Tax Competition: Theoretical Analysis beyond Nash Equilibrium

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Wirtschafts- wissenschaft

Dissertation

Robert Philipowski

Tax Competition

Theoretical Analysis beyond

Nash Equilibrium

Tax Competition – Theoretical Analysis beyond Nash Equilibrium

Dissertation

zur Erlangung des Doktorgrades (Dr. rer. pol.) der Fakult¨at f¨ur Wirtschaftswissenschaft

der Fernuniversit¨at Hagen

vorgelegt von Robert Philipowski aus Bonn

August 2017

Erstgutachter: Univ.-Prof. Dr. Thomas Eichner Zweitgutachter: Univ.-Prof. Dr. Joachim Grosser Tag der Disputation: 2. Mai 2018

Contents

1 Introduction 5

1.1 Maximization of relative performance . . . 6

1.2 Tax havens as Stackelberg followers . . . 7

1.3 Efficient taxation under perfect mobility . . . 8

2 Comparison of Nash and evolutionarily stable equilibrium in asymmetric tax competition 9 2.1 Introduction . . . 9

2.2 Nash equilibrium and evolutionarily stable tax rate in an abstract setting . 11 2.3 Application to capital tax competition . . . 15

2.3.1 Description of the model . . . 16

2.3.2 Quadratic production and linear utility function (Bucovetsky, 2009) 16 2.3.3 Two jurisdictions, linear utility and absentee capital owners (Laussel and Le Breton, 1998) . . . 18

2.4 Application to commodity tax competition . . . 20

2.5 Concluding remark . . . 20

3 Spiteful behavior can make everybody better off 21 4 Irrelevance of the strategic variable in the case of relative performance maximization 25 4.1 Introduction . . . 25

4.2 Games with two possibilities for the strategic variable . . . 26

4.3 Evolutionarily stable strategies . . . 26

4.4 Main results . . . 27

4.5 Relative performance maximization and aggregate-taking behavior . . . 29

4.6 Extension to asymmetric games . . . 31

4.7 Concluding remark . . . 31

5 Should profit shifting be prohibited? The importance of timing 33 5.1 Introduction . . . 33

5.2 Nash vs Stackelberg in the context of profit shifting . . . 34

5.3 Concluding remark . . . 35

6 Locational efficiency in a federal system without land rent taxation 37

6.1 Introduction . . . 37

6.2 The model . . . 38

6.3 First-order conditions for an efficient allocation . . . 38

6.4 Market behavior . . . 39

6.5 Efficient taxation . . . 40

6.5.1 Necessary and sufficient conditions . . . 40

6.5.2 Explicit computation of efficient tax rates in the case of two (types of) regions . . . 41

6.6 The case of more than two types of regions . . . 42

6.7 Concluding remark . . . 43

Appendix A: Proof of Proposition 6.1 . . . 43

Appendix B: Derivatives ofL with respect toτ_Kⁱ ,L_i and K_i . . . 44

7 Conclusion 47

Bibliography 49

Chapter 1

Introduction

In the tax competition literature it is usually assumed that governments choose their tax rates

• simultaneously

• and in such a way as to maximize their own payoffs (e.g., the utility of a represen- tative citizen).

It is then natural to study the Nash equilibrium of the tax competition game. However, these standard assumptions can be questioned in several ways:

1. Instead of maximizing their payoffs in the absolute sense, governments might try to maximize their relative performance compared to other jurisdictions. Intuitively, relative performance maximization makes tax competition more aggressive, and accordingly Sano (2012) and Wagener (2013) showed that in symmetric tax com- petition it leads to lower tax rates and lower welfare than the usual Nash behavior.

Moreover, Wagener (2016) showed that in contrast to the case of absolute per- formance maximization the equilibrium under relative performance maximization does not depend on whether tax rates or expenditure levels serve as the strategic variable. These results naturally lead to the following questions:

Question 1. Can the result that relative performance maximization leads to lower tax rates and lower welfare be extended to asymmetric tax com- petition?

Question 2. Does the (intuitively very plausible) result that relative per- formance maximization leads to lower (or equal) payoffs hold in complete generality, or are there counterexamples?

Question 3. Can the result that the equilibrium under relative per- formance maximization does not depend on the choice of the strategic variable be generalized?

2. In asymmetric situations the assumption of simultaneous action becomes less natu- ral. Small countries, in particular tax havens, are typically much more flexible than large countries, so that one might expect them to act as Stackelberg followers. It is then natural to investigate the influence of this timing issue on the desirability of public policies. In particular, we ask:

Question 4. In which way does the desirability of anti-profit-shifting measures (such as transfer pricing or thin-capitalization rules) depend on

whether tax havens act as Stackelberg followers or make their policy deci- sions simultaneously with normal countries?

3. Suppose that the population of a federation is homogeneous and perfectly mobile.

Then the utility levels in all regions necessarily coincide, so that local governments have a common objective and therefore a strong incentive to cooperate – even if they only care about the utility of their own inhabitants. Wildasin (1980) showed that in this case efficiency can be achieved by a tax system consisting of land and wage taxes. However, in view of the drawbacks of massive land rent taxation it is natural to ask:

Question 5. Can efficiency be achieved by taxing capital instead of land?

In the remaining part of this introductory chapter we discuss these issues in greater detail.

A full treatment will then be given in Chapters 2–6.¹

1.1 Maximization of relative performance

An obvious approach to formalize the idea of relative performance maximization is to consider the Nash equilibrium in a game whose payoff functions are utility differences.

This approach, however, is problematic because utility differences are not economically meaningful: if we subject a utility function to a strictly increasing transformation it still represents the same preference relation, but the utility differences we obtain after such a transformation can lead to a completely different Nash equilibrium.

One should therefore try to formalize the idea of relative performance maximization without using utility differences. In the context of theoretical biology Schaffer (1988) showed that this can be done – at least in symmetric situations – using the concept of an evolutionarily stable strategy.

To understand this concept in the context of tax competition consider a federal state consisting of several identical regions. Imagine that you are in charge of choosing the first region’s tax rate and that you want to maximize its relative performance. Suppose further that all other regions choose one and the same tax rate, sayt^E. Would you then chooset^E as well?

The answer depends on whether there exists a tax rate tsuch that

u₁(t, t^E, . . . , t^E)> u₂(t, t^E, . . . , t^E). (1.1)

• If you choose such a tax rate the first region will be better off than the other regions.

• If instead you choose the tax ratet^E, then, due to symmetry, all regions will perform equally well.

Hence if (1.1) holds and you want to maximize relative performance you would not chooset^E. These considerations motivate the following definition: A tax rate t^E is called evolutionarily stableif the case considered above does not occur, i.e., if for all tax rates t

u1(t, t^E, . . . , t^E)≤u2(t, t^E, . . . , t^E).

Sano (2012) and Wagener (2013) studied this concept in the context of the symmetric Zodrow-Mieszkowski (1986) capital tax competition model. One of their main results is that the evolutionarily stable tax rate t^E is strictly lower than t^N, the tax rate in

1In slightly modified form Chapters 2, 3, 5 and 6 have appeared in refereed journals, see Philipowski (2015, 2016a, 2016b, 2017).

symmetric Nash equilibrium, and that moreover u₁(t^E, . . . , t^E) < u₁(t^N, . . . , t^N). That is, under relative performance maximization both tax rates and utility levels are strictly lower than under absolute performance maximization. Moreover, Wagener (2016) showed that – contrary to the Nash equilibrium – the evolutionarily stable equilibrium does not depend on whether tax rates or expenditure levels serve as the strategic variable.

Unfortunately, Sano (2012) and Wagener (2013, 2016) considered identical regions only. Moreover, they obtained their findings only in the context of a specific model, so that it remained unclear whether or not they are special cases of more general results.

The purpose of Chapters 2–4 is to deal with these issues:

In Chapter 2 we show that the finding that relative performance maximization leads to lower tax rates and lower welfare can be extended to asymmetric tax competition.

Our result is not restricted to a particular model, but holds in considerable generality, requiring only a set of rather natural assumptions. Although these assumptions are not satisfied in all tax competition models, we show that they hold in the well-known models of Bucovetsky (2009), Laussel and Le Breton (1998) and Nielsen (2001).

It is then natural to ask whether relative performance maximization always leads to lower (or equal) payoffs than the usual Nash behavior. Quite surprisingly, this is not the case: In Chapter 3 we present examples of games admitting a unique Nash equilibrium and a unique evolutionarily stable strategy and such that the payoff in the evolutionarily stable equilibrium is strictly higher than the Nash payoff. We hence show that spiteful behavior can make everybody better off.

Unfortunately, our examples are not of immediate economic significance, so that it remains an open problem to either find economically relevant examples or to show that in all economically relevant situations relative performance maximization leads to lower (or equal) payoffs than the usual Nash behavior.

In Chapter 4 we study games with two possibilities for the strategic variable. Our motivation for this chapter mainly comes from Wildasin (1988, 1991) and Wagener (2016):

While Wildasin (1988, 1991) observed that the Nash equilibrium in tax competition can heavily depend on whether governments choose tax rates or expenditure levels as their strategic variable, Wagener (2016) showed that the evolutionarily stable equilibrium in the symmetric Zodrow-Mieszkowski (1986) model does not depend on this choice.

However, the analysis in Wagener’s paper strongly depends on modeling details, and consequently Wagener (2016, Section 4) himself wrote that his “results clearly do not imply that the choice of the policy variable never matters in evolutionary play for any type of fiscal interactions”. In contrast, we show that this finding is of a very general nature and does not depend on any modeling details.

1.2 Tax havens as Stackelberg followers

Profit shifting by multinational firms has become a serious problem. Governments are nevertheless not helpless. Effective measures against profit shifting include transfer pricing and thin-capitalization rules. In Germany, for example, the arm’s length principle is implemented in Section 1 AStG², and an interest cap is codified in Section 4h EStG³ and Section 8a KStG⁴.

However, while empirical research has shown that such measures are indeed effective, Langenmayr (2015) pointed out that they impose compliance costs even on firms that are not engaged in profit shifting. In view of these costs it is necessary to carefully deliberate on whether and under which conditions anti-profit-shifing measures are desirable.

2AStG = Außensteuergesetz = Foreign Tax Act

3EStG = Einkommensteuergesetz = Income Tax Act

4KStG = K¨orperschaftsteuergesetz = Corporation Tax Act

An aspect that has not yet been discussed in this context is the timing of decisions.

While most tax competition papers – including Langenmayr (2015) – assume that govern- ments take their decisions simultaneously, this assumption should not be taken for granted:

tax havens are typically much more flexible than normal countries and might therefore be expected to act as Stackelberg followers. It is therefore natural to study the influence of the timing of decisions on the desirability of anti-profit-shifting measures. In Chapter 5 we find that under the more realistic Stackelberg assumption the desirability of anti- profit-shifting measures is even more questionable than under the usual Nash assumption.

Our considerations are of a rather general nature, independent of specific assumptions concerning preferences, industry structure or other economic data.

1.3 Efficient taxation under perfect mobility

In Chapter 6 we consider a federal state whose population is homogenous and perfectly mobile within the federation. Under this assumption the sort of competition studied in the other chapters does not take place: perfect mobility implies that the utility level must be the same in all regions, and consequently all local governments have a strong incentive to cooperate – even if each of them only cares about the utility level in its own region.

In spite of this encouraging observation the question remains which kind of taxes are needed to obtain an efficient allocation. The standard view so far (Wildasin, 1980;

Wellisch, 2000, Chapter 2) has been that efficiency requires the use of both location-based taxes (e.g. head or wage taxes) and undistortive land taxes: the location-based taxes are used to internalize congestion costs, and the land taxes to finance the efficient level of public good provision. We show, however, that this view is not completely correct: If the federation consists of two types of regions such that the regions of each type are identical to each other efficiency can be obtained by taxing capital instead of land.

In the general case we study the resulting second-best problem and find that in the second-best optimum the level of public good provision is efficient, but neither capital nor labor is allocated efficiently.

Chapter 2

Comparison of Nash and

evolutionarily stable equilibrium in asymmetric tax competition

We show that under suitable assumptions the evolutionarily stable tax rate in asymmetric tax competition is strictly lower than all tax rates obtained in Nash equilibrium, generalizing in this way a result by Sano (2012) and Wagener (2013) obtained in the context of symmetric tax competition. Our assumptions are satisfied in several models of capital and commodity tax competition.

2.1 Introduction

In the tax competition literature it is usually assumed that governments try to maximize utility of their jurisdiction in the absolute sense. If moreover they determine their tax rates simultaneously, this assumption leads to the well-known concept ofNash equilibrium.

Recently, however, Sano (2012) and Wagener (2013) suggested that instead of maximizing the utility of their jurisdiction in the absolute sense, governments might try to maximize theirrelativeperformance compared to other jurisdictions. In symmetric tax competition this assumption leads to the following definition of anevolutionarily stabletax rate (Sano, 2012, Definition 1; Wagener, 2013, Definition 1): Denoting byu_i(t₁, . . . , t_n) the utility of jurisdictioniif the tax rates are t1, . . . , tn, a tax rate t^E is evolutionarily stable if for any tax ratet

u1(t, t^E, . . . , t^E)≤u2(t, t^E, . . . , t^E).

(Because of symmetry it suffices to consider the first and second jurisdictions.)¹ The inter- pretation of this notion is as follows: Suppose that all jurisdictions except the first one choose the tax ratet^E. Then the government of the first jurisdiction will only choose t^E as well if there is no tax ratet satisfying u₁(t, t^E, . . . , t^E) > u₂(t, t^E, . . . , t^E), because by setting such a tax rate the first jurisdiction would be better off than the other ones.

The assumption that governments try to maximize relative performance can be justified by the theory of yardstick competition (Besley and Case, 1995; Wrede, 2001). According to this theory voters observe policy outcomes in other jurisdictions and compare them to the situation at home. As a consequence, reelection probabilities of governments increase

1Note that this definition is a special case of the concept of a finite population evolutionarily stable strategy (ESS) introduced by Schaffer (1988) which, contrary to the usual ESS introduced by Maynard Smith and Price (1973), is not a Nash equilibrium in general. For background on evolutionary game theory we refer to the book by Vega-Redondo (1996).

with relative performance, so that for a government that wants to be reelected it is very natural to maximize relative performance.

Moreover, as shown by Sano (2012, Section 5) and Wagener (2013, Section 2.5), the evolutionarily stable tax rate does not only result from the attempt to maximize relative performance, but can also arise as the result of policy mimicking. Such a mimicking behavior has been observed on the local level in the USA (Ladd, 1992), Belgium (Heyndels and Vuchelen, 1998), England (Revelli, 2001), Spain (Sol´e Oll´e, 2003; Delgado and Mayor, 2011), Switzerland (Schaltegger, 2004) and the Netherlands (Allers and Elhorst, 2005).

Moreover, under the name “Open method of co-ordination” policy mimicking officially belongs to the means of government of the European Union, see e.g. Borr´as and Jacobsson (2004) or Kerber and Eckhardt (2007).

One of the main results of Sano and Wagener (Sano, 2012, Proposition 3; Wagener, 2013, Result 2) obtained in the context of the symmetric Zodrow-Mieszkowski (1986) capital tax competition model is the following: Under evolutionarily stable behavior both tax rate and welfare are strictly lower than under Nash behavior. Intuitively, this is due to the fact that relative performance maximization makes tax competition more aggressive.

The above considerations suggest that the concept of an evolutionarily stable tax rate is of greatest practical importance. Unfortunately, however, Sano (2012) and Wagener (2013) obtained their result only under the highly unrealistic assumption of identical jurisdictions. In view of the importance of asymmetries in tax competition (Bucovetsky, 1991; Wilson, 1991), the following two questions naturally arise:

Question 2.1. Does the concept of evolutionarily stable tax rate make sense also for non-identical jurisdictions?

Question 2.2. If yes, can Sano and Wagener’s result be extended to asymmetric tax competition, i.e. are tax rates and utilities still strictly lower under evolutionarily stable than under Nash behavior?

Concerning Question 2.1, one can easily see that – contrary to the concept of Nash equilibrium – the concept of evolutionarily stable tax rate does not make sense in complete generality. However, it does make sense if the following condition is satisfied: Whenever two jurisdictions choose the same tax rate, their utilities coincide,

t_i=t_j ⇒ u_i(t₁, . . . , t_n) =u_j(t₁, . . . , t_n). (2.1) Namely, in this case we define:

Definition 2.1. A tax ratet^E isevolutionarily stableif for alli∈ {1, . . . , n}, all tax rates t_i and some (and then all, because of (2.1))j 6=i,

ui(t^E, . . . , t^E, ti, t^E, . . . , t^E)≤uj(t^E, . . . , t^E, ti, t^E, . . . , t^E) (2.2) (where on both sides of the inequalityti appears at the i-th position).

The interpretation is the same as in the symmetric case: Suppose that all jurisdictions except thei-th one choose the tax ratet^E and that (2.2) does not hold. Then by choosing t^E as well, jurisdiction iwould obtain the same utility as all other jurisdictions (because of (2.1)), while by choosing a tax ratet_i that violates (2.2) jurisdictioniwould be better off than the other jurisdictions. Consequently, jurisdiction i will not choose t^E as well, and in this senset^E is not evolutionarily stable.

Of course (2.1) cannot be expected to hold in general, but in Section 2.3.1 we will see that it does hold in capital tax competition if jurisdictions may differ in size, but are otherwise equal, i.e. have the same production and utility function and the same capital

endowment per inhabitant. Since many federations consist of jurisdictions which are of very different sizes, but rather similar in other economic respects, requiring (2.1) instead of full symmetry considerably enlarges the scope of the concept of evolutionarily stable tax rate. Moreover, in Section 2.4 we will see that (2.1) also holds in Nielsen’s (2001) model of commodity tax competition in the context of cross-border shopping.

Concerning Question 2.2, we will give an affirmative answer under suitable assumptions and show that these assumptions are satisfied in several models of capital and commodity tax competition.

Before stating our main results, let us note that the definition of an evolutionarily stable tax rate can be reformulated as follows:

Lemma 2.1. A tax ratet^E is evolutionarily stable if and only if for all i∈ {1, . . . , n}and some (and then all)j 6=i

t^E ∈arg max

ti

(ui−uj)(t^E, . . . , t^E, ti, t^E, . . . , t^E). (2.3) Proof. Ift^E is evolutionarily stable, we have (u_i−u_j)(t^E, . . . , t^E, t_i, t^E, . . . , t^E)≤0 for all tax ratesti and, because of (2.1), (ui−uj)(t^E, . . . , t^E) = 0. Hence t^E satisfies (2.3).

If however t^E is not evolutionarily stable, there exists a tax rate ti satisfying (u_i−u_j)(t^E, . . . , t^E, t_i, t^E, . . . , t^E)>0, so that t^E does not satisfy (2.3).

Lemma 2.1 shows that an evolutionarily stable tax rate is the same as a symmetric Nash equilibrium in a game whose payoff functions areutility differences. Formalizing the idea of relative performance maximization in this way does not require condition (2.1), which, therefore, might seem to be dispensable. Some care should be taken, however:

1. If there are more than two jurisdictions, instead of maximizing ui −uj for some (arbitrary) j 6= i, it seems more reasonable to maximize the difference of ui and a weighted average of the values u_j, j6=i.

2. Since utility is an ordinal, not a cardinal concept, the economic meaning of utility differences is questionable. If we subject a utility function to a strictly increasing transformation, it still represents the same preference relation. The resulting utility differences after such a transformation, however, can lead to a completely different Nash equilibrium.

In spite of these objections, when (2.1) (or a weaker version, see Remark 2.2 below) does not hold (namely when jurisdictions differ in other economic aspects than population size), studying a game whose payoff functions are utility differences (or quotients) seems to be the only approach to formalize the idea of relative utility maximization. Here, however, we assume (2.1) and leave the study of situations where (2.1) does not hold to future research. In this thesis utility differences do not appear in definitions or theorems, so that their lack of economic significance is not problematic.

The chapter is organized as follows: In Section 2.2 we formulate general conditions which ensure that under evolutionarily stable behavior tax rates and utility levels of all jurisdictions are strictly lower than in Nash equilibrium. In Sections 2.3 and 2.4 we show that these conditions are satisfied in several models of capital and commodity tax competition. Finally, Section 2.5 concludes.

2.2 Nash equilibrium and evolutionarily stable tax rate in an abstract setting

In this section we formulate general assumptions which, as we will show, imply the follow- ing results:

1. There exists a unique Nash equilibrium (t^N₁ , . . . , t^N_n). Moreover, starting from any combination of tax rates, the usual tˆatonnement process converges to the Nash equilibrium. See Theorem 2.1 below.

2. If the tˆatonnement process is started at the vector (t^E, . . . , t^E), wheret^Eis any evolu- tionarily stable tax rate, tax rates and utilities are strictly increasing. Consequently, under evolutionarily stable behavior tax rates and utilities of all jurisdictions are strictly lower than in Nash equilibrium. See Theorem 2.2 and Corollary 2.1 below.

Our considerations are of a very general nature, not restricted to any particular model.

We consider a federation consisting ofn≥2 jurisdictions engaged in tax competition. Let u_i(t₁, . . . , t_n) be the utility of jurisdiction iif tax rates aret₁, . . . , t_n. We assume that the set of admissible tax rates isR+ for each jurisdiction.² In order to obtain a great level of generality, we leave the utility functions ui :Rⁿ₊ → R unspecified, but assume that they are twice differentiable, satisfy (2.1) and have the following properties:

Assumption 2.1. For eachi∈ {1, . . . , n} we have

∂²u_i

∂t²_i (t₁, . . . , t_n)<0

for all combinations of tax rates (t₁, . . . , t_n) at which the first-order condition

∂ui/∂ti(t1, . . . , tn) = 0 is satisfied.

This is a standard assumption. It ensures that for each combinationt−i = (t1, . . . , ti−1, t_i+1, . . . , t_n) of tax rates of the other (not i) jurisdictions the function t_i 7→ u_i(t_i, t−i) is strictly quasiconcave. Consequently, for eacht−ithere is at most one solutiont_iof the first- order condition∂ui/∂ti(ti, t−i) = 0, and this solution, if it exists, is the only maximizer of the functiont_i7→u_i(t_i, t−i).

Assumption 2.2. For each i ∈ {1, . . . , n} and all combinations t−i of tax rates of the other (not i) jurisdictions we have ∂ui/∂ti(0, t−i) > 0, and there exists a tax rate ti

satisfying∂ui/∂ti(ti, t−i)<0.

Together with Assumption 2.1, Assumption 2.2 ensures that for each t−i there exists a unique best response Ri(t−i), which moreover is strictly positive and therefore satisfies the first-order condition∂ui/∂ti(Ri(t−i), t−i) = 0. (Without Assumption 2.2, the function t_i 7→ u_i(t_i, t−i) might have no maximizer, or the maximizer might be on the boundary (i.e. be equal to 0) and not satisfy the usual first-order condition.) Economically, Assump- tion 2.2 is very natural: If a jurisdiction’s current tax rate is zero, this jurisdiction can increase its utility by choosing instead a positive tax rate, and if its tax rate is very high, the jurisdiction can increase its utility by reducing its tax rate.

Assumption 2.3. There exists a constantC <1 such that X

j6=i

∂²ui/∂ti∂tj

∂²u_i/∂t²_i (t1, . . . , tn)

≤C

for each i ∈ {1, . . . , n} and all t₁, . . . , t_n ≥ 0 at which the first-order condition

∂ui/∂ti(t1, . . . , tn) = 0 is satisfied.

2As will become clear from the proofs, our results also hold (with appropriately modified assumptions) if negative tax rates are allowed.

Since by the implicit function theorem

∂Ri

∂t_j =−∂²ui/∂ti∂tj

∂²u_i/∂t²_i ,

Assumption 2.3 ensures strong contractivity (w.r.t. to the maximum norm) of the function R:Rⁿ+ →Rⁿ+ obtained by patching together the various best response functions, so that Banach’s fixed point theorem implies the existence of a unique Nash equilibrium and moreover the convergence of the usual tˆatonnement process to it. See Theorem 2.1 and its proof for details. Economically, Assumption 2.3 means that jurisdictions’ reactions to changes of the other jurisdictions’ tax rates are not too strong. For example, ifn= 2, it means that the slopes of the reaction functions are uniformly less than 1.

Assumption 2.4. We have

∂²ui

∂t_i∂t_j(t1, . . . , tn)>0 (2.4) for alli 6=j and all (t1, . . . , tn) at which the first-order condition ∂ui/∂ti(t1, . . . , tn) = 0 is satisfied.

As the proof of Theorem 2.2 will show, this assumption is crucial. It is a weak form of strategic complementarity (or supermodularity).³ As noticed by (among others) Keen and Konrad (2013, p. 267), strategic complementarity of tax rates is difficult to establish theo- retically in most models of tax competition. However, empirical evidence (e.g. Brueckner and Saavedra, 2001; Rork, 2003; Allers and Elhorst, 2005) indicates that in reality the assumption of strategic complementarity is often satisfied.⁴

Assumption 2.5. For alli6=j we have

∂uj

∂t_i ≥0.

Moreover, the inequality is strict whenever all tax rates are equal and the first-order condition∂ui/∂ti = 0 is satisfied, i.e.

∂uj

∂ti

(t, . . . , t)>0 whenever∂u_i/∂t_i(t, . . . , t) = 0.

Assumption 2.5 essentially means that tax rates are plain complements. It is frequently satisfied in tax competition: By reducing its tax rate, a jurisdiction attracts capital from other jurisdictions, so that utility in the other jurisdictions decreases.⁵

We can now formulate our main results:

Theorem 2.1. Under Assumptions 2.1–2.3 there exists a unique Nash equilibrium (t^N₁ , . . . , t^N_n). Moreover it can be obtained as the result of the following tˆatonnement pro- cess: Starting from an arbitrary combinationt⁰ = (t⁰₁, . . . , t⁰_n) of tax rates in period 0, in

3Strategic complementarity in the usual sense would mean that (2.4) holds foralltax rates (t1, . . . , tn), not only those at which the first-order condition∂ui/∂ti(t1, . . . , tn) = 0 is satisfied. For background on the concept of strategic complementarity and its use in economics we refer to the survey papers by Amir (2005) and Vives (2005).

4But note that according to Parchet (2014) in Switzerland local tax rates are strategic substitutes in most cases. For a survey of the empirical literature on tax competition we refer to Devereux and Loretz (2013).

5There do however exist models where Assumption 2.5 does not necessary hold, e.g. capital tax com- petition with publicly provided inputs. See Zodrow and Mieszkowski (1986, Section III) and also Noiset (1995), who corrected a serious error of Zodrow and Mieszkowski. See also Remark 2.1 below.

each further periodk≥1 each jurisdictioni chooses its tax rate t^k_i as the best response to the tax rates t^k−1_−i of the other jurisdictions in the previous period: t^k_i = Ri(t^k−1_−i ). Then the sequence (t^k)k∈N0 = (t^k₁, . . . , t^k_n)k∈N0 defined by this procedure converges to the Nash equilibrium(t^N₁ , . . . , t^N_n) as k→ ∞.

Theorem 2.2. Suppose that Assumptions 2.1–2.5 are satisfied, that there exists an evo- lutionarily stable tax rate t^E and that the tˆatonnement process described in Theorem 2.1 starts from the vector (t^E, . . . , t^E). Then for each i ∈ {1, . . . , n} the sequences (t^k_i)k∈N0

and(u_i(t^k₁, . . . , t^k_n))k∈N0 are strictly increasing.⁶

Corollary 2.1. If Assumptions 2.1–2.5 are satisfied and there exists an evolutionarily stable tax rate t^E, we have

t^E < t^N_i and

ui(t^E, . . . , t^E)< ui(t^N₁ , . . . , t^N_n)

for alli∈ {1, . . . , n}, i.e. under evolutionarily stable behavior tax rates and utilities of all jurisdictions are strictly lower than in Nash equilibrium.

Proof of Theorem 2.1. Fixi∈ {1, . . . , n}for the moment. As already mentioned, Assump- tions 2.1 and 2.2 guarantee that for all combinationst−i of tax rates of the other (noti) jurisdictions there exists exactly one best response R_i(t−i), which moreover is strictly positive and therefore satisfies the first-order condition ∂u_i/∂t_i(R_i(t−i), t−i) = 0. Con- sequently, the implicit function theorem implies that

∂R_i

∂tj

(t−i) =−∂²u_i/∂t_i∂t_j

∂²ui/∂t²_i (R_i(t−i), t−i) (2.5) fori6=j. We now patch the various best response functions together to obtain a function R:Rⁿ+→Rⁿ+. Clearly, a combination (t₁, . . . , t_n) of tax rates is a Nash equilibrium if and only if it is a fixed point ofR. Since Ri does not depend on ti, we have ∂Ri/∂ti = 0, so that (2.5) together with Assumption 2.3 implies that

n

X

j=1

∂Ri

∂t_j

≤C <1

and hence|R(t)−R(t⁰)|_∞ ≤ C|t−t⁰|_∞ for all t,t⁰ ∈ Rⁿ+, i.e. R is a strong contraction with respect to the distanced(t,t⁰) := |t−t⁰|_∞.⁷ The result now follows from Banach’s fixed point theorem.

Proof of Theorem 2.2. We first show by induction that for each i ∈ {1, . . . , n} and all k≥1 we have

t^k_i > t^k−1_i . (2.6)

k= 1: If t^E >0, (2.3) implies that t^E satisfies the first-order condition

∂ui

∂t_i(t^E, . . . , t^E) = ∂uj

∂t_i(t^E, . . . , t^E) (j6=i), (2.7)

6We have not been able to prove the existence of an evolutionarily stable tax rate using our set of assumptions. As the proof of Sano (2012, Proposition 3) shows, this problem is far from trivial. In the examples presented in Sections 2.3 and 2.4 however, existence and uniqueness of an evolutionarily stable strategy can be obtained by elementary explicit computations.

7As usual,|x|∞:= maxⁿ_i=1|xi|denotes the maximum norm of a vectorx∈Rⁿ.

which together with Assumption 2.5 implies that

∂ui

∂t_i(t^E, . . . , t^E)>0. (2.8) If t^E = 0, (2.7) need not hold, but in this case (2.8) follows immediately from Assump- tion 2.2. In both cases, (2.8) implies that t^E is on the increasing branch of the function t_i 7→u_i(t^E, . . . , t^E, t_i, t^E, . . . , t^E), so that t¹_i > t^E =t⁰_i.

k−1→k, k≥2: Since by (2.5) and Assumption 2.4 ∂Ri/∂tj > 0 for all j 6= i, the induction hypothesis implies that

t^k_i =Ri(t^k−1₁ , . . . , t^k−1_n )> Ri(t^k−2₁ , . . . , t^k−2_n ) =t^k−1_i .

We now use (2.6), Assumption 2.5 and the fact thatt^k_i is the unique best response tot^k−1_−i to obtain

u_i(t^k₁, . . . , t^k_n) ≥ u_i(t^k_i, t^k−1_−i )

> ui(t^k−1₁ , . . . , t^k−1_n ) for alli∈ {1, . . . , n}and all k≥1.

Remark 2.1. The proof of Theorem 2.2 shows that if the inequalities in Assumption 2.5 are reversed, the sequences (t^k_i)k∈N0 are strictly decreasing, while the sequences (ui(t^k₁, . . . , t^k_n))k∈N0 are still strictly increasing. As a consequence, in this case we have t^E > t^N_i and u_i(t^E, . . . , t^E) < u_i(t^N₁ , . . . , t^N_n) for all i∈ {1, . . . , n}, i.e. the evolutionarily stable tax rate is strictly higher than all Nash tax rates, while utilities of all jurisdictions are strictly lower in evolutionary than in Nash equilibrium.

If however the inequality in Assunption 2.4 is reversed (i.e. if tax rates are strategic substitutes), monotonicity of the sequences (t^k_i)k∈N0 and (u_i(t^k₁, . . . , t^k_n))k∈N0 breaks down.

Remark 2.2. In the spirit of Tanaka (1999), Definition 2.1 can be generalized to situa- tions where (2.1) is replaced with the following weaker condition: The jurisdictions can be arranged intokgroups (each of which contains at least two jurisdictions) such that when- ever two jurisdictionsof the same group choose the same tax rate, their utilities coincide.

In this case it is reasonable to assume that local governments are mainly interested in their relative performance compared to other jurisdictionsbelonging to the same group. Con- sequently, instead of just one evolutionarily stable tax ratet^E, this generalization involves different tax rates for the various groups or, in other words, the notion of anevolutionarily stable vector(t^E₁, . . . , t^E_k).

One can easily check that all our arguments work in this more general context as well (requiring much more cumbersome notation, however). In particular, given Assumptions 2.1–2.5, it is still true that under under evolutionarily stable behavior tax rates and utilities of all jurisdictions are strictly lower than in Nash equilibrium.

2.3 Application to capital tax competition

In this section we apply our results of Section 2.2 to the well-known Zodrow-Mieszkowski (1986) capital tax competition model, assuming that jurisdictions may differ in size, but are otherwise equal (i.e. have the same production and utility function and the same capital endowment per inhabitant). While in general Assumptions 2.1–2.5 are not satisfied in this model, we will show that they are satisfied in the following special cases:

1. The production function is quadratic and utility is linear, as in Bucovetsky (2009).

2. The production function is rather general (with f⁰⁰ < 0 and f⁰⁰⁰ ≥ 0), but there are only two jurisdictions, capital owners are absent, and utility is just the sum of private income and tax revenue. This variant of the Zodrow-Mieszkowski model was introduced by Laussel and Le Breton (1998).

2.3.1 Description of the model

Since the Zodrow-Mieszkowski model is well-known, our description here is rather brief.

For more details we refer to the survey by Keen and Konrad (2013, Section 2.1.1).

We assume that the utility of the residents of jurisdiction iis given by

u_i=U(c_i, g_i) (2.9)

wherec_i is private income per capita andg_i is tax revenue per capita from a source-based capital tax. Each individual in each jurisdiction supplies one unit of labor inelastically.

Output is produced from this labor, and from capital, using a production function f. Capital is perfectly mobile among jurisdictions. It earns a gross return in each jurisdiction equal to its marginal productf⁰(k_i), but this return is subject to a source-based unit taxt_i. Perfect mobility of capital implies that its net return is the same in all jurisdictions. That net return equals

ρ=f⁰(k_i)−t_i. (2.10)

The population of jurisdictions may differ. Lets_i be the share of the federation’s popula- tion that lives in in jurisdiction i (so that si >0 for all i and Pn

i=1si = 1). We assume that each jurisdiction has the same endowment of capital per capita, ¯k, and that capital can neither flow into nor out of the federation, so that

n

X

i=1

siki = ¯k. (2.11)

Private income is the sum of labor income f(ki)−f⁰(ki)ki and a fraction θ ∈ [0,1]⁸ of capital incomeρk, i.e.¯

c_i =f(k_i)−f⁰(k_i)k_i+θρ¯k, (2.12) and tax revenue equals

gi =tiki. (2.13)

Note that ifti =tj, it follows thatf⁰(ki) =ρ+ti=ρ+tj =f⁰(kj) and henceui =uj, so that (2.1) is satisfied.

2.3.2 Quadratic production and linear utility function (Bucovetsky, 2009) In this subsection we assume that the production functionf is quadratic,

f(k) =ak− b

2k², (2.14)

and that the utility functionU is linear,

U(c, g) =c+ (1 +ε)g, (2.15)

whereε≥0 is a fixed parameter.⁹

8As explained by Wildasin (1988, p. 232) and Laussel and Le Breton (1998, p. 285), assumingθ <1 is justified if either a part of the capital income accrues to people not present elsewhere in the model, or if fiscal policy focuses on the median voter (whose capital income is typically lower than the average capital income). See also Kempf and Rota-Graziosi (2010, p. 770) and Taugourdeau and Ziad (2011, p. 440).

9As pointed out by Bucovetsky (2009, p. 730), “1 +εrepresents the marginal cost of public funds”.

The quadratic form (2.14) of the production function allows to compute the interest rateρ and the distribution of capital explicitly:

ρ = a−bk¯−t,¯ k_i = ¯k+t¯−t_i

b , where ¯t := Pn

i=1siti is the average tax rate in the federation. Combining (2.9), (2.15), (2.12) and (2.13) we obtain

ui = b

2k_i²+θρk¯+ (1 +ε)tiki. Elementary calculations now imply that

∂ui

∂t_i = [ε+ (1−θ)si]¯k+ (ε+si)

b (¯t−ti)−(1 +ε)(1−si)

b ti,

which is strictly positive for ti = 0 (unlessε= 0, θ= 1 and all tax rates vanish) and negative for sufficiently large values of ti,

∂²u_i

∂t²_i = −(1 + 2ε+s_i)(1−s_i)

b <0,

∂²u_i

∂ti∂tj

= (ε+s_i)s_j

b >0 ifi6=j, X

j6=i

∂²u_i/∂t_i∂t_j

∂²ui/∂t²_i

= X

j6=i

(ε+s_i)s_j (1 + 2ε+si)(1−si)

= ε+si

1 + 2ε+s_i ≤ 1 2,

∂uj

∂t_i = s_i(1−θ)¯k+si(¯t+εtj)

b ≥0 ifi6=j,

and this inequality is strict unless θ= 1 and all tax rates vanish.

Hence, noting that∂ui/∂ti(0, . . . ,0) = (ε+ (1−θ)si)¯k, Assumptions 2.1–2.5 are satisfied unlessε= 0 and θ= 1.

Remark 2.3. In this variant of the Zodrow-Mieszkowski model one can compute Nash equilibrium and evolutionarily stable tax rate explicitly: Let

Z :=

n

X

i=1

s_i[ε+ (1−θ)s_i] 1 + 2ε−εsi

! , _n X

i=1

s_i(1−s_i) 1 + 2ε−εsi

! .

(Note that in the caseθ = 1 our constant Z is related to Bucovetsky’s (2009, Section 3)

“index of tax competition” ζ by the formula Z = (1 +ε)ζ.) Then by elementary com- putations (similar to those in Bucovetsky, 2009, p. 733) one obtains that

t^N_i = b¯k 1 +ε

(1 +ε)[ε+ (1−θ)s_i] +Z(ε+s_i) 1 + 2ε−εsi

and

t^E = bεk¯ 1 +ε.

Since Z > ε (unless ε= 0 and θ = 1) it follows that t^E < t^N_i for all i∈ {1, . . . , n}, and this confirms our general result obtained in Section 2.2. If howeverε = 0 andθ = 1, we

obtain t^E =t^N_i = 0 for all i∈ {1, . . . , n}, i.e. in this case evolutionarily stable and Nash equilibrium coincide.

Note also that in the competitive limit (i.e. maxⁿ_i=1s_i → 0 as n → ∞) we obtain that Z → ε and consequently max_i=1,...,n|t^N_i −t^E| → 0. That is, in the competitive limit Nash and evolutionarily stable tax rate coincide. This finding extends Sano (2012, Proposition 2) and Wagener (2013, Result 2) in two ways: to asymmetric tax competition and to the caseθ <1.

2.3.3 Two jurisdictions, linear utility and absentee capital owners (Laus- sel and Le Breton, 1998)

In this subsection we assume that there are only two jurisdictions, that θ = 0 and that utility is just the sum of private income and tax revenue, U(c, g) = c+g. In contrast, the production functionf is no longer assumed to be quadratic; instead we assume that f isC³ and satisfiesf⁰>0,f⁰⁰ <0 andf⁰⁰⁰ ≥0.

Our calculations are similar to those of Laussel and Le Breton (1998, Section 3) and Kempf and Rota-Graziosi (2010, Appendix A.1), the difference being that these authors restrict themselves to the case s1 = s2 = 1/2. In the whole subsection we use the sub- scriptj to denote the other (noti) jurisdiction.

From (2.10) and (2.11) it follows that

∂ρ

∂ti

= − sif⁰⁰(kj)

sif⁰⁰(kj) +sjf⁰⁰(ki) <0,

∂k_i

∂t_i = s_j

s_if⁰⁰(k_j) +s_jf⁰⁰(k_i) <0,

∂kj

∂ti

= − si

sif⁰⁰(kj) +sjf⁰⁰(ki) >0, j6=i.

Combining (2.9), (2.12) and (2.13) we obtain

ui =f(ki)−f⁰(ki)ki+tiki, so that

∂u_i

∂ti

= −f⁰⁰(k_i)k_i∂k_i

∂ti

+k_i+t_i∂k_i

∂ti

=

ti−f⁰⁰(ki)ki+ki(sif⁰⁰(kj) +sjf⁰⁰(ki)) s_j

∂ki

∂t_i

=

ti+ s_i sj

kif⁰⁰(kj) ∂k_i

∂ti

(2.16)

= Φi(ti, tj)∂ki

∂t_i, where

Φi(ti, tj) :=ti+ si

s_jkif⁰⁰(kj).

Clearly, Φi(0, tj)<0, and Φi(ti, tj)>0 for sufficiently large values ofti. Moreover, since

∂Φi

∂ti

= 1 + si

sj

f⁰⁰(kj)∂ki

∂ti

+ si

sj

kif⁰⁰⁰(kj)∂k_j

∂ti

= 1 + s_i

s_jf⁰⁰(k_j)−s²_i

s²_jk_if⁰⁰⁰(k_j)

!∂k_i

∂t_i

> 1

and

∂Φ_i

∂tj

= s_j sj

f⁰⁰(k_j)∂k_i

∂tj

+ s_i sj

k_if⁰⁰⁰(k_j)∂k_j

∂tj

= −si

s_jf⁰⁰(k_j) +s²_i

s²_jk_if⁰⁰⁰(k_j)

!∂ki

∂t_i

< 0, it follows that

∂²u_i

∂t²_i = ∂Φ_i

∂ti

∂k_i

∂ti

+ Φ_i∂²k_i

∂t²_i

= ∂Φ_i

∂ti

∂k_i

∂ti

< 0 and

∂²ui

∂t_i∂t_j = ∂Φi

∂t_j

∂ki

∂t_i + Φi

∂²ki

∂t_i∂k_j

= ∂Φ_i

∂tj

∂k_i

∂ti

> 0

whenever the first-order condition∂u_i/∂t_i= 0 (and hence Φ_i= 0) is satisfied. Moreover,

∂²u_i/∂t_i∂t_j

∂²u_i/∂t²_i = ∂Φ_i/∂t_j

∂Φi/∂ti

=

−^s_sⁱ

jf⁰⁰(kj) +^s_s²ⁱ2 j

kif⁰⁰⁰(kj)

∂ki

∂ti

1 +

sj

sjf⁰⁰(kj)−^s_s²ⁱ2

jkif⁰⁰⁰(kj)

∂ki

∂ti

,

so that

∂²ui/∂ti∂tj

∂²u_i/∂t²_i

≤ A

1 +A <1, where

A:= sup

ki,kj

−si

s_jf⁰⁰(k_j) +s²_i

s²_jk_if⁰⁰⁰(k_j)

! sj

s_if⁰⁰(k_j) +s_jf⁰⁰(k_i)

is finite because the supremum is taken over the compact set{(k_i, kj)∈R²+|s_iki+sjkj = ¯k}.

Finally,

∂u_j

∂ti

=

t_j −f⁰⁰(k_j)k_j∂k_j

∂ti

(2.17)

> 0.

Hence Assumptions 2.1–2.5 are satisfied.

Remark 2.4. In this variant of the Zodrow-Mieszkowski model one cannot compute the Nash equilibrium explicitly (unless s1 = s2 = 1/2). However, from (2.16) and (2.17) it follows thatt^N_i = −_s^si

jk_if⁰⁰(k_j) >0 and t^E = 0, which confirms again our general result obtained in Section 2.2.

2.4 Application to commodity tax competition

In this section we apply our results of Section 2.2 to Nielsen’s (2001) model of commodity tax competition in the context of cross-border shopping. In this model there are two coun- tries which are together represented by the interval [−1,1]. Population density equals 1 in both countries. The first country extends from−1 to some border parameter b∈(−1,1), while the second country extends fromb to 1. Lett_i be the commodity tax rate in coun- try i. Inhabitants of each country purchase goods at home or abroad depending on the difference in tax rates and on their respective transportation cost. Transport cost per unit is denoted byd >0. As shown by Nielsen (2001, Eq. (2)), these specifications imply that per capita tax revenues in the two countries equal

u1(t1, t2) =t1

1 + t2−t1

(1 +b)d

, u2(t1, t2) =t2

1 + t1−t2

(1−b)d

. Clearly, ift₁=t₂, we have u₁=u₂, so that (2.1) is satisfied. Moreover,

∂u1

∂t₁ = 1 + t2

(1 +b)d− 2t1

(1 +b)d,

which is strictly positive for t1 = 0

and negative for sufficiently large values of t₁,

∂²u₁

∂t²₁ = − 2

(1 +b)d<0,

∂²u₁

∂t₁∂t₂ = 1

(1 +b)d >0,

∂²u₁/∂t₁∂t₂

∂²u1/∂t²₁

= 1

2,

∂u2

∂t₁ = t2

(1−b)d ≥0,

and this inequality is strict unlesst₂ = 0.

By symmetry, analogous formulae hold if we interchange the indices 1 and 2. Hence, taking into account that∂u1/∂t1(0,0) = 1>0, Assumptions 2.1–2.5 are satisfied.

Remark 2.5. We can compute Nash equilibrium and evolutionarily stable tax rate explicitly:

t^N₁ =d+bd/3, t^N₂ =d−bd/3, t^E = (1 +b)(1−b)d

2 .

Since t^N_i > 2d/3 and t^E ≤ d/2 it follows that t^E < t^N_i for all i ∈ {1, . . . , n}, and this confirms again our general result obtained in Section 2.2.

2.5 Concluding remark

Sano and Wagener’s result that tax rate and welfare are strictly lower under evolutionarily stable than under Nash behavior extends to the much more realistic case of asymmetric tax competition, provided that some natural assumptions are satisfied. We have shown that these assumptions are satisfied in several models of capital and commodity tax com- petition. Further research should either

• show that our result also holds in greater generality,

• or exhibit situations where it does not hold.¹⁰

10The examples presented in Chapter 3 do not seem to allow for a sensible interpretation in terms of tax competition.

Chapter 3

Spiteful behavior can make everybody better off

We present examples of symmetric two-player games admitting a unique Nash equilibrium and a unique evolutionarily stable strategy (ESS) and such that the ESS payoff is strictly higher than the Nash payoff. In this sense we show that spiteful behavior can make everybody better off.

In economics one usually assumes that agents (households, firms, local governments, etc.) behaveegoisticallyin the sense that they try to maximize their own payoff (utility, profit, social welfare, etc.) in the absolute sense, i.e. without caring about the payoffs of the others. If moreover they choose their strategic variables simultaneously, this assumption leads to the well-known concept ofNash equilibrium.

However, as argued by Schaffer (1989) and many subsequent authors (e.g. Al´os-Ferrer and Ania, 2005, and the references therein), in certain contexts it seems reasonable that instead of maximizing their payoff in the absolute sense, agents care about their relative performance compared to their peers. In symmetric games the appropriate equilibrium concept for this kind of behavior is that of afinite population evolutionarily stable strategy (ESS) introduced by Schaffer (1988):¹

Definition 3.1. Let (X, u) be a symmetricn-player game, whereX is the strategy set of each player and u :Xⁿ → Rⁿ the payoff function, which is assumed to be symmetric in the sense thatu_i(s_π(1), . . . , s_π(n)) =u_π(i)(s₁, . . . , s_n) for alli∈ {1, . . . n}, all permutations πof{1, . . . , n}and all strategy combinations (s1, . . . , sn)∈Xⁿ. We then say that a pure² strategys^E ∈X isevolutionarily stable if for everys∈X

u₁(s, s^E, . . . , s^E)≤u₂(s, s^E, . . . , s^E).

The interpretation of this notion is as follows: Suppose that all players except the first one choose the strategy s^E. Then the first player will only choose s^E as well if there is no strategys satisfying u₁(s, s^E, . . . , s^E)> u₂(s, s^E, . . . , s^E), because by choosing such a strategy the first player would be better off than the others.

By Lemma 2.1 a strategys^E is evolutionarily stable if and only if s^E ∈arg max

s∈X

(u1−u2)(s, s^E, . . . , s^E). (3.1) Hence, ESS strategists behavespitefullyin the sense that they do not maximize their own payoff, but the difference between own and other players’ payoffs.

1The original ESS concept introduced by Maynard Smith and Price (1973) is only suitable for infinite populations.

2The definition can be extended to mixed strategies in the obvious way.