- Email: [email protected]
Interregional competition, spillovers and attachment in a federation
Journal of Urban Economics 67 (2010) 219–225
Contents lists available at ScienceDirect
Journal of Urban Economics www.elsevier.com/locate/jue
Interregional competition, spillovers and attachment in a federation Emilson C.D. Silva a,*, Chikara Yamaguchi b a b
School of Economics, Georgia Institute of Technology, Atlanta, GA 30332-0695, USA Faculty of Economic Sciences, Hiroshima Shudo University, 1-1-1, Ozukahigashi, Asaminani-ku, Hiroshima 731-3195, Japan
a r t i c l e
i n f o
Article history: Received 8 November 2006 Revised 7 May 2009 Available online 6 September 2009 JEL Classification: D6 D7 H4 H7 Q2 R5
a b s t r a c t We examine decentralized environmental policy making in a federation characterized by decentralized leadership and imperfect labor mobility due to attachment to regions. Energy consumption generates positive consumption benefits, but energy supply generates federal air pollution. Regional authorities regulate energy supply by controlling supplies of pollution permits. Energy and pollution permits are traded in interregional markets. The center redistributes incomes after it observes regional supplies of pollution permits. Regions are populated by mobile and immobile households and profits are expatriated. We show that the subgame perfect equilibrium for the federal policy game played by regional and central authorities is socially optimal. Ó 2009 Elsevier Inc. All rights reserved.
Keywords: Federation Interregional spillovers Redistribution Labor attachment
1. Introduction According to conventional wisdom in fiscal federalism, regional public goods that generate interregional spillovers – that is, goods whose economic jurisdiction exceeds the political jurisdiction of a regional or local government – should be subjected to a Pigouvian subsidy determined by a higher-level government, namely, the central government (see, e.g., Oates, 1972). Local governments should provide local public goods as far as economic and political jurisdictions coincide, but local public goods that generate interregional spillovers should be provided by a supra-local level of government. Recent studies, however, demonstrate that there are circumstances under which competing regional governments may not only provide public goods that generate transboundary spillovers efficiently, but also implement interregional transfers to each other so that a socially efficient population distribution is obtained.1 In Wellisch (1994), competing regional governments fully internalize * Corresponding author. Fax: +1 404 894 1890. E-mail addresses: [email protected] (E.C.D. Silva), [email protected] (C. Yamaguchi). 1 See Myers (1990) for the demonstration that the Nash equilibrium for the game played by decentralized regional governments is efficient provided they control instruments to make income transfers to each other. 0094-1190/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jue.2009.09.002
externalities associated with provision of public goods in their regions whenever households are not attached to regions. Caplan et al. (2000) demonstrate that decentralized provision of a pure public good – a good whose economic benefit is available for an entire federation – may be efficient for a game where regional governments are policy leaders and a central government is a policy follower. Each regional government decides how much of the pure public good to provide in anticipation of the redistributive income policy implemented by the center. The efficiency result does not depend on the degree of household attachment. In this paper, we extend the framework advanced by Caplan et al. (2000) in three significant ways. First, we consider commodities that generate regional-specific benefits as well as a federal benefit (good) or cost (bad). Consider, for example, provision of energy. Energy generates private consumption benefits as well as federal damages, since its consumption or production typically yields emissions of pollutants in the atmosphere (e.g., carbon dioxide). Regional governments are concerned with controlling emissions of carbon dioxide in their own regions. We postulate that such a control is done through setting up quotas of pollution permits that can be sold in a federal market for pollution permits. We also assume that the energy commodity is traded within and across regions. Thus, unlike Caplan et al. (2000), we consider an economy featuring tradable commodities and an impure public bad.
220
E.C.D. Silva, C. Yamaguchi / Journal of Urban Economics 67 (2010) 219–225
The second extension is that we examine the actions of large government authorities which fully recognize that their policies affect prices of all commodities traded in the economy. In our framework, there are three commodities which are traded globally: numeraire and energy goods and pollution permits. The third extension comes from the fact that we depart from the unrealistic assumption used in Caplan et al. (2000) that each household owns an equal share of the resources available in the region in which it resides. We consider instead a more general and realistic setting where two types of households, mobile and immobile, reside in each region. Each region’s immobile population owns the regionally fixed productive resource, which we shall assume to be land. The mobile population in this economy is its population of workers. Landowners earn land rents and workers earn wages. In addition, both types of households possess positive shares of profits produced in both regions, earn income from the sale of pollution permits and receive or pay centrally determined income transfers. As in Caplan et al. (2000), we are interested in analyzing the allocation of resources for a federal economy characterized by imperfect labor mobility and decentralized leadership. Our example of such a federal economy coincides with theirs, namely, the European Union (EU). The governments of the member nations are endowed with considerable economic and political powers vis-à-vis the center concerning most types of policies, including environmental policies. Accordingly, in our setting, the regional governments select the regional quantities of pollution permits to be supplied in the pollution permit market prior to the center’s decisions of how to redistribute income in the federation. This paper is organized as follows. Section 2 presents the model and determines the competitive equilibrium given the set of policy instruments. The problems faced by the decentralized agents (consumers and producers) yield the relevant indirect utility and profit functions that are taken into account by the governmental authorities when they make their decisions. Section 3 examines a setting in which the center controls all policy instruments and demonstrates that the implied allocation is socially optimal. Section 4 describes and solves the two-stage federal policy game played by regional and central authorities. We demonstrate that the subgame perfect equilibrium for this game is socially optimal by showing that it is isomorphic to the center’s most preferable allocation. Section 5 concludes the paper. 2. The model Consider a federation consisting of two regions, two regional governments and a central government. There are two types of households, workers and owners of a fixed regional resource, which we shall consider to be land. Landowners are immobile. Workers are imperfectly mobile due to regional attachment benefits. Let nij and nm j denote respectively the populations of immobile and mobile households who reside in region j, j = 1, 2. We assume that nij > 0, j = 1, 2. Let nj nij þ nm j be the population of region j and Nh > 0 be the total population of households of type h in the federation, h = i,m. A household of type h who lives in region j consumes xhj units of a numeraire good, ehj units of energy and is harmed by the aggregate amount of air pollution, E, that is produced in the federation. For simplicity, we assume that one unit of energy produced in each region leads to the emission of one unit of air pollution. Hence, E also corresponds to the aggregate supply of energy in the economy. Let uðxhj ; ehj ; EÞ denote the utility derived by every household of type h in region j. We assume that u() is increasing in the first two arguments, decreasing in the last, strictly concave, and satisfies 2 h h h h uhj xe @ u=@xj @ej P 0 for all xj > 0 and ej > 0. The last assumption ensures that demands for private goods are normal.
Every household h in region j chooses nonnegative fxhj ; ehj g to maximize uðxhj ; ehj ; EÞ subject to xhj þ pehj ¼ yhj ; where p is the price of energy and yhj is the household’s income. Each household treats the price of energy, its income and the level of air pollution in the federation as exogenous variables. In the game played by government authorities, however, household income levels, the price of energy and the level of air pollution are endogenous variables. Assuming that the solution to the consumer’s maximization problem is interior, the solution satisfies the budget constraint hj hj h and the tangency condition, uhj e =ux ¼ p, where ue @uðxj ; h h h ehj ; EÞ=@ehj and uhj x @uðxj ; ej ; EÞ=@xj , h = i, m, j = 1, 2. In any region, each household’s marginal rate of substitution between energy and the numeraire good equals the price of energy. Since u() is strictly concave, the solution for each household’s problem is unique. We let xhj ðp; yhj ; EÞ and ehj ðp; yhj ; EÞ denote the demand functions and v ðp; yhj ; EÞ uðxhj ðp; yhj ; EÞ; ehj ðp; yhj ; EÞ; EÞ denote the indirect utility function of a household of type h in region j, h = i, m and j = 1, 2. For future reference, we establish that
v hjp v hjy mhjE
@ v ðp; yhj ; EÞ @p @ v ðp; yhj ; EÞ @yhj @ v ðp; yhj ; EÞ @E
hj hj hj hj hj hj hj h ¼ uhj x xp þ ue ep ¼ ux ðxp þ pep Þ ¼ ux ej < 0;
hj hj hj hj hj hj hj ¼ uhj x xy þ ue ey ¼ ux ðxy þ pey Þ ¼ ux > 0;
hj hj hj hj hj hj hj hj hj ¼ uhj x xE þ ue eE þ uE ¼ ux ðxE þ peE Þ þ uE ¼ uE < 0:
In addition, v ðp; yhj ; EÞ is strictly concave in yhj because uðxhj ; ehj ; EÞ h h is strictly concave and uhj xe P 0 for all xj > 0 and ej > 0. Mobile households are heterogeneous with respect to their idiosyncratic attachment benefits as in Mansoorian and Myers (1993), Wellisch (1994) and Caplan et al. (2000). These households are distributed uniformly on the interval [0, Nm]. If a mobile household nm resides in region 1, its total utility is v ðp; ym 1 ; EÞþ aðN m nm Þ. If it resides in region 2 instead, its total utility is v ðp; ym2 ; EÞ þ anm . For household nm, the term a(Nm nm) represents the attachment benefit of residing in region 1 and the term anm represents the attachment benefit of residing in region 2. The parameter a > 0 is the attachment intensity. m m In the migration equilibrium, v ðp; ym 1 ; EÞ þ aðN n1 Þ ¼ v ðp; m m m y2 ; EÞ þ an1 : This condition provides a solution n1 2 ð0; N m Þ; where 2 nm 1 is the mobile household that is indifferent between regions. Every household nm 2 ½0; nm Þ strictly prefers to live in region 1 and 1 m every household nm 2 ðnm 1 ; N strictly prefers to live in region 2. Let m m m m Dðnm ; p; y ; y ; EÞ v ðp; y ; EÞ þ aðN 2nm 1 1 2 1 1 Þ v ðp; y2 ; EÞ denote the surplus of household nm 1 of residing in region 1 relative to residing in region 2. In equilibrium, this surplus is zero. Since Dn @ D=@nm 1 ¼ 2a < 0, the migration equilibrium is stable.3 The equilibrium condim tion implicitly defines nm1 ðp; ym 1 ; y2 ; EÞ, the migration response funcm1 m2 tion. The marginal migration responses are nm1 p ¼ ðv p v p Þ=2a, m1 m1 m2 m1 m1 m2 ¼ v =2a > 0, n ¼ v =2a < 0 and n ¼ ð v v Þ=2a. nm1 m m y y E E E y1 y2 Let us now examine the supply side of the federal economy, starting with the numeraire-good industry. In what follows, we follow the literature in assuming that factor markets (i.e., labor and land) are regional.4 In region j, the labor supply is nm j and the land supply 2 We make assumptions below that guarantee that each region is populated with workers in equilibrium. 3 See Boadway (1982) and Stiglitz (1977) on issues pertaining to the stability of the migration equilibrium in economies with public goods. 4 Caplan et al. (2000), Mansoorian and Myers (1993) and Wellisch (1994) implicitly assume that there is a labor market in each region. The standard model utilized by these authors does not include regional land markets. It is assumed that regional residents (workers) have equal ownership shares of regional resources and that the aggregate supply of land in each region is fixed. Since in our model we allow landowners and workers to possess different shares of regional resources and we wish to clearly demonstrate how the factor prices are derived in equilibrium, we must examine the market-clearing conditions for the factor markets.
221
E.C.D. Silva, C. Yamaguchi / Journal of Urban Economics 67 (2010) 219–225
is Lj. Each worker is endowed with one unit of labor, which is supplied in the region in which the worker resides. All workers and land units are employed in the production of the numeraire good. There are jj > 0 firms in the numeraire-good industry in region j. Each of these firms employs zj workers and lj units of land to produce gj(zj, lj) units of output. We assume that gj() is increasing in each argument and strictly concave. We also assume that gj() satisfies the Inada conditions limzj !0 g jz ¼ 1 and limlj !0 g jl ¼ 1. The Inada condition with respect to the labor input guarantees that each region is populated with workers in the migration equilibrium. Each firm chooses nonnegative {zj, lj} to maximize gj(zj, lj) wjzj rjlj, where wj > 0 and rj > 0 are the wage and land rent faced by each firm located in region j, respectively. Firms take factor prices as given. The first order conditions for profit maximization are g jz ¼ wj and g jl ¼ r j . These conditions define the continuous and differentiable factor demand functions, zj(wj, rj) and lj(wj, rj). Since in region j the labor supply is nm j for all wj P 0 and the land supply is Lj for all r j P 0; the wage and land rent levels that clear j the regional factor markets solve jj zj ðwj ; rj Þ ¼ nm j and jjl (wj, rj) = Lj, j = 1, 2, respectively. The clearing conditions for the regional factor j m markets in region j enable us to obtain wj ðnm j ; Lj Þ and r ðnj ; Lj Þ, j = 1, 2; that is, the regional factor prices as functions of regional factor supplies. j j j j m j m j m j m ^j ðnm Let g j ; Lj Þ g ðz ðw ðnj ; Lj Þ; r ðnj ; Lj ÞÞ; l ðw ðnj ; Lj Þ; r ðnj ; Lj ÞÞÞ j m j m ^ ðnj ; Lj Þ. The latter is region j’s output of the and f ðnj ; Lj Þ jj g numeraire good expressed as a function of the regional factor supj ¼ g jzz =jj < 0. Thus, fj() is plies. Note that fnj ¼ g jz ¼ wj > 0 and fnn m increasing and strictly concave in nj . We can now define the indirect profit function for the numeraire-good industry in region j as pXj ðnmj ; Lj Þ f j ðnmj ; Lj Þ nmj wj ðnmj ; Lj Þ Lj rj ðnmj ; Lj Þ. The numeraire good is used for consumption and as input in the production of energy. The energy industry in region j produces Ej units of energy at a cost of Cj(Ej) units of the numeraire good, where Cj() is increasing and strictly convex. We allow for technological differences in the production of energy across regions. The energy industry in each region is mandated to purchase one pollution permit per unit of pollution it emits. Hence, in addition to the production cost, the energy industry in region j incurs a regulatory cost equal to sEj, where s is the price of a pollution permit that clears the federal market for permits. The energy industry in region j chooses nonnegative Ej to maximize (p s)Ej Cj(Ej), taking p and s as given. Let q p s denote the net effective energy price. Assuming that q > 0, the first order condition is q ¼ C jE . Since Cj() is strictly convex, the first order condition is both necessary and sufficient for a unique solution for each j. We denote by Ej(q) and pEj(q) qEj(q) Cj(Ej(q)) the supply and indirect profit functions of the energy industry in region j, respectively. Note that Ejq ¼ 1=C jEE > 0 and pEj q ¼ Ej > 0; namely, regional energy supplies and regional energy indirect profits increase with the effective energy price. Having considered the demand and supply side of each final good, we are now ready to examine the clearing conditions for the permit and energy markets. The clearing condition for the P2 P2 j interregional permit market is j¼1 E ðqÞ EðqÞ ¼ j¼1 Q j Q , since E(q) is both the aggregate supply of energy and the aggregate demand for pollution permits. This condition implicitly defines s(p, Q), the permit price function. The permit price responses are sp = 1 and sQ = 1/Eq < 0. Since qp = 1 sp and qQ = sQ, the permit price responses imply that qp = 0 and qQ = 1/Eq > 0. Thus, the energy effective price is not a function of the energy price, responding only to changes in the federal pollution quota. This implies that we may rewrite the clearing condition for the permit market as E(q(Q)) = Q. The results of the analysis above indicate that the aggregate supply of energy is perfectly inelastic with respect to the price of energy, p, and that it rises with an expansion in the aggregate pol-
lution quota, because the effective energy price, q. rises as the pollution quota expands. The clearing condition for the federal energy market is 2 X m mj m ½nij eij ðp; yij ; EðqðQ ÞÞÞ þ nmj ðp; ym 1 ; y2 ; EðqðQ ÞÞÞe ðp; yj ; EðqðQ ÞÞÞ j¼1
¼ EðqðQ ÞÞ: This condition allows us to see that the price of energy is affected by household incomes and the aggregate pollution quota, m fyi1 ; yi2 ; ym 1 ; y2 ; Q g, and that the energy-price effects associated with changes in household incomes originate from the demand side of the market. Assuming that the law of demand holds in the energy market, the market-clearing condition above enables us to obtain m pðyi1 ; yi2 ; ym 1 ; y2 ; Q Þ, the energy price that clears the market as a function of household incomes and the aggregate pollution quota. The energy price responses are as follows:
nij eijy
pij ¼ P2
i ij j¼1 ðnj ep
pmj ¼ P2
mj m m1 m þ nm j ep Þ þ np ðe1 e2 Þ
mj m1 m m nm j ey þ nym ðe1 e2 Þ
i ij j¼1 ðnj ep
pQ ¼
> 0;
j
mj m m1 m þ nm j ep Þ þ np ðe1 e2 Þ
;
P mj m1 m m 1 2j¼1 ðnij eijE þ nm j eE Þ nE ðe1 e2 Þ ; P2 i ij m mj m m m1 j¼1 ðnj ep þ nj ep Þ þ np ðe1 e2 Þ
where pij @p=@yij , pmj @p=@ym j , j = 1, 2 and pQ o p/ o Q. The denominator of each energy price response is negative given our assumption that the law of demand holds. Having examined the market-clearing conditions and how prices react to changes in household incomes and the aggregate pollution quota, we now consider the determinants of household incomes. As we shall demonstrate below, household income levels and the regional pollution quotas are the key strategic variables controlled by regional and central government authorities. Every landowner in region j earns rental income equal to Lj rj =nij : Each worker in region j earns wage income equal to wj. These two types of households also have other sources of income; namely, sources from the private and public sectors. From the private sector, the income sources are profit shares. Let rhjk > 0 be the share of the regional profits produced in region k owned by a household of type h in region j, for h = i, m and j, k = 1, 2. The distribution of profP m k ¼ 1; 2. it shares must satisfy 2j¼1 ðnij rijk þ nm j rjk Þ ¼ 1; As for the income sources from the public sector, one comes from the regional government and the other comes from the central government. Regional government j supplies Qj pollution permits to the residents of region j. Let qhj P 0 denote the amount of pollution permits held by household h in region j. We assume that qhj is exogenous. However, the distribution of permits in region j m must satisfy nij qij þ nm j qj ¼ Q j . Since household h of region j earns an income amount equal to sqhj if it sells its endowment of permits in the federal market for pollution permits, it will certainly do so. The central government redistributes income in the federation. Let shj be the income transfer received (if positive) or paid (if negative) by household h of region j. The total income of a landowner P in region j is yij ¼ 2k¼1 rijk Pk þ sqij þ Lj rj =nij þ sij , and the total inP2 m m m come of a worker in region j is ym k¼1 rjk Pk þ sqj þ wj þ sj , j ¼ where Pk is the total profit generated in region k, k = 1, 2. The income transfer mechanism implemented by the central P2 i i m m government satisfies j¼1 ðnj sj þ nj sj Þ ¼ 0; the constraint for redistributive income transfers. As it can be observed from the conditions that determine each household’s income, the center’s ability of transferring income to households implies that it is capable of determining each household’s final income level. A unit increase (decrease) in the transfer received by a household implies that this
222
E.C.D. Silva, C. Yamaguchi / Journal of Urban Economics 67 (2010) 219–225
household’s income increases (decreases) by one unit. Since the analysis of the federal policy games examined below becomes more transparent and easier to follow if we postulate that the center chooses household income levels rather than income transfers, we will adopt the more straightforward route here. The constraint for redistributive income transfers can now be rewritten in terms of household incomes and the aggregate pollum tion quota, fyi1 ; yi2 ; ym 1 ; y2 ; Q g, as follows:
(
þ pmj
m m m ðpðyi1 ; yi2 ; ym 1 ; y2 ; Q Þ; y1 ; y2 ; Eð
( 2 X
þ pij
a
v
þ
hm k
v
mk p Þ
l
h
ðym 1
þ w2
ym 2
w1 Þnm1 p
E
E
)
) i
k¼1
¼ 0; ð2aÞ
a
v
v
l
problem. We will later show that indeed this is the case. Given this assumption, Eq. (2a), (2b), (2c) become
aj hij v ijy ¼ lnij ; m m1 m aj hmj v mj ðy1 þ w2 ym y ¼ l½nj þ nym 2 w1 Þ; j 2 X
ð3aÞ ð3bÞ
j m m m1 ak ðhik v ikE þ hmk v mk ð3cÞ E Þ ¼ l½C E þ ðy1 þ w2 y2 w1 ÞnE p:
k¼1
Combining Eqs. (3a)–(3c) 2 X
nik
k¼1
v ikE m v mk þ nk Emk v iky vy
nm1 E
nm1 ym 1
! m þ p ¼ C jE þ ðym 1 þ w2 y2 w1 Þ
v m1 v m2 E E nm1 y m1 vy v m2 y
!
m 2
:
ð4Þ
m1 m1 m1 m2 Since nm1 ¼ v m1 ¼ v m2 y =2a, nym y =2a and nE ¼ ðv E v E Þ=2a, Eq. ym 1 2 (4) reduces to
2 X
nik
k¼1
v ikE m v mk þ nk Emk v iky vy
! þ p ¼ C jE :
ð5Þ
In sum, the solution to the center’s optimization problem is given by Eqs. (1), (3a), (3b), and (5). Note that Eqs. (3a), (3b), and (5) are equivalent to the following conditions for j = 1, 2:
aj hij uijx ¼ lnij ; m m1 m aj hmj umj ððym x ¼ l½nj þ nym 1 w1 Þ ðy2 w2 ÞÞ; j k¼1
ik p
m m m1 ak ðhik v ikp þ hmk v mk p Þ l½ðy1 þ w2 y2 w1 Þnp E
q
We first determine the allocation that the center implements if it controls all policy instruments. This is the center’s most preferred allocation. We later show that this allocation corresponds to the social optimum for the economy. m The center chooses nonnegative fyi1 ; yi2 ; ym 1 ; y2 ; Q g to maximize P2 i m i m a ðh v ðpðÞ; y ; EðÞÞ þ h v ðpðÞ; y ; EðÞÞÞ subject to Eq. (1), j j¼1 j j j j m ; y ; Q Þ and E() E( q (Q)). Let l be the where pðÞ pðyi1 ; yi2 ; ym 1 2 Lagrangian multiplier associated with constraint (1). Assuming an interior solution, the first order conditions are the constraint and the following equations for j = 1, 2:
i k ðhk
þ w2
w1 Þnm1 p
ð2cÞ
2 X
( 2 X
l
) ym 2
where in writing Eqs. (2a)–(2c) we make use of the facts that fnj ¼ wj and in writing Eq. (2c) we also make use of the facts that C 1E ¼ C 2E P2 i ik m mk m and Eq Q = 1. Suppose that k¼1 k ðhk p þ hk p Þ ½ðy1 þ w2 m m1 y2 w1 Þnp E ¼ 0 in the solution to the center’s optimization
3. Centralized policy making and the social optimum
aj hij v ijy lnij
v
½ðym 1
k¼1
where P ðÞ p ðn qðQ ÞÞÞ; Lj Þ þ pEj ðqðQ ÞÞ. The first equality says that the sum of household incomes must be equal to the sum of functional incomes (profits, rents and wages) plus the total revenue income generated with the sale of pollution permits. The rationale for the second equality follows from the facts that producers of the numeraire good must pay wages and rents, energy producers must purchase pollution permits and the aggregate demand for permits equals the aggregate supply of permits. Thus, the total factor costs equal total factor incomes and the total regulatory costs faced by energy producers equal the total revenue generated with the sale of pollution permits. It follows that the sum of functional incomes plus the total revenue income generated with the sale of pollution permits reduces to the sum of industrial revenues net of the total costs of producing energy. Let us now discuss the regional and central government objective functions. As it is standard in the literature, we assume that regional governments wish to maximize regional welfare and that regional welfare does not directly account for the idiosyncratic attachment benefits of the mobile population that resides in the region. The regional welfare function is a weighted sum of consumption utilities of representative immobile and mobile households who reside in the region. We also assume that the center wishes to maximize social welfare and that social welfare is simply a weighted sum of regional welfares. Hence, welfare in region j is hij v ðpðÞ; yij ; EðÞÞþ P2 i m hm j v ðpðÞ; yj ; EðÞÞ and the social welfare function is j¼1 aj ðhj v m i i m ðpðÞ; yj ; EðÞÞ þ hj v ðpðÞ; yj ; EðÞÞÞ. The regional weights, hj > 0 and 1 2 hm j > 0, and the social weights, a > 0 and a > 0, are exogenously determined. mj
þ
mk p Þ
¼ 0;
ð1Þ Xj
v
hm k
j m m m1 ak ðhik v ikE þ hmk v mk E Þ l½C E þ ðy1 þ w2 y2 w1 ÞnE p
þ pQ
j¼1
j
a
ik p
ð2bÞ
k¼1
2 X f j ðnmj ðÞ; Lj Þ þ pðÞEj ðÞ C j ðEj ðÞÞ ;
i k ðhk
¼ 0;
j¼1
¼
2 X
i
k¼1
2 X
2 2 X X nij yij þ nmj ðÞym Pj ðÞ þ sðÞQ j þ Lj rj ðÞ þ nmj ðÞwj ðÞ ¼ j j¼1
h
m m1 m aj hmj v mj ðy1 þ w2 ym y l nj þ nym 2 w1 Þ j
uik nik Eik ux
þ
umk E nm k umk x
þ
uhj e uhj x
¼ C jE ;
ð6aÞ ð6bÞ ð6cÞ
hj hj hj hj hj since v hj y ¼ ux , v E ¼ uE and p ¼ ue =ux . Eq. (6a) inform us that the center’s optimal redistribution policy equates the marginal social benefit of transferring income to an immobile household in region j to the marginal social cost of doing so. The marginal social benefit of the transfer to a representative immobile household in region j is equal to the marginal social utility of income for this type of household, aj hij uijx . The marginal social cost of the transfer to a representative immobile household in region j is equal to the shadow price of transferring income in the federation, l, times the population of immobile households in region j, nij . Eq. (6b) tell us that the center’s optimal redistribution policy also equates the marginal social benefit of transferring income to a representative mobile household in region j to the marginal social cost of doing so. The marginal social benefit of the transfer to a representative mobile household in region j is equal to the marginal social utility of income for this type of household,
223
E.C.D. Silva, C. Yamaguchi / Journal of Urban Economics 67 (2010) 219–225
aj hmj umj x . The marginal social cost of the transfer to a representative mobile household in region j is equal to the shadow price of transferring income in the federation times the population of mobile m1 m ððym households in region j, l½nm 1 w1 Þ ðy2 w2 ÞÞ. The j þ nym j term inside of the bracket represents the population of mobile households in region j after the center’s mechanism is implemented and the mobile households choose their residential locations. Eq. (6c) shows that the center’s optimal pollution control policy equates the marginal social benefit of energy consumption to the marginal social cost of producing energy. The marginal social benefit of energy consumption is equal to the private marginal benefit of energy consumption minus the marginal social pollution damage, where the private marginal benefit of energy consumption is the marginal rate of substitution between energy consumption and the numeraire good and the marginal social pollution damage is the sum of the marginal rates of substitution between the total amount of pollution and the numeraire good. Eq. (6c) makes it clear that the energy good is an impure public bad, since its consumption yields a private and positive marginal benefit but also a pure and negative marginal social benefit. P2 i ik m mk m We now show that k¼1 ak ðhk v p þ hk v p Þ ¼ l½ðy1 þ w2 m1 w Þn E in the optimum. Utilizing Eqs. (3a) and (3b), we ym 1 2 p have 2 X
2 X ik m m1 m m ak ðhik v ikp þ hmk v mk ½nik ðv ik p Þ ¼ l p =v y Þ þ ðnk þ nym ðy1 þ w2 y2
k¼1
tions for an interior solution are the constraints and the following equations for j, k = 1, 2:
aj hij uijx ¼ cnij ; m m m m ^ m1 ½cðxm aj hmj umj x ¼ cnj þ nxm 1 þ w2 x2 w1 Þ þ kðe1 e2 Þ; j
ð9bÞ
aj hij uije ¼ knij ; m m m m ^ m1 ½cðxm aj hmj umj e ¼ knj þ nem 1 þ w2 x2 w1 Þ þ kðe1 e2 Þ; j
ð9dÞ
2 X
w1 ÞÞðv
m m m m a1 hm1 ¼ ðcnm1 =um1 x Þ þ f½cðx1 þ w2 x2 w1 Þ þ kðe1 e2 Þ=2ag; ð10aÞ m m m2 m m m a2 h2 ¼ ðcn2 =ux Þ f½cðx1 þ w2 x2 w1 Þ þ kðe1 em2 Þ=2ag; ð10bÞ m m m m a1 hm1 ¼ ðknm1 =um1 e Þ þ f½cðx1 þ w2 x2 w1 Þ þ kðe1 e2 Þ=2ag; ð10cÞ m m m2 m m m a2 h2 ¼ ðkn2 =ue Þ f½cðx1 þ w2 x2 w1 Þ þ kðe1 em2 Þ=2ag: ð10dÞ m1 Combining Eqs. (10a) and (10c), we have um1 ¼ k=c. Similarly, e =ux m2 =u ¼ k=c. Now, note combining Eqs. (10b) and (10d) yields um2 e x that conditions (9a), (9e), (10a) and (10b) imply that for k = 1, 2
c
2 X
v
2 X m m mk mk l ½nik ðv ikp =v iky Þ þ ðnmk þ nm1 ym ðy1 þ w2 y2 w1 ÞÞðv p =v y Þ
j¼1
k
¼ l½E ¼l
½ðym 1
þ w2
þ w2
ym 2
w1 Þðv
w1 Þnm1 p
m1 p
v
m2 p Þ=2a
E;
P2 i ik m mk m1 m2 m since nm1 p ¼ ðv p v p Þ=2a. Thus, k¼1 ak ðhk v p þ hk v p Þ ¼ l½ðy1 þ m1 w Þn E. In sum, the center’s optimal redistributive w2 ym 1 2 p policies nullify all energy-price effects. If such effects were not nullified by the center’s optimal redistributive mechanism, the center’s most preferred allocation would not maximize social welfare. As we demonstrate below, the center’s most preferred allocation, characterized by conditions (1), (6a)–(6c), corresponds to the social optimum for the federal economy. Consider now the problem faced by a social planner. The probi m lem is the choice of nonnegative fxik ; xm k ; ek ; ek ; Ek gk¼1;2 to maximize social welfare 2 X
aj ðhij uðxij ; eij ; EÞ þ hmj uðxmj ; emj ; EÞÞ
ð7Þ
j¼1
subject to: E ¼
P2
k¼1 Ek
and
2 X
2 X
j¼1
j¼1
j ^ mj ðÞxm ½nij xij þ n j þ C ðEj Þ ¼
2 2 X X ^ mj ðÞem ½nij eij þ n Ej ; j ¼ j¼1
uijE uijx
^ mj ðÞ; Lj Þ; f j ðn
þ
nm j
umj E umj x
#
m1 um2 E uE ^ m1 ½cðxm ¼ cC kE k þ n E þ 1 2a ð11Þ
m1 m2 ^ m1 Since n E ¼ ðuE uE Þ=2a, Eq. (11) reduces to
m 2
ym 2
nij
m m þ w2 xm 2 w1 Þ þ kðe1 e2 Þ:
2 X
ðym 1
"
j¼1
mk mk p = y Þ:
m 1
k¼1
k m m m m ^ m1 aj ðhij uijE þ hmj umj E Þ ¼ cC E k þ nE ½cðx1 þ w2 x2 w1 Þ þ kðe1 e2 Þ: ð9eÞ
Combining Eqs. (9a) and (9c) yields uije =uijx ¼ k=c, j = 1, 2. Since ^ m1 ¼ um2 ^ m1 ¼ um1 ^ m1 ¼ um2 ^ m1 ¼ um1 n x =2a, nxm x =2a, nem e =2a and nem e =2a, xm 1 2 1 2 conditions (9b) and (9d) yield
h hk m1 m1 m1 m2 v hk p ¼ ek v y , ny ¼ v y =2a, ny ¼ v y =2a, we obtain
As
ð9cÞ
j¼1
k
k¼1
ð9aÞ
ð8aÞ ð8bÞ
j¼1
m m m ^ m2 ðÞ ¼ Nm n ^ m1 ðÞ and n ^ m1 ðÞ n ^ m1 ðxm where n 1 ; x2 ; e1 ; e2 ; EÞ is defined implicitly by the migration equilibrium condition, m m m m m m uðxm 1 ; e1 ; EÞ þ aðN n1 Þ ¼ uðx2 ; e2 ; EÞ þ an1 . Eqs. (8a) and (8b) are the resource constraints for the numeraire and energy goods, respectively. Let c and k be the Lagrangian multipliers associated with constraints (8a) and (8b), respectively. The first order condi-
"
nij
# mj uhj e m uE þ hj ¼ C kE ; þ n j ij mj ux ux ux uijE
k ¼ 1; 2;
ð12Þ
hj where in writing Eqs. (12) we used the fact that uhj e =ux ¼ k=c, h = i, m and j = 1, 2. The equivalence between the social optimum and the center’s most preferred allocation follows from the fact that the conditions that determine the social optimum are equivalent to the conditions that determine the center’s most preferred allocation. First, note hj that Eqs. (12) are identical to Eq. (6c). Since uhj e =ux ¼ k=c, h = i, m and j = 1, 2, one realizes that the center’s optimal allocation satishj fies these conditions because uhj e =ux ¼ p, and thus k=c ¼ p. In addition, once one also observes that c = l, one also realizes that Eqs. (6a) and (6b) correspond to Eqs. (9a) and (9b), respectively. Finally, notice that Eq. (1) is identical to Eq. (8a) and the market-clearing condition for the energy good is identical to Eq. (8b). Therefore, the center’s optimal allocation is socially optimal.
4. Decentralized leadership: the federal policy game Regional and central authorities play a two-stage game as follows. In the first stage, region j chooses Q j P 0 to maximize m m where pðÞ pðyi1 ; yi2 ; ym hij v ðpðÞ; yij ; EðÞÞ þ hm j v ðpðÞ; yj ; EðÞÞ, 1 ; y2 ; Q Þ, E() E(q(Q)) and Q = Q1 + Q2, taking Qk, k – j, as given. In the second stage, the center observes {Q1, Q2} and then chooses P2 i m i m i to maximize fyi1 ; ym 1 ; y2 ; y2 g j¼1 aj ðhj v ðpðÞ; yj ; EðÞÞ þ hj v ðpðÞ; ; EðÞÞÞ subject to (1), the federation’s redistributive constraint. ym j The equilibrium concept used is subgame perfection. We should first consider the second stage. Since the center’s problem is the same as the center solved in the previous section with respect to the choices of household income levels, the interior solution to its problem satisfies constraint (1) and first order conditions (6a) and (6b). Close inspection of these equations reveals that the optimal household incomes depend on the aggregate quota of pollution permits, Q, rather than on the separate regional quotas, Qj, j = 1, 2. Let yhj(Q) denote the optimal income function of
224
E.C.D. Silva, C. Yamaguchi / Journal of Urban Economics 67 (2010) 219–225
household of type h in region j defined implicitly by the system of first order conditions that characterize the optimal redistribution scheme, namely, Eqs. (1), (6a), and (6b). ~j ðQ Þ Ej ðqðQ ÞÞ, ~ðQ Þ pðyi1 ðQ Þ; yi2 ðQ Þ; ym1 ðQ Þ; ym2 ðQ Þ; Q Þ, E Let p mj mj ~ m1 ~ ÞÞ. ~ Þ P2 E ~j ðQ Þ and n ~ ðQ Þ n ð p ðQ Þ; y ðQ Þ; ym2 ðQ Þ; EðQ EðQ j¼1 Inserting these functions and the optimal household income functions into the federation’s redistributive constraint yields 2 X
m
~ m1 ðQÞym1 ðQ Þ þ ðN n ~ m1 ðQ ÞÞym2 ðQÞ f 1 ðn ~ m1 ðQ ÞÞ nij yij ðQÞ þ n
j¼1 2 X ~j ðQÞ C j ðE ~j ðQÞÞ: ~ m1 ðQ ÞÞ þ ~ðQ ÞE ½p þ f 2 ðNm n
ð13Þ
j¼1
Differentiating identity (13) with respect to Q, we have 2 X mj m m ~ m1 ~ ½nij yijQ þ nm j yQ þ ½ðy1 w1 Þ ðy2 w2 ÞnQ pQ E ¼ s
ð14Þ
j¼1
since 2 2 X X j ~j ðQ Þ þ p ~j ¼ p ~j ~j C j E ~Q E ~ðQ ÞE ~ E ½p E þ ðp C Þ Q E E Q Q Q j¼1
j¼1
~Q E þ p ¼p
C jE
~ Q E þ s: ¼p
The first equality in the last set of equalities follows from the fact that C 1E ¼ C 2E . For future use, one should also note that m1 ~ m1 m1 m1 m2 m1 ~ m1 n Q np pQ þ nym yQ þ nym yQ þ nE : 1
ð15Þ
2
Consider now the first stage of the game. Assuming interior solutions, the first order conditions that characterize the Nash equilibrium are as follows for j = 1, 2: mj mj ~ mj ij ~Q þ v ijy yijQ þ v ijE Þ þ hm hij ðv ijp p j ðv p pQ þ v y yQ þ v E Þ ¼ 0:
ð16Þ
hj hj Given the early results for v hj p , v y and v E , and Eqs. (6a) and (6b), Eqs. (16) can be written as follows for j = 1, 2:
m X
" nhj
h¼i
"
uhj E uhj x
# þ
ymj Q þ
nhj yhj Q
umj E umj x
~Q nhj ehj p
m m þ nm1 ym ðy1 w1 Þ ðy2 w2 Þ j
# ~ em j pQ ¼ 0:
ð17Þ
Adding Eqs. (17) and utilizing Eq. (14), we obtain: 2 X
"
uijE
nij
uijx
j¼1
þ
nm j
umj E
#
umj x
" ! # 2 X umj mj m m1 m~ m1 E ~ n w Þ ðy w Þ n y þ e þ ðym p m 1 2 1 2 yj j Q Q Q umj x j¼1 ¼ s: ð18Þ
We now demonstrate that Using identity (15), we have 2 X j¼1
nm1 ym j
ymj Q
þ
umj E umj x
mj uE mj m1 m~ ~ m1 n Q ¼ 0. j¼1 nym yQ þ mj ej pQ
P2
!
~ em j pQ
ux
j
~m1 n Q
¼
2 X j¼1
nm1 ym j
umj E umj x
!
~ em j pQ
~ nm1 p pQ
nm1 E : Given the results for nm1 , nm1 and nm1 we have p E , ym P2 m1 umj j m~ m1 ~ m1 E n n e n ¼ 0. Thus, Eq. (18) becomes p p Q Q mj p E j¼1 ym j ux
j
2 X j¼1
" nij
uijE uijx
þ
nm j
umj E umj x
# ¼ s:
ð19Þ
In sum, the subgame perfect equilibrium is given by Eqs. (1), hj (6a), (6b), and (19). Since s ¼ p C kE , k = 1, 2, and uhj e =ux ¼ p, h = i,m, j = 1, 2, Eq. (19) corresponds to Eqs. (12). Thus, the subgame perfect equilibrium is isomorphic to the center’s most preferred allocation, which in turn is socially optimal. Proposition 1 gathers these results: Proposition 1. The subgame perfect equilibrium for the federal policy game is socially optimal. Proposition 1 makes it clear that combining decentralized control of federal pollution with an interregional income redistribution scheme implemented by the center induces the regional governments to behave efficiently provided that energy and pollution permits are traded within and across regions.5 This is true even though the regional authorities are fully aware that their policy choices influence both energy and permit prices. Both regional governments agree on the total amount of pollution permits they together should supply. They make choices so that the permit price that clears the market equals the sum of marginal regional pollution damages. Our main contributions to the literature is to extend the efficiency result of Caplan et al. (2000) to a more realistic setting which has the following features: (i) there is an impure public bad; (ii) commodities (energy and pollution permits) are traded within and across regions; (iii) the regional governments are large in the sense that they do not take prices as given; (iv) there are mobile and immobile households; (v) households in each region possess shares of profits produced in both regions; and (vi) there are regional factor markets. Proposition 1 informs us that our framework provides an ideal efficiency benchmark for federal structures characterized by decentralized leadership, imperfect labor mobility, heterogeneous households and profit expatriation. These are all key characteristics of the European Union. It is tempting, therefore, to conclude that our main result is good news for the European Union. Undoubtedly, the governments of the member nations in the European Union have considerably greater economic and political powers than the central government and are autonomous to implement policies that have transboundary consequences. It is also true that the center is endowed with financial resources, derived from the member nations, which enable it to implement income redistribution within the federation. Structural and Cohesion Funds provide good examples of centrally determined redistributive schemes. But, the central authority in the European Union is far weaker than the central authority in our model. The volume of resources controlled by the center in the European Union is limited, and the limitations stem from the fact that the center is politically and economically weak vis-à-vis the national authorities of the member countries. The center in our model is ‘‘weak” only to the extent that it is unable to credibly commit to punish or reward the regional governments for their actions. This weakness is formally modeled by postulating that the center is a follower in the federal policy game. Unlike in the European Union, the center’s redistribution scheme here is not limited by the regional authorities. The center is capable of transferring income from any household to any other 5 In an earlier version of this paper, we demonstrated that hierarchical federal policy making yields an inefficient subgame perfect equilibrium if the regions do not trade energy and pollution permits with each other. It is also straightforward to demonstrate that if there is an interregional market for energy or pollution permits only, the subgame perfect equilibrium for the federal policy game is inefficient. In addition to equalization of marginal social utilities of income, efficiency requires equalization of marginal rates of substitution between energy and the numeraire good across households, equalization of marginal costs of energy production across regions and satisfaction of the Samuelson condition for federal pollution reduction.
E.C.D. Silva, C. Yamaguchi / Journal of Urban Economics 67 (2010) 219–225
household as it wishes. It determines each household’s final income after it observes the actions taken by the regional authorities. Indeed, it is the center’s ability of determining household income coupled with the anticipation of the optimal redistribution conditions that induce the regional governments to behave efficiently. With limited income redistribution, one can imagine that even though the qualitative incentive features of the redistribution scheme would remain, the income transfer amounts would not be sufficiently high to promote efficient behavior. 5. Conclusion This paper analyzes decentralized regulation of energy supply, an impure public bad. Energy generates positive consumption benefits and pollution damages. Our federal structure is characterized by decentralized leadership and imperfect labor mobility. We show that decentralized environmental policy is efficient if it is combined with an optimal redistribution policy carried out by the center provided that energy and pollution permits are traded within and across regions and the center redistributes income after the regional authorities choose regional supplies of pollution permits. We show that this result holds in an economy featuring heterogeneous households and profit expatriation. Our main finding may be applicable to many situations in which the consumption benefits of regionally provided public goods are not restricted to the residents of the regions that provide the goods. Examples of such goods abound. Public infrastructure, forest conservation and disease eradication are just a few examples. A central government may be able to induce regional governments to adopt efficient policies in the provision of impure public goods provided that it implements redistributive income transfers and federal markets are built for the relevant commodities.
225
One word of caution is in order, however. Our setting should be viewed as an efficient benchmark for federations characterized by decentralized leadership. The center in our model is weak only to the extent that it is unable to credibly punish or reward the regional governments for their actions. The center does not face any limits from the regional governments in implementing its most desirable income redistribution scheme. Were such limitations exist, the incentives promoted by income transfers for efficiency would be limited as well. This is an interesting avenue for future work. Acknowledgments We greatly benefited from valuable comments made by the editor, Jan Brueckner, and two anonymous referees. We also thank Jun-ichi Itaya, Se-il Mun and the participants of a seminar at Hokkaido University and the Japanese Economic Association Meetings held at Oita University for helpful comments on earlier versions. References Boadway, R., 1982. On the method of taxation and the provision of local public goods: comment. American Economic Review 72, 846–851. Caplan, A.J., Cornes, R.C., Silva, E.C.D., 2000. Pure public goods and income redistribution in a federation with decentralized leadership and imperfect labor mobility. Journal of Public Economics 77, 265–284. Mansoorian, A., Myers, G.M., 1993. Attachment to home and efficient purchases of population in a fiscal externality economy. Journal of Public Economics 52, 117–132. Myers, G.M., 1990. Optimality, free mobility, and the regional authority in a federation. Journal of Public Economics 43, 107–121. Oates, W.E., 1972. Fiscal Federalism. Harcourt Brace Jonavonich, New York. Stiglitz, J.E., 1977. The theory of local public goods. In: Feldstein, M., Inman, R.P. (Eds.), The Economics of Public Services. Macmillan, London, pp. 274–333. Wellisch, D., 1994. Interregional spillovers in the presence of perfect and imperfect household mobility. Journal of Public Economics 55, 167–184.
Contents lists available at ScienceDirect
Journal of Urban Economics www.elsevier.com/locate/jue
Interregional competition, spillovers and attachment in a federation Emilson C.D. Silva a,*, Chikara Yamaguchi b a b
School of Economics, Georgia Institute of Technology, Atlanta, GA 30332-0695, USA Faculty of Economic Sciences, Hiroshima Shudo University, 1-1-1, Ozukahigashi, Asaminani-ku, Hiroshima 731-3195, Japan
a r t i c l e
i n f o
Article history: Received 8 November 2006 Revised 7 May 2009 Available online 6 September 2009 JEL Classification: D6 D7 H4 H7 Q2 R5
a b s t r a c t We examine decentralized environmental policy making in a federation characterized by decentralized leadership and imperfect labor mobility due to attachment to regions. Energy consumption generates positive consumption benefits, but energy supply generates federal air pollution. Regional authorities regulate energy supply by controlling supplies of pollution permits. Energy and pollution permits are traded in interregional markets. The center redistributes incomes after it observes regional supplies of pollution permits. Regions are populated by mobile and immobile households and profits are expatriated. We show that the subgame perfect equilibrium for the federal policy game played by regional and central authorities is socially optimal. Ó 2009 Elsevier Inc. All rights reserved.
Keywords: Federation Interregional spillovers Redistribution Labor attachment
1. Introduction According to conventional wisdom in fiscal federalism, regional public goods that generate interregional spillovers – that is, goods whose economic jurisdiction exceeds the political jurisdiction of a regional or local government – should be subjected to a Pigouvian subsidy determined by a higher-level government, namely, the central government (see, e.g., Oates, 1972). Local governments should provide local public goods as far as economic and political jurisdictions coincide, but local public goods that generate interregional spillovers should be provided by a supra-local level of government. Recent studies, however, demonstrate that there are circumstances under which competing regional governments may not only provide public goods that generate transboundary spillovers efficiently, but also implement interregional transfers to each other so that a socially efficient population distribution is obtained.1 In Wellisch (1994), competing regional governments fully internalize * Corresponding author. Fax: +1 404 894 1890. E-mail addresses: [email protected] (E.C.D. Silva), [email protected] (C. Yamaguchi). 1 See Myers (1990) for the demonstration that the Nash equilibrium for the game played by decentralized regional governments is efficient provided they control instruments to make income transfers to each other. 0094-1190/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jue.2009.09.002
externalities associated with provision of public goods in their regions whenever households are not attached to regions. Caplan et al. (2000) demonstrate that decentralized provision of a pure public good – a good whose economic benefit is available for an entire federation – may be efficient for a game where regional governments are policy leaders and a central government is a policy follower. Each regional government decides how much of the pure public good to provide in anticipation of the redistributive income policy implemented by the center. The efficiency result does not depend on the degree of household attachment. In this paper, we extend the framework advanced by Caplan et al. (2000) in three significant ways. First, we consider commodities that generate regional-specific benefits as well as a federal benefit (good) or cost (bad). Consider, for example, provision of energy. Energy generates private consumption benefits as well as federal damages, since its consumption or production typically yields emissions of pollutants in the atmosphere (e.g., carbon dioxide). Regional governments are concerned with controlling emissions of carbon dioxide in their own regions. We postulate that such a control is done through setting up quotas of pollution permits that can be sold in a federal market for pollution permits. We also assume that the energy commodity is traded within and across regions. Thus, unlike Caplan et al. (2000), we consider an economy featuring tradable commodities and an impure public bad.
220
E.C.D. Silva, C. Yamaguchi / Journal of Urban Economics 67 (2010) 219–225
The second extension is that we examine the actions of large government authorities which fully recognize that their policies affect prices of all commodities traded in the economy. In our framework, there are three commodities which are traded globally: numeraire and energy goods and pollution permits. The third extension comes from the fact that we depart from the unrealistic assumption used in Caplan et al. (2000) that each household owns an equal share of the resources available in the region in which it resides. We consider instead a more general and realistic setting where two types of households, mobile and immobile, reside in each region. Each region’s immobile population owns the regionally fixed productive resource, which we shall assume to be land. The mobile population in this economy is its population of workers. Landowners earn land rents and workers earn wages. In addition, both types of households possess positive shares of profits produced in both regions, earn income from the sale of pollution permits and receive or pay centrally determined income transfers. As in Caplan et al. (2000), we are interested in analyzing the allocation of resources for a federal economy characterized by imperfect labor mobility and decentralized leadership. Our example of such a federal economy coincides with theirs, namely, the European Union (EU). The governments of the member nations are endowed with considerable economic and political powers vis-à-vis the center concerning most types of policies, including environmental policies. Accordingly, in our setting, the regional governments select the regional quantities of pollution permits to be supplied in the pollution permit market prior to the center’s decisions of how to redistribute income in the federation. This paper is organized as follows. Section 2 presents the model and determines the competitive equilibrium given the set of policy instruments. The problems faced by the decentralized agents (consumers and producers) yield the relevant indirect utility and profit functions that are taken into account by the governmental authorities when they make their decisions. Section 3 examines a setting in which the center controls all policy instruments and demonstrates that the implied allocation is socially optimal. Section 4 describes and solves the two-stage federal policy game played by regional and central authorities. We demonstrate that the subgame perfect equilibrium for this game is socially optimal by showing that it is isomorphic to the center’s most preferable allocation. Section 5 concludes the paper. 2. The model Consider a federation consisting of two regions, two regional governments and a central government. There are two types of households, workers and owners of a fixed regional resource, which we shall consider to be land. Landowners are immobile. Workers are imperfectly mobile due to regional attachment benefits. Let nij and nm j denote respectively the populations of immobile and mobile households who reside in region j, j = 1, 2. We assume that nij > 0, j = 1, 2. Let nj nij þ nm j be the population of region j and Nh > 0 be the total population of households of type h in the federation, h = i,m. A household of type h who lives in region j consumes xhj units of a numeraire good, ehj units of energy and is harmed by the aggregate amount of air pollution, E, that is produced in the federation. For simplicity, we assume that one unit of energy produced in each region leads to the emission of one unit of air pollution. Hence, E also corresponds to the aggregate supply of energy in the economy. Let uðxhj ; ehj ; EÞ denote the utility derived by every household of type h in region j. We assume that u() is increasing in the first two arguments, decreasing in the last, strictly concave, and satisfies 2 h h h h uhj xe @ u=@xj @ej P 0 for all xj > 0 and ej > 0. The last assumption ensures that demands for private goods are normal.
Every household h in region j chooses nonnegative fxhj ; ehj g to maximize uðxhj ; ehj ; EÞ subject to xhj þ pehj ¼ yhj ; where p is the price of energy and yhj is the household’s income. Each household treats the price of energy, its income and the level of air pollution in the federation as exogenous variables. In the game played by government authorities, however, household income levels, the price of energy and the level of air pollution are endogenous variables. Assuming that the solution to the consumer’s maximization problem is interior, the solution satisfies the budget constraint hj hj h and the tangency condition, uhj e =ux ¼ p, where ue @uðxj ; h h h ehj ; EÞ=@ehj and uhj x @uðxj ; ej ; EÞ=@xj , h = i, m, j = 1, 2. In any region, each household’s marginal rate of substitution between energy and the numeraire good equals the price of energy. Since u() is strictly concave, the solution for each household’s problem is unique. We let xhj ðp; yhj ; EÞ and ehj ðp; yhj ; EÞ denote the demand functions and v ðp; yhj ; EÞ uðxhj ðp; yhj ; EÞ; ehj ðp; yhj ; EÞ; EÞ denote the indirect utility function of a household of type h in region j, h = i, m and j = 1, 2. For future reference, we establish that
v hjp v hjy mhjE
@ v ðp; yhj ; EÞ @p @ v ðp; yhj ; EÞ @yhj @ v ðp; yhj ; EÞ @E
hj hj hj hj hj hj hj h ¼ uhj x xp þ ue ep ¼ ux ðxp þ pep Þ ¼ ux ej < 0;
hj hj hj hj hj hj hj ¼ uhj x xy þ ue ey ¼ ux ðxy þ pey Þ ¼ ux > 0;
hj hj hj hj hj hj hj hj hj ¼ uhj x xE þ ue eE þ uE ¼ ux ðxE þ peE Þ þ uE ¼ uE < 0:
In addition, v ðp; yhj ; EÞ is strictly concave in yhj because uðxhj ; ehj ; EÞ h h is strictly concave and uhj xe P 0 for all xj > 0 and ej > 0. Mobile households are heterogeneous with respect to their idiosyncratic attachment benefits as in Mansoorian and Myers (1993), Wellisch (1994) and Caplan et al. (2000). These households are distributed uniformly on the interval [0, Nm]. If a mobile household nm resides in region 1, its total utility is v ðp; ym 1 ; EÞþ aðN m nm Þ. If it resides in region 2 instead, its total utility is v ðp; ym2 ; EÞ þ anm . For household nm, the term a(Nm nm) represents the attachment benefit of residing in region 1 and the term anm represents the attachment benefit of residing in region 2. The parameter a > 0 is the attachment intensity. m m In the migration equilibrium, v ðp; ym 1 ; EÞ þ aðN n1 Þ ¼ v ðp; m m m y2 ; EÞ þ an1 : This condition provides a solution n1 2 ð0; N m Þ; where 2 nm 1 is the mobile household that is indifferent between regions. Every household nm 2 ½0; nm Þ strictly prefers to live in region 1 and 1 m every household nm 2 ðnm 1 ; N strictly prefers to live in region 2. Let m m m m Dðnm ; p; y ; y ; EÞ v ðp; y ; EÞ þ aðN 2nm 1 1 2 1 1 Þ v ðp; y2 ; EÞ denote the surplus of household nm 1 of residing in region 1 relative to residing in region 2. In equilibrium, this surplus is zero. Since Dn @ D=@nm 1 ¼ 2a < 0, the migration equilibrium is stable.3 The equilibrium condim tion implicitly defines nm1 ðp; ym 1 ; y2 ; EÞ, the migration response funcm1 m2 tion. The marginal migration responses are nm1 p ¼ ðv p v p Þ=2a, m1 m1 m2 m1 m1 m2 ¼ v =2a > 0, n ¼ v =2a < 0 and n ¼ ð v v Þ=2a. nm1 m m y y E E E y1 y2 Let us now examine the supply side of the federal economy, starting with the numeraire-good industry. In what follows, we follow the literature in assuming that factor markets (i.e., labor and land) are regional.4 In region j, the labor supply is nm j and the land supply 2 We make assumptions below that guarantee that each region is populated with workers in equilibrium. 3 See Boadway (1982) and Stiglitz (1977) on issues pertaining to the stability of the migration equilibrium in economies with public goods. 4 Caplan et al. (2000), Mansoorian and Myers (1993) and Wellisch (1994) implicitly assume that there is a labor market in each region. The standard model utilized by these authors does not include regional land markets. It is assumed that regional residents (workers) have equal ownership shares of regional resources and that the aggregate supply of land in each region is fixed. Since in our model we allow landowners and workers to possess different shares of regional resources and we wish to clearly demonstrate how the factor prices are derived in equilibrium, we must examine the market-clearing conditions for the factor markets.
221
E.C.D. Silva, C. Yamaguchi / Journal of Urban Economics 67 (2010) 219–225
is Lj. Each worker is endowed with one unit of labor, which is supplied in the region in which the worker resides. All workers and land units are employed in the production of the numeraire good. There are jj > 0 firms in the numeraire-good industry in region j. Each of these firms employs zj workers and lj units of land to produce gj(zj, lj) units of output. We assume that gj() is increasing in each argument and strictly concave. We also assume that gj() satisfies the Inada conditions limzj !0 g jz ¼ 1 and limlj !0 g jl ¼ 1. The Inada condition with respect to the labor input guarantees that each region is populated with workers in the migration equilibrium. Each firm chooses nonnegative {zj, lj} to maximize gj(zj, lj) wjzj rjlj, where wj > 0 and rj > 0 are the wage and land rent faced by each firm located in region j, respectively. Firms take factor prices as given. The first order conditions for profit maximization are g jz ¼ wj and g jl ¼ r j . These conditions define the continuous and differentiable factor demand functions, zj(wj, rj) and lj(wj, rj). Since in region j the labor supply is nm j for all wj P 0 and the land supply is Lj for all r j P 0; the wage and land rent levels that clear j the regional factor markets solve jj zj ðwj ; rj Þ ¼ nm j and jjl (wj, rj) = Lj, j = 1, 2, respectively. The clearing conditions for the regional factor j m markets in region j enable us to obtain wj ðnm j ; Lj Þ and r ðnj ; Lj Þ, j = 1, 2; that is, the regional factor prices as functions of regional factor supplies. j j j j m j m j m j m ^j ðnm Let g j ; Lj Þ g ðz ðw ðnj ; Lj Þ; r ðnj ; Lj ÞÞ; l ðw ðnj ; Lj Þ; r ðnj ; Lj ÞÞÞ j m j m ^ ðnj ; Lj Þ. The latter is region j’s output of the and f ðnj ; Lj Þ jj g numeraire good expressed as a function of the regional factor supj ¼ g jzz =jj < 0. Thus, fj() is plies. Note that fnj ¼ g jz ¼ wj > 0 and fnn m increasing and strictly concave in nj . We can now define the indirect profit function for the numeraire-good industry in region j as pXj ðnmj ; Lj Þ f j ðnmj ; Lj Þ nmj wj ðnmj ; Lj Þ Lj rj ðnmj ; Lj Þ. The numeraire good is used for consumption and as input in the production of energy. The energy industry in region j produces Ej units of energy at a cost of Cj(Ej) units of the numeraire good, where Cj() is increasing and strictly convex. We allow for technological differences in the production of energy across regions. The energy industry in each region is mandated to purchase one pollution permit per unit of pollution it emits. Hence, in addition to the production cost, the energy industry in region j incurs a regulatory cost equal to sEj, where s is the price of a pollution permit that clears the federal market for permits. The energy industry in region j chooses nonnegative Ej to maximize (p s)Ej Cj(Ej), taking p and s as given. Let q p s denote the net effective energy price. Assuming that q > 0, the first order condition is q ¼ C jE . Since Cj() is strictly convex, the first order condition is both necessary and sufficient for a unique solution for each j. We denote by Ej(q) and pEj(q) qEj(q) Cj(Ej(q)) the supply and indirect profit functions of the energy industry in region j, respectively. Note that Ejq ¼ 1=C jEE > 0 and pEj q ¼ Ej > 0; namely, regional energy supplies and regional energy indirect profits increase with the effective energy price. Having considered the demand and supply side of each final good, we are now ready to examine the clearing conditions for the permit and energy markets. The clearing condition for the P2 P2 j interregional permit market is j¼1 E ðqÞ EðqÞ ¼ j¼1 Q j Q , since E(q) is both the aggregate supply of energy and the aggregate demand for pollution permits. This condition implicitly defines s(p, Q), the permit price function. The permit price responses are sp = 1 and sQ = 1/Eq < 0. Since qp = 1 sp and qQ = sQ, the permit price responses imply that qp = 0 and qQ = 1/Eq > 0. Thus, the energy effective price is not a function of the energy price, responding only to changes in the federal pollution quota. This implies that we may rewrite the clearing condition for the permit market as E(q(Q)) = Q. The results of the analysis above indicate that the aggregate supply of energy is perfectly inelastic with respect to the price of energy, p, and that it rises with an expansion in the aggregate pol-
lution quota, because the effective energy price, q. rises as the pollution quota expands. The clearing condition for the federal energy market is 2 X m mj m ½nij eij ðp; yij ; EðqðQ ÞÞÞ þ nmj ðp; ym 1 ; y2 ; EðqðQ ÞÞÞe ðp; yj ; EðqðQ ÞÞÞ j¼1
¼ EðqðQ ÞÞ: This condition allows us to see that the price of energy is affected by household incomes and the aggregate pollution quota, m fyi1 ; yi2 ; ym 1 ; y2 ; Q g, and that the energy-price effects associated with changes in household incomes originate from the demand side of the market. Assuming that the law of demand holds in the energy market, the market-clearing condition above enables us to obtain m pðyi1 ; yi2 ; ym 1 ; y2 ; Q Þ, the energy price that clears the market as a function of household incomes and the aggregate pollution quota. The energy price responses are as follows:
nij eijy
pij ¼ P2
i ij j¼1 ðnj ep
pmj ¼ P2
mj m m1 m þ nm j ep Þ þ np ðe1 e2 Þ
mj m1 m m nm j ey þ nym ðe1 e2 Þ
i ij j¼1 ðnj ep
pQ ¼
> 0;
j
mj m m1 m þ nm j ep Þ þ np ðe1 e2 Þ
;
P mj m1 m m 1 2j¼1 ðnij eijE þ nm j eE Þ nE ðe1 e2 Þ ; P2 i ij m mj m m m1 j¼1 ðnj ep þ nj ep Þ þ np ðe1 e2 Þ
where pij @p=@yij , pmj @p=@ym j , j = 1, 2 and pQ o p/ o Q. The denominator of each energy price response is negative given our assumption that the law of demand holds. Having examined the market-clearing conditions and how prices react to changes in household incomes and the aggregate pollution quota, we now consider the determinants of household incomes. As we shall demonstrate below, household income levels and the regional pollution quotas are the key strategic variables controlled by regional and central government authorities. Every landowner in region j earns rental income equal to Lj rj =nij : Each worker in region j earns wage income equal to wj. These two types of households also have other sources of income; namely, sources from the private and public sectors. From the private sector, the income sources are profit shares. Let rhjk > 0 be the share of the regional profits produced in region k owned by a household of type h in region j, for h = i, m and j, k = 1, 2. The distribution of profP m k ¼ 1; 2. it shares must satisfy 2j¼1 ðnij rijk þ nm j rjk Þ ¼ 1; As for the income sources from the public sector, one comes from the regional government and the other comes from the central government. Regional government j supplies Qj pollution permits to the residents of region j. Let qhj P 0 denote the amount of pollution permits held by household h in region j. We assume that qhj is exogenous. However, the distribution of permits in region j m must satisfy nij qij þ nm j qj ¼ Q j . Since household h of region j earns an income amount equal to sqhj if it sells its endowment of permits in the federal market for pollution permits, it will certainly do so. The central government redistributes income in the federation. Let shj be the income transfer received (if positive) or paid (if negative) by household h of region j. The total income of a landowner P in region j is yij ¼ 2k¼1 rijk Pk þ sqij þ Lj rj =nij þ sij , and the total inP2 m m m come of a worker in region j is ym k¼1 rjk Pk þ sqj þ wj þ sj , j ¼ where Pk is the total profit generated in region k, k = 1, 2. The income transfer mechanism implemented by the central P2 i i m m government satisfies j¼1 ðnj sj þ nj sj Þ ¼ 0; the constraint for redistributive income transfers. As it can be observed from the conditions that determine each household’s income, the center’s ability of transferring income to households implies that it is capable of determining each household’s final income level. A unit increase (decrease) in the transfer received by a household implies that this
222
E.C.D. Silva, C. Yamaguchi / Journal of Urban Economics 67 (2010) 219–225
household’s income increases (decreases) by one unit. Since the analysis of the federal policy games examined below becomes more transparent and easier to follow if we postulate that the center chooses household income levels rather than income transfers, we will adopt the more straightforward route here. The constraint for redistributive income transfers can now be rewritten in terms of household incomes and the aggregate pollum tion quota, fyi1 ; yi2 ; ym 1 ; y2 ; Q g, as follows:
(
þ pmj
m m m ðpðyi1 ; yi2 ; ym 1 ; y2 ; Q Þ; y1 ; y2 ; Eð
( 2 X
þ pij
a
v
þ
hm k
v
mk p Þ
l
h
ðym 1
þ w2
ym 2
w1 Þnm1 p
E
E
)
) i
k¼1
¼ 0; ð2aÞ
a
v
v
l
problem. We will later show that indeed this is the case. Given this assumption, Eq. (2a), (2b), (2c) become
aj hij v ijy ¼ lnij ; m m1 m aj hmj v mj ðy1 þ w2 ym y ¼ l½nj þ nym 2 w1 Þ; j 2 X
ð3aÞ ð3bÞ
j m m m1 ak ðhik v ikE þ hmk v mk ð3cÞ E Þ ¼ l½C E þ ðy1 þ w2 y2 w1 ÞnE p:
k¼1
Combining Eqs. (3a)–(3c) 2 X
nik
k¼1
v ikE m v mk þ nk Emk v iky vy
nm1 E
nm1 ym 1
! m þ p ¼ C jE þ ðym 1 þ w2 y2 w1 Þ
v m1 v m2 E E nm1 y m1 vy v m2 y
!
m 2
:
ð4Þ
m1 m1 m1 m2 Since nm1 ¼ v m1 ¼ v m2 y =2a, nym y =2a and nE ¼ ðv E v E Þ=2a, Eq. ym 1 2 (4) reduces to
2 X
nik
k¼1
v ikE m v mk þ nk Emk v iky vy
! þ p ¼ C jE :
ð5Þ
In sum, the solution to the center’s optimization problem is given by Eqs. (1), (3a), (3b), and (5). Note that Eqs. (3a), (3b), and (5) are equivalent to the following conditions for j = 1, 2:
aj hij uijx ¼ lnij ; m m1 m aj hmj umj ððym x ¼ l½nj þ nym 1 w1 Þ ðy2 w2 ÞÞ; j k¼1
ik p
m m m1 ak ðhik v ikp þ hmk v mk p Þ l½ðy1 þ w2 y2 w1 Þnp E
q
We first determine the allocation that the center implements if it controls all policy instruments. This is the center’s most preferred allocation. We later show that this allocation corresponds to the social optimum for the economy. m The center chooses nonnegative fyi1 ; yi2 ; ym 1 ; y2 ; Q g to maximize P2 i m i m a ðh v ðpðÞ; y ; EðÞÞ þ h v ðpðÞ; y ; EðÞÞÞ subject to Eq. (1), j j¼1 j j j j m ; y ; Q Þ and E() E( q (Q)). Let l be the where pðÞ pðyi1 ; yi2 ; ym 1 2 Lagrangian multiplier associated with constraint (1). Assuming an interior solution, the first order conditions are the constraint and the following equations for j = 1, 2:
i k ðhk
þ w2
w1 Þnm1 p
ð2cÞ
2 X
( 2 X
l
) ym 2
where in writing Eqs. (2a)–(2c) we make use of the facts that fnj ¼ wj and in writing Eq. (2c) we also make use of the facts that C 1E ¼ C 2E P2 i ik m mk m and Eq Q = 1. Suppose that k¼1 k ðhk p þ hk p Þ ½ðy1 þ w2 m m1 y2 w1 Þnp E ¼ 0 in the solution to the center’s optimization
3. Centralized policy making and the social optimum
aj hij v ijy lnij
v
½ðym 1
k¼1
where P ðÞ p ðn qðQ ÞÞÞ; Lj Þ þ pEj ðqðQ ÞÞ. The first equality says that the sum of household incomes must be equal to the sum of functional incomes (profits, rents and wages) plus the total revenue income generated with the sale of pollution permits. The rationale for the second equality follows from the facts that producers of the numeraire good must pay wages and rents, energy producers must purchase pollution permits and the aggregate demand for permits equals the aggregate supply of permits. Thus, the total factor costs equal total factor incomes and the total regulatory costs faced by energy producers equal the total revenue generated with the sale of pollution permits. It follows that the sum of functional incomes plus the total revenue income generated with the sale of pollution permits reduces to the sum of industrial revenues net of the total costs of producing energy. Let us now discuss the regional and central government objective functions. As it is standard in the literature, we assume that regional governments wish to maximize regional welfare and that regional welfare does not directly account for the idiosyncratic attachment benefits of the mobile population that resides in the region. The regional welfare function is a weighted sum of consumption utilities of representative immobile and mobile households who reside in the region. We also assume that the center wishes to maximize social welfare and that social welfare is simply a weighted sum of regional welfares. Hence, welfare in region j is hij v ðpðÞ; yij ; EðÞÞþ P2 i m hm j v ðpðÞ; yj ; EðÞÞ and the social welfare function is j¼1 aj ðhj v m i i m ðpðÞ; yj ; EðÞÞ þ hj v ðpðÞ; yj ; EðÞÞÞ. The regional weights, hj > 0 and 1 2 hm j > 0, and the social weights, a > 0 and a > 0, are exogenously determined. mj
þ
mk p Þ
¼ 0;
ð1Þ Xj
v
hm k
j m m m1 ak ðhik v ikE þ hmk v mk E Þ l½C E þ ðy1 þ w2 y2 w1 ÞnE p
þ pQ
j¼1
j
a
ik p
ð2bÞ
k¼1
2 X f j ðnmj ðÞ; Lj Þ þ pðÞEj ðÞ C j ðEj ðÞÞ ;
i k ðhk
¼ 0;
j¼1
¼
2 X
i
k¼1
2 X
2 2 X X nij yij þ nmj ðÞym Pj ðÞ þ sðÞQ j þ Lj rj ðÞ þ nmj ðÞwj ðÞ ¼ j j¼1
h
m m1 m aj hmj v mj ðy1 þ w2 ym y l nj þ nym 2 w1 Þ j
uik nik Eik ux
þ
umk E nm k umk x
þ
uhj e uhj x
¼ C jE ;
ð6aÞ ð6bÞ ð6cÞ
hj hj hj hj hj since v hj y ¼ ux , v E ¼ uE and p ¼ ue =ux . Eq. (6a) inform us that the center’s optimal redistribution policy equates the marginal social benefit of transferring income to an immobile household in region j to the marginal social cost of doing so. The marginal social benefit of the transfer to a representative immobile household in region j is equal to the marginal social utility of income for this type of household, aj hij uijx . The marginal social cost of the transfer to a representative immobile household in region j is equal to the shadow price of transferring income in the federation, l, times the population of immobile households in region j, nij . Eq. (6b) tell us that the center’s optimal redistribution policy also equates the marginal social benefit of transferring income to a representative mobile household in region j to the marginal social cost of doing so. The marginal social benefit of the transfer to a representative mobile household in region j is equal to the marginal social utility of income for this type of household,
223
E.C.D. Silva, C. Yamaguchi / Journal of Urban Economics 67 (2010) 219–225
aj hmj umj x . The marginal social cost of the transfer to a representative mobile household in region j is equal to the shadow price of transferring income in the federation times the population of mobile m1 m ððym households in region j, l½nm 1 w1 Þ ðy2 w2 ÞÞ. The j þ nym j term inside of the bracket represents the population of mobile households in region j after the center’s mechanism is implemented and the mobile households choose their residential locations. Eq. (6c) shows that the center’s optimal pollution control policy equates the marginal social benefit of energy consumption to the marginal social cost of producing energy. The marginal social benefit of energy consumption is equal to the private marginal benefit of energy consumption minus the marginal social pollution damage, where the private marginal benefit of energy consumption is the marginal rate of substitution between energy consumption and the numeraire good and the marginal social pollution damage is the sum of the marginal rates of substitution between the total amount of pollution and the numeraire good. Eq. (6c) makes it clear that the energy good is an impure public bad, since its consumption yields a private and positive marginal benefit but also a pure and negative marginal social benefit. P2 i ik m mk m We now show that k¼1 ak ðhk v p þ hk v p Þ ¼ l½ðy1 þ w2 m1 w Þn E in the optimum. Utilizing Eqs. (3a) and (3b), we ym 1 2 p have 2 X
2 X ik m m1 m m ak ðhik v ikp þ hmk v mk ½nik ðv ik p Þ ¼ l p =v y Þ þ ðnk þ nym ðy1 þ w2 y2
k¼1
tions for an interior solution are the constraints and the following equations for j, k = 1, 2:
aj hij uijx ¼ cnij ; m m m m ^ m1 ½cðxm aj hmj umj x ¼ cnj þ nxm 1 þ w2 x2 w1 Þ þ kðe1 e2 Þ; j
ð9bÞ
aj hij uije ¼ knij ; m m m m ^ m1 ½cðxm aj hmj umj e ¼ knj þ nem 1 þ w2 x2 w1 Þ þ kðe1 e2 Þ; j
ð9dÞ
2 X
w1 ÞÞðv
m m m m a1 hm1 ¼ ðcnm1 =um1 x Þ þ f½cðx1 þ w2 x2 w1 Þ þ kðe1 e2 Þ=2ag; ð10aÞ m m m2 m m m a2 h2 ¼ ðcn2 =ux Þ f½cðx1 þ w2 x2 w1 Þ þ kðe1 em2 Þ=2ag; ð10bÞ m m m m a1 hm1 ¼ ðknm1 =um1 e Þ þ f½cðx1 þ w2 x2 w1 Þ þ kðe1 e2 Þ=2ag; ð10cÞ m m m2 m m m a2 h2 ¼ ðkn2 =ue Þ f½cðx1 þ w2 x2 w1 Þ þ kðe1 em2 Þ=2ag: ð10dÞ m1 Combining Eqs. (10a) and (10c), we have um1 ¼ k=c. Similarly, e =ux m2 =u ¼ k=c. Now, note combining Eqs. (10b) and (10d) yields um2 e x that conditions (9a), (9e), (10a) and (10b) imply that for k = 1, 2
c
2 X
v
2 X m m mk mk l ½nik ðv ikp =v iky Þ þ ðnmk þ nm1 ym ðy1 þ w2 y2 w1 ÞÞðv p =v y Þ
j¼1
k
¼ l½E ¼l
½ðym 1
þ w2
þ w2
ym 2
w1 Þðv
w1 Þnm1 p
m1 p
v
m2 p Þ=2a
E;
P2 i ik m mk m1 m2 m since nm1 p ¼ ðv p v p Þ=2a. Thus, k¼1 ak ðhk v p þ hk v p Þ ¼ l½ðy1 þ m1 w Þn E. In sum, the center’s optimal redistributive w2 ym 1 2 p policies nullify all energy-price effects. If such effects were not nullified by the center’s optimal redistributive mechanism, the center’s most preferred allocation would not maximize social welfare. As we demonstrate below, the center’s most preferred allocation, characterized by conditions (1), (6a)–(6c), corresponds to the social optimum for the federal economy. Consider now the problem faced by a social planner. The probi m lem is the choice of nonnegative fxik ; xm k ; ek ; ek ; Ek gk¼1;2 to maximize social welfare 2 X
aj ðhij uðxij ; eij ; EÞ þ hmj uðxmj ; emj ; EÞÞ
ð7Þ
j¼1
subject to: E ¼
P2
k¼1 Ek
and
2 X
2 X
j¼1
j¼1
j ^ mj ðÞxm ½nij xij þ n j þ C ðEj Þ ¼
2 2 X X ^ mj ðÞem ½nij eij þ n Ej ; j ¼ j¼1
uijE uijx
^ mj ðÞ; Lj Þ; f j ðn
þ
nm j
umj E umj x
#
m1 um2 E uE ^ m1 ½cðxm ¼ cC kE k þ n E þ 1 2a ð11Þ
m1 m2 ^ m1 Since n E ¼ ðuE uE Þ=2a, Eq. (11) reduces to
m 2
ym 2
nij
m m þ w2 xm 2 w1 Þ þ kðe1 e2 Þ:
2 X
ðym 1
"
j¼1
mk mk p = y Þ:
m 1
k¼1
k m m m m ^ m1 aj ðhij uijE þ hmj umj E Þ ¼ cC E k þ nE ½cðx1 þ w2 x2 w1 Þ þ kðe1 e2 Þ: ð9eÞ
Combining Eqs. (9a) and (9c) yields uije =uijx ¼ k=c, j = 1, 2. Since ^ m1 ¼ um2 ^ m1 ¼ um1 ^ m1 ¼ um2 ^ m1 ¼ um1 n x =2a, nxm x =2a, nem e =2a and nem e =2a, xm 1 2 1 2 conditions (9b) and (9d) yield
h hk m1 m1 m1 m2 v hk p ¼ ek v y , ny ¼ v y =2a, ny ¼ v y =2a, we obtain
As
ð9cÞ
j¼1
k
k¼1
ð9aÞ
ð8aÞ ð8bÞ
j¼1
m m m ^ m2 ðÞ ¼ Nm n ^ m1 ðÞ and n ^ m1 ðÞ n ^ m1 ðxm where n 1 ; x2 ; e1 ; e2 ; EÞ is defined implicitly by the migration equilibrium condition, m m m m m m uðxm 1 ; e1 ; EÞ þ aðN n1 Þ ¼ uðx2 ; e2 ; EÞ þ an1 . Eqs. (8a) and (8b) are the resource constraints for the numeraire and energy goods, respectively. Let c and k be the Lagrangian multipliers associated with constraints (8a) and (8b), respectively. The first order condi-
"
nij
# mj uhj e m uE þ hj ¼ C kE ; þ n j ij mj ux ux ux uijE
k ¼ 1; 2;
ð12Þ
hj where in writing Eqs. (12) we used the fact that uhj e =ux ¼ k=c, h = i, m and j = 1, 2. The equivalence between the social optimum and the center’s most preferred allocation follows from the fact that the conditions that determine the social optimum are equivalent to the conditions that determine the center’s most preferred allocation. First, note hj that Eqs. (12) are identical to Eq. (6c). Since uhj e =ux ¼ k=c, h = i, m and j = 1, 2, one realizes that the center’s optimal allocation satishj fies these conditions because uhj e =ux ¼ p, and thus k=c ¼ p. In addition, once one also observes that c = l, one also realizes that Eqs. (6a) and (6b) correspond to Eqs. (9a) and (9b), respectively. Finally, notice that Eq. (1) is identical to Eq. (8a) and the market-clearing condition for the energy good is identical to Eq. (8b). Therefore, the center’s optimal allocation is socially optimal.
4. Decentralized leadership: the federal policy game Regional and central authorities play a two-stage game as follows. In the first stage, region j chooses Q j P 0 to maximize m m where pðÞ pðyi1 ; yi2 ; ym hij v ðpðÞ; yij ; EðÞÞ þ hm j v ðpðÞ; yj ; EðÞÞ, 1 ; y2 ; Q Þ, E() E(q(Q)) and Q = Q1 + Q2, taking Qk, k – j, as given. In the second stage, the center observes {Q1, Q2} and then chooses P2 i m i m i to maximize fyi1 ; ym 1 ; y2 ; y2 g j¼1 aj ðhj v ðpðÞ; yj ; EðÞÞ þ hj v ðpðÞ; ; EðÞÞÞ subject to (1), the federation’s redistributive constraint. ym j The equilibrium concept used is subgame perfection. We should first consider the second stage. Since the center’s problem is the same as the center solved in the previous section with respect to the choices of household income levels, the interior solution to its problem satisfies constraint (1) and first order conditions (6a) and (6b). Close inspection of these equations reveals that the optimal household incomes depend on the aggregate quota of pollution permits, Q, rather than on the separate regional quotas, Qj, j = 1, 2. Let yhj(Q) denote the optimal income function of
224
E.C.D. Silva, C. Yamaguchi / Journal of Urban Economics 67 (2010) 219–225
household of type h in region j defined implicitly by the system of first order conditions that characterize the optimal redistribution scheme, namely, Eqs. (1), (6a), and (6b). ~j ðQ Þ Ej ðqðQ ÞÞ, ~ðQ Þ pðyi1 ðQ Þ; yi2 ðQ Þ; ym1 ðQ Þ; ym2 ðQ Þ; Q Þ, E Let p mj mj ~ m1 ~ ÞÞ. ~ Þ P2 E ~j ðQ Þ and n ~ ðQ Þ n ð p ðQ Þ; y ðQ Þ; ym2 ðQ Þ; EðQ EðQ j¼1 Inserting these functions and the optimal household income functions into the federation’s redistributive constraint yields 2 X
m
~ m1 ðQÞym1 ðQ Þ þ ðN n ~ m1 ðQ ÞÞym2 ðQÞ f 1 ðn ~ m1 ðQ ÞÞ nij yij ðQÞ þ n
j¼1 2 X ~j ðQÞ C j ðE ~j ðQÞÞ: ~ m1 ðQ ÞÞ þ ~ðQ ÞE ½p þ f 2 ðNm n
ð13Þ
j¼1
Differentiating identity (13) with respect to Q, we have 2 X mj m m ~ m1 ~ ½nij yijQ þ nm j yQ þ ½ðy1 w1 Þ ðy2 w2 ÞnQ pQ E ¼ s
ð14Þ
j¼1
since 2 2 X X j ~j ðQ Þ þ p ~j ¼ p ~j ~j C j E ~Q E ~ðQ ÞE ~ E ½p E þ ðp C Þ Q E E Q Q Q j¼1
j¼1
~Q E þ p ¼p
C jE
~ Q E þ s: ¼p
The first equality in the last set of equalities follows from the fact that C 1E ¼ C 2E . For future use, one should also note that m1 ~ m1 m1 m1 m2 m1 ~ m1 n Q np pQ þ nym yQ þ nym yQ þ nE : 1
ð15Þ
2
Consider now the first stage of the game. Assuming interior solutions, the first order conditions that characterize the Nash equilibrium are as follows for j = 1, 2: mj mj ~ mj ij ~Q þ v ijy yijQ þ v ijE Þ þ hm hij ðv ijp p j ðv p pQ þ v y yQ þ v E Þ ¼ 0:
ð16Þ
hj hj Given the early results for v hj p , v y and v E , and Eqs. (6a) and (6b), Eqs. (16) can be written as follows for j = 1, 2:
m X
" nhj
h¼i
"
uhj E uhj x
# þ
ymj Q þ
nhj yhj Q
umj E umj x
~Q nhj ehj p
m m þ nm1 ym ðy1 w1 Þ ðy2 w2 Þ j
# ~ em j pQ ¼ 0:
ð17Þ
Adding Eqs. (17) and utilizing Eq. (14), we obtain: 2 X
"
uijE
nij
uijx
j¼1
þ
nm j
umj E
#
umj x
" ! # 2 X umj mj m m1 m~ m1 E ~ n w Þ ðy w Þ n y þ e þ ðym p m 1 2 1 2 yj j Q Q Q umj x j¼1 ¼ s: ð18Þ
We now demonstrate that Using identity (15), we have 2 X j¼1
nm1 ym j
ymj Q
þ
umj E umj x
mj uE mj m1 m~ ~ m1 n Q ¼ 0. j¼1 nym yQ þ mj ej pQ
P2
!
~ em j pQ
ux
j
~m1 n Q
¼
2 X j¼1
nm1 ym j
umj E umj x
!
~ em j pQ
~ nm1 p pQ
nm1 E : Given the results for nm1 , nm1 and nm1 we have p E , ym P2 m1 umj j m~ m1 ~ m1 E n n e n ¼ 0. Thus, Eq. (18) becomes p p Q Q mj p E j¼1 ym j ux
j
2 X j¼1
" nij
uijE uijx
þ
nm j
umj E umj x
# ¼ s:
ð19Þ
In sum, the subgame perfect equilibrium is given by Eqs. (1), hj (6a), (6b), and (19). Since s ¼ p C kE , k = 1, 2, and uhj e =ux ¼ p, h = i,m, j = 1, 2, Eq. (19) corresponds to Eqs. (12). Thus, the subgame perfect equilibrium is isomorphic to the center’s most preferred allocation, which in turn is socially optimal. Proposition 1 gathers these results: Proposition 1. The subgame perfect equilibrium for the federal policy game is socially optimal. Proposition 1 makes it clear that combining decentralized control of federal pollution with an interregional income redistribution scheme implemented by the center induces the regional governments to behave efficiently provided that energy and pollution permits are traded within and across regions.5 This is true even though the regional authorities are fully aware that their policy choices influence both energy and permit prices. Both regional governments agree on the total amount of pollution permits they together should supply. They make choices so that the permit price that clears the market equals the sum of marginal regional pollution damages. Our main contributions to the literature is to extend the efficiency result of Caplan et al. (2000) to a more realistic setting which has the following features: (i) there is an impure public bad; (ii) commodities (energy and pollution permits) are traded within and across regions; (iii) the regional governments are large in the sense that they do not take prices as given; (iv) there are mobile and immobile households; (v) households in each region possess shares of profits produced in both regions; and (vi) there are regional factor markets. Proposition 1 informs us that our framework provides an ideal efficiency benchmark for federal structures characterized by decentralized leadership, imperfect labor mobility, heterogeneous households and profit expatriation. These are all key characteristics of the European Union. It is tempting, therefore, to conclude that our main result is good news for the European Union. Undoubtedly, the governments of the member nations in the European Union have considerably greater economic and political powers than the central government and are autonomous to implement policies that have transboundary consequences. It is also true that the center is endowed with financial resources, derived from the member nations, which enable it to implement income redistribution within the federation. Structural and Cohesion Funds provide good examples of centrally determined redistributive schemes. But, the central authority in the European Union is far weaker than the central authority in our model. The volume of resources controlled by the center in the European Union is limited, and the limitations stem from the fact that the center is politically and economically weak vis-à-vis the national authorities of the member countries. The center in our model is ‘‘weak” only to the extent that it is unable to credibly commit to punish or reward the regional governments for their actions. This weakness is formally modeled by postulating that the center is a follower in the federal policy game. Unlike in the European Union, the center’s redistribution scheme here is not limited by the regional authorities. The center is capable of transferring income from any household to any other 5 In an earlier version of this paper, we demonstrated that hierarchical federal policy making yields an inefficient subgame perfect equilibrium if the regions do not trade energy and pollution permits with each other. It is also straightforward to demonstrate that if there is an interregional market for energy or pollution permits only, the subgame perfect equilibrium for the federal policy game is inefficient. In addition to equalization of marginal social utilities of income, efficiency requires equalization of marginal rates of substitution between energy and the numeraire good across households, equalization of marginal costs of energy production across regions and satisfaction of the Samuelson condition for federal pollution reduction.
E.C.D. Silva, C. Yamaguchi / Journal of Urban Economics 67 (2010) 219–225
household as it wishes. It determines each household’s final income after it observes the actions taken by the regional authorities. Indeed, it is the center’s ability of determining household income coupled with the anticipation of the optimal redistribution conditions that induce the regional governments to behave efficiently. With limited income redistribution, one can imagine that even though the qualitative incentive features of the redistribution scheme would remain, the income transfer amounts would not be sufficiently high to promote efficient behavior. 5. Conclusion This paper analyzes decentralized regulation of energy supply, an impure public bad. Energy generates positive consumption benefits and pollution damages. Our federal structure is characterized by decentralized leadership and imperfect labor mobility. We show that decentralized environmental policy is efficient if it is combined with an optimal redistribution policy carried out by the center provided that energy and pollution permits are traded within and across regions and the center redistributes income after the regional authorities choose regional supplies of pollution permits. We show that this result holds in an economy featuring heterogeneous households and profit expatriation. Our main finding may be applicable to many situations in which the consumption benefits of regionally provided public goods are not restricted to the residents of the regions that provide the goods. Examples of such goods abound. Public infrastructure, forest conservation and disease eradication are just a few examples. A central government may be able to induce regional governments to adopt efficient policies in the provision of impure public goods provided that it implements redistributive income transfers and federal markets are built for the relevant commodities.
225
One word of caution is in order, however. Our setting should be viewed as an efficient benchmark for federations characterized by decentralized leadership. The center in our model is weak only to the extent that it is unable to credibly punish or reward the regional governments for their actions. The center does not face any limits from the regional governments in implementing its most desirable income redistribution scheme. Were such limitations exist, the incentives promoted by income transfers for efficiency would be limited as well. This is an interesting avenue for future work. Acknowledgments We greatly benefited from valuable comments made by the editor, Jan Brueckner, and two anonymous referees. We also thank Jun-ichi Itaya, Se-il Mun and the participants of a seminar at Hokkaido University and the Japanese Economic Association Meetings held at Oita University for helpful comments on earlier versions. References Boadway, R., 1982. On the method of taxation and the provision of local public goods: comment. American Economic Review 72, 846–851. Caplan, A.J., Cornes, R.C., Silva, E.C.D., 2000. Pure public goods and income redistribution in a federation with decentralized leadership and imperfect labor mobility. Journal of Public Economics 77, 265–284. Mansoorian, A., Myers, G.M., 1993. Attachment to home and efficient purchases of population in a fiscal externality economy. Journal of Public Economics 52, 117–132. Myers, G.M., 1990. Optimality, free mobility, and the regional authority in a federation. Journal of Public Economics 43, 107–121. Oates, W.E., 1972. Fiscal Federalism. Harcourt Brace Jonavonich, New York. Stiglitz, J.E., 1977. The theory of local public goods. In: Feldstein, M., Inman, R.P. (Eds.), The Economics of Public Services. Macmillan, London, pp. 274–333. Wellisch, D., 1994. Interregional spillovers in the presence of perfect and imperfect household mobility. Journal of Public Economics 55, 167–184.