1 Tax competition in the presence of trade unions in each country

Technical Appendix to:

Do trade unions trigger or relax tax competition?

Nelly Exbrayat^y, Carl Gaigné^z: and Stéphane Riou^x{

September, 2011

Abstract

This …le contains robustness checks regarding the timing of events, that we left out of the submitted version of our paper because of space constraints.

Contents

1 Tax competition in the presence of trade unions in each country 2 1.1 Trade unions act as Stackelberg leaders . . . . 2 1.2 Trade unions and governments play Nash . . . . 4 2 Tax competition between a unionized and a non-unionized coun-

try 5

2.1 Trade union acts as a Stackelberg leader . . . . 5 2.2 Trade union and governments play Nash . . . . 7

Available on the author’s homepage.

yUMR CNRS GATE Lyon-Saint-Etienne, Jean Monnet University, Saint-Etienne.

zINRA, UMR1302, SMART, 4, allée Bobierre, F-35000 Rennes.

xUMR CNRS GATE Lyon-Saint-Etienne, Jean Monnet University, Saint-Etienne.

{Corresponding author. Stéphane Riou. E-mail: stephane.riou@univ-st-etienne.fr. Ad- dress: GATE Lyon-Saint-Etienne, 6 rue basse des rives 42023 Saint-Etienne, cedex 02, France.

1 Tax competition in the presence of trade unions in each country

In this section, we assume that labor markets are unionized in each country i = H; F, and the trade union in country F has an identical or a stronger preference for wages than the trade union of country H ( _{H F}). The model is a four-stage game with perfect information, where the players are workers, …rms, trade unions and governments. In the paper, we consider that governments act as Stackelberg leaders (stage 1), before trade unions set wages (stage 2), capital owners choose the location of their capital investment (stage 3) and …rms and residents make their respective production, consumption and labor allocation choices (stage 4). In this appendix, we consider two alternative con…gurations: trade unions act as Stackelberg leaders (section 1.1), or trade unions and governments play Nash (section 1.2). The game is solved by a sub- game perfect Nash equilibrium, involving backward induction beginning with the last stage. We refer the reader to the paper for the resolution of stages 3 and 4, and we present the outcome regarding stages 1 and 2 to show the robustness of our results (Lemma 1, Propositions 1, 2 and 5).

1.1 Trade unions act as Stackelberg leaders

Equilibrium taxes (stage 2) Solving the …rst-order condition for each government, we obtain the following Nash tax in countryH:

t^uu_H = ~t^cc+1

4fw_F(2n+ 3) w_H(10n+ 13)

4n+ 5 (1)

and a symmetric expression holds for countryF.

Equilibrium wage rates (stage 1) We obtain the following equilibrium wage rate in countryH:

~

w^uu_H = ~w^cc+2ln ² _H(4n+ 5) + 2ln ² _{H F}(4n+ 5) f(n+ 1)²(1 _{H F})

and a symmetric expression holds for countryF. We easily verify thatdw~^uu_i =d >

0for i=H; F, anddw~_F^uu=d > dw~^uu_H=d >0 for all _F > _H. Thus,Lemma 1remains valid when trade unions act as Stackelberg leaders.

Incorporating these equilibrium wages in (1), we obtain the expression of capital taxes at the subgame perfect Nash equilibrium (hereafter, SPNE):

t~^uu_H = ~t^cc f

2w~^cc+ _H( _H; _F; ) andt~^uu_F = ~t^cc f

2w~^cc+ _F( _H; _F; ) with

H( _H; _F; ) = 1

2ln ^{2 F}(2n+ 3) _H(10n+ 13) 2 _{H F}(4n+ 5) (n+ 1)²(1 _{H F})

F( _H; _F; ) = 1

2ln ^{2 F}(10n+ 13) _H(2n+ 3) + 2 _{H F}(4n+ 5) (n+ 1)²(1 _{H F})

Clearly, capital taxes in the presence of trade unions are lower than~t^ccwhen trade unions share identical preferences across countries ( _H = _F). This is no longer the case when trade unions di¤er with respect to their preference for wages. Assuming, for example, that _F > _H, we verify that:

~t^uu_F < t~^ccfor all _F

~t^uu_H > t~^ccwhen _H < 2n+ 3

10n+ 13 + 2 _F(4n+ 5) ^F ~ andw~^cc< 2

f ^H( _H; _F; )

~t^uu_H < t~^ccwhen _H >~ 2(0; _F):

Thus, we obtain qualitatively similar results as those described inProposition 1.

Moreover, observe that the impact of trade integration on capital taxes is qualitatively similar. Precisely, trade costs in‡uence taxes identically and in a non-monotonic way in both countries, through the term~t^cc. They also in‡uence capital taxes by modifying the labor market characteristics. The latter e¤ect is captured by the term i( _H; _F; ), that is always negative in countryF and is respectively positive or negative in countryH when _H7 F = ~.

Importantly, we also verify that trade liberalization reduces the capital tax and wage di¤erentials at the SPNE:

~t^uu_H ~t^uu_F = 2ln ² (3n+ 4) ( _{F H})

(n+ 1)²(1 _{F H}) andw~_H^uu w~^uu_F = 2ln ² (4n+ 5) ( _{F H}) f(n+ 1)²( _{F H} 1) with~t^uu_H >~t^uu_F andw~^uu_H <w~^uu_F for all >0and0< _H< _F <1:

Inserting equilibrium taxes and wages in the location equilibrium, we obtain:

~^uu= 1 2

1 2

H F

1 _{H F} 2(1=2;1) for all0< _H< _F <1 Thus,Proposition 2 still holds.

Evaluating the national welfare at the SPNE when trade unions share iden- tical preferences in both countries, we obtain:

W~_i^uu= ~W_i^cc 1

2ln^{2 2} 4n+ 5

(n+ 1)²(1 ) <W~_i^cc

When _F > _H, the welfare di¤erential between countryHand countryFreads W~_H^uu W~_F^uu= 1

2ln^{2 2}( _{F H}) 4n+ 5

(n+ 1)²(1 _{F H}) >0 Consequently,proposition 5 still holds.

1.2 Trade unions and governments play Nash

By solving the maximization programs of monopoly unions and governments, we obtain the following equilibrium taxes:

t~^uu_H = ~t^cc f

2w~^cc+ H( _H; _F; ) andt~^uu_F = ~t^cc f

2w~^cc+ F( _H; _F; ) with

H( _H; _F; ) = ln ² 2 (n+ 1)

F(2 _H+ 1) (2n+ 3) _H(10n+ 13) 4n+ 5 (3n+ 4) ( _H+ _F) + _{H F}(2n+ 3)

F( _H; _F; ) = ln ² 2 (n+ 1)

H(2 _F+ 1) (2n+ 3) _F(10n+ 13) 4n+ 5 (3n+ 4) ( _H+ _F) + _{H F}(2n+ 3) When _H = _F, we verify that ~t^uu_i < t~^cc 8 i = H; F. Assuming that preferences for wages are higher in countryF ( _F > _H), we obtain:

~t^uu_F < t~^ccfor all _H

~t^uu_H > t~^ccwhen _F > ^H 2 _H+ 1

10n+ 13

2n+ 3 ^ andw~^cc< 2

f ^H( _H; _F; ) and~t^uu_H < t~^ccwhen _F <^

Results are thus qualitatively similar to those described in Proposition 1.

Moreover, observe that the impact of trade integration on capital taxes is qual- itatively similar to its impact when monopoly unions act as Stackelberg.

Importantly, we also con…rm that trade liberalization reduces the business tax and wage di¤erentials at the SPNE. Indeed, the tax di¤erential is described as follows:

t~^uu_H ~t^uu_F = ln ² 2 (n+ 1)

4 (3n+ 4) ( _{F H})

4n+ 5 (3n+ 4) ( _H+ _F) + _{H F}(2n+ 3) with ~t^uu_H > ~t^uu_F for all > 0 and 0 < _H < _F < 1:Besides, the equilibrium wage in a countryH reads:

~

w_H^uu= ~w^cc 2ln ² _H ^F(2n+ 3) (4n+ 5)

f(n+ 1) (4n+ 5 (3n+ 4) ( _H+ _F) + _{H F}(2n+ 3)) (2) and a symmetric expressions holds for the equilibrium wage in country F. Again,Lemma 1remains valid as dw~^uu_i =d >0 fori=H; F and dw~^uu_F =d >

dw~_H^uu=d >0for all _F > _H. We resulting wage di¤erential is:

~

w_H^uu w~^uu_F = 2ln ²(4n+ 5) ( _{H F})

f(n+ 1) (4n+ 5 (3n+ 4) ( _H+ _F) + _{H F}(2n+ 3)) withw~_H^uu<w~_F^uu for all >0 and0< _H < _F <1.

Finally, the location equilibrium at the SPNE is given by:

~^uu=1 2

1

2(n+ 1) ^{H F}

4n+ 5 (3n+ 4) ( _H+ _F) + _{H F}(2n+ 3) with~^uu2(1=2;1)for all0< _H< _F <1. Thus,Proposition 2 still holds.

Substituting all the welfare variables at their equilibrium values in (19) for symmetric preferences and comparing the resulting expression with the national welfare when both countries adopt competitive labor markets, we get:

W~_i^uu= ~W_i^cc 1 2ln^{2 2}

(1 ) (n+ 1) <W~_i^cc Moreover, the welfare di¤erential for _F > _H is given by

W~_H^uu W~_F^uu= ln^{2 2}( _{F H})

2 (n+ 1) (4n 4 _F 4 _H 3n _F 3n _H+ 3 _{F H}+ 2n _{F H}+ 5)² with = 4 3 _{F H} ²_H+ _F ²_H+ 4 n²

+ 12 _F ²_H 9 _H 12 ²_H 31 _F + 40 n+ 9 _F ²_H 5 _H 9 ²_H 20 _F + 25 . Using the fact that parameters _H and _F lie in the interval (0;1), simu- lations show that each term in bracket is positive. Hence, is positive for all admissible values of trade union preferences. We can conclude thatW~_H^uu W~_F^uu is also positive for all0< _H< _F <1. Therefore,Proposition 5 still holds.

2 Tax competition between a unionized and a non-unionized country

In this section, we assume that the labor market is competitive in countryHand the national wage is set by a monopoly union in countryF. Moreover, we con- sider two sequences of events: the trade union in countryFacts as a Stackelberg leader (section 2.1), or the trade union and governments play Nash (section 2.2).

The game is solved by a sub-game perfect Nash equilibrium, involving backward induction beginning with the last stage. We present the results regarding stages 1 and 2 to show the robustness of our results (Propositions 3 and 4) and refer the reader to the paper for the resolution of stages 3 and 4.

2.1 Trade union acts as a Stackelberg leader

Equilibrium taxes (stage 2) Solving the …rst-order condition for each government, we obtain the Nash tax equilibrium:

~t^uu_H = ~t^cc+ _H(w^cu_F; ) andt~^uu_F = ~t^cc+ _F(w_F^cu; )

with

H(w^cu_F; ) = l ²f(2n+ 3) (lw^cu_F f n) 4 (4n+ 5)l^{2 2}+f² (n+ 1)²

F(w^cu_F; ) = f 4l

(1 +n)²f²( nf 3lw^cu_F) +l^{2 2}[(2n+ 3) nf (10n+ 13)lw^cu_F] f²(1 +n)²+l^{2 2}(4n+ 5)

We verify thatt^cu_F <~t^cc as long asw_F^cu>w~^cc. In other words, a shift from a ‡exible labor market to a unionized labor market in a single country (here countryF) lowers the capital tax rate in this country. By contrast,~t^cu_H ?~t^cc if and only ifw^cu_F ?2 ~w^cc. Finally, we verify that if the wage rate in country F is set above the competitive wage rate (w^cu_F >w~^cc), capital taxation is higher in the country with the ‡exible labor market (~t^cu_H >~t^cu_F).

Equilibrium wage rate (stage 1) Maximizing the union’s objective function with respect tow_F^cu, we obtain:

~

w^cu_F = (1 _F) ~w^cc+n _F2l^{2 2}(4n+ 5) + 3f² (n+ 1)² f l(n+ 1)²

Observe that Lemma 1remains valid as dw~^cu_F=d > 0. Moreover, w~_F^cu >w~^cc for all _F 2(0;1). Thus, at the SPNE, we verify thatt~^cu_H >~t^cu_F and~t^cu_F <~t^cc. By contrast, inserting this equilibrium wage in Nash capital taxes and replacing

~

w^ccby its value, we obtain:

~t^cu_H ?~t^ccwhen ^F

F+ 1 ?f² (n+ 1)² 2

h2l^{2 2}(4n+ 5) + 3f² (n+ 1)²i

=

This result is qualitatively similar toProposition 3.

We now turn to the location of …rms at the SPNE. By inserting the equilib- rium level ofw~_F^cu into the location equilibrium, we obtain:

~^cu= 1 2 +1

8 h

4l²(4n+ 5) ²+ 5f² (n+ 1)²i

F f² (n+ 1)² f² (1 +n)²+l^{2 2}(4n+ 5)

We verify that

~^cu? 1

2 when _F ? = f² (n+ 1)²

4l²(4n+ 5) ²+ 5f² (n+ 1)² 2(0;1=5) Recalling that~t^cu_H >~t^cu_F, we thus obtain a qualitatively similar result as the one described inProposition 4.

2.2 Trade union and governments play Nash

By solving the maximization program of the monopoly union in countryF and the maximization program of governments, we obtain:

~t^uu_H = ~t^cc+ H( _F; ) andt~^uu_F = ~t^cc+ F( _F; ) with

H(w^cu_F; ) = nl ^{2 2n+3}_2(n+1)4l2[4n+5^{4 F}_F^l2(3n+4)]^{2+f2 (22}+f^F21)(n+1)(n+1)²(4 3 _F)

F(w^cu_F; ) = _2l(n+1)^{n 2(n+1)3(3 F+1)f4+l2} (n+1)[6n+7+2 _F(10n+11)]f^{2 2}+4l⁴ _F(10n+13) ⁴ 4l²[4n+5 _F(3n+4)] ²+f² (n+1)²(4 3 _F)

and

~

w_F^cu= (1 _F) ~w^cc+ _Fn^{3 2(n+1)3( F 3)f4+4f2 2l2 (n+1)( F(3n+4) 13n 15) 16 4l4(4n+5)}

2f l(n+1)[^{4 2l2(} F(3n+4) 4n 5)+f² (n+1)²(3 _F 4)] Several comments are in order. First, we verify that dw~^uu_i =d > 0 in both countries so thatLemma 1 remains valid. Second,~t^cu_F <~t^ccfor all _F 2(0;1) while

~t^cu_H ?~t^ccwhen _F ? _ = 1 2

f² (n+ 1)

2l^{2 2}+f² (n+ 1) 2(0;1=2) Hence,Proposition 3still holds.

This yields the following tax wedge:

~t^cu_H ~t^cu_F =nf² (1 +n) + 4l^{2 2} 2 (1 +n)l

f² (n+ 1)²(3 _F+ 1) + 4l^{2 2}(3n+ 4) _F

4l²[4n+ 5 _F(3n+ 4)] ²+f² (n+ 1)²(4 3 _F) >0:

We thus obtain the location equilibrium at the SPNE:

~^cu =1 2 +1

2(n+ 1) f² (2 _F 1) (n+ 1) + 4 _Fl^{2 2}

4l²[4n+ 5 _F(3n+ 4)] ²+f² (n+ 1)²(4 3 _F) with

~^cu ?1

2 when _F ? _ Proposition 4still holds.