than full ntegraton) states have some monopoly power over the moble factor and the hrng cartel effect has a postve mpact on the tax/subsdy. States wll wsh to use ther monopoly power over the moble factor to tax t and redstrbute n favor of mmoble domestc labor. Whether states levy a tax or subsdy depends on the nteracton between the externalty and the hrng cartel effects. Therefore, the zero moble factor tax result n prevous papers s a specal case n our model where the externalty s zero and the regonal common factor market s fully ntegrated wth the world factor market. Allowng for mgraton frctons across the common external border, and externaltes, one derves the result that states wll levy nonzero moble factor taxes/subsdes. 4

9 The thrd result s to show that the magntude of the tax/subsdy depends upon the value of ε, the ntegraton parameter. Specfcally, hgher degrees of ntegraton mply a smaller hrng cartel effect. As states are exposed to more nternatonal competton for the moble factor, ther monopoly power dmnshes. In the lmt, where there s full ntegraton, the tax/subsdy s determned only by the externalty. Fnally, we are able to generate some new nsghts nto the effcency of nterstate competton wthn a regonal unon. It s mportant to note here that our treatment of effcency s somewhat dfferent to the standard one. The usual approach, as dscussed n Wellsch (2000), has been to examne whether publc goods are under or over provded by states that tax a moble factor. Rather than pursue ths ssue, we choose to examne how the tax/subsdy polcy of the states affects effcency n the regonal dstrbuton and total regonal supply of the moble factor. It s shown that f ε s less than nfnte, the monoply power of the states, exercsed through the (non-zero) tax/subsdy, creates a dstorton n the dstrbuton of the moble factor wthn the regon, and neffcency n ts total supply. The presence of ths dstorton depends on the states choosng a non-zero tax/subsdy, and ths, n tself, s a result of the adopton of our more general factor mgraton condton wth mperfect moblty. In a sense then, we dentfy a new source of neffcency assocated wth polcy competton between states. However, n the specal case where ε s nfnte (full ntegraton), the monopoly power of the states dsappears (the tax reflects only the externalty) and the Nash equlbrum results n an effcent dstrbuton of the moble factor and total supply. The paper s organzed as follows. Secton 2 develops the basc set up of the model. Secton 3 examnes the nature of the Nash equlbrum. It provdes the general proof of exstence and characterzes the (non-zero) equlbrum choce of the tax/subsdy. Secton 4 examnes the effect of greater ntegraton on the tax/subsdy polces adopted by states. In Secton 5, the effcency of these polces s examned whle conclusons are presented n Secton 6. 5

10 2. The Set Up of the Model Consder a regonal unon (a federaton or confederaton) of two ndependent states =1,2 each wth a populaton of mmoble resdents wth such hgh attachment to place that they never mgrate between states, or between the regon and the rest of the world 4. These resdents own the land, fxed captal and means of producton (frms) n state, and they each supply one unt of labor. Hence, they can be thought of as domestc workers, captalsts or landowners. From now on we refer to them, for presentatonal convenence, as homefolks, by vrtue of ther "stay at home" mentalty. There s also a generc moble factor of producton n state, whch can be thought of as captal, or labor (e.g. guest or tnerant workers). Unlke the homefolks, ths factor mgrates between states, as well as between the regon and the rest of the world. The quantty of the factor present n state s denoted as n. Snce the number of homefolks (and ther labor supply), the quantty of land and the amount of (fxed) captal are gven, the strctly quas concave producton functon of state can be wrtten as f(n) (1) where f(n) > 0 and < f(n) 0. Each unt of the moble factor receves a return, w, whch s equal to the value of ts margnal product, f (n ). Implct here s the assumpton that there are many frms n each state, and hence, that the market for the moble factor s perfectly compettve. The homefolks n state receve the resdual from producton, whch mght nclude rent from any of the fxed factors, or the return to ther labor nput (ths return can be aggregated wth rents when homefolk labor s nelastcally suppled). The total resdual s R(n) = f(n) w(n)n. (2) The moble factor has a strctly quas concave contnuous payoff functon, ( ) p x,q, (3) 4 The results generalze to k states. 6

11 where q s a local publc good provded by state for the beneft of the moble factor, x = w (n ) t s the net monetary reward accrung to each unt of the moble factor and t s a lump sum tax leved on each unt of the moble factor. In the case of captal, q can be thought of as economc nfrastructure offered by state (e.g. port and ral facltes). For mgrant labor, q mght nclude the provson of servces such as health, educaton and welfare. The tax, whch can be negatve (a subsdy), zero or postve, s a generc nstrument desgned to capture the polces that states use to mpose monetary costs on mgratng factors, or provde them wth subsdes. Homefolks and the moble factor contrbute to the cost of provdng the publc good. The per unt contrbuton of the moble factor n state s va the tax/subsdy dscussed above. If the tax s postve then the moble factor makes a fnancal contrbuton to the publc good but when the tax s negatve (a subsdy) the moble factor benefts from the publc good and s provded wth a monetary subsdy. The contrbuton of the homefolks n state s T = q tn. (4) Therefore, the total monetary reward to the homefolks n state, denoted as X (n ), s equal to the resdual together wth the contrbuton from the moble factor (whch s postve, zero or negatve) less expendture on the publc good, X(n) = R(n) + nt q. (5) There are multple consequences from the nward mgraton of the moble factor nto any partcular state. For example, the homefolks are favored by an ncrease n the economc resdual, snce R (n ) s ncreasng n n. However, there may be broader external benefts and costs from n and out mgraton of the moble factor. In the case of mgratng labor, mmgrants may brng wth them cultural deas that can enhance the qualty of lfe n the host country and sklls that add to overall productvty. But the cultural dfferences can be annoyances to the homefolks and mmgraton can also cause ncreased congeston. Also, whle mported captal produces hgher ncome for the 7

12 homefolks, and s a beneft to them, ndustralzaton may stran socal coheson and envronmental qualty can be compromsed by ncreases n captal ntensty. We capture these benefts and costs wth a net externalty functon, E(n), where E, the net externalty value n state, s a strctly quas concave contnuous functon of the quantty of the moble factor present. The homefolks addtvely separable pay-off functon, P (n ), depends on the monetary reward and the net externalty 5 P(n) = X(n) + E(n). (6) As observed n the ntroductory comments, federatons (at least democratc ones) have common nternal markets n whch factors of producton are free to move between states wthout artfcal restrcton. Confederatons often also have common nternal captal markets and, as wth the declared am of the European Unon, an emergng common labor market n whch ctzenshp of one state guarantees regon wde ctzenshp. To capture ths feature of regons, we assume that the two states share a common factor market n whch the moble factor moves wthout restrcton between states to maxmze ts payoff. As noted earler, ths s a standard way to model common regonal factor markets. The mplcaton s that, for a gven regonal moble factor supply, f the pay-off s hgher n state than j then the moble factor wll move from j to. Ths depresses the return n state and ncreases t n state j untl payoffs are equal (there are no regonal arbtrage opportuntes). Therefore, n equlbrum, ( ) ( j j j j) p w(n) t,q = p w(n) t,q. (7) Another mportant feature of regons that share a common factor market between member states, ncludng federatons and confederatons, s that labor and captal mgrate between the regon and the rest of the world. There are varous ways to capture ths process. In many studes, ncludng Oates and Schwab, and Wellsch and Wldasn, the 5 We assume that the homefolks derve no beneft from the publc good, whch s provded purely for the beneft of the moble factor. It would also be possble to allow the homefolks to generate externaltes for the mgrant factor. However, such an extenson would add some complexty wthout changng the results n any materal way. 8

13 moble factor equlbrum s acheved when the payoff s the same nternally, as t s n the rest of the world. Specfed as a condton on relatve payoffs, f p* s the world factor payoff, the equlbrum requres p/p* = 1. Ths equlbrum condton s the result of assumptons, ether explctly or mplctly made; frst, that the moble factor can move costlessly across all borders, and second, the moble factor s homogeneous, both n ts productvty, and regonal preferences. We mantan the productvty homogenety condton mplct n ths approach, but drop the assumpton that the moble factor s alke n all other ways. Whle movement between states, wthn the regon, s assumed costless, mgraton n and out of the regon may not be costless. Factor owners may have regonal preferences. For labor there s a strong pull towards home. Famlar customs, language, as well as a desre for famlal propnquty, nfluence the readness of labor to mgrate. Owners of captal also dsplay an nclnaton to nvest n famlar states. The desre to avod uncertantes assocated wth foregn nvestment s thought to explan the captal market home bas: nvestors wll accept a lower return from nvestments n ther own country than those taken abroad. The market manfestatons of a home bas n both captal and labor markets are equlbra wth a dfference between regonal payoffs and those receved n the rest of the world. The potental for dfferences n tastes for the regon, and/or dfferences n the ease of movement n and out of the regon s modeled here as a reservaton payoff, denoted as ρ. Ths s the value of the payoff requred by an ndvdual unt of the moble factor, relatve to the world value, to locate wthn the regon rather than outsde t. Snce the nternatonal return s fxed, for convenence, and wthout loss of generalty, t s assumed to be one (p* = 1). Wth ths condton, hence forth, we wll be concerned only wth the local equlbrum payoff, p. The reservaton value s expressed as a locatonal choce condton: a factor s located n the regon f the specfc reservaton value s no larger than the local payoff: ρ p. If ρ > p, the factor wll ether emgrate from or reman out of the regon. 9

14 It s assumed that ρ s a contnuous random varable wth a dstrbuton f( ρ ). The proporton of the populaton of the moble factor that ether remans, or mgrates to, the regon s p F(p) = f ( ρ)dρ (8) The regonal moble factor supply s thus 0 n1 + n 2 = NF(p) (9) where N s the world supply. The regonal moble factor supply s potentally a very complcated functon of p. It mght depend on the hgher order moments of the dstrbuton, as well as ts mean. For that reason, t s convenent for purposes of analyss to choose a one-parameter dstrbuton. The Webull dstrbuton s often employed n the analyss of extreme values. If ρ s dstrbuted as Webull, the regonal moble factor supply functon s ε αp n1 n2 N(1 e ) + = (10) Whle not strctly an elastcty, n the common use of the term, ε performs much the same functon: the supply response to changes n the relatve payoffs, depends on ε. The response s not proportonal to t, as t s wth a true elastcty, but t nonetheless vares postvely wth ε. We also thnk of the elastcty parameter ε as a measure of the degree of ntegraton between the regons market for the moble factor, and the world market. In ths scheme, ε= 0 mples no ntegraton, 0 <ε< mples partal ntegraton whle ε= ndcates full ntegraton (the factor s perfectly moble between the regon and the world). In the dscusson n Secton 4, we wll examne two polar cases; ε = 0 and ε =. For ε = 0, the moble factor does not move n or out of the regon and from (10) t can be seen that total regonal supply of the moble factor s fxed at N(1 e ). The parameter α s determned by the relatve sze of the regonal populaton. A small value of α s consstent wth the assumpton that the regon s small compared to the rest of the world (and vce versa). For ε =, the entre mass of the dstrbuton s concentrated at ρ = 1. α 10

15 Ths means that f p s smaller than 1, the regon s denuded entrely of populaton, and f p s larger than one, the entre world resdes n the regon. In ths regard, the parameter ε s smlar to the common supply elastcty. As wll be dscussed later, when ε =, the only equlbrum s one n whch the regonal moble factor payoff s equal to the gven world payoff (as dscussed, ths s assumed to be one). Ths, n turn, mples that the total regonal supply of the moble factor s fxed at N(1 e ) when ε s nfnte. α 3. The Nature of an Equlbrum We now characterze an equlbrum where t s supposed that the government of state s elected by the homefolks who mantan a numercal majorty. 6 Therefore, the nterests of state are synonymous wth those of the homefolks and qand t are chosen by state to maxmze P, as defned by (6). There are two types of polcy choce nterdependences between states. Frst, through the nternal equlbrum condton (7) the polces of state affect ts own moble factor supply n and the moble factor supply n state j (.e. of n j ). Snce state js net ncome and moble factor externalty are functons n j ths means that state s polces affect the welfare of the homefolks n state j. Thus, the model captures the drect effect that ndependent polces n a regonal unon of states have on one another. Second, because state polces also affect the total supply of the moble factor through (10) there s an addtonal source of nterdependence. Interdependence rases the prospect that states wll act strategcally. To capture ths, we suppose that when makng polcy choces, states take as gven the polces of the other states (Nash behavoral conjectures). However, each state s assumed to assess (accurately) the mpact of ts polces on the supply of the moble factor wthn ts own borders, condtonal on the (supposed) fxed value of the other states polces. The state takes account of these moble factor supply changes n so far as they affect the well beng 6 Ths mght be due to mgraton outcomes or consttutonal provsons that gve a votng franchse to the homefolks (e.g. landowners) only. 11

16 of the states homefolks. In other words, each state wthn the regon pursues ts own selfnterest and gnores the broader regonal effects of ts polces Exstence of Equlbrum Snce t makes lttle sense to analyze the nature of a non-exstent outcome, we frst examne the queston of the exstence of a Nash equlbrum of the non-cooperatve game mplct n the states polcy choces. The dffculty we see for the problem as stated s that the objectve functons of the states may not be concave n the strateges. Therefore, whle there may be an optmum for both states, condtonal on ts neghbor s polcy choces, the set of optma may be nether contnuous nor convex. Whle ths potental problem does not rule out exstence, we are not able to prove that, n general, or for our partcular problem, an equlbrum exsts n the game descrbed wth tax/subsdy and expendture polces as strateges. However, we fnd that we can prove the exstence of a Nash equlbrum n a game that s the dual of the game n polcy strateges. The dualty s such that the exstence of equlbrum of the dual game, so descrbed, mples exstence of a Nash equlbrum of the prmal game n whch the strateges are tax/subsdy and publc good provson polces. We start wth the recognton that states, whle they can make choces from the entre set of feasble strateges, wll choose only those that are least expensve and acheve a desred supply of the moble factor. Whle there may be many combnatons of taxes and publc good levels that wll nduce a chosen n, states wll choose only that polcy combnaton that has the smallest cost. We show, n what follows, that state specfc moble factor supply s a strctly quas-concave functon of the polcy varables. Ths beng the case, t follows that there s a unque mnmum cost combnaton of publc 7 The mpact that one state s choce of polcy has on another state s welfare s often termed a fscal externalty (see Wldasn (1988)). Our model set up mples that there s both tax/subsdy and polcy competton between states. Wldasn (1988) characterzes equlbra n a fscal competton model where states have one strategc varable, ether publc expendture (a Z equlbrum), or a tax (a T equlbrum). Further dscusson of fscal competton models wth two strategc varables s provded n Wldasn and Wlson (1991) and Wldasn (1991b). 12

17 polces that wll produce a gve n response, gven the desred level of moble factor n the neghbor state, n -. The dual game s the one wth desred moble factor supples as strateges. We show what restrctons are necessary to nsure that the state objectve functons are strctly quas-concave n n. Snce the set of feasble n s s convex, t follows that a Nash equlbrum of the dual game wth n s as strateges, exsts. Snce there s a unque mappng from desred n to state polces, t follows that an equlbrum n the prmal polcy game exsts as well. In the model structure above, ndvdual states have two polcy nstruments a tax, t, and a publc good, q 8. As stated, the tax can be ether postve or negatve, n whch case t s a subsdy. For the dscusson of exstence, we fnd t more natural to thnk of the monetary nstrument as a subsdy, s. When s s postve, t enhances the moble factor s payoff and, when negatve, t dmnshes t. There are three fundamental elements of the model developed above: () Moble factor payoff s a strctly quas concave contnuous functon of ts ncome and the publc good. Income s the wage, a declnng functon of moble factor populaton, plus the state subsdy, p = p(w (n ) s, q ). (11) + () A state s moble factor supply s an ncreasng functon of ts own factor payoff. The nternal mgraton equlbrum condton (7), can be used to derve state-specfc moble factor supply as a functon of ts polces, condtonal on ts neghbor s polces, () n = n (s,q s,q ). (12) The payoff to a state s the sum of two strctly quas concave functons of n Π n ) = R (n ) E (n ) (13) ( + less ts expendture on the publc good, q, and ts subsdy payout (tax collected), s. 8 The proof that follows s restrcted to but one, constant margnal cost, non-monetary polcy. It would apply to a vector of publcly suppled goods and servces, wth constant or rsng margnal cost, that are ether uncrowded or crowded. 13

18 We start our development of the dual game by showng that n s a strctly concave functon of s and q. From (11), one can show that p s a strctly quasconcave contnuous functon of s and q for a constant value of n 9. p = v (s,q, n s,q ) (14) We wll use the concavty of v (.) wth constant n to prove that the functon n ( ) s, tself, strctly quas concave. For every value of n, defne ts upper contour set as UC (n s,q ) {s,q n (s, q s,q ) n}. Lemma The set UC (n s -,q - ) s convex. Proof: Choose polcy values s, q and s, q on the boundary of UC ( n s -, q - ) λ λ Defne s,q = λ(s,q ) + (1 λ)(s,q ), 0 < λ 1. Because n s monotoncally < 0 0 ncreasng n 1 1 v(s,q,n s,q ) v(s,q,n s =,q ), and because v (.) s strctly quas 0 0 concave, λ λ v(s,q,n s,q ) < v(s,q,n s,q ). The polcy choce ( s, λ λ q ) s on the boundary of an upper contour set of value of n greater than n. Thus, UC (n s -,q - ) s convex. // Corollary: n (s,q s -,q - ) s a strctly quas-concave functon of s,q. Proof: Ths follows mmedately from the convexty of ts upper contour set.// 9 Ths can be shown as follows (where the subscrpts are deleted). For a constant n = ~ n, choose any two 0 0 pars of values, sq, and sq , such that p(w(n) + s,q ) = p(w(n) + s,q ). For 0 λ 1, defne ~ λ λ ~ 0 0 w( n) s,q (w( n) s,q ) (1 )(w( ~ = λ + + λ n) + s,q ). The strct quas concavty of the moble factor payoff functon mples that ~ λ λ ~ 0 0 p(w( n) + s,q ) p(w( n) + s,q ). Thus, for fxed n the functon p(.) s strctly quas concave n s and q. 14

19 The analyss above s characterzed n Fg. 1. The p functon s drawn for a fxed value of moble factor supply of n = n. p s the moble factor payoff that nduces a moble factor supply of n. The lne p p represents the combnatons of s and q (gven n ) that yeld the (constant) payoff p. The ponts s 0 0 q and sq 1 1 are ponts on the so-n lne wth the convex combnaton sq λ λ represented by the dashed lne between. Fg. 1: Moble Factor Payoff for n = n p p sq 0 0 sq 1 1 p q s 0 For a gven n, the cost or prce of each unt of the monetary subsdy s just $1. Smlarly, the cost or prce of a unt of the publc good s a constant amount, n ths case, $1. However, snce a sngle unt of the publc good benefts all unts of the moble factor 15

20 equally, ts cost per unt of moble factor s, polcy choce wth a factor supply of n s 1/n. 10 Thus, the total cost of a partcular q +. In s,q space, these costs are (s )n n represented by a lnear so-cost lne that has a slope of n. The so cost lne s dsplayed n Fg. 2, together wth the convex so-n curve from Fg. 1. The pont of tangency between the two yelds the combnaton of polces that acheves n = n at mnmum cost. Fg. 2: Iso Cost Lne and Iso n Curve q * q n = n 0 * s s The strct quas concavty of n (s,q s -,q - ) means that there s a unque mnmum cost polcy choce for every value of n. The effcent nput combnaton (the nputs are the subsdy and publc good polces), and thus the mnmum cost, depends on the desred factor supply, n -, of the neghbor. The mnmum cost s then a functon of the desred choce of each state 10 If q s not produced at a constant margnal cost, and t s crowded, ts per unt factor prce would be MC(q)/B(n) where B(n) represents crowdng. 16

21 C = C (n n ). (15) We have now transformed the prmal game wth polcy choces as strateges to ts dual n whch the quantty of the moble factor s the strategc varable for state. In a sense, the transformaton restrcts the orgnal strategy set of polcy varables to be only those that produce the target level of factor supply at mnmum cost. Ths s a plausble restrcton snce t s always n a state s nterests to make cost mnmzng choces n order to maxmze the payoff to ctzens. The set of possble choces of n, Ω, s convex and bounded. Where N s the world moble factor populaton, Ω= { n R n [0,N]}. In the dual game, the objectve of each state s a functon of n alone P (n n 1 ) = Π (n ) C (n n ). (16) A Nash equlbrum, (n,n ) Ω Ω, s defned n the usual way, namely, * * P(n n ) P(n n ) n Ω. (17) * * * Theorem (Exstence): If C (n, n - ) s a convex functon of n, a Nash equlbrum to the polcy game exsts. Proof: Snce Π (.) s strctly quas concave, the convexty of C (.) mples that each state s objectve functon P (n n - ) s contnuous and quas concave. Ths game s played by the choce of n from a compact convex set to maxmze a strctly quas-concave objectve functon. Ths s suffcent for the exstence of a Nash equlbrum 11 [see Rosen (1965), Border (1985), p. 90, Mas-Colell, et.al. (1995), p. 253]. It s well known that wth constant prces, the cost functon s convex f n (.) s strctly quas concave. In the case here, even though n (.) s strctly concave n s and q, the prce of the publc good declnes wth n. Thus, as a state moves from an equlbrum at q,s (Fg. 2) and acheves a hgher value of n, the slope of the so-cost * * 11 Ths Theorem s an example of the well-known Gale and Mas- Colell (1975, 79) Theorem. 17

22 lne changes n such a way that reduces the relatve prce of usng the publc good versus the subsdy. As the sze of the moble factor populaton ncreases, there wll be a tendency for states to substtute publc goods for subsdes, or decreasng taxes. Wth a suffcent moble factor response to ncreases n publc good supply, the cost functon may not be convex. Nevertheless, a Nash equlbrum may stll exst, even f the cost functon s not convex. Concavty of the objectve functon s a suffcent, not a necessary condton for exstence. Furthermore, the objectve functon Π n ) can be concave n moble factor supply f the concavty of R (.) outweghs the non-concavty of C (.), that s f R > C for all n. ( 3.2 Unqueness of Equlbrum We have not been able to demonstrate unqueness n general. However, there s one value of the ntegraton parameter, ε, where t s possble to show that the equlbrum s unque. Ths s the case where ε s nfnte (full ntegraton). In ths nstance, equlbrum mposes an addtonal constrant on states n the sense that the moble factor must receve a monetary return that s exactly equal to the world payoff (assumed to be one). If the regonal return to the moble factor s less than the world return, the entre factor supply wll leave. If the regonal return s hgher, the entre world supply wll flow nto the regon. Thus, as long as the state producton functons exhbt decreasng returns to scale, t s mpossble for a state to sustan a moble factor return that s hgher or lower than the world value f ε s nfnte. Snce there s only one moble factor supply n each state consstent wth the world payoff, there can only be a sngle Nash equlbrum f ε s nfnte. Interestngly, ths s the case found n the exstng lterature. However, for the more general case where ε s less than nfnte, and the moble factor s mperfectly moble between the regon and the world, there may be more than one Nash equlbrum. Thus, what we have shown s exstence and unqueness for the more usual model where the moble factor mgrates to equate ts regonal return wth the world return, and 18

23 exstence and possble multple equlbra for the more general case of mperfect ntegraton where the world and regonal payoffs dverge. An mplcaton s that t would be extremely dffcult to undertake comparatve statc analyss, except n the polar case of ε= where we are guaranteed to have both exstence and unqueness. 3.3 The Tax/Subsdy As noted n the Introducton, a major focus of the paper s to examne whether states wthn regonal unons wll adopt non-zero tax/subsdy polces towards a factor of producton that s perfectly moble wthn the regon, but mperfectly moble between the regon and the outsde world. For ths part of our analyss, we fnd t more convenent to contnue developng the model n terms of ts prmal verson, where state chooses tand qto maxmze (6), condtonal on j tandq. j Development of the model n ths way requres knowledge of the moble factor supply responses to changes n t and q. Snce our ntenton s to examne the Nash equlbrum tax/subsdy, rather than the publc good, whch we have shown n the dscusson of exstence to be provded n equlbrum at mnmum cost, we need only derve the moble factor supply responses to changes n t. They can be found by dfferentatng (7) and (10) wth respect to t, n p w p w t 1 = px, (19) b x xj j 1 bp n x w j 1 t ε 1 αp where b = N( αεp e ) 0. ε In the prmal verson of the polcy game, state chooses ts polces, t and q, to maxmze (6), subject to ts neghbor s polcy choces, t,q. j j Wth ths set up, there are two necessary condtons, one for t and one for q, that must be satsfed at any Nash equlbrum. Snce our focus s on the tax/subsdy, we only requre the necessary condton for t, namely, 19

24 n ( t w n E ) + + n = 0 (20) t From (19), the moble factor response to a change n state s tax s n p x = (1 bp x w j ) < 0 j (21) t j where = px w + px w j j bpx p x w j wj < 0. Substtutng (21) nto (20) yelds an expresson for the tax/subsdy that s satsfed n any Nash equlbrum, n ether the dual or prmal versons of the game, p w t E (n ) n ( ). (22) p [1 bp w ] * xj j = x xj j The frst pont to note s that, unlke the result obtaned n the papers dscussed n the Introducton, n general the equlbrum tax/subsdy adopted by state s non-zero. The reason for ths can be seen f one examnes the rght hand sde of the tax/subsdy expresson. The frst part, E (n ), s the change n the externalty resultng from an ncrease n the quantty of the moble factor present n state. We refer to ths from now on as the "margnal externalty value". As dscussed, ths term can be postve, zero or negatve n sgn. The second part of the equlbrum tax/subsdy, the term ( ) j j n p w /p (1 bp w ) x j x x j, we label the hrng cartel component (the ratonale for ths termnology s gven below). We know that b 0, and that, because of the concavty of the producton functon, w j s non-postve. Therefore, the hrng cartel part s always negatve. However, when the mnus sgn n front of the cartel term s taken nto account one can thnk of the cartel component as non-negatve. Thus, the hrng cartel effect tends to make t * postve. The hrng cartel component of the tax/subsdy requres further explanaton. In ths regard, t should be recalled that there are many perfectly compettve frms producng a homogeneous product n both states. As a consequence, the moble factor s pad a return equal to the value of ts margnal product. Were there but one frm, or were 20

25 frms able to form a statewde hrng cartel, homefolks could mprove ther poston, vsà-vs the perfectly compettve outcome, by payng the moble factor somethng less than ts margnal product. A hrng cartel could explot ts monopoly poston to the advantage of the frm-ownng homefolks. Clearly, there s no cartel n our model. However, effectvely the state government acts n the place of such a cartel - t acts on behalf of homefolks snce t represents ther nterests - and uses ts taxng power to explot the potental monopoly gans and ncrease the ncome of the homefolks. The revenue rased from the tax on the moble factor s redstrbuted to homefolks va a reduced monetary contrbuton to the publc good (see (4)) and drect monetary awards. The result s that frms and homefolks, through the polces of the government of a state, pay the moble factor a return that s less than ts margnal product. It s for ths reason that we have used the words "hrng cartel" to descrbe ths part of the tax/subsdy equaton: the state essentally acts as a hrng cartel whch has the opton of payng the moble factor somethng other than ts margnal product. When wll states tax the moble factor and when wll they provde t wth a subsdy? It s clear that the hrng cartel effect always nduces state to tax the moble factor, that s, to try to explot the moble factor for the advantage of the homefolks. If the margnal externalty value s also negatve (n the case of labor ths mght be so f the costs of more guest workers outweghs the benefts) then the tendency to explot the moble factor s ntensfed and the tax s hgher than otherwse. However, f the margnal externalty value s postve t wll tend to offset the desre to explot the moble factor. If suffcently large (and postve), the margnal externalty value may encourage a state to provde the moble factor wth a monetary subsdy. Hence, the model explans the tax/subsdy polces of states as the result of an nteracton between the desre to redstrbute from moble factors for the beneft of homefolks and the externaltes generated by mgratng factors. It also explans dfferences n the polces adopted by states; specfcally, states wll have dfferent margnal externalty values and dfferent hrng cartel effects, and hence wll choose to 21

26 adopt dverse tax/subsdy polces n equlbrum. For example, a state that perceves that t has a relatvely low, though postve, margnal externalty value wth respect to mgratng guest workers may wsh to tax them. Ths mght be a state that already has relatvely large numbers of guest workers and hence s at a pont on ts externalty functon for whch the margnal value of extra guest workers s low. Alternatvely, a state that s relatvely small n terms of moble populaton may have a much hgher margnal externalty value and choose to subsdze moble labor. 4. Regonal Integraton and the Tax/Subsdy We now examne how the degree of ntegraton between the regons factor market and the world factor market, as captured by the parameter ε, nfluences the (nonzero) tax/subsdy chosen by a state wthn a regonal unon. Snce ε also captures the degree of competton faced by the regon as a whole n competng nternatonally for the moble factor, we can also vew ths as an analyss of how changng nternatonal competton affects the polces of states wthn the regon. As part of the dscusson, t s also possble to consder the mpact of wthnregon competton (drectly between the states). Ths s an nterestng queston to ponder because the proponents of compettve federalsm, such as Breton (1984), have argued that competton between states s desrable n the sense that t reduces the taxng power of governments. We tackle ths ssue by examnng what we call the monopoly state n whch the regon conssts of only one state. The analyss s undertaken for two polar cases. In the frst, we suppose that the regon s fully solated from the world ( ε= 0 ), whle n the second case we suppose that the regon s fully ntegrated ( ε= ). It has not been possble to obtan analytcal results on the ntermedate case, where ε s less than nfnte, but more than zero, because b s a functon of ε and vares n an ndetermnate way as ε changes. Before dscussng each case, t s useful to note that, as wth the compettve twostate case, a monopoly state chooses ts polces to maxmze the payoff to the homefolks, but faces only one factor market constrant, specfcally, ε αp n N(1 e ) = where we drop 22

27 the subscrpt when consderng the monopoly state. Snce the state has no fully open border wth a neghborng state, there s no nternal free mgraton condton such as (7). The state faces only nternatonal competton for the moble factor. In ths case, n bp = x. (23) t 1 bpx w There s also a dfference between the tax/subsdy expresson when state s a monopoly state, and the tax/subsdy expresson f t faces competton for the moble factor wthn the regon. Ths can be seen by notng that f a state s a monopoly state ts tax/subsdy s * n t = E (n ) +. (24) bp x Comparng (24) wth (22), the equlbrum tax/subsdy for the compettve case, t s clear that the hrng cartel effect dffers, dependng on whether a state faces competton from wthn the regon. Now consder the polar case where the regon s fully ntegrated wth the world factor market (ε = ) and the regonal moble factor payoff s equal to the world payoff. For the compettve case, we know from the dscusson of unqueness n Secton 3.2 that the Nash equlbrum s unque. One can also show that b = when ε=. Hence, the hrng cartel effect s zero for both the monopoly and compettve state cases and the tax/subsdy s t = E (n ). (25) Thus, f the regon s fully ntegrated a monopoly state chooses the same tax/subsdy as a compettve state. The mplcaton s that the compettve federalsm outcome s no dfferent to the monopoly outcome when the regon s fully ntegrated smply because the monopoly power of the state s completely constraned by foregn competton for the moble factor. Now consder the other polar case, a fully solated regon (ε = 0) where states face no outsde competton for the moble factor. Here, we have b = 0 and the hrng cartel 23

28 component of the tax/subsdy expresson for the compettve state case s compettve equlbrum tax/subsdy s t = E (n ) n w. (26) * j n w j. The Because we do not know whether or not there are multple Nash equlbra n the nterstate polcy game, we are unable to compare the monopoly and compettve state outcomes n general. However for the polar cases of ε =, we know that the monopoly and the compettve outcomes are smlar. In both cases, the hrng cartel effects go to zero and the tax/subsdy reflects only the externaltes. An unambguous comparson s also possble for the fully solated regon (ε = 0). It s clear, from a comparson of (24) and (26) that for a regon of smlar sze the sngle state moble factor tax s hgher than the equlbrum value of the compettve state tax. As ε (and thus b) goes to 0 the expresson n (24) becomes nfntely large. However, a state cannot tax wthout lmt, and ths ndcates that the monopoly state tax s fully explotve: t wll equal the full prce w(n) pad to the moble factor. Furthermore, the monopoly state wll supply no publc good. However, full explotaton cannot be an equlbrum outcome for the compettve state, as long as the margnal externalty s not a large negatve value. If one state were to be fully explotve, ts neghbor could nduce the entre moble factor supply nto ts state and enjoy the returns from t wth a small reducton n ts tax, or a small ncrease n ts supply of the publc good. Snce the resdual, R, s an ncreasng functon of n t wll be n ts nterest to make the change f the moble factor-created externalty s not negatve and large. The mplcaton s that when ε= 0 there s a dfference between a compettve federalsm outcome and a monopoly outcome, namely, the compettve outcome wll nvolve a lower tax/subsdy on the moble factor. Ths begs the queston of what determnes the extent to whch the compettve state tax s lower than the monopoly tax? In answerng ths, one mght suspect (rghtly, as t turns out) that the more compettve s the regonal factor market, the smaller s the cartel effect. One can gan some nsght here by consderng the case of a two state 24

29 regon where both states have the same technology and the equlbrum s one n whch each state has half the total moble factor supply (whch does not change snce ε= 0 ). The value of w for each state s also the same n the equlbrum. Snce each state has the same factor supply, the steeper s the wage functon (the larger n absolute value s w) the hgher s the equlbrum tax. At frst sght ths s curous but t ponts vvdly to the role of ntra regonal competton n nfluencng the tax chosen by states. Specfcally, we know that the value of the moble factor response to changes n state taxes n the compettve case s determned by (19). When ε = b = 0, the response n the compettve case s n/ t= 1/(w+ w). Clearly, the larger n absolute value s w, the smaller s the 1 2 moble factor loss to a state for any ncrease n ts tax rate. If the margnal value of the wage functon s nfntely steep, a state can ncrease ts factor tax wth no change n factor supply. The lmtng case s smlar to the fully solated sngle state n whch the state possesses a complete monopoly over ts own mgratng factor. Thus, the magntude of the dfference between the compettve and monopoly moble factor tax when ε s zero depends on the steepness of the wage functons. Snce we take the steepness of the wage functon to be a measure of the degree of nterdependence, and hence competton between states, the mplcaton s that greater ntra regonal competton results n a smaller dfference between the compettve and monopoly moble factor tax. 5. Sub-Optmal Supply and Dstrbuton of the Moble Factor We now turn to the fnal part of the paper, the dscusson of effcency. Ths ssue has been studed extensvely n models of regons that levy taxes on moble factors, for example, the tax competton lterature. As noted n the Introducton, the man ssue addressed s whether the publc good s over or under provded relatve to an optmal supply. A comprehensve survey of these deas, and the man results, can be found n Chapter 4 of Wellsch (2000). Rather than follow ths path here, we choose to explore a dfferent queston: the nteracton between mgraton frctons, as captured by the ε parameter n the moble 25