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On the existence of Nash equilibria in an asymmetric tax competition game
Regional Science and Urban Economics 41 (2011) 439–445
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Regional Science and Urban Economics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / r e g e c
On the existence of Nash equilibria in an asymmetric tax competition game☆ Emmanuelle Taugourdeau a,⁎, Abderrahmane Ziad b a b
CNRS CES, Paris School of Economics, France CREM, CNRS, University of Caen (UCBN), Esplanade de la Paix, 14032 Caen Cadex, France
a r t i c l e
i n f o
Article history: Received 21 June 2010 Received in revised form 22 February 2011 Accepted 22 February 2011 Available online 7 March 2011 JEL classification: C72 H21 H42 R50
a b s t r a c t In this methodological paper, we prove that the famous tax competition game introduced by Zodrow and Mieszkowski (1986) and Wildasin (1988) in which the capital is completely owned by foreigners possesses a Nash equilibrium even when the assumption of symmetric jurisdictions is dropped. The normality of both private and public goods is all that is needed concerning restrictions on preferences when a peculiar regime of taxation is ruled out. Moreover, we show that conditions about technology allowing for the existence of a Nash equilibrium are satisfied by most of the widely-used production functions. © 2011 Elsevier B.V. All rights reserved.
Keywords: Tax competition Nash equilibrium
1. Introduction In the established literature on tax competition the existence of a Nash equilibrium is assumed (see Zodrow and Mieszkowski, 1986; Wilson, 1985, 1986 and Wildasin, 1988). These studies focus on the comparative statics of Nash equilibria, and demonstrate that public services are provided at inefficiently low levels in equilibrium. However, little attention has been devoted to the question of whether such equilibria do exist. This is for the most part because the demonstration is very difficult, as noted by Laussel and Le Breton (1998): “Both the existence and uniqueness issues are difficult in general and have not been up to now dealt with in the literature. It seems however of primary interest to solve them in order to understand the comparative statics of the equilibrium.” Some results have already been established in the literature for particular cases. Firstly, Bucovetsky (1991) demonstrated the existence of a Nash equilibrium in tax rates in the case of two regions and quadratic production functions. A second and important result was highlighted by Laussel and Le Breton (1998) who proved the existence of a symmetric Nash equilibrium when private and public goods are perfect substitutes and when capital is not owned by
☆ We are indebted to David Wildasin, Maurice Salles, Yves Zenou and two anonymous referees for their substantial and helpful comments. ⁎ Corresponding author at: CNRS CES, University of Paris 1, ENS Cachan, Laplace 309, 94235 Cachan Cedex, France. E-mail addresses: [email protected] (E. Taugourdeau), [email protected] (A. Ziad). 0166-0462/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.regsciurbeco.2011.02.003
residents. In addition, this framework enables the authors to prove the uniqueness of the equilibrium, which is the primary purpose of their paper. In a more recent paper, Bayindir-Upmann and Ziad (2005) apply a weaker concept than the standard Nash equilibrium – the concept of a second-order locally consistent equilibrium (2-LCE) – which is a local Nash equilibrium (i.e., a small deviation is undesirable). With this tool, the authors are able to show both the existence and uniqueness of a symmetric equilibrium in tax rates when regions are homogeneous and when either (i) there are only two regions, (ii) capital demand curves are convex, or firms apply (iii) CES, (iv) Cobb–Douglas, or (v) logistic production functions. More recently, Dhillon et al. (2007) investigate the existence of a Nash equilibrium in a symmetric tax competition model where the public good enters the production function. Rothstein (2007) analyses the fiscal competition game as a game with discontinuous payoff and demonstrates the existence of a pure strategy Nash equilibrium for this kind of game under several assumptions respecting the production function. Rothstein moves away from the standard fiscal competition game “à la Wildasin” by assuming: first, an ad valorem tax; and second, that the aggregate amount of mobile capital is fixed in all regions. Finally, Petchey and Shapiro (2009) examine the problem of the existence of a Nash equilibrium in a tax competition model when governments are no longer benevolent but only make constrained efficient choices. A key point of this paper is that we deal with asymmetric regions. The literature on asymmetric tax competition is mainly based on two articles by Wilson (1991) and Bucovetsky (1991). Both assume that regions differ in their population and show that the “small” region
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may benefit from the tax competition by attracting capital from the “large” region thanks to the tax competition mechanism. In the present paper, we retain the methodological question regarding the existence of a Nash equilibrium in the tax competition model “à la Wildasin”. We depart from this seminal model by assuming that capital is owned by foreigners and we extend it to regions with different production functions.1 In doing so, we extend the existing literature by proving the existence of a Nash equilibrium in a more general framework. Our paper is in line with Laussel and Le Breton (1998) and Bayindir-Upmann and Ziad (2005), but we basically depart from their analysis by relaxing the assumption of symmetry. We also depart from the paper by Rothstein, firstly by considering proportional taxes, whereas Rothstein uses an ad valorem tax; secondly, by establishing a weaker condition of existence than the quasiconcavity condition of Rothstein, and thirdly, by directly deriving our result in the tax competition model. To prove our results, we use several assumptions: that goods are normal; that the demand for capital is convex; and that elasticity of the demand for capital is non increasing in absolute value. Moreover, we use an additional condition on preferences which enables us to rule out the particular regime of taxation in which the return on capital is zero and capital owners limit their capital supply. This paper is organized as follows. The second section outlines the tax competition model for a given number of regions, notation and description of the model being taken for the most part from Bayindir-Upmann and Ziad (2005). In Section 3 we establish a general theorem for the existence of the Nash equilibrium in fiscal competition. In Section 4 we introduce several examples of the commonly-used production functions and check the tractability of the result in Section 3. The final section summarizes our conclusions. 2. The model
i
i
Xi = f ðKi Þ−pi Ki + θi ρi K
Pi = ti Ki :
ð1Þ
where pi denotes the after-tax price of capital in jurisdiction i. Equating the price of capital to the value of its marginal product, fki(Ki) = pi determines region i's capital demand as a function of the corresponding after-tax price of capital, K ̂ i ðpi Þ. Let U(Xi, Pi) be the utility that the representative household of jurisdiction i derives from the provision of the public good, Pi, and from the consumption of the private good, Xi, produced by the firms. The utility function U(Xi, Pi) is twice-continuously differentiable and monotonically increasing. The source of the household's income is twofold: one part from the provision of the fixed factor, which is exclusively owned by local residents; and one part from their initial capital endowment. Let θi ∈ [0, 1] denote region i's share of the fixed national capital stock K, and pi, the net return of capital 1 In his paper, Wildasin (1988) uses also the assumption of absentee owners of capital in the jurisdictions to derive his results in the special case of identical jurisdictions.
ð3Þ
When choosing the level of the tax rate, each local government acts as a benevolent one and aims to maximize its representative resident's utility. In doing so, each local authority behaves non-cooperatively and treats its specific tax on capital ti, i = 1, …, n, as the strategic variable. This leads to a tax competition game between jurisdictions. The capital market clearing condition implies that aggregate demand for capital must equal capital supply: n
∑ Ki = K;
ð4Þ
i=1
for some exogenously given capital supply K. Capital being freely mobile across regions, the arbitrage condition equals the net return of capital in each jurisdiction: i
i = 1; …; n
ð2Þ
for each jurisdiction i = 1, …, n. In the following of the paper, we assume θi = 0 ∀ i. One explanation is that capital is owned by agents outside the jurisdictions under consideration, as in Wildasin (1988), Laussel and Le Breton (1998) and Rothstein (2007).2 Another explanation is to consider that the income of the resident capital owners does not affect the utility of the representative agent in each jurisdiction.3 Each local government provides a public good that it finances by taxing the mobile capital at a tax rate ti ≥ 0.4 The budget constraint of jurisdiction i is given by
ρ = fk ðKi Þ−ti ð = ρi Þ;
Consider n (n ≥ 2) jurisdictions inhabited by a given number of homogeneous residents that we normalize to one without loss of generality. A fixed number of competitive firms produce a homogeneous output in each jurisdiction using capital and some fixed factor(s) (land or labour). Aggregating production over all firms in each region allows us to treat the industry of one jurisdiction as one competitive firm. Let f i be the production function of the firm in jurisdiction i, f i is assumed to be monotonically increasing and strictly concave in capital, Ki. Fixed factors as explicit arguments of f i are suppressed so that the production function is expressed in terms of capital only. The jurisdiction i firm's profit can be written as Πi = f ðKi Þ−pi Ki ;
in region i. The private budget constraint of the consumer in region i amounts to
∀i = 1; …; n:
ð5Þ
Let t = (t1, …, tn) be the profile of tax rates, t− i = (t1, …, ti − 1, ti + 1, …, tn) be the profile of all tax rates except ti; whereas (ti, t− i) stands for (t1, …, ti − 1, ti, ti + 1, …, tn). Both Eqs. (4) and (5) define the equilibrium allocation of capital and the equilibrium of the net return of capital, i.e. K1(t),…, Kn(t) and ρ(t). Capital supply K is assumed exogenously fixed, whenever the solution of Eqs. (4) and (5) requires that capital earns a non-negative net return, ρ ≥ 0. When ρ b 0, we suppose that capital owners limit capital supply. The resulting allocation of capital is a vector (K1, K2, …, Kn) with ∑ni = 1 Ki = K 4 b K, for which fki(Ki) = ti. We assume that if for some profile t = (t1, t2, …, tn) we have ρ(t) = ρi(t) = 0 for each i, and if a region j raises its tax rate from tj to tj + Δtj such that ρj(t− j, tj + Δtj) b 0 while the other regions keep their tax rate unchanged, then capital owners limit supply only in region j until ρj = 0. As a result, as in Bucovetsky (1991), we have three regimes: 1. Positive return regime: ∑ni = 1 Ki = K; ρ N 0, 2. Excess supply regime: ∑ni = 1 Ki = K 4 b K; ρ = 0, 3. Boundary between regimes: ∑ni = 1 Ki = K; ρ = 0.
∂Ki changes ∂ti discontinuously at any tax vector (t1, t2, …, tn) at which ρ = 0 and ∑ni = 1 Ki = K, i.e. in the boundary between regimes. In any regime It is shown in Bucovetsky (1991) that the derivative
2 Alternatively, Zodrow and Mieszkowski (1986), Bucovetsky (1991) or BayindirUpmann and Ziad (2005) assumed that owners of capital own an equal share of the total capital stock. 3 This could be the case if we consider two groups of agents in each jurisdiction, one which does not own capital but controls decision making, say the “poor/middle class”, and one which owns capital and passively consumes its net income of capital, say the “rich class”. This comment has been suggested by one of the referees. 4 Capital subsidies cannot be funded in this model since there are no other taxes available.
E. Taugourdeau, A. Ziad / Regional Science and Urban Economics 41 (2011) 439–445
the utility function is continuous everywhere but may be not differentiable at some tax rates ti for which ρ(ti) = 0. At this stage of the analysis, the utility function may have multiple maxima and/or minima. We will show later that the net return of capital is a continuous and non increasing function with respect to the tax rates. Let us assume that there exists t 0i to be the first tax rate such that ρ(t 0i , t− i) = 0. For each ti N ti0 we have ρ(ti, t− i) = 0 (capital owners limit the capital supply). We will show that under some reasonable conditions, the utility function cannot reach any maximum with ti N t 0i (the excess supply regime) and that for each ti b t 0i (the positive return regime), the utility function is differentiable.5 The following formulas are derived from the case of a positive return regime. The capital market equilibrium enables us to determine the response of the net return of capital to an increase in jurisdiction i's tax rate: K ̂j′ pj ∂ρðtÞ : =− ∂tj ∑i K ̂i′ðpi Þ
∂Pi ∂ρ = ti K ̂i′ N0 ∂tj ∂tj
Eq. (8) states that the private good is clearly decreasing with the local tax rate whereas nothing can be said about the reaction of the public good. Both Eqs. (6) and (7) enable us to write an indirect utility function Vi(t) = Ui(Xi(t), Pi(t)), which directly relates tax policy to welfare. Maximizing the indirect utility function of jurisdiction i with respect to the tax rate ti yields the following first-order condition: ∂Vi ðtÞ ∂Ui ∂Xi ∂Ui ∂Pi = ðtÞ + ðtÞ = 0; ∂Xi ∂ti ∂Pi ∂ti ∂ti
∂Ui = ∂Pi −∂Xi = ∂ti = ∂Ui = ∂Xi ∂Pi = ∂ti and by Eqs. (8) and (11) we have:
∂Ui = ∂Pi MRSXi ;Pi := = ∂Ui = ∂Xi
Differentiating Kj(t) and Ki(t) with respect to tj, we obtain the overall impact of a variation of the tax rate of region j on capital demand: ! ∂Kj ðtÞ ∂ρðtÞ = K ̂j′ pj 1 + b 0; ∂tj ∂tj ∂Ki ðtÞ ∂ρðtÞ = K ̂i′ð pi Þ N 0; ∀i≠j; ∂tj ∂tj with pj = ρ(t) + tj, ∀ j. Using pi = ρ + ti and substituting Ki(t) and ρ(t) into the private and the public budget constraint Eqs. (2) and (3), we obtain Xi ðtÞ = f ðKi ðtÞÞ−ðti + ρðtÞÞKi ðtÞ;
ð6Þ
Pi ðtÞ = ti Ki ðtÞ:
ð7Þ
Differentiating Eqs. (6) and (7) with respect to ti in the positive regime yields7: ∂Xi ∂ρ = −Ki 1 + b 0; ∂ti ∂ti
ð8Þ
2 2 ∂ Xi ∂K ∂ρ ∂ ρ =− i 1+ −Ki 2 ; 2 ∂ti ∂ti ∂ti ∂ti
ð9Þ
∂Xi ∂ρ = Ki b0 ∂tj ∂tj
∀i≠j;
ð10Þ
∂Pi ∂Ki = Ki + t; ∂ti ∂ti i
ð11Þ
∂2 Pi ∂K ∂2 K = 2 i + ti 2 i ; 2 ∂ti ∂ti ∂ti
ð12Þ
5
The utility may reach a maximum at ti = t 0i . These expressions are perfectly in line with the standard results of the tax competition literature such as Keen and Kotsogiannis (2002) for instance. 7 Note that all functions are continuously differentiable with respect to the strategic variables. 6
ð13Þ
which implies the relationship
and since each region's capital demand ! monotonically decreases with 1 ∂ρðtÞ ′ ̂ ∈½−1; 0, which the price of capital K j pj = j b 0 , we have ∂tj fkk ! ∂ρðtÞ is equivalent to 1 + ∈½0; 1.6 ∂tj
i
∀i≠j:
441
∂ρ Ki + Ki −∂Xi = ∂ti ∂ti = = MRTXi ;Pi : ∂ρ ∂Pi = ∂ti Ki + 1 + ti K ̂i′ ∂ti ð14Þ
Each government equates the marginal rate of substitution (MRS) to the marginal rate of fiscal transformation (MRT) between the private and the public goods. In what follows, we are interested in the existence of a Nash equilibrium, a profile of strategies t* such that t i* maximizes Vi(ti, t− * i) with respect to ti for each i. Before developing the analysis which will enable us to prove the existence of a Nash equilibrium in the fiscal game, we focus on the two regimes which are the excess supply regime and the boundary between regimes. To deal with those regimes, we need the following condition: (C1). Marginal utility of private good dominates marginal utility of public good when capital return is fully taxed away: MRS( f i(Ki) − Ki fki(Ki) ; Ki fki(Ki)) ≤ 1 for each i. This condition was first introduced by Bucovetsky (1991), and says that when capital return is fully taxed away, the share of the public good is higher than the optimal share (for which MRS = 1). The result is that agents are less inclined to give up private goods in exchange for one unit of public good because a high tax rate implies a low level of capital and a low level of private good. It is proved in Bucovetsky (1991, Lemma 4) that if (C1) is satisfied, no Nash equilibrium exists within the excess supply regime with a quadratic production function.8 In our model, this condition is useful to show that if the utility function reaches a maximum (global or not) within the excess supply regime or the boundary between regimes, this maximum is obviously the last one. Under condition (C1), the following lemma precludes the existence of a maximum under the excess supply regime. Recall that t0i is the first value of ti ≥ 0 for which ρ(t 0i , t− i) = 0. 8 Bucovetsky (1991) proposes an alternative assumption on the production function that also guarantees that the excess supply regime can be ruled out. In our model this − fki ðKi Þ f i ðKi Þ−Ki fki ðKi Þ condition writes: (C′1): σi ≥ 1 − αi ∀ i where σi = stands for i ðK Þ Ki f i ðKi Þ fkk i Ki fki ðKi Þ for the capital's share. the elasticity of substitution in production, and αi = i f ðKi Þ This condition is equivalent to Ki fki(Ki) increasing in Ki. It is easy to check that all the
production functions that will be used in the following part of the paper satisfy this condition.
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E. Taugourdeau, A. Ziad / Regional Science and Urban Economics 41 (2011) 439–445
Lemma 1. [No maximum within the excess supply regime] There is no maximum with ti N t 0i within the excess supply regime if (C1) is satisfied. Proof: Suppose that we can reach a maximum at some tax rate ti N t 0i within the excess supply regime then t′i = fki(Ki′) ∀ ti′ ≥ t 0i . The utility function Ui(Xi, Pi) is assumed to be increasing in (Xi, Pi), with Xi = f i(Ki) − Ki f ik(Ki), and Pi = tiKi = fki(Ki)Ki. Lowering the tax rate ti increases Ki. The impact of a rise in Ki on the utility is given by: ∂Ui ðXi ; Pi Þ ∂Ui ðXi ; Pi Þ ∂Xi ∂Ui ðXi ; Pi Þ ∂Pi = + ∂Ki ∂Ki ∂Ki ∂Xi ∂Pi =
i ∂Ui ðXi ; Pi Þ h i i : Ki fkk ðKi Þ:ðMRSi −1Þ + MRSi ⋅ fk ðKi Þ ∂Xi
□ which is strictly positive according to (C1). At this stage of the analysis, Lemma 1 stipulates that in the case of multiple minima and/or maxima, the point defined by t 0i such that ρ(t 0i , t− i) = 0 can only be the last maximum that the utility function can reach. Before this point, the function Vi(ti, t− i) is differentiable everywhere.9 3. The existence of a Nash equilibrium In this section, we present our first main result concerning the existence of Nash equilibria in the fiscal game described above. To obtain useful and clear results, in addition to condition (C1), we postulate the following assumptions: ∂ (C2). Normality of goods: For each i and for all Xi, Pi N 0: MRSXi ;Pi ≥0 ∂X i ∂ and MRSXi ;Pi ≤ 0. ∂Pi (C3). Marginal product of capital falls at diminishing rate: The third i ≥ 0 for each i. derivative of the production functions is positive, f kkk (C4). Non increasing elasticity demand for capital (in For absolute −1 value): fki ðKi Þ dlnfk i = each i the elasticity of demand for capital k = i ðK Þ dlnK Ki fkk i must be non decreasing (non increasing in absolute value) with respect i i 2 10 i i to the capital. This condition is equivalent to fk f kk + Ki fkkk ≤ Ki fkk . Condition (C2), already postulated by Bucovetsky (1991) and Bayindir-Upmann and Ziad (2005), requires that the marginal rate of substitution between the private and the public goods is nondecreasing in the first argument and non-increasing in the second argument. This is equivalent to stating that both the private and the public goods are normal goods, which is regarded as standard in economics. Condition (C3) was first introduced by Laussel and Le Breton (1998) and later used by Rothstein (2007) and Petchey and Shapiro (2009). It stipulates that the marginal product of capital is always a convex function of the amount of invested capital. More precisely, the slope of the marginal production is flattening out as the capital rises. The use of this condition will be explained in Lemma 4. Usual production functions such as Cobb–Douglas, CES and Quadratic functions satisfy (C3). Condition (C4) is a new condition which states that the elasticity of demand for capital must be non increasing in absolute value with respect to the capital: the higher the level of capital in the economy, the lower the demand for capital reaction to an inflow of capital. This property applies perfectly for linear demand curves that are infinitely elastic at zero quantities of capital and have zero elasticities at their horizontal intercepts. More generally, most of the standard produc9 Following from the differentiability of the functions U(⋅), f(⋅) and then K(⋅) and finally X(⋅) and P(⋅). 10 Recall that fik is the inverse demand for capital.
tion functions satisfy (C4) (see later examples). In particular, this condition is obvious for the quadratic production functions (since i f kkk = 0). This condition is looser than the conditions introduced by Rothstein (2007) who firstly assumes that the elasticity of the marginal product of capital εik must be higher than (− 1) (assumption i 2 i 8ii), and secondly, that 2 fkk −fki fkkk ≥ 0 (assumption 9iii). Both conditions (C3) and (C4) bound the third derivative of the i 1−εik fkk i production functions since (C4) can be rewritten as fkkk ≤ or, i εk Ki fi i more strongly, fkkk ≤ − kk . Thus, conditions (C3) and (C4) stipulate Ki i that f kkk must be positive but not “too big” relative to the second i derivative f kk . At this stage of the reasoning, we are able to set that in both the positive return regime and the boundary between regimes, the tax rate belongs to [0, Ti] (see the Appendix for a demonstration of the existence of the upper bound Ti). In order to prove the existence of at least one Nash equilibrium, we consider the best response of each jurisdiction and prove that they are functions and not a correspondence. Then by continuity assumptions, we can use a fixed point theorem (Brouwer's theorem) to prove the existence of a Nash equilibrium. The principal general result of the paper is as follows:
Theorem 1. When consumers' preferences are represented by twicecontinuously differentiable, monotonically increasing utility functions Ui(.,.) in (Xi, Pi) and assumptions (C1), (C2), (C3) and (C4) are satisfied, then the asymmetrical fiscal game possesses a Nash equilibrium. To prove our result, we have to introduce several Lemmas, which are of great importance in understanding the tax competition mechanisms at work in the economy. Lemma 2. [Capital allocation] For any given t, the capital allocation K1(t), …, Kn(t) and the net ∂Kj ðtÞ ∂K ðtÞ N 0∀j≠i, and return of capital ρ(t) are unique, with i b 0, ∂t ∂ti i ∂ρðtÞ b 0 ∀i. ∂ti Lemma 3. [Increasing branch of the Laffer curve] For any given tax vector t− i, a utility local-maximizing (respectively ∂Pi local minimizing) strategy of region i requires ðtÞ N 0 on the left ∂ti (respectively the right) of any local extremum point. Proof: As the private good Xi is a decreasing function in ti, and the utility function Ui(.,.) is an increasing function in (Xi, Pi), in the neighbourhood of any local maximum (resp. minimum) t 0i , the public good must be an increasing function on the left (respectively on the right) of t 0i .11 □ Lemma 4. Decreasing marginal rate of substitution Let condition (C2) hold. For any tax vector t, the marginal rate of substitution between the private and the public goods is falling on the left or on the right of any local extremum. Proof: The derivative of the marginal rate of substitution is ∂MRSXi ;Pi ∂Xi ðtÞ ∂MRSXi ;Pi ∂Pi ðtÞ ∂ MRSXi ;Pi ðtÞ = + : ∂Xi ∂Pi ∂ti ∂ti ∂ti ∂ From condition (C2) (goods are normal) we have MRSXi ;Pi ≥ 0 ∂X i ∂ ∂Xi ðtÞ ∂Pi ðtÞ MRSXi ;Pi ≤ 0. Since b 0 and N 0 (Lemma 3), then and ∂Pi ∂ti ∂ti ∂ MRSXi ;Pi ðtÞ b 0. □ ∂ti 11
Not on the boundary.
E. Taugourdeau, A. Ziad / Regional Science and Urban Economics 41 (2011) 439–445
Lemma 5. [Convexity of the net return of capital function] Condition (C3) is sufficient to ensure the convexity of the net return of capital ρðtÞ. Proof: The second derivative of the net return function is written as: ∂Kj ðtÞ 2
∂ ρðtÞ = ∂ti2
∂Ki ðtÞ ∂ti
i fkkk i 2 fkk
∑j
1 j fkk
1
∑j
1 ∑j i fkk !2
−
2
∂ ρðtÞ = ∂t i2
∂Ki ðtÞ ∂ti
−
∂2 V ðt−i ; ti Þ b 0; is to prove that ∂ti2
∂ MRTXi ;Pi ðti ; t−i Þ b 0: ∂ti
0 1 ! ∂Xi −2 2 2 ∂ @ ∂ti ðtÞA ∂Pi ∂ Xi ∂Pi ∂Xi ∂ Pi = − : ∂ti ∂ti ∂ti2 ∂ti ∂Pi ðtÞ ∂ti2 ∂ti
:
∂ti
j fkk
Then
∂Kj ðtÞ j ∂ρðtÞ = f Kj . But ρ(t) = fkj(Kj) − tj, then for j ≠ i, we have ∂ti ∂ti kk We obtain
i f kkk i 2 fkk
A sufficient condition to check
Let
j
f ∂ti kkk 2 j fkk
443
∂Ki ðtÞ i f kkk 1 1 ∂ρðtÞ ∂t ∑j j − i ∑j≠i 3 − i i 3 j ∂ti f kk f kk fkk f kk !2 1 ∑j j fkk
3 2 ∂2 Xi ∂Pi ∂Xi ∂2 Pi ∂Ki ∂2 ρ ∂K i 2∂ ρ − = −t fkk −Ki −ti Ki 2 i i 2 ∂t 2 2 ∂ti ∂ti ∂ti ∂ti ∂ti ∂ti ∂ti i + Ki
j f kkk
and i j f kkk ∂ρðtÞ f kkk 1 ∂ρðtÞ 1 1+ ∑ i 3 ∑j≠i j − j≠i i j 3 ∂ti ∂ti fkk f kk f kk 2 f kk ∂ ρðtÞ = !2 ∂ti2 1 ∑j j f kk
∂Ki ∂ti
2
i
f kk + ti Ki
∂Ki ∂2 Ki i f : ∂ti ∂ti2 kk
Replacing ∂ρðtÞ 1+ ∂Ki ðtÞ ∂ti = i ∂ti f kk and ! ∂2 ρðtÞ i 2 ∂ρðtÞ 2 i fkkk fkk − 1 + 2 ∂ti ∂ti i 3 fkk
∂2 Ki ðtÞ = ∂ti2
j
which is positive as soon as f kkk ≥ 0 (condition (C3). □ As mentioned by Laussel and Le Breton (1998) “for a jurisdiction the advantage of taxing capital is in the induced reduction of the equilibrium net rate of return of capital (i.e. of the equilibrium “price of capital” which must be paid to capital holders who invest in the jurisdiction)”. Condition (C3) implies that this reduction occurs at an increasing rate as ti increases. In other words, condition (C3) guarantees that the marginal benefit from taxing capital diminishes when the level of capital is large. Lemma 6. [Uniqueness of the best reply functions] In the positive return regime, conditions (C2), (C3) and (C4) are sufficient to ensure the local-concavity of the indirect utility function in the neighborhood of any tax rate satisfying the first order condition. Proof: If the first order condition is satisfied for some ti we have: 2
3
∂Xi ∂V ðt−i ; ti Þ ∂U ðXi ; Pi Þ ∂Pi ðt−i ; ti Þ 4 ∂ti ðt−i ; ti Þ + MRSXi ;Pi ðt−i ; ti Þ5 = 0 = ∂Pi ∂ti ∂Xi ∂ti ðt−i ; ti Þ ∂ti
we get ! 2 2 ∂ρðtÞ 3 ∂ρðtÞ 2 i i i i ∂ ρ −ti f kk + Ki fkkk + Ki fkk 1+ −Ki fkk 1+ 2 ∂ti ∂ti ∂ti ∂ Xi ∂Pi ∂Xi ∂ Pi − = : i 3 ∂ti ∂ti2 ∂ti2 ∂ti fkk 2
2
Then ∂2 Xi ∂Pi ∂Xi ∂2 Pi − ∂ti ∂ti2 ∂ti2 ∂ti is negative if and only if
i ti fkk
+
i Ki fkkk
! 2 2 ∂ρðtÞ 3 ∂ρðtÞ 2 i i ∂ ρ 1+ b Ki fkk 1+ −Ki fkk 2 : ∂ti ∂ti ∂ti ð15Þ
∂2 ρ N 0 from Lemma 5. ∂ti2 i i • If fkk + Ki fkkk b 0, Eq. (15) is always satisfied i i • If fkk + Ki fkkk ≥ 0, i i b 0, fkkk N 0 from (C3) and With fkk
i ∂V ðt−i ; ti Þ ∂U ðXi ; Pi Þ ∂Pi ðt−i ; ti Þ h = MRTXi ;Pi ðt−i ; ti Þ + MRSXi ;Pi ðt−i ; ti Þ = 0: ∂ti ∂Xi ∂ti
And the second derivative of the indirect utility function for government i is: 2 ∂ V ðt−i ; ti Þ ∂U ðXi ; Pi Þ ∂Pi ðt−i ; ti Þ ∂ ∂ = − MRTXi ;Pi ðt−i ; ti Þ + MRSXi ;Pi ðt−i ; ti Þ : ∂ti ∂Xi ∂ti ∂ti ∂ti
While from Lemma 4 we have: ∂ 0 MRSXi ;Pi ti ; t−i b 0: ∂ti
as we work in the positive regime ti ≤ fki(Ki(t1, …, tn)) for each ti, a sufficient condition to check Eq. (15) is then ! 2 2 ∂ρðtÞ 3 ∂ρðtÞ 2 i i i i i ∂ ρ f k f kk + Ki f kkk 1 + b Ki f kk 1+ −Ki f kk 2 ; ∂ti ∂ti ∂ti ! i i 2 f kk + Ki f kkk f ki ∂ρðtÞ 3 ∂ρðtÞ 2 i ∂ ρ 1 + b 1 + −K f : ð16Þ i kk i i ∂ti ∂ti ∂ti2 f kk Ki f kk
444
E. Taugourdeau, A. Ziad / Regional Science and Urban Economics 41 (2011) 439–445
i i fi f kk + Ki f kkk This condition is verified since k i ≤1 by condition i 3 K2i fkk 2 f kk ∂ρðtÞ ∂ρðtÞ ∂ ρ (C4), 1 + b 1+ and 2 N 0 by Lemma 5 . □ ∂ti ∂ti ∂ti Lemma 6 precludes the existence of any minimum in the positive ∂V 2 ðti ; t−i Þ return regime since any minimum requires N 0 for any ti ∂ti2 ∂V ðti ; t−i Þ satisfying = 0. The absence of a minimum in the positive ∂ti return regime precludes also the existence of multiple maxima. Lemmas 1 to 6 enable us to establish the Proof of Theorem 1: Proof of Theorem 1. Given Lemma 1 that rules out the possibility of any maximum (global or not) in the excess supply regime, we concentrate our analysis on the positive return regime and the boundary between regimes. Let t− i = (t1, …, ti − i, ti + 1, …, tn) be the profile of all tax rates except ti, our first step is to look at the best reply correspondence of jurisdiction i. ∂V • If the first order condition is never satisfied, that is i ðti ; t−i Þ≠0, ∂ti then ∂Vi - either ðti ; t−i Þ N 0 on [0, Ti] and the best reply is unique (ti = Ti) ∂ti ∂V - either i ðti ; t−i Þb 0 on [0, Ti] and the best reply is unique (ti = 0) ∂ti - either there exists a unique maximum in the boundary regime: Lemma 1 ensures that there is no maximum for any ti N t 0i whereas Lemma 6 ensures that there is no maximum for any ti b t 0i . The best response is then unique and the maximum is a peak reached at ti = t 0i . • If the first order condition is verified in the positive regime, or if the first order condition is verified on the boundary regime that ∂V is, i t i0 ; t−i = 0 and ρ(t 0i , t− i) = 0 with t 0i = Ti, since all functions ∂ti are continuous and differentiable (U(⋅), f(⋅)and K(⋅)and then X(⋅) and P(⋅)) for any ti b t 0i , Lemma 6 ensures the existence of a unique best reply. Let us now introduce the best reply correspondence of jurisdiction i, by n
BRi ðt−i Þ = ti ∈½0; Ti : Vi ðti ; t−i Þ = Supt′∈½0;T Vi t i′; t−i Þ i
i
and define the Cartesian product of the different reply correspondences BRi by BRðt1 ; t2 ; …; tn Þ = BRi ðt−1 Þ × BRi ðt−2 Þ × … × BRi ðt−n Þ Both the single value of the best reply and the continuity of the indirect utility function (following from the differentiability and the continuity of the functions U(⋅), f(⋅) and then K(⋅) and finally X(⋅) and P(⋅)), imply that we can state that the best reply is a function defined on [0, Ti]n into [0, Ti]n, which is continuous. From Brouwer's theorem, the function BR(.,., …,.) possesses a fixed point. Then, it is obvious to show that the fixed point of BR(.,., …,.) is a Nash equilibrium of the fiscal game under study. □ Let us summarize the role of the different conditions in the proof: (C1) is useful to rule out the excess supply regime in which the net return of capital is fully taxed and the capital owners limit their capital supply. (C2) implies that the marginal rate of substitution between the private and the public goods is a non increasing function of the tax rate. (C3) guarantees that the marginal benefit of taxing more is diminishing for a high level of capital. Finally, both conditions (C3) and (C4) ensure that the marginal rate of the fiscal transformation between the private and the public goods is increasing with the tax rate. We crucially depart from Rothstein's result in several ways: firstly, we stipulate a weaker condition than those defined in Assumptions 8 and 9 of Rothstein's
paper, secondly, our proof is direct while Rothstein's proof is much more complicated and uses Reny's (1999) result, and, lastly, we consider a unit tax whereas Rothstein uses an ad valorem tax. Our theorem applies for asymmetric or symmetric production functions. In that sense, our paper extends the result by Laussel and Le Breton (1998) by considering more than two jurisdictions and by allowing for asymmetric regions. It also complements the results of local equilibria by Bayindir-Upmann and Ziad (2005) which apply to symmetric production functions with national capital ownerships. 4. Applications to usual production functions In this section, we establish our second main result about the existence of equilibria in our fiscal game by introducing several examples. Our purpose is firstly to show the tractability of our method with commonly-used production functions and secondly to state that most of the commonly-used economic models of tax competition possess a Nash equilibrium: • Cobb–Douglas production function f(K) = K α, with fk(K) = αK α − 1, fkk(K) = α(α − 1)K α − 2, fkkk(K) = α(α − 1)(α − 2)K α − 3 N 0. a • Quadratic production function f(K) = (a − bK)K, with K b , so 2b f k (K) = a − 2bK, f kk (K) = − 2b. • Logarithm production function f(K) = ln(1 + βK) with fk ðK Þ = β β2 , f ðK Þ = − : 1 + βK kk ð1 + βK Þ2 • Exponential production function f(K) = 1 − exp(− βK) with fk(K) = βexp(− βK), fkk(K) = − β2exp(− βK). 1 1 • Logistic production function f ðK Þ = − , with x = x(K) : = 1 + e−x 2 Rα e x R2α ð1−e x Þe x α KR , so fk ðK Þ = , fkk ðK Þ = . ð1 + e x Þ2 ð1 + e x Þ3 1 • CES production function f ðK Þ = αK ψ + ð1 − αÞRψ ψ where R stands for the quantity of the fixed factor, 0 b α b 1, ψ≤1 and ψ≠0, so f k ðK Þ =
αK ψ f ðK Þ αð1 − αÞð1 − ψÞðKRÞψ f ðK Þ , f ðK Þ = − . 2 αK ψ + ð1 − αÞRψ K kk K2 ðαK ψ + ð1 − αÞRψ Þ
Theorem 2. Assume that consumers' preferences are represented by twice-continuously differentiable, monotonically increasing utility functions Ui(.,.) in (Xi, Pi) and that conditions (C1) and (C2) are satisfied, that is the private and public goods are normal goods. If for each jurisdiction i, the production function fi ∈ {Cobb–Douglas, Quadratic, Logarithmic, Exponential, Logistic, CES} under further conditions for the parameters for the logistic and CES production functions, then the fiscal game possesses a Nash equilibrium. Proof: For each of the production functions above, we have to show that conditions (C3) and (C4) are verified: • Cobb–Douglas production function: fkkk ðK Þ = αðα−1Þðα−2ÞK
α−3
f ki N 0 and i Ki f kk
i i fkk + Ki f kkk i f kk
= 1:
• Quadratic production function: fki fkkk ðK Þ = 0 and i Ki fkk
i i f kk + Ki f kkk i fkk
=
a−2bK b 0: −2bK
• Logarithm production function:
2β3 fki N 0 and fkkk ðK Þ = 3 i ð1 + βK Þ Ki fkk
i i fkk + Ki fkkk i fkk
= 1−
1 b 1: βK
E. Taugourdeau, A. Ziad / Regional Science and Urban Economics 41 (2011) 439–445
• Exponential production function: 3
fkkk ðK Þ = β expð−βK Þ N 0 and
fki i Ki fkk
i i fkk + Ki fkkk i fkk
Appendix A. Boundary of the tax rates = 1−
1 b 1: βK
• Logistic production function: fkkk ðK Þ =
R3α 1−4e x + e 2x e x ð1 +
e x Þ4
445
i h pffiffiffii f + Ki f i N 0 b = N x N ln 2 + 3 and kk i kkk = fkk
ð1 + e x Þð1−e x Þ + x 1−4ex + e2x where g(x) : = x(1 − 4e x + e 2x) + (1 + x x ð1 + e Þð1−e Þ
e x)(1 − e x) is non positive since g(0) = 0 and gk′ (x) = (1 − e2x) − (4 + i i 2x)ex b 0. Therefore f kk + Ki fkkk b 0 and the marginal rate of fiscal transformation is increasing. • CES production function: αð1 − αÞð1 − ψÞðKRÞψ f ðK Þ αð1 + ψÞK ψ + ð1 − αÞ 3 K3 ðαK ψ + ð1 − αÞRψ Þ i i 2 αψK ψ i + Ki fkkk = , −Ki fkk ð2 − ψÞRψ ÞN 0 a n d fki fkk ψ ð 1−α ÞRψ + αK i i 2 i i i i which implies fk fkk + Ki fkkk −Ki fkk b 0 for ψ b 0 and fk fkk + i 2 i Ki fkkk Þ−Ki fkk N 0 for ψ N 0. In the last case, the sufficient condition fkkk ðK Þ =
of theorem 1 is not satisfied. However a closer examination of inequality i (16)i in the Proof of Theorem 1 shows that the term fki fkk + Ki fkkk may be higher than 1 and still respect (15). This is i i Ki fkk fkk the case for some sufficiently low values for α or for ψ, for instance. Then for Cobb–Douglas, Quadratic, Logarithmic, Exponential, Logistic and CES production functions, a Nash equilibrium exists under conditions (C1) and (C2) (from Theorem 1) even with asymmetries in technology. For instance, production functions can write K α1, K α2, …, K αn respectively with αi ≠ αj for each i ≠ j. 5. Conclusion In this methodological paper we have determined conditions that enable the existence of a Nash equilibrium in a standard tax competition game. We show that our result is general and simple to use by applying our method to standard production functions. Our result extends the existing literature by considering more than two jurisdictions and by assuming that these jurisdictions may have asymmetric production functions. A further step for this analysis would be to concentrate on the uniqueness of the equilibrium in such a general framework or on the case of capital owned by residents. This would be a challenging task. In parallel, we may consider the fiscal game levying the Wildasin's assumption that capital is owned by agents outside the considered jurisdictions. Another objective is to conduct the same exercise (proof of the existence of a Nash equilibrium) in a model of trade with interactions between fiscal policies “à la Turnovsky” (1988). Our final objective is to combine both frameworks which would form a further interesting challenge. In addition, this framework will enable us to test the impact of trade upon public good provision in a tax competition model.
The following argument is valid for both the positive return regime and the boundary between regimes. Consider the profile ˜t = ð0; 0; …; 0; 0Þ, then ρ ˜t = fki Ki ˜t = j þ ̃ ˜ ˜ f k Kj t , ∀i,j and Ki t N 0. Fix some i and choose t i N 0 small enough + þ ˜ ̃ ˜ ̃ such that Kiþ = Ki t þ i ; t−i N 0 with t−i = ð0; …; 0Þ. Such t i and Ki ̃ ˜ exist by continuity of the utility function. Finally, with the couple t þ i ; t−i i + + i + + + + ̃ þ we obtain P iþ = t þ i K i and X i =f (K i )−f k(K i )K i , with P i ,X i N 0. Now pick a very small capital stock K si such that 0 b f i(K si ) b min(P + i , i i s X+ i ). Such K i exists since f (0) =0 and f is a continuous increasing function.12
1. Let set Ti = f ki(K is). If ti N Ti, then Ki(ti, t− i) b K is, ∀ t− i, since fk(Ki) ≥ ti N Ti = f ki(K is). It follows that f(Ki) b fi(K is), regardless of the tax rates in the other jurisdictions. Since Pi ≤ f i(Ki) and Xi ≤ f i(Ki), therefore, if + + + ti N Ti, then Xi b X+ i and Pi b Pi and Ui(Xi, Pi) b Ui(Xi , Pi ). þ þ 2. But setting ti = t ̃ guarantees K˜ i = Ki t ̃ ; t−i ≥K þ t þ̃ ; ˜t−i since i
i
i
i
∂Ki ðtÞ þ ̃ ˜ N 0, regardless of the vector t− i, so that Pi = t þ i K i ≥ Pi and ∂tj Xi = f i K˜ i −fki K˜ i K˜ i ≥Xiþ , regardless of what the other jurisdictions do.
As a conclusion of 1 and 2, jurisdiction i will never select a tax rate ̃ 13 greater than Ti, since it yields lower quantities of Xi, Pi than for ti = t þ i . References Bayindir-Upmann, T., Ziad, A., 2005. Existence of second-order locally consistent equilibria in fiscal competition models. Regional Sciences and Urban Economics 35, 1–22. Bucovetsky, S., 1991. Asymmetric tax competition. Journal of Urban Economics 30, 167–181. Dhillon, A., Wooders, M., Zissimos, B., 2007. Tax competition reconsidered. Journal of Public Economic Theory 9 (3), 391–423 06. Keen, M.J., Kotsogiannis, C., 2002. Does federalism lead to excessively high taxes? American Economic Review 92, 363–370. Laussel, D., Le Breton, M., 1998. Existence of Nash equilibria in fiscal competition models. Regional Science and Urban Economics 28, 283–296. Petchey, J., Shapiro, P., 2009. Equilibrium in fiscal competition games from the point of view of the dual. Regional Sciences and Urban Economics 39, 97–108. Reny, P.J., 1999. On the existence of pure and mixed strategy Nash equilibria in discontinuous games. Econometrica 67, 1029–1056. Rothstein, 2007. Discontinuous payoffs, shared resources, and games of fiscal competition: existence of pure strategy Nash equilibrium. Journal of Public Economic Theory 9 (2), 335–368. Turnovsky, S.J., 1988. The gains from fiscal cooperation in the two-commodity real trade model. Journal of International Economics 25, 111–121. Wildasin, D.E., 1988. Nash equilibria in models of fiscal competition. Journal of Public Economics 35, 229–240. Wilson, J.D., 1985. Optimal property taxation in the presence of interregional capital mobility. Journal of Urban Economics 17, 73–89. Wilson, J.D., 1986. A Theory of interregional tax competition. Journal of Urban Economics 19, 296–315. Wilson, John D., 1991. Tax competition with interregional differences in factor endowments. Regional Science and Urban Economics 21 (3), 423–451. Zodrow, G.R., Mieszkowski, P., 1986. Pigou, Tiebout, property taxation, and the underprovision of local public goods. Journal of Urban Economics 19, 356–370.
1
12
For a CES production function f ðK Þ = αK ψ + ð1 − αÞRψ ψ where R stands for the 1
quantity of the fixed factor, 0 b α b 1, ψ ≤ 1 and ψ ≠ 0, when K = 0, f ðK Þ = ð1 − αÞψ R. The value of R has no role in the analysis, so without restriction of generality, we can 1
choose R very small such that 0 b f ð0Þ = ð1 − αÞψ R b min Piþ ; Xiþ . Therefore there + exists a very small capital stock K is such that 0 b f i(K is) b min(P+ i , Xi ), for a CES
production function. 13 This demonstration has been suggested by one of the referees.
Contents lists available at ScienceDirect
Regional Science and Urban Economics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / r e g e c
On the existence of Nash equilibria in an asymmetric tax competition game☆ Emmanuelle Taugourdeau a,⁎, Abderrahmane Ziad b a b
CNRS CES, Paris School of Economics, France CREM, CNRS, University of Caen (UCBN), Esplanade de la Paix, 14032 Caen Cadex, France
a r t i c l e
i n f o
Article history: Received 21 June 2010 Received in revised form 22 February 2011 Accepted 22 February 2011 Available online 7 March 2011 JEL classification: C72 H21 H42 R50
a b s t r a c t In this methodological paper, we prove that the famous tax competition game introduced by Zodrow and Mieszkowski (1986) and Wildasin (1988) in which the capital is completely owned by foreigners possesses a Nash equilibrium even when the assumption of symmetric jurisdictions is dropped. The normality of both private and public goods is all that is needed concerning restrictions on preferences when a peculiar regime of taxation is ruled out. Moreover, we show that conditions about technology allowing for the existence of a Nash equilibrium are satisfied by most of the widely-used production functions. © 2011 Elsevier B.V. All rights reserved.
Keywords: Tax competition Nash equilibrium
1. Introduction In the established literature on tax competition the existence of a Nash equilibrium is assumed (see Zodrow and Mieszkowski, 1986; Wilson, 1985, 1986 and Wildasin, 1988). These studies focus on the comparative statics of Nash equilibria, and demonstrate that public services are provided at inefficiently low levels in equilibrium. However, little attention has been devoted to the question of whether such equilibria do exist. This is for the most part because the demonstration is very difficult, as noted by Laussel and Le Breton (1998): “Both the existence and uniqueness issues are difficult in general and have not been up to now dealt with in the literature. It seems however of primary interest to solve them in order to understand the comparative statics of the equilibrium.” Some results have already been established in the literature for particular cases. Firstly, Bucovetsky (1991) demonstrated the existence of a Nash equilibrium in tax rates in the case of two regions and quadratic production functions. A second and important result was highlighted by Laussel and Le Breton (1998) who proved the existence of a symmetric Nash equilibrium when private and public goods are perfect substitutes and when capital is not owned by
☆ We are indebted to David Wildasin, Maurice Salles, Yves Zenou and two anonymous referees for their substantial and helpful comments. ⁎ Corresponding author at: CNRS CES, University of Paris 1, ENS Cachan, Laplace 309, 94235 Cachan Cedex, France. E-mail addresses: [email protected] (E. Taugourdeau), [email protected] (A. Ziad). 0166-0462/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.regsciurbeco.2011.02.003
residents. In addition, this framework enables the authors to prove the uniqueness of the equilibrium, which is the primary purpose of their paper. In a more recent paper, Bayindir-Upmann and Ziad (2005) apply a weaker concept than the standard Nash equilibrium – the concept of a second-order locally consistent equilibrium (2-LCE) – which is a local Nash equilibrium (i.e., a small deviation is undesirable). With this tool, the authors are able to show both the existence and uniqueness of a symmetric equilibrium in tax rates when regions are homogeneous and when either (i) there are only two regions, (ii) capital demand curves are convex, or firms apply (iii) CES, (iv) Cobb–Douglas, or (v) logistic production functions. More recently, Dhillon et al. (2007) investigate the existence of a Nash equilibrium in a symmetric tax competition model where the public good enters the production function. Rothstein (2007) analyses the fiscal competition game as a game with discontinuous payoff and demonstrates the existence of a pure strategy Nash equilibrium for this kind of game under several assumptions respecting the production function. Rothstein moves away from the standard fiscal competition game “à la Wildasin” by assuming: first, an ad valorem tax; and second, that the aggregate amount of mobile capital is fixed in all regions. Finally, Petchey and Shapiro (2009) examine the problem of the existence of a Nash equilibrium in a tax competition model when governments are no longer benevolent but only make constrained efficient choices. A key point of this paper is that we deal with asymmetric regions. The literature on asymmetric tax competition is mainly based on two articles by Wilson (1991) and Bucovetsky (1991). Both assume that regions differ in their population and show that the “small” region
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may benefit from the tax competition by attracting capital from the “large” region thanks to the tax competition mechanism. In the present paper, we retain the methodological question regarding the existence of a Nash equilibrium in the tax competition model “à la Wildasin”. We depart from this seminal model by assuming that capital is owned by foreigners and we extend it to regions with different production functions.1 In doing so, we extend the existing literature by proving the existence of a Nash equilibrium in a more general framework. Our paper is in line with Laussel and Le Breton (1998) and Bayindir-Upmann and Ziad (2005), but we basically depart from their analysis by relaxing the assumption of symmetry. We also depart from the paper by Rothstein, firstly by considering proportional taxes, whereas Rothstein uses an ad valorem tax; secondly, by establishing a weaker condition of existence than the quasiconcavity condition of Rothstein, and thirdly, by directly deriving our result in the tax competition model. To prove our results, we use several assumptions: that goods are normal; that the demand for capital is convex; and that elasticity of the demand for capital is non increasing in absolute value. Moreover, we use an additional condition on preferences which enables us to rule out the particular regime of taxation in which the return on capital is zero and capital owners limit their capital supply. This paper is organized as follows. The second section outlines the tax competition model for a given number of regions, notation and description of the model being taken for the most part from Bayindir-Upmann and Ziad (2005). In Section 3 we establish a general theorem for the existence of the Nash equilibrium in fiscal competition. In Section 4 we introduce several examples of the commonly-used production functions and check the tractability of the result in Section 3. The final section summarizes our conclusions. 2. The model
i
i
Xi = f ðKi Þ−pi Ki + θi ρi K
Pi = ti Ki :
ð1Þ
where pi denotes the after-tax price of capital in jurisdiction i. Equating the price of capital to the value of its marginal product, fki(Ki) = pi determines region i's capital demand as a function of the corresponding after-tax price of capital, K ̂ i ðpi Þ. Let U(Xi, Pi) be the utility that the representative household of jurisdiction i derives from the provision of the public good, Pi, and from the consumption of the private good, Xi, produced by the firms. The utility function U(Xi, Pi) is twice-continuously differentiable and monotonically increasing. The source of the household's income is twofold: one part from the provision of the fixed factor, which is exclusively owned by local residents; and one part from their initial capital endowment. Let θi ∈ [0, 1] denote region i's share of the fixed national capital stock K, and pi, the net return of capital 1 In his paper, Wildasin (1988) uses also the assumption of absentee owners of capital in the jurisdictions to derive his results in the special case of identical jurisdictions.
ð3Þ
When choosing the level of the tax rate, each local government acts as a benevolent one and aims to maximize its representative resident's utility. In doing so, each local authority behaves non-cooperatively and treats its specific tax on capital ti, i = 1, …, n, as the strategic variable. This leads to a tax competition game between jurisdictions. The capital market clearing condition implies that aggregate demand for capital must equal capital supply: n
∑ Ki = K;
ð4Þ
i=1
for some exogenously given capital supply K. Capital being freely mobile across regions, the arbitrage condition equals the net return of capital in each jurisdiction: i
i = 1; …; n
ð2Þ
for each jurisdiction i = 1, …, n. In the following of the paper, we assume θi = 0 ∀ i. One explanation is that capital is owned by agents outside the jurisdictions under consideration, as in Wildasin (1988), Laussel and Le Breton (1998) and Rothstein (2007).2 Another explanation is to consider that the income of the resident capital owners does not affect the utility of the representative agent in each jurisdiction.3 Each local government provides a public good that it finances by taxing the mobile capital at a tax rate ti ≥ 0.4 The budget constraint of jurisdiction i is given by
ρ = fk ðKi Þ−ti ð = ρi Þ;
Consider n (n ≥ 2) jurisdictions inhabited by a given number of homogeneous residents that we normalize to one without loss of generality. A fixed number of competitive firms produce a homogeneous output in each jurisdiction using capital and some fixed factor(s) (land or labour). Aggregating production over all firms in each region allows us to treat the industry of one jurisdiction as one competitive firm. Let f i be the production function of the firm in jurisdiction i, f i is assumed to be monotonically increasing and strictly concave in capital, Ki. Fixed factors as explicit arguments of f i are suppressed so that the production function is expressed in terms of capital only. The jurisdiction i firm's profit can be written as Πi = f ðKi Þ−pi Ki ;
in region i. The private budget constraint of the consumer in region i amounts to
∀i = 1; …; n:
ð5Þ
Let t = (t1, …, tn) be the profile of tax rates, t− i = (t1, …, ti − 1, ti + 1, …, tn) be the profile of all tax rates except ti; whereas (ti, t− i) stands for (t1, …, ti − 1, ti, ti + 1, …, tn). Both Eqs. (4) and (5) define the equilibrium allocation of capital and the equilibrium of the net return of capital, i.e. K1(t),…, Kn(t) and ρ(t). Capital supply K is assumed exogenously fixed, whenever the solution of Eqs. (4) and (5) requires that capital earns a non-negative net return, ρ ≥ 0. When ρ b 0, we suppose that capital owners limit capital supply. The resulting allocation of capital is a vector (K1, K2, …, Kn) with ∑ni = 1 Ki = K 4 b K, for which fki(Ki) = ti. We assume that if for some profile t = (t1, t2, …, tn) we have ρ(t) = ρi(t) = 0 for each i, and if a region j raises its tax rate from tj to tj + Δtj such that ρj(t− j, tj + Δtj) b 0 while the other regions keep their tax rate unchanged, then capital owners limit supply only in region j until ρj = 0. As a result, as in Bucovetsky (1991), we have three regimes: 1. Positive return regime: ∑ni = 1 Ki = K; ρ N 0, 2. Excess supply regime: ∑ni = 1 Ki = K 4 b K; ρ = 0, 3. Boundary between regimes: ∑ni = 1 Ki = K; ρ = 0.
∂Ki changes ∂ti discontinuously at any tax vector (t1, t2, …, tn) at which ρ = 0 and ∑ni = 1 Ki = K, i.e. in the boundary between regimes. In any regime It is shown in Bucovetsky (1991) that the derivative
2 Alternatively, Zodrow and Mieszkowski (1986), Bucovetsky (1991) or BayindirUpmann and Ziad (2005) assumed that owners of capital own an equal share of the total capital stock. 3 This could be the case if we consider two groups of agents in each jurisdiction, one which does not own capital but controls decision making, say the “poor/middle class”, and one which owns capital and passively consumes its net income of capital, say the “rich class”. This comment has been suggested by one of the referees. 4 Capital subsidies cannot be funded in this model since there are no other taxes available.
E. Taugourdeau, A. Ziad / Regional Science and Urban Economics 41 (2011) 439–445
the utility function is continuous everywhere but may be not differentiable at some tax rates ti for which ρ(ti) = 0. At this stage of the analysis, the utility function may have multiple maxima and/or minima. We will show later that the net return of capital is a continuous and non increasing function with respect to the tax rates. Let us assume that there exists t 0i to be the first tax rate such that ρ(t 0i , t− i) = 0. For each ti N ti0 we have ρ(ti, t− i) = 0 (capital owners limit the capital supply). We will show that under some reasonable conditions, the utility function cannot reach any maximum with ti N t 0i (the excess supply regime) and that for each ti b t 0i (the positive return regime), the utility function is differentiable.5 The following formulas are derived from the case of a positive return regime. The capital market equilibrium enables us to determine the response of the net return of capital to an increase in jurisdiction i's tax rate: K ̂j′ pj ∂ρðtÞ : =− ∂tj ∑i K ̂i′ðpi Þ
∂Pi ∂ρ = ti K ̂i′ N0 ∂tj ∂tj
Eq. (8) states that the private good is clearly decreasing with the local tax rate whereas nothing can be said about the reaction of the public good. Both Eqs. (6) and (7) enable us to write an indirect utility function Vi(t) = Ui(Xi(t), Pi(t)), which directly relates tax policy to welfare. Maximizing the indirect utility function of jurisdiction i with respect to the tax rate ti yields the following first-order condition: ∂Vi ðtÞ ∂Ui ∂Xi ∂Ui ∂Pi = ðtÞ + ðtÞ = 0; ∂Xi ∂ti ∂Pi ∂ti ∂ti
∂Ui = ∂Pi −∂Xi = ∂ti = ∂Ui = ∂Xi ∂Pi = ∂ti and by Eqs. (8) and (11) we have:
∂Ui = ∂Pi MRSXi ;Pi := = ∂Ui = ∂Xi
Differentiating Kj(t) and Ki(t) with respect to tj, we obtain the overall impact of a variation of the tax rate of region j on capital demand: ! ∂Kj ðtÞ ∂ρðtÞ = K ̂j′ pj 1 + b 0; ∂tj ∂tj ∂Ki ðtÞ ∂ρðtÞ = K ̂i′ð pi Þ N 0; ∀i≠j; ∂tj ∂tj with pj = ρ(t) + tj, ∀ j. Using pi = ρ + ti and substituting Ki(t) and ρ(t) into the private and the public budget constraint Eqs. (2) and (3), we obtain Xi ðtÞ = f ðKi ðtÞÞ−ðti + ρðtÞÞKi ðtÞ;
ð6Þ
Pi ðtÞ = ti Ki ðtÞ:
ð7Þ
Differentiating Eqs. (6) and (7) with respect to ti in the positive regime yields7: ∂Xi ∂ρ = −Ki 1 + b 0; ∂ti ∂ti
ð8Þ
2 2 ∂ Xi ∂K ∂ρ ∂ ρ =− i 1+ −Ki 2 ; 2 ∂ti ∂ti ∂ti ∂ti
ð9Þ
∂Xi ∂ρ = Ki b0 ∂tj ∂tj
∀i≠j;
ð10Þ
∂Pi ∂Ki = Ki + t; ∂ti ∂ti i
ð11Þ
∂2 Pi ∂K ∂2 K = 2 i + ti 2 i ; 2 ∂ti ∂ti ∂ti
ð12Þ
5
The utility may reach a maximum at ti = t 0i . These expressions are perfectly in line with the standard results of the tax competition literature such as Keen and Kotsogiannis (2002) for instance. 7 Note that all functions are continuously differentiable with respect to the strategic variables. 6
ð13Þ
which implies the relationship
and since each region's capital demand ! monotonically decreases with 1 ∂ρðtÞ ′ ̂ ∈½−1; 0, which the price of capital K j pj = j b 0 , we have ∂tj fkk ! ∂ρðtÞ is equivalent to 1 + ∈½0; 1.6 ∂tj
i
∀i≠j:
441
∂ρ Ki + Ki −∂Xi = ∂ti ∂ti = = MRTXi ;Pi : ∂ρ ∂Pi = ∂ti Ki + 1 + ti K ̂i′ ∂ti ð14Þ
Each government equates the marginal rate of substitution (MRS) to the marginal rate of fiscal transformation (MRT) between the private and the public goods. In what follows, we are interested in the existence of a Nash equilibrium, a profile of strategies t* such that t i* maximizes Vi(ti, t− * i) with respect to ti for each i. Before developing the analysis which will enable us to prove the existence of a Nash equilibrium in the fiscal game, we focus on the two regimes which are the excess supply regime and the boundary between regimes. To deal with those regimes, we need the following condition: (C1). Marginal utility of private good dominates marginal utility of public good when capital return is fully taxed away: MRS( f i(Ki) − Ki fki(Ki) ; Ki fki(Ki)) ≤ 1 for each i. This condition was first introduced by Bucovetsky (1991), and says that when capital return is fully taxed away, the share of the public good is higher than the optimal share (for which MRS = 1). The result is that agents are less inclined to give up private goods in exchange for one unit of public good because a high tax rate implies a low level of capital and a low level of private good. It is proved in Bucovetsky (1991, Lemma 4) that if (C1) is satisfied, no Nash equilibrium exists within the excess supply regime with a quadratic production function.8 In our model, this condition is useful to show that if the utility function reaches a maximum (global or not) within the excess supply regime or the boundary between regimes, this maximum is obviously the last one. Under condition (C1), the following lemma precludes the existence of a maximum under the excess supply regime. Recall that t0i is the first value of ti ≥ 0 for which ρ(t 0i , t− i) = 0. 8 Bucovetsky (1991) proposes an alternative assumption on the production function that also guarantees that the excess supply regime can be ruled out. In our model this − fki ðKi Þ f i ðKi Þ−Ki fki ðKi Þ condition writes: (C′1): σi ≥ 1 − αi ∀ i where σi = stands for i ðK Þ Ki f i ðKi Þ fkk i Ki fki ðKi Þ for the capital's share. the elasticity of substitution in production, and αi = i f ðKi Þ This condition is equivalent to Ki fki(Ki) increasing in Ki. It is easy to check that all the
production functions that will be used in the following part of the paper satisfy this condition.
442
E. Taugourdeau, A. Ziad / Regional Science and Urban Economics 41 (2011) 439–445
Lemma 1. [No maximum within the excess supply regime] There is no maximum with ti N t 0i within the excess supply regime if (C1) is satisfied. Proof: Suppose that we can reach a maximum at some tax rate ti N t 0i within the excess supply regime then t′i = fki(Ki′) ∀ ti′ ≥ t 0i . The utility function Ui(Xi, Pi) is assumed to be increasing in (Xi, Pi), with Xi = f i(Ki) − Ki f ik(Ki), and Pi = tiKi = fki(Ki)Ki. Lowering the tax rate ti increases Ki. The impact of a rise in Ki on the utility is given by: ∂Ui ðXi ; Pi Þ ∂Ui ðXi ; Pi Þ ∂Xi ∂Ui ðXi ; Pi Þ ∂Pi = + ∂Ki ∂Ki ∂Ki ∂Xi ∂Pi =
i ∂Ui ðXi ; Pi Þ h i i : Ki fkk ðKi Þ:ðMRSi −1Þ + MRSi ⋅ fk ðKi Þ ∂Xi
□ which is strictly positive according to (C1). At this stage of the analysis, Lemma 1 stipulates that in the case of multiple minima and/or maxima, the point defined by t 0i such that ρ(t 0i , t− i) = 0 can only be the last maximum that the utility function can reach. Before this point, the function Vi(ti, t− i) is differentiable everywhere.9 3. The existence of a Nash equilibrium In this section, we present our first main result concerning the existence of Nash equilibria in the fiscal game described above. To obtain useful and clear results, in addition to condition (C1), we postulate the following assumptions: ∂ (C2). Normality of goods: For each i and for all Xi, Pi N 0: MRSXi ;Pi ≥0 ∂X i ∂ and MRSXi ;Pi ≤ 0. ∂Pi (C3). Marginal product of capital falls at diminishing rate: The third i ≥ 0 for each i. derivative of the production functions is positive, f kkk (C4). Non increasing elasticity demand for capital (in For absolute −1 value): fki ðKi Þ dlnfk i = each i the elasticity of demand for capital k = i ðK Þ dlnK Ki fkk i must be non decreasing (non increasing in absolute value) with respect i i 2 10 i i to the capital. This condition is equivalent to fk f kk + Ki fkkk ≤ Ki fkk . Condition (C2), already postulated by Bucovetsky (1991) and Bayindir-Upmann and Ziad (2005), requires that the marginal rate of substitution between the private and the public goods is nondecreasing in the first argument and non-increasing in the second argument. This is equivalent to stating that both the private and the public goods are normal goods, which is regarded as standard in economics. Condition (C3) was first introduced by Laussel and Le Breton (1998) and later used by Rothstein (2007) and Petchey and Shapiro (2009). It stipulates that the marginal product of capital is always a convex function of the amount of invested capital. More precisely, the slope of the marginal production is flattening out as the capital rises. The use of this condition will be explained in Lemma 4. Usual production functions such as Cobb–Douglas, CES and Quadratic functions satisfy (C3). Condition (C4) is a new condition which states that the elasticity of demand for capital must be non increasing in absolute value with respect to the capital: the higher the level of capital in the economy, the lower the demand for capital reaction to an inflow of capital. This property applies perfectly for linear demand curves that are infinitely elastic at zero quantities of capital and have zero elasticities at their horizontal intercepts. More generally, most of the standard produc9 Following from the differentiability of the functions U(⋅), f(⋅) and then K(⋅) and finally X(⋅) and P(⋅). 10 Recall that fik is the inverse demand for capital.
tion functions satisfy (C4) (see later examples). In particular, this condition is obvious for the quadratic production functions (since i f kkk = 0). This condition is looser than the conditions introduced by Rothstein (2007) who firstly assumes that the elasticity of the marginal product of capital εik must be higher than (− 1) (assumption i 2 i 8ii), and secondly, that 2 fkk −fki fkkk ≥ 0 (assumption 9iii). Both conditions (C3) and (C4) bound the third derivative of the i 1−εik fkk i production functions since (C4) can be rewritten as fkkk ≤ or, i εk Ki fi i more strongly, fkkk ≤ − kk . Thus, conditions (C3) and (C4) stipulate Ki i that f kkk must be positive but not “too big” relative to the second i derivative f kk . At this stage of the reasoning, we are able to set that in both the positive return regime and the boundary between regimes, the tax rate belongs to [0, Ti] (see the Appendix for a demonstration of the existence of the upper bound Ti). In order to prove the existence of at least one Nash equilibrium, we consider the best response of each jurisdiction and prove that they are functions and not a correspondence. Then by continuity assumptions, we can use a fixed point theorem (Brouwer's theorem) to prove the existence of a Nash equilibrium. The principal general result of the paper is as follows:
Theorem 1. When consumers' preferences are represented by twicecontinuously differentiable, monotonically increasing utility functions Ui(.,.) in (Xi, Pi) and assumptions (C1), (C2), (C3) and (C4) are satisfied, then the asymmetrical fiscal game possesses a Nash equilibrium. To prove our result, we have to introduce several Lemmas, which are of great importance in understanding the tax competition mechanisms at work in the economy. Lemma 2. [Capital allocation] For any given t, the capital allocation K1(t), …, Kn(t) and the net ∂Kj ðtÞ ∂K ðtÞ N 0∀j≠i, and return of capital ρ(t) are unique, with i b 0, ∂t ∂ti i ∂ρðtÞ b 0 ∀i. ∂ti Lemma 3. [Increasing branch of the Laffer curve] For any given tax vector t− i, a utility local-maximizing (respectively ∂Pi local minimizing) strategy of region i requires ðtÞ N 0 on the left ∂ti (respectively the right) of any local extremum point. Proof: As the private good Xi is a decreasing function in ti, and the utility function Ui(.,.) is an increasing function in (Xi, Pi), in the neighbourhood of any local maximum (resp. minimum) t 0i , the public good must be an increasing function on the left (respectively on the right) of t 0i .11 □ Lemma 4. Decreasing marginal rate of substitution Let condition (C2) hold. For any tax vector t, the marginal rate of substitution between the private and the public goods is falling on the left or on the right of any local extremum. Proof: The derivative of the marginal rate of substitution is ∂MRSXi ;Pi ∂Xi ðtÞ ∂MRSXi ;Pi ∂Pi ðtÞ ∂ MRSXi ;Pi ðtÞ = + : ∂Xi ∂Pi ∂ti ∂ti ∂ti ∂ From condition (C2) (goods are normal) we have MRSXi ;Pi ≥ 0 ∂X i ∂ ∂Xi ðtÞ ∂Pi ðtÞ MRSXi ;Pi ≤ 0. Since b 0 and N 0 (Lemma 3), then and ∂Pi ∂ti ∂ti ∂ MRSXi ;Pi ðtÞ b 0. □ ∂ti 11
Not on the boundary.
E. Taugourdeau, A. Ziad / Regional Science and Urban Economics 41 (2011) 439–445
Lemma 5. [Convexity of the net return of capital function] Condition (C3) is sufficient to ensure the convexity of the net return of capital ρðtÞ. Proof: The second derivative of the net return function is written as: ∂Kj ðtÞ 2
∂ ρðtÞ = ∂ti2
∂Ki ðtÞ ∂ti
i fkkk i 2 fkk
∑j
1 j fkk
1
∑j
1 ∑j i fkk !2
−
2
∂ ρðtÞ = ∂t i2
∂Ki ðtÞ ∂ti
−
∂2 V ðt−i ; ti Þ b 0; is to prove that ∂ti2
∂ MRTXi ;Pi ðti ; t−i Þ b 0: ∂ti
0 1 ! ∂Xi −2 2 2 ∂ @ ∂ti ðtÞA ∂Pi ∂ Xi ∂Pi ∂Xi ∂ Pi = − : ∂ti ∂ti ∂ti2 ∂ti ∂Pi ðtÞ ∂ti2 ∂ti
:
∂ti
j fkk
Then
∂Kj ðtÞ j ∂ρðtÞ = f Kj . But ρ(t) = fkj(Kj) − tj, then for j ≠ i, we have ∂ti ∂ti kk We obtain
i f kkk i 2 fkk
A sufficient condition to check
Let
j
f ∂ti kkk 2 j fkk
443
∂Ki ðtÞ i f kkk 1 1 ∂ρðtÞ ∂t ∑j j − i ∑j≠i 3 − i i 3 j ∂ti f kk f kk fkk f kk !2 1 ∑j j fkk
3 2 ∂2 Xi ∂Pi ∂Xi ∂2 Pi ∂Ki ∂2 ρ ∂K i 2∂ ρ − = −t fkk −Ki −ti Ki 2 i i 2 ∂t 2 2 ∂ti ∂ti ∂ti ∂ti ∂ti ∂ti ∂ti i + Ki
j f kkk
and i j f kkk ∂ρðtÞ f kkk 1 ∂ρðtÞ 1 1+ ∑ i 3 ∑j≠i j − j≠i i j 3 ∂ti ∂ti fkk f kk f kk 2 f kk ∂ ρðtÞ = !2 ∂ti2 1 ∑j j f kk
∂Ki ∂ti
2
i
f kk + ti Ki
∂Ki ∂2 Ki i f : ∂ti ∂ti2 kk
Replacing ∂ρðtÞ 1+ ∂Ki ðtÞ ∂ti = i ∂ti f kk and ! ∂2 ρðtÞ i 2 ∂ρðtÞ 2 i fkkk fkk − 1 + 2 ∂ti ∂ti i 3 fkk
∂2 Ki ðtÞ = ∂ti2
j
which is positive as soon as f kkk ≥ 0 (condition (C3). □ As mentioned by Laussel and Le Breton (1998) “for a jurisdiction the advantage of taxing capital is in the induced reduction of the equilibrium net rate of return of capital (i.e. of the equilibrium “price of capital” which must be paid to capital holders who invest in the jurisdiction)”. Condition (C3) implies that this reduction occurs at an increasing rate as ti increases. In other words, condition (C3) guarantees that the marginal benefit from taxing capital diminishes when the level of capital is large. Lemma 6. [Uniqueness of the best reply functions] In the positive return regime, conditions (C2), (C3) and (C4) are sufficient to ensure the local-concavity of the indirect utility function in the neighborhood of any tax rate satisfying the first order condition. Proof: If the first order condition is satisfied for some ti we have: 2
3
∂Xi ∂V ðt−i ; ti Þ ∂U ðXi ; Pi Þ ∂Pi ðt−i ; ti Þ 4 ∂ti ðt−i ; ti Þ + MRSXi ;Pi ðt−i ; ti Þ5 = 0 = ∂Pi ∂ti ∂Xi ∂ti ðt−i ; ti Þ ∂ti
we get ! 2 2 ∂ρðtÞ 3 ∂ρðtÞ 2 i i i i ∂ ρ −ti f kk + Ki fkkk + Ki fkk 1+ −Ki fkk 1+ 2 ∂ti ∂ti ∂ti ∂ Xi ∂Pi ∂Xi ∂ Pi − = : i 3 ∂ti ∂ti2 ∂ti2 ∂ti fkk 2
2
Then ∂2 Xi ∂Pi ∂Xi ∂2 Pi − ∂ti ∂ti2 ∂ti2 ∂ti is negative if and only if
i ti fkk
+
i Ki fkkk
! 2 2 ∂ρðtÞ 3 ∂ρðtÞ 2 i i ∂ ρ 1+ b Ki fkk 1+ −Ki fkk 2 : ∂ti ∂ti ∂ti ð15Þ
∂2 ρ N 0 from Lemma 5. ∂ti2 i i • If fkk + Ki fkkk b 0, Eq. (15) is always satisfied i i • If fkk + Ki fkkk ≥ 0, i i b 0, fkkk N 0 from (C3) and With fkk
i ∂V ðt−i ; ti Þ ∂U ðXi ; Pi Þ ∂Pi ðt−i ; ti Þ h = MRTXi ;Pi ðt−i ; ti Þ + MRSXi ;Pi ðt−i ; ti Þ = 0: ∂ti ∂Xi ∂ti
And the second derivative of the indirect utility function for government i is: 2 ∂ V ðt−i ; ti Þ ∂U ðXi ; Pi Þ ∂Pi ðt−i ; ti Þ ∂ ∂ = − MRTXi ;Pi ðt−i ; ti Þ + MRSXi ;Pi ðt−i ; ti Þ : ∂ti ∂Xi ∂ti ∂ti ∂ti
While from Lemma 4 we have: ∂ 0 MRSXi ;Pi ti ; t−i b 0: ∂ti
as we work in the positive regime ti ≤ fki(Ki(t1, …, tn)) for each ti, a sufficient condition to check Eq. (15) is then ! 2 2 ∂ρðtÞ 3 ∂ρðtÞ 2 i i i i i ∂ ρ f k f kk + Ki f kkk 1 + b Ki f kk 1+ −Ki f kk 2 ; ∂ti ∂ti ∂ti ! i i 2 f kk + Ki f kkk f ki ∂ρðtÞ 3 ∂ρðtÞ 2 i ∂ ρ 1 + b 1 + −K f : ð16Þ i kk i i ∂ti ∂ti ∂ti2 f kk Ki f kk
444
E. Taugourdeau, A. Ziad / Regional Science and Urban Economics 41 (2011) 439–445
i i fi f kk + Ki f kkk This condition is verified since k i ≤1 by condition i 3 K2i fkk 2 f kk ∂ρðtÞ ∂ρðtÞ ∂ ρ (C4), 1 + b 1+ and 2 N 0 by Lemma 5 . □ ∂ti ∂ti ∂ti Lemma 6 precludes the existence of any minimum in the positive ∂V 2 ðti ; t−i Þ return regime since any minimum requires N 0 for any ti ∂ti2 ∂V ðti ; t−i Þ satisfying = 0. The absence of a minimum in the positive ∂ti return regime precludes also the existence of multiple maxima. Lemmas 1 to 6 enable us to establish the Proof of Theorem 1: Proof of Theorem 1. Given Lemma 1 that rules out the possibility of any maximum (global or not) in the excess supply regime, we concentrate our analysis on the positive return regime and the boundary between regimes. Let t− i = (t1, …, ti − i, ti + 1, …, tn) be the profile of all tax rates except ti, our first step is to look at the best reply correspondence of jurisdiction i. ∂V • If the first order condition is never satisfied, that is i ðti ; t−i Þ≠0, ∂ti then ∂Vi - either ðti ; t−i Þ N 0 on [0, Ti] and the best reply is unique (ti = Ti) ∂ti ∂V - either i ðti ; t−i Þb 0 on [0, Ti] and the best reply is unique (ti = 0) ∂ti - either there exists a unique maximum in the boundary regime: Lemma 1 ensures that there is no maximum for any ti N t 0i whereas Lemma 6 ensures that there is no maximum for any ti b t 0i . The best response is then unique and the maximum is a peak reached at ti = t 0i . • If the first order condition is verified in the positive regime, or if the first order condition is verified on the boundary regime that ∂V is, i t i0 ; t−i = 0 and ρ(t 0i , t− i) = 0 with t 0i = Ti, since all functions ∂ti are continuous and differentiable (U(⋅), f(⋅)and K(⋅)and then X(⋅) and P(⋅)) for any ti b t 0i , Lemma 6 ensures the existence of a unique best reply. Let us now introduce the best reply correspondence of jurisdiction i, by n
BRi ðt−i Þ = ti ∈½0; Ti : Vi ðti ; t−i Þ = Supt′∈½0;T Vi t i′; t−i Þ i
i
and define the Cartesian product of the different reply correspondences BRi by BRðt1 ; t2 ; …; tn Þ = BRi ðt−1 Þ × BRi ðt−2 Þ × … × BRi ðt−n Þ Both the single value of the best reply and the continuity of the indirect utility function (following from the differentiability and the continuity of the functions U(⋅), f(⋅) and then K(⋅) and finally X(⋅) and P(⋅)), imply that we can state that the best reply is a function defined on [0, Ti]n into [0, Ti]n, which is continuous. From Brouwer's theorem, the function BR(.,., …,.) possesses a fixed point. Then, it is obvious to show that the fixed point of BR(.,., …,.) is a Nash equilibrium of the fiscal game under study. □ Let us summarize the role of the different conditions in the proof: (C1) is useful to rule out the excess supply regime in which the net return of capital is fully taxed and the capital owners limit their capital supply. (C2) implies that the marginal rate of substitution between the private and the public goods is a non increasing function of the tax rate. (C3) guarantees that the marginal benefit of taxing more is diminishing for a high level of capital. Finally, both conditions (C3) and (C4) ensure that the marginal rate of the fiscal transformation between the private and the public goods is increasing with the tax rate. We crucially depart from Rothstein's result in several ways: firstly, we stipulate a weaker condition than those defined in Assumptions 8 and 9 of Rothstein's
paper, secondly, our proof is direct while Rothstein's proof is much more complicated and uses Reny's (1999) result, and, lastly, we consider a unit tax whereas Rothstein uses an ad valorem tax. Our theorem applies for asymmetric or symmetric production functions. In that sense, our paper extends the result by Laussel and Le Breton (1998) by considering more than two jurisdictions and by allowing for asymmetric regions. It also complements the results of local equilibria by Bayindir-Upmann and Ziad (2005) which apply to symmetric production functions with national capital ownerships. 4. Applications to usual production functions In this section, we establish our second main result about the existence of equilibria in our fiscal game by introducing several examples. Our purpose is firstly to show the tractability of our method with commonly-used production functions and secondly to state that most of the commonly-used economic models of tax competition possess a Nash equilibrium: • Cobb–Douglas production function f(K) = K α, with fk(K) = αK α − 1, fkk(K) = α(α − 1)K α − 2, fkkk(K) = α(α − 1)(α − 2)K α − 3 N 0. a • Quadratic production function f(K) = (a − bK)K, with K b , so 2b f k (K) = a − 2bK, f kk (K) = − 2b. • Logarithm production function f(K) = ln(1 + βK) with fk ðK Þ = β β2 , f ðK Þ = − : 1 + βK kk ð1 + βK Þ2 • Exponential production function f(K) = 1 − exp(− βK) with fk(K) = βexp(− βK), fkk(K) = − β2exp(− βK). 1 1 • Logistic production function f ðK Þ = − , with x = x(K) : = 1 + e−x 2 Rα e x R2α ð1−e x Þe x α KR , so fk ðK Þ = , fkk ðK Þ = . ð1 + e x Þ2 ð1 + e x Þ3 1 • CES production function f ðK Þ = αK ψ + ð1 − αÞRψ ψ where R stands for the quantity of the fixed factor, 0 b α b 1, ψ≤1 and ψ≠0, so f k ðK Þ =
αK ψ f ðK Þ αð1 − αÞð1 − ψÞðKRÞψ f ðK Þ , f ðK Þ = − . 2 αK ψ + ð1 − αÞRψ K kk K2 ðαK ψ + ð1 − αÞRψ Þ
Theorem 2. Assume that consumers' preferences are represented by twice-continuously differentiable, monotonically increasing utility functions Ui(.,.) in (Xi, Pi) and that conditions (C1) and (C2) are satisfied, that is the private and public goods are normal goods. If for each jurisdiction i, the production function fi ∈ {Cobb–Douglas, Quadratic, Logarithmic, Exponential, Logistic, CES} under further conditions for the parameters for the logistic and CES production functions, then the fiscal game possesses a Nash equilibrium. Proof: For each of the production functions above, we have to show that conditions (C3) and (C4) are verified: • Cobb–Douglas production function: fkkk ðK Þ = αðα−1Þðα−2ÞK
α−3
f ki N 0 and i Ki f kk
i i fkk + Ki f kkk i f kk
= 1:
• Quadratic production function: fki fkkk ðK Þ = 0 and i Ki fkk
i i f kk + Ki f kkk i fkk
=
a−2bK b 0: −2bK
• Logarithm production function:
2β3 fki N 0 and fkkk ðK Þ = 3 i ð1 + βK Þ Ki fkk
i i fkk + Ki fkkk i fkk
= 1−
1 b 1: βK
E. Taugourdeau, A. Ziad / Regional Science and Urban Economics 41 (2011) 439–445
• Exponential production function: 3
fkkk ðK Þ = β expð−βK Þ N 0 and
fki i Ki fkk
i i fkk + Ki fkkk i fkk
Appendix A. Boundary of the tax rates = 1−
1 b 1: βK
• Logistic production function: fkkk ðK Þ =
R3α 1−4e x + e 2x e x ð1 +
e x Þ4
445
i h pffiffiffii f + Ki f i N 0 b = N x N ln 2 + 3 and kk i kkk = fkk
ð1 + e x Þð1−e x Þ + x 1−4ex + e2x where g(x) : = x(1 − 4e x + e 2x) + (1 + x x ð1 + e Þð1−e Þ
e x)(1 − e x) is non positive since g(0) = 0 and gk′ (x) = (1 − e2x) − (4 + i i 2x)ex b 0. Therefore f kk + Ki fkkk b 0 and the marginal rate of fiscal transformation is increasing. • CES production function: αð1 − αÞð1 − ψÞðKRÞψ f ðK Þ αð1 + ψÞK ψ + ð1 − αÞ 3 K3 ðαK ψ + ð1 − αÞRψ Þ i i 2 αψK ψ i + Ki fkkk = , −Ki fkk ð2 − ψÞRψ ÞN 0 a n d fki fkk ψ ð 1−α ÞRψ + αK i i 2 i i i i which implies fk fkk + Ki fkkk −Ki fkk b 0 for ψ b 0 and fk fkk + i 2 i Ki fkkk Þ−Ki fkk N 0 for ψ N 0. In the last case, the sufficient condition fkkk ðK Þ =
of theorem 1 is not satisfied. However a closer examination of inequality i (16)i in the Proof of Theorem 1 shows that the term fki fkk + Ki fkkk may be higher than 1 and still respect (15). This is i i Ki fkk fkk the case for some sufficiently low values for α or for ψ, for instance. Then for Cobb–Douglas, Quadratic, Logarithmic, Exponential, Logistic and CES production functions, a Nash equilibrium exists under conditions (C1) and (C2) (from Theorem 1) even with asymmetries in technology. For instance, production functions can write K α1, K α2, …, K αn respectively with αi ≠ αj for each i ≠ j. 5. Conclusion In this methodological paper we have determined conditions that enable the existence of a Nash equilibrium in a standard tax competition game. We show that our result is general and simple to use by applying our method to standard production functions. Our result extends the existing literature by considering more than two jurisdictions and by assuming that these jurisdictions may have asymmetric production functions. A further step for this analysis would be to concentrate on the uniqueness of the equilibrium in such a general framework or on the case of capital owned by residents. This would be a challenging task. In parallel, we may consider the fiscal game levying the Wildasin's assumption that capital is owned by agents outside the considered jurisdictions. Another objective is to conduct the same exercise (proof of the existence of a Nash equilibrium) in a model of trade with interactions between fiscal policies “à la Turnovsky” (1988). Our final objective is to combine both frameworks which would form a further interesting challenge. In addition, this framework will enable us to test the impact of trade upon public good provision in a tax competition model.
The following argument is valid for both the positive return regime and the boundary between regimes. Consider the profile ˜t = ð0; 0; …; 0; 0Þ, then ρ ˜t = fki Ki ˜t = j þ ̃ ˜ ˜ f k Kj t , ∀i,j and Ki t N 0. Fix some i and choose t i N 0 small enough + þ ˜ ̃ ˜ ̃ such that Kiþ = Ki t þ i ; t−i N 0 with t−i = ð0; …; 0Þ. Such t i and Ki ̃ ˜ exist by continuity of the utility function. Finally, with the couple t þ i ; t−i i + + i + + + + ̃ þ we obtain P iþ = t þ i K i and X i =f (K i )−f k(K i )K i , with P i ,X i N 0. Now pick a very small capital stock K si such that 0 b f i(K si ) b min(P + i , i i s X+ i ). Such K i exists since f (0) =0 and f is a continuous increasing function.12
1. Let set Ti = f ki(K is). If ti N Ti, then Ki(ti, t− i) b K is, ∀ t− i, since fk(Ki) ≥ ti N Ti = f ki(K is). It follows that f(Ki) b fi(K is), regardless of the tax rates in the other jurisdictions. Since Pi ≤ f i(Ki) and Xi ≤ f i(Ki), therefore, if + + + ti N Ti, then Xi b X+ i and Pi b Pi and Ui(Xi, Pi) b Ui(Xi , Pi ). þ þ 2. But setting ti = t ̃ guarantees K˜ i = Ki t ̃ ; t−i ≥K þ t þ̃ ; ˜t−i since i
i
i
i
∂Ki ðtÞ þ ̃ ˜ N 0, regardless of the vector t− i, so that Pi = t þ i K i ≥ Pi and ∂tj Xi = f i K˜ i −fki K˜ i K˜ i ≥Xiþ , regardless of what the other jurisdictions do.
As a conclusion of 1 and 2, jurisdiction i will never select a tax rate ̃ 13 greater than Ti, since it yields lower quantities of Xi, Pi than for ti = t þ i . References Bayindir-Upmann, T., Ziad, A., 2005. Existence of second-order locally consistent equilibria in fiscal competition models. Regional Sciences and Urban Economics 35, 1–22. Bucovetsky, S., 1991. Asymmetric tax competition. Journal of Urban Economics 30, 167–181. Dhillon, A., Wooders, M., Zissimos, B., 2007. Tax competition reconsidered. Journal of Public Economic Theory 9 (3), 391–423 06. Keen, M.J., Kotsogiannis, C., 2002. Does federalism lead to excessively high taxes? American Economic Review 92, 363–370. Laussel, D., Le Breton, M., 1998. Existence of Nash equilibria in fiscal competition models. Regional Science and Urban Economics 28, 283–296. Petchey, J., Shapiro, P., 2009. Equilibrium in fiscal competition games from the point of view of the dual. Regional Sciences and Urban Economics 39, 97–108. Reny, P.J., 1999. On the existence of pure and mixed strategy Nash equilibria in discontinuous games. Econometrica 67, 1029–1056. Rothstein, 2007. Discontinuous payoffs, shared resources, and games of fiscal competition: existence of pure strategy Nash equilibrium. Journal of Public Economic Theory 9 (2), 335–368. Turnovsky, S.J., 1988. The gains from fiscal cooperation in the two-commodity real trade model. Journal of International Economics 25, 111–121. Wildasin, D.E., 1988. Nash equilibria in models of fiscal competition. Journal of Public Economics 35, 229–240. Wilson, J.D., 1985. Optimal property taxation in the presence of interregional capital mobility. Journal of Urban Economics 17, 73–89. Wilson, J.D., 1986. A Theory of interregional tax competition. Journal of Urban Economics 19, 296–315. Wilson, John D., 1991. Tax competition with interregional differences in factor endowments. Regional Science and Urban Economics 21 (3), 423–451. Zodrow, G.R., Mieszkowski, P., 1986. Pigou, Tiebout, property taxation, and the underprovision of local public goods. Journal of Urban Economics 19, 356–370.
1
12
For a CES production function f ðK Þ = αK ψ + ð1 − αÞRψ ψ where R stands for the 1
quantity of the fixed factor, 0 b α b 1, ψ ≤ 1 and ψ ≠ 0, when K = 0, f ðK Þ = ð1 − αÞψ R. The value of R has no role in the analysis, so without restriction of generality, we can 1
choose R very small such that 0 b f ð0Þ = ð1 − αÞψ R b min Piþ ; Xiþ . Therefore there + exists a very small capital stock K is such that 0 b f i(K is) b min(P+ i , Xi ), for a CES
production function. 13 This demonstration has been suggested by one of the referees.