Int Tax Public Finance (2012) 19:203–215DOI 10.1007/s10797-011-9180-1

Labor markets and capital tax competition

Thomas Eichner · Thorsten Upmann

Published online: 21 June 2011© Springer Science+Business Media, LLC 2011

Abstract Ogawa et al. (J. Urban Econ. 60:350, 2006) analyze capital tax compe-tition in a fixed-wage approach and show that the original results of Zodrow andMieszkowski (J. Urban Econ. 19:356, 1986) are not preserved in the presence of un-employment. In the present paper, we challenge this view and investigate capital taxcompetition for some arbitrary institutional setting of the labor market. We find thatif the labor market is characterized by some efficient bargaining solution, the resultsof Zodrow and Mieszkowski (J. Urban Econ. 19:356, 1986) are preserved.

Keywords Capital tax competition · Unemployment · Efficient bargains · Labortax · Under and overprovision of public goods

JEL Classification H21 · H71 · J51

1 Introduction

In their seminal paper, Zodrow and Mieszkowski (1986), hereafter ZM, explore theconsequences of capital tax competition when regions are small and one factor ofproduction, interpreted as land, is fixed. They find that (i) the optimal capital tax rateis zero when head taxes are available, and that (ii) tax competition results in under-provision of public goods (along with positive capital tax rates) when governmentsare restricted from using head taxes.

T. Eichner (�)Department of Economics, University of Hagen, Universitätsstr. 41, 58097 Hagen, Germanye-mail: Thomas.Eichner@FernUni-Hagen.de

T. UpmannMercator School of Management, University Duisburg-Essen, Lotharstr. 65, 47057 Duisburg,Germanye-mail: Thorsten.Upmann@uni-duisburg-essen.de

204 T. Eichner, T. Upmann

Recently, Ogawa et al. (2006), hereafter OST, introduce labor as a second variablefactor of production in the model of ZM and analyze capital tax competition in thepresence of unemployment. Using a fixed-wage approach, OST derive the followingresults: (i) when head taxes on immobile residents are available, the optimal capitaltax rate is generically not zero, and (ii) when restrictions are imposed on head taxes,capital tax competition may result in overprovision of public goods. In addition, OSTpoint out that increases in the capital tax rate affect employment, which is the drivingforce for the above mentioned results. Clearly, employment effects are absent in ZM,even if we are willing to interpret the fixed factor as labor (instead of as land). In sum,OST find that the results of ZM are not preserved when introducing labor marketimperfections.

Since OST use a highly stylized labor market model—a fixed-wage approachalong with a zero reservation wage (due to the absence of any utility from leisure)—itis important to scrutinize the robustness of their results with respect to modificationsof the institutional setting on the labor market. OST write in their concluding re-marks (p. 355): “Alternative models, such as the unemployment models of efficiencywage, trade union, and job research, can be presented. However, these extensions willshow that, even though we allow for several types of labor market imperfections, thepropositions in the paper are preserved.”

The present paper challenges this view. To this end, we use the basic ZM model,together with the specification of the utility function as applied by OST comple-mented by utility from leisure time, and introduce unemployment in a general fash-ion. Residents either are employed and receive wage income, or they are unemployedand enjoy full leisure time. The wage rate and the employment level are determinedon the labor market, which in order to achieve at greatest generality is left unspeci-fied. In this framework, the labor market may be distorted, driving a wedge betweenmarginal productivity of labor and the reservation wage rate, which reflects the socialopportunity cost of labor. It is shown that the capital tax rate is used to internalizethe distortion on the labor market and, in this way, indirectly does the job of a labortax. However, for some institutional settings of the labor market, this distortion maybe absent and employment is chosen such that the marginal productivity of labor isequal to the reservation wage rate. In this case, we get the original results of ZM: acapital tax rate equal to zero when head taxes are available, and underprovision ofpublic goods when head taxes are unavailable. More specifically, we prove that thelabor-market effect vanishes, and we thus arrive at the classical ZM capital-tax for-mula, whenever the wage rate and the employment level are negotiated according tosome efficient solution concept, that is, if the outcome lies on the (vertical) Paretocurve. Moreover, we find that there is little scope for overprovision of public goodseven if the employment level is inefficiently low.

In Sect. 2, we set up our basic model of capital tax competition, where we leavethe question of the specification of the functioning of the labor market open. Theimplications for different specifications for the labor market model are then analyzedin Sect. 3. A possible extension of our model lies in the introduction of a labor tax asa second source of tax revenue, which we discuss in Sect. 4. Finally, in Sect. 5, wesummarize our results.

Labor markets and capital tax competition 205

2 Capital tax competition

Consider tax competition between n identical jurisdictions. In each jurisdiction,whose economy is small relative to the rest of the world, a representative consumeris endowed with capital k, and with f ≥ 1 units of time. Capital is perfectly mobile,while labor is assumed to be immobile. The consumer’s preferences are representedby the utility function

u = U(x,f,g) = x + U (f ) + V (g). (1)

Equation (1) presupposes that the consumer derives utility from private consump-tion, x, from public-good consumption, g, and from leisure time, f . The subutilityfunctions U and V are increasing and strictly concave in their arguments.1

We normalize the mass of consumers to unity and distinguish two types ofconsumers: employed workers and unemployed workers. For institutional reasons,consumers can only obtain full employment contracts, the working time of whichwe normalize to unity, so that each consumer is either unemployed with maximalleisure time f , or has a full-time job with leisure time f − 1. Accordingly, an em-ployed worker receives the wage rate w, whereas each unemployed worker doesnot receive any labor income. Households are only willing to give up leisure ifw + U (f − 1) > U(f ). Setting U (f − 1) ≡ 0 and U (f ) ≡ u, consumers are indif-ferent between working and being unemployed at the reservation wage rate w ≡ u.Next to labor consumers have additional sources of income: They receive part of theirincome in the form of untaxed dividends paid out of firms’ profits, π , (to be speci-fied below) and in the form of capital income. With this information the utility of theemployed (e) and unemployed (ue) workers are given by

uj ={

π + w + rk − h + V (g) if j = e,

π + u + rk − h + V (g) if j = ue,(2)

respectively, where h denotes a head tax, and r the net return of capital.In each jurisdiction, firms use capital, k, and labor, �, to produce their output ac-

cording to some increasing and strictly concave production function F(k, �). Theprice-taking firms face an output price equal to one, a wage rate w, a net return ofcapital r and a (source-based) tax on capital τ . Then profits are given by

π = Π(k, �,w) ≡ F(k, �) − w� − pk, (3)

1As shall become clear soon, we could have specified some more general utility function, because all weneed is the separability between private and public goods and the linearity of U(·, f − 1, ·) in the firstargument. In principle, we could have dropped both features, but decided to maintain them for the purposeof best comparability with the model of OST, who specified residents’ utility as U(x,g) = x + V (g).Without linearity in income, the Pareto-curve (see Sect. 3) would have been a strictly increasing function,not a vertical; without separability, V ′ had to be replaced by the marginal rate of substitution betweenprivate and public goods without having some effect on the results of this section, while the labor marketnegotiations in Sect. 3 were parameterized in the provision level of public goods g.

206 T. Eichner, T. Upmann

where p ≡ r + τ denotes the after-tax price of capital. Irrespective of the institutionalsetting of the labor market, we assume that firms equate the marginal productivity ofcapital to its cost

Fk(k, �) = r + τ. (4)

Equation (4) together with two equilibrium conditions of the labor market, one forthe wage rate and one for the employment level, determine the equilibrium values ofcapital, employment and the wage rate as functions of the capital tax rate (which fornotational simplicity we continue to denote by k, � and w for the rest of the paper):2

k = K(τ), � = L(τ) and w = W(τ). (5)

Since we wish to demonstrate how some noncompetitive labor market institution gen-erally affects capital tax competition, it suffices to keep the labor market model un-specified and to assume merely that there are conditions uniquely determining theequilibrium wage and employment level. We reasonably assume that investments aredecreasing in the capital tax rate, while the signs of the derivatives L′ and W ′ areleft open. In the fixed wage model, employment is determined by the first-order con-dition of the firm’s profit maximization F� = w, where w is the exogenously given(fixed) wage rate. OST point out that the functions K , L and W from (5) exhibit theproperties

sgnK ′ = sgnF�� < 0, sgnL′ = sgnFk�, and W ′ = 0. (6)

Output can be transformed one-to-one into private consumption or into publicconsumption. The jurisdictional government finances the provision of the public goodby means of the revenues from the head tax and the capital tax. Then the fiscal budgetconstraint reads as

τk + h = g. (7)

The local government takes the net return of capital as given but realizes that theequilibrium variables depend on the capital tax rate according to (5). It applies theutilitarian welfare function

�U(xe, f e, g) + (1 − �)U(xue, f ue, g), (8)

where � and 1 − � are the shares of employed and unemployed workers, respectively.Inserting (1)–(3), (5) and (7) into (8), the welfare function can be written as

Ω(τ,h) := (1−L(τ)

)u + rk − h + F

(K(τ),L(τ)

) − (r+τ)K(τ) + V(τK(τ)+h

).

2For example, if the labor market is competitive, the conditions F�(k, �) = w and w = w together with (4)

determine the equilibrium values of kC, �C , and wC as functions of τ (for a given value of r). A similarargument applies for other specifications, and thus other equilibrium conditions, of the labor market. InSect. 3, we explore various concrete labor market models.

Labor markets and capital tax competition 207

Maximizing Ω with respect to h and τ yields the first-order conditions

Ωh = V ′ − 1 = 0, (9a)

Ωτ = (F� − u)L′ + [Fk − (r + τ)

]︸ ︷︷ ︸

=0, from (4)

K ′ − k + (k + τK ′)V ′ = 0. (9b)

Following ZM and OST, we investigate the following two tax scenarios.

Head taxes If head taxes are available, the government can provide the efficientlevel of public goods, which is characterized by the well-known Samuelson ruleV ′ = 1. Using this, rearranging (9b) yields the optimal capital tax rate

τ = − (F� − u)L′

K ′ . (10)

From (10), we conclude the following proposition.

Proposition 1 Suppose that jurisdictional governments can use a head tax on immo-bile residents. Then they choose a zero capital tax rate if and only if (i) F� = u or(ii) L′ = 0.

Production efficiency is reached if, along with condition (2), employment is de-termined such that the marginal product of labor equals its marginal social cost (op-portunity cost of labor), i.e., if F�(k

C, �C) = wC = w ≡ u, where (wC, �C) denotesthe competitive labor market equilibrium. Hence, any distortion on the labor marketis reflected by the difference between the marginal product of labor and the reserva-tion wage rate F� − u, —and according to (10), the capital tax serves to reduce thisdistortion: As long as F� differs from u and the capital tax rate has some employ-ment effects (L′ �= 0), the optimal capital tax rate is nonzero. For example, supposeemployment falls along with a higher capital tax, L′ < 0, and employment is ineffi-ciently low, F� > u (i.e., below the competitive level �C ), then capital is subsidizedto attract more capital in order to promote employment. The capital tax thus playsthe role of an indirect tax/subsidy on labor, which is absent in this framework. (Wediscuss the consequences of an availability of a labor tax in Sect. 4.) It is importantto realize at this instance that the capital tax rate is determined by the labor marketmechanism.

Note that Proposition 1 reproduces the ZM results where employment effects ofthe tax rate are absent (L′ = 0) and, at the same time, extends their results to a per-fectly competitive labor market. However, as will be shown below (see Sect. 3), thereexist other institutional settings of the labor market, different from perfect competi-tion, at which production efficiency prevails in equilibrium, i.e., where F� = u holds.Proposition 1 is also in accordance with OST who do not consider leisure and ac-cordingly assume that the reservation wage is equal to zero, i.e., u ≡ wC ≡ 0. Hence,for F� > u ≡ 0, they get that the capital tax rate is zero if and only if L′ = 0.3 (Recall

3However, the dependence of the tax rate on the functioning of the labor market is somewhat veiled inOST; see (10) and Proposition 1 therein, for the tax rate is expressed in terms of the properties of the

208 T. Eichner, T. Upmann

that in OST sgnLτ = − sgnFk�, see (6), and L′ therefore generically differs fromzero.)

Distortionary taxation Next, we assume that head taxes are not at the government’sdisposal. Rearranging (9b) then yields

V ′ = 1 − 1k(F� − u)L′

1 + τkK ′ = 1 − 1

k(F� − u)L′

1 + τr+τ

ε(p), (11)

where ε(p) ≡ pK ′(p)/K(p) denotes the elasticity of capital demand with respect tothe (after-tax) price of capital, and K denotes the (ordinary) capital demand function.Equation (11) represents the allocation rule for the local public good when it is exclu-sively financed by a distorting capital tax rate, and may be viewed as a generalizationof ZM’s public-good provision formula (see (8) therein). The second term of the de-nominator shows the capital tax rate effects on investments, while the second term ofthe numerator captures the capital tax rate effects on labor market distortions. Anyincrease in the capital tax rate induces capital to flight out of the jurisdiction, K ′ < 0.Throughout the rest of the paper, we follow ZM and assume that governments acton the left-side of the Laffer curve, formally 1 + τ

kK ′ > 0.4 Then (11) provides the

following information:

V ′ � 1 ⇐⇒ τK ′ � −(F� − u)L′. (12)

In view of condition (12), we get V ′ > 1, and hence the original ZM underprovi-sion result, if either the capital tax rate has no impact on employment (L′ = 0), or ifF� = u holds in the labor market equilibrium, that is if � = �C is satisfied. Clearly,according to condition (12), there are further constellations which are sufficient forunderprovision. In contrast, since K ′ < 0 and τ must be positive, overprovision oflocal public goods may only emerge, if (F� − u)L′ > 0, that is, if the marginal pro-ductivity of labor is smaller [greater] than the competitive wage rate wC = u andan increase in the capital tax rate hampers [fosters] employment. These findings aresummarized in the following proposition.

Proposition 2 Suppose that jurisdictional governments are restricted from using ahead tax.

(a) Then the capital tax rate is inefficiently low and local public goods are underpro-vided if (F� − u)L′ ≤ 0.

(b) A necessary condition for inefficiently high capital tax rates and overprovision ofpublic goods is (F� − u)L′ > 0.

production function. Since in their fixed-wage model the labor market equilibrium is determined by thelabor demand side, and thus by the properties of the production function alone, one can hardly distinguishbetween properties of the production function and the functioning of the labor market.4Note that this constitutes a necessary condition for the optimality of the tax rate, and thus for an equilib-rium.

Labor markets and capital tax competition 209

Proposition 2 illustrates how the optimal (distortionary) tax rates are affected bythe labor market mechanism. For any choice of a labor market model, we are nowable to accommodate for the labor market effects and to calculate the optimal capitaltax rate and the resulting provision level of public services. In particular, it followsimmediately from part (a) that public goods are underprovided if either F� = u orL′ = 0, that is, if we have either production efficiency or if the employment levelis independent of the capital tax rate. In the next section, we explore in more detailunder which labor market scenarios public goods are in fact under or overprovided.

Proposition 2 contains as special case Proposition 2 of OST. As previously men-tioned in the fixed wage model, the marginal product of labor is greater than thereservation wage (F� > u ≡ 0) and the employment function exhibits the propertysgnL′ = − sgnFk�. Hence, in the fixed wage model, one may get overprovision ifincreases in the capital tax rate reduce investments and raise employment in the ju-risdiction (formally if sgnL′ = − sgnFk� > 0). In the more plausible case, though,where sgnL′ = − sgnFk� < 0 tax competition results in underprovision of local pub-lic goods.

3 Labor market scenarios

A variety of different, plausible labor market institutions has been applied in the pub-lic economics literature, e.g., the monopolistic union (Richter and Schneider 2001),the right-to manage approach (Fuest and Huber 1999; Aronsson and Wehke 2008) inthe context of optimal taxation of small economies, efficient Nash bargains (Bayındır-Upmann and Raith 2003, 2005) in the analysis of environmental tax reforms. Sincethe results of the capital tax competition hinge upon (F� − u)L′, we now take a closerlook at this term for these institutional settings of the labor market.

Figure 1 illustrates different labor market scenarios. F� represents the labor-demand curve, Ls the labor-supply curve and C the contract (or Pareto) curve.5

The intersection of the labor demand curve and the labor supply curve at pointC ≡ (�C,wC) determines the equilibrium of a perfectly competitive labor market. Forthe right-to-manage approach, the equilibrium is at point R ≡ (�R,wR), and for themonopoly union the equilibrium is at point M ≡ (�M,wM), both of which lie on thelabor demand curve. In comparison to the right-to-manage equilibrium, the monopolyunion equilibrium is characterized by a higher wage rate but also by a lower employ-ment level (wM ≥ wR, �M ≤ �R). At any equilibrium on the labor demand curve,the marginal product of labor (weakly) exceeds its social cost, i.e., F� − u ≥ 0.

Any efficient bargaining outcome, however, lies on the contract curve, and thusleads to an employment level equal to �C satisfying F� = u. This holds in particularfor the frequently applied Nash solution (McDonald and Solow 1981; Creedy andMcDonald 1991),6 the equilibrium outcome of which is depicted in Fig. 1 as N ≡

5Note that for the quasilinear specification of the utility function (see (1)) the Pareto curve is vertical at �C .6Other well-known efficient bargaining solutions are the Kalai–Smorodinsky solution (Kalai andSmorodinsky 1975), the egalitarian solution (Kalai 1977), or the equal loss solution (Chun 1988).

210 T. Eichner, T. Upmann

Fig. 1 Different labor marketscenarios

(�N ,wN). With F� = u, however, the optimal tax formula (11) collapses to the well-known tax formula of ZM. Hence, we conclude with the following proposition.7

Proposition 3 If the wage rate and the employment are determined by efficient bar-gains on the labor market, the ZM results are preserved.

Given the quasilinear specification of the utility function (1), we are not aware ofany labor market scenario at which F� < u, i.e., for which � > �C holds. Hence, ifthe comparative statics of the labor market equilibrium yields L′ < 0, then there is noscope for overprovision of public goods (Proposition 2(a)). To put it differently, pro-vided that F� ≥ u at some equilibrium of the labor market, a necessary (albeit not suf-ficient) condition for overprovision is that employment benefits from the tax-inducedcapital flight, i.e., L′ > 0. But this condition to be fulfilled should be considered as anexception rather than as the rule, for otherwise governments could typically alleviatetheir employment problems by driving capital out of its jurisdiction—an economi-cally strange scenario. Hence, there is little scope for overprovision of public goodsin the classical ZM-framework, even if we allow for a noncompetitive labor market.

4 Extension of the model: a labor tax

One may ask to what extent the results of the previous sections are due to the fact thatwe have considered capital taxation exclusively. To that end, we will now explorehow the results established so far are affected by introducing a labor tax as a second(third) source of public income.

7A formal proof of Proposition 3 for Nash bargains can be found in the Appendix.

Labor markets and capital tax competition 211

Suppose the government levies next to a capital tax also a tax on the firm’s laborinput at rate t . The profit of the firm then changes to

π = F(k, �) − (w + t)� − (r + τ)k. (3′)

As a consequence, the equilibrium values of capital, employment, and the wage ratenow depend on both the capital tax rate and the labor tax rate, formally

k = K(τ, t), � = L(τ, t) and w = W(τ, t), (5′)

and the fiscal budget constraint is given by

τk + t� + h = g. (7′)

Replacing (3) by (3′), (5) by (5′), and (7) by (7′), the welfare function now reads as

Ω(τ, t, h) := (1 − L(τ, t)

)u + rk − h + F

(K(τ, t),L(τ, t)

)− (r + τ)K(τ, t) − tL(τ, t) + V

(τK(τ, t) + tL(τ, t) + h

)and the first-order conditions with respect to h, τ and t are given by

Ωh = V ′ − 1 = 0, (13a)

Ωτ = (F� − t − u)Lτ − k + (k + τKτ + tLτ )V′ = 0, (13b)

Ωt = (F� − t − u)Lt − � + (� + τKt + tLt )V′ = 0. (13c)

In the following, we assume that investments are decreasing in the capital tax rate,Kτ < 0, that employment is decreasing in the labor tax rate Lt < 0, and that gov-ernments operate on the left side of the Laffer curve corresponding to the capital taxrevenues and labor tax revenues, respectively, formally 1 + τ

kKτ + t

kLτ > 0, and

1 + τ�Kt + t

�Lt > 0.

Head taxes When head taxes are available, governments choose the efficient provi-sion of public goods (V ′ = 1) and (13b) and (13c) simplify to

τKτ = −(F� − u)Lτ , (14a)

τKt = −(F� − u)Lt . (14b)

First, suppose that the marginal product of labor equals its marginal social cost (op-portunity cost of labor), i.e., F� = u; then in view of (14a) and (14b) τ must beequal to zero. Observe that the competitive employment level is now characterizedby F� − t = w = u. As a consequence, for any efficient labor market model both, thecapital tax and the labor tax rate, must be equal to zero, as this guarantees productionefficiency along with conditions (14a) and (14b). As a consequence, both the laborand the capital market are undistorted.

For those labor markets, though, for which F� > u holds (evaluated at t = 0),that is, for equilibria on the labor demand curve, the labor tax can be adjusted so as to

212 T. Eichner, T. Upmann

shift the labor demand curve appropriately, usually outward by means of a subsidy onlabor, up to the point where F� = u holds. The cost of subsidizing labor is covered byan appropriate head tax so that τ is kept zero and (14a) and (14b) are again matched.

If however, F� − u = 0 is infeasible, as in case of the fixed wage model discussedby OST (where u = 0), the optimal labor tax/subsidy invalidates the fixed wage. Fora fixed wage exceeding the competitive wage, the labor subsidy covers the differencebetween the fixed wage and the competitive wage. Then subsidized labor demand ab-sorbs aggregate labor supply completely and full employment ensues. At this point,however, any further increase in the labor supply would not change equilibrium em-ployment, i.e., Lt = 0. In addition, in the fixed wage model with labor taxes/subsidiesgovernments refrain from using capital taxes. Formally, (14a) and (14b) are satisfiedby Lt = 0 and τ = 0.

Distortionary taxation Next, we turn to the more relevant case where head taxes areunavailable and governments must rely on distortionary taxation. Rearranging (13b)and (13c), we obtain

V ′ = 1 − 1k(F� − t − u)Lτ

1 + τkKτ + t

kLτ

= 1 − 1�(F� − t − u)Lt

1 + t�Lt + τ

�Kt

.

Manipulating this condition, we find

V ′ � 1 ⇐⇒ τKτ � −(F� − u)Lτ (15a)

⇐⇒ τKt � −(F� − u)Lt . (15b)

We now explore whether public services may arguably be overprovided in equi-librium or overprovision must be considered as a pathological case. To this end, wepresume that equilibrium demand of each factor is decreasing in its own price, i.e., Kτ

and Lt are both negative; also, as argued above, we focus on labor market institutionssuch that overemployment (relative to the competitive level) does not materialize, sothat F� − u is always nonnegative. As a consequence, −(F� − u)Lt is nonnegative,and overprovision can thus only emerge if, as a necessary condition, τKt is positive.This, however, requires either

(i) τ > 0 and Kt > 0 or(ii) τ < 0 and Kt < 0.

In the first case with τ > 0, and thus τKτ < 0, it follows from (15a) that Lτ must bepositive for overprovision to be possible. But this is an unplausible equilibrium effectas argued above (see p. 210). Only in the second case in which the left-hand sides ofboth condition (15a) and (15b) are positive, may overprovision come about. But thiscan only occur when capital is subsidized in equilibrium and the tax revenue from thelabor tax thus suffices to cover both: overprovision of public goods and subsidizingcapital. We summarize these results in the following proposition.

Proposition 4 Suppose that jurisdictional governments may tax labor along withcapital.

Labor markets and capital tax competition 213

(a) If governments can use a head tax on immobile residents, they choose a zero cap-ital tax rate and subsidize labor (if necessary) to establish production efficiencyon the labor market, i.e., F� = u.

(b) If governments are restricted from using a head tax, a necessary condition foroverprovision of local public goods is either

τ > 0 ∧ Kt > 0 or τ < 0 ∧ Kt < 0.

Proposition 4 shows that our previous results are, with some modifications ofcourse, also valid when the government has, in addition to the capital tax, also alabor tax at hand. Also, the results obtained in this section are fully in line with thoseof ZM. When head taxes are available, capital is not taxed and public goods are pro-vided efficiently. When head taxes are not at the governments’ disposal, public goodsare underprovided, since overprovision seems to be a pathological case. Finally, ourapproach covers a broader class of labor market models than the fixed wage modelof OST, which, of course, recovers as a special case; however, their discussion ofthe possibility of overprovision becomes somewhat suspect, as this outcome requirespathological behavior of factor demand.

5 Concluding remarks

The present paper aims at answering the question whether or not, and if so underwhich conditions, the results of Zodrow and Mieszkowski (1986) are preserved inthe presence of unemployment. To this end, we elaborate a generalized ZM frame-work which accommodates for any arbitrary labor market model, and show how theoriginal ZM public-good provision formula ((8) therein) may be generalized. WhileOgawa et al. (2006) point out that this tax formula is not preserved in a fixed-wagelabor market model, we show that the ZM formula continues to hold for any efficientlabor market model. More precisely, we show how the result of OST, who employ afixed-wage approach along with a zero reservation wage, is to be modified to accom-modate for a general, unspecific labor market model: Whenever labor differs fromits competitive level and capital taxation causes employment effects, the public-goodprovision formula of ZM must account for labor market effects. This is in particulartrue for the right-to-manage model and the monopoly union. In contrast, if at someequilibrium of the labor market the marginal productivity of labor equals the reserva-tion wage, which in our framework holds for any efficient bargaining solution, thenthe ZM formula and hence their economic results are strictly preserved.

It is worth mentioning that both our and OST’s results depend on the quasi-linearspecification of the utility function. Allowing for more general utility functions, thehead tax and the capital tax rate cause distributional effects between employed andunemployed workers. However, since ZM who deal with a representative residentalso abstract from distributional effects, our (and OST’s) approach is fully in thespirit of ZM. Finally, incorporating distributional effects requires a modification ofthe Samuelson rule for the provision of public goods, even if head taxes were avail-able. Nevertheless, augmenting our generalized tax competition model with invol-untary unemployment by distributional considerations may constitute an interestingissue for future research.

214 T. Eichner, T. Upmann

Acknowledgement We are grateful to three anonymous referees for valuable comments and sugges-tions.

Appendix

Proof of Proposition 3 At efficient Nash bargains, the negotiations on the wage rateand employment take place simultaneously. On the supply side of the labor market,there is an (encompassing) labor union which acts on behalf of all laborer’s interests.On the demand side, the firms’ interests are represented by an employers’ association.While an employed worker obtains the wage rate w, an unemployed worker enjoysleisure time. When the labor union maximizes the sum of its members’ utility itsobjective function may be written as �w + (1 − �)u, or after subtracting the statusquo utility u as,8

Ψ (w,�) = �(w − u). (A.1)

The employers’ association aims to maximize the firms’ profits. Applying the (asym-metric) Nash solution, the outcome of the bargaining process is obtained as the max-imand of the Nash product:

Ψ (w,�)μ · Π(k, �,w)1−μ, (A.2)

where μ ∈ [0,1]. The bargaining parties take into account that once they have agreedon some tuple (w,L) firms subsequently choose capital according to (4), which de-termines the capital demand function k = κ(�, τ ). Then maximizing (A.2) subject tok = κ(�, τ ) yields the first-order conditions

w

w − u= w

w − F�(k, �), (A.3a)

w = μF(k, �) − (r + τ)k

�+ (1 − μ)F�(k, �). (A.3b)

Equation (A.3a) constitutes the Pareto curve which is the set of all employment-wage combinations such that both parties’ indifference curves are tangent to eachother. Equation (A.3b) is the Nash curve and determines the bargained wage as theweighted average of the average and marginal productivity of labor, net of the cost ofother factors. From (A.3a), we immediately infer F� = u.

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Labor markets and capital tax competition 215

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