Vertical externalities with lump-sum taxes: how much difference does unemployment make?

Vertical externalities with lump-sum taxes: how much di¤erence does unemployment make? Diego Martinez Department of Economics, University Pablo de Olavide Tomas Sjögren Department of Economics, Umeå University This version: August 7, 2012 Abstract This paper analyses how the existence of unemployment a¤ects the conventional approach to vertical externalities. We discuss the optimality rule for the provision of public inputs both in an unitary and a federal country. Our …ndings show that decentralizing the spending responsability on public inputs can bring its optimality rule closer to the production e¢ ciency condition. Moreover, we describe the inability of the federal government, behaving as Stackelberg leader, to replicate the unitary outcome, unless to have new policy instruments at government’s disposal. Keywords: Public inputs, unemployment, vertical externalities. JEL Classi…cation: J2, H4, H7 The authors would like to thank the Spanish Ministry of Science (Projects ECO2010-15553 and ECO201021706), Junta de Andalucia (Projects SEJ-02479 and SEJ-6882), the Bank of Sweden Tercentenary Foundation (Stiftelsen Riksbankens Jubileumsfond), the Swedish Council for Working Life and Social Research (FAS) and the National Tax Board (Skatteverket) for research grants. The views expressed by the authors are not necessarily those of the institutions they are a¢ liated with. Corresponding author: Diego Martinez, Department of Economics, University Pablo Olavide, Ctra. Utrera Km. 1, 41013 Seville. Spain. Email: [email protected] 1 1 Introduction The usual approach to vertical externalities establishes that sharing taxes between di¤erent levels of government has an impact on e¢ ciency. From the seminar contribution by Keen (1998), a number a papers has dealt with this issue, o¤ering various solutions to internalize this problem as well (see, for instance, Boadway and Tremblay, 2006). A common issue in all these contributions is assuming distortionary taxation. In fact, it is clear that vertical tax externalities only appear as households’decisions are in‡uenced by distorting taxes; otherwise, the marginal cost of public funds is not a¤ected by lump-sum taxes decided by one level of government and, consequently, the impact of …scal policies across di¤erent tiers of government does not take place. Another common feature in this literature is that the labor market is competitive, with the labor force matching exactly the demand for labor. Papers such as Dahlby and Wilson (2003) and Kotsogiannis and Martinez (2008) give a central role to the labor supply and demand for labor in determining equilibria but always with labor market clearing. In such a world, there is no scope for one of the conventional …scal policies aimed at …ghting against unemployment, namely the provision of public inputs. In fact, to the best of our knowledge, no paper so far has dealt with vertical expenditure externalities (caused by the provision of productivity-enhancing public expenditures in a federal context) in the presence of unemployment. This has not been the case when horizontal externalities are involved; Ogawa et al (2006) study the implications of labor market imperfections on capital tax competition at the same level of governments. This paper precisely combines vertical externalities and labor market imperfections in a single model. Indeed, we build a theoretical framework in which the federal government is in charge of unemployment bene…ts and the states provide a public input with positive e¤ects on demand for labor. Taxes are assumed to be lump-sum because we are interesting in focussing on the e¢ ciency implications derived from the expenditure side of government decisions rather than on vertical tax externalities. Anyway, we will show that ignoring distortionary taxation as a policy variable may play a crucial role for correcting the vertical externality. The following contributions can be summarized from our results. Firstly, we prove that, in spite of using exclusively lump-sum taxes to …nance governments (and thus no space for tax externalities), a vertical expenditure externality arises when unemployment exists. This con…rms a previous result found in the literature (Dahlby and Wilson, 2003; Martinez, 2008), namely, that both vertical (tax and expenditure) externalities are independent of each other. The provision of public inputs creates a positive vertical impact on federal revenues as long 2 as this type of public spending increases the demand for labor and, therefore, it reduces the resources needed at federal level for paying unemployment bene…ts. And this occurs without the co-occupancy of elastic tax bases. Moreover, we also see how the optimality rule for the provision of public inputs at state level is closer to the production e¢ ciency condition than the optimal condition in a unitary country with a non-clearing labor market. In a sense, one could say that more federalism does not necessarily leads to more ine¢ ciency. Particularly, in the presence of a distortion (in the labor market, resulting in unemployment), it could be positive for e¢ ciency to bring in a new distortion (that coming from the vertical expenditure externality). Secondly, we have studied whether the federal government is able to replicate the equilibrium of an unitary country. As usual, we have assumed that the upper level of government knows the states’reaction functions and, behaving as Stackelberg leader, tries to achieve the centralized outcome. Our result deviates from previous papers as long as we conclude that the policy variables available for the federal government are not e¤ective instruments to get the unitary equilibrium. We guess here that the fact of using exclusively lump-sum taxes prevents from a¤ecting decisions taken by governments and households, in an attempt to internalize the e¤ects from states’policy. In a sense, this result can be placed on the discussion initiated by Sato (2000) about the capability of federal government to replicate second-best results depending on the federal instruments available. Precisely, as result of taking into consideration a new policy instrument, i. e., a public input provided by the federal government that is complement to that o¤ered by the states, the upper level of government is able to replicate the second-best outcome of an unitary country. The structure of the paper is as follows. Section 2 describes the main features of the model and the di¤erent versions of the optimality rule for the provision of public inputs, taking account whether the country is federal or not. Sections 3 and 4 evaluate the ability of the federal government to replicate the unitary outcome with the policy instruments available and with a complementary public input, respectively. Finally, section 5 concludes. 2 The Basic Model This section aims to show two points. First, to characterize the equilibrium in a centralized country with unemployment; this will allow us not only to see how the optimal rule for the 3 provision of public inputs must be modi…ed with respect to a situation with full employment, but also having a benchmark scenario to compare with federal equilibria. Second, to highlight that the …scal decisions taken by one level of government (particularly that with spending responsabilities on public inputs) will a¤ect other levels of government; consequently, vertical expenditure externalities will arise despite of using exclusively lump-sum taxes. The theoretical framework consists of …rms, households and two di¤erent tiers of government: the federal level and ksubnational states. Firms are identical across the country and, for the sake of simplicity, we assume that their number is normalized to one in each state. All of them produce a single good on the basis of the following production function: F(N; K; G) = NK1G;(1) where Nis labor, Ka …xed factor and Ga public input. Such a production technology allows us to qualify the public input as factor-augmenting1. In this context, the public spending will increase the return to the …xed production factor K, which we normalized to one, in which case the pro…t can be expressed as:2 =F(N; G)wN; (2) where wis the wage rate. Pro…t maximization implies to de…ne the …rst-order condition w= FN(N; G), that implictly de…nes the following function for labor demand: N(w; G) = 1 1G 1w1 1(3) Combining equations (2) and (3), the pro…t function can be obtained: (w; G)(4) We consider that all households have the same preferences for consumption cacross the federation and described by a utility function u(c), which is increasing in c. Each state is populated by three types of consumers: a …rm-owner, employed and unemployed workers, 1An alternative approach would imply a production function with constant returns to scale in all the inputs (private and public). This would be the case of …rm-augmenting public input. It would create economic rents that, in terms of the model we develop here, would not exhibit substantial di¤erences with respect to what we obtain below. 2The return to labor is not a¤ected by the public input, although this would be the normal situation with factor-augmenting public inputs. This is not the case here because we are interested in considering the impact of the public input on employment, and the demand for labor we obtain below implies that the wage rate is independent of G. In a model with full-employment, however, we should set up w(M; g). 4 which are denoted by superindices "f", "e" and "u", respectively. The …rm-owner, endowed with the production factor K, is who receives the pro…t in return for hiring the …xed factor to the …rm. His budget-constraint is de…ned by cf=f, where fis a lump-sum tax. Regarding the other two types of consumers, we insert here a distinction between the total labor force available for working Mand the number of households that e¤ectively are employed N. Obsviously, full employment is characterized by M=N. The budget constraint for an employed worker is ce=we, where eis a lump-sum tax, while workers without jobs faces cu=b, where bdenotes a net of tax unemployment bene…t. In a centralized country, for the policy variables f; e; b; G, the government maximizes a utilitarian welfare function W=kNue+k[MN]uu+kuf(5) subject to the following budget constraint: kNe+kfkG k[MN]b= 0 (6) In a situation where there is no unemployment, the …rst-order conditions are as follows: FOC f:=uf0 (7) FOC (e) : = (ue)0(8) FOC (G) : FG= 1 (9) FOC () : Ne+fG= 0;(10) where is the Lagrange multiplier. The two …rst equations show the usual result from optimization with lump-sum taxes and transfers: private marginal utility (of each type of consumer) must be equal to social welfare cost of taxation, which is represented here by the Lagrange multiplier of government budget constraint. The equation (9) is the standard production ef- …ciency condition in the provision of public inputs. Finally, (10) is the budget constraint of central government, where the last term of LHS in (6) has been dropped as M=N. Let us turn to the equilibrium with unemployment. For institutional reasons (i. e., the existence of a minimum wage), the rate wage is assumed to exceed the market-clearing wage and, consequently, M > N. Things dramatically change for the optimal provision of Gwhen unemployment appears; additionally, the …rst-order condition for the unemployment bene…t b also must be taken into consideration: FOC (b) : = (uu)0(11) 5 FOC (G) : NG(ueuu) +NGe+NGb+FG= 1:(12) Let us consider now the case of di¤erent tiers of governments. We assume that the federal level is in charge of providing the unemployment bene…t while the states provide the public inputs3. Both levels of government share the tax on employed workers (with the tax rates Teand techosen by the federal and states governments, respectively; e=Te+te). The revenues collected from the tax on pro…ts are assigned in a proportion (which is exogenously determined) to the states (0 1), while the tax rate fis exclusively decided by the federal government. Under such a framework, let us assume that the states behave as Nash players, that is, each subnational government ignores the impact of its …scal decisions on federal revenues. Therefore, the optimization problem to be solved by the states is: Max W =Nue(we)+(MN)ub(b) + uff(13) s:t: Nte+fG+S= 0 N=N(w; G) wo> we; where Sis a vertical lump-sum from the federal government to states. Last inequality refers to the distorsion existing in the labor market, which is the reason for unemployment. First-order conditions for te,Gand give: FOC (te) : = (ue)0(14) FOC (G) : NG(ueuu) +NGte+FG1 = 0 (15) FOC () : Nte+fG+S= 0 (16) Expression (14) sets up an identical rule for chosing the optimal tax rate on employed workers in a centralized country than in a world with two tiers of government. This is a direct consequence of using lump-sum taxes. Even in the presence of tax sharing between di¤erent levels of government, if the households’behavior is not a¤ected by taxes, there is no scope for vertical tax externalities. By contrast, and leaving aside the discussion on the optimal levels of G(see Martinez and Sjongren (2009) for a further analysis), expression (15) shows the main di¤erence by comparing 3This distribution of spending responsabilities is not crucial for the results, which would be symmetric with an inverse vertical assignment of public expenditures. Anyway, the scheme we follow here is in line with the mainstream of theory of …scal federalism. 6 it to the expression (12). The term NGtedi¤ers from its equivalent in (12), namely, NG(e+b). As long as the federal government sets up a non-negative tax rate Teon employed workers, the fact of having states deciding on Gleads to reduce the overprovision bias that the presence of unemployment creates in the provision of public inputs. In other words, expression (15) is closer to (9) than equation (12).4 In this regard, and contrary to the conventional view in previous literature on vertical externalities, we guess here that more federalism may lead to more e¢ ciency in the design of …scal policies. To see this in an extreme case, assume that all rent taxes accrue to the states (= 1); the federal government needs to be …nanced by a negative …scal grant (from states) and/or by charging a positive tax rate Teon workers. This latter solution involves an optimal rule for the provision of public inputs closer to the production e¢ ciency condition, minimizing the di¤erential e¤ect that the presence of unemployment creates in the discussion on optimality. Consequently, the behavior of federal government becomes a crucial issue to determine the e¤ect of unemployment on the achivement of production e¢ ciency condition in the provision of public inputs. This is what we study in the next section. 3 The ability of federal government to replicate the centralized outcome A usual way of correcting vertical (tax and expenditure) externalities is assuming a federal government behaving as Stackelberg leader. In such a context, the sequence of the game is as follows. Firstly, the federal government decides on Te,f,Sand, residually, on b, taking into consideration the states’reaction to changes in federal policy variables. Secondly, the states choose Gand te, taken as exogenous all the decision variables of the upper-level of government. 4It is straightforward to show that with full employment no vertical (tax and expenditure) externalities appear. 7 Consequently, the optimization problem of the federal government is: Max W =kNue(we) + k(MN)ub(b) + kuff(17) s:t: kNTe+k(1 )fk(MN)bkS = 0 G=G(Te; ; f; S; M; N)(18) te=te(Te; ; f; S; M; N)(19) N=N(w; G) wo> we: Expressions (18) and (19) are the states’reaction functions. Therefore, for solving the federal problem, some information on comparative statics of these reaction functions is required. To do that, we start from the …rst-order conditions of states (15) and (16)5. Di¤erentiating totally and (and ignoring superindex "e" for sake of simplicity in the notation), we have: GdG + tdt + TdT + SdS + fdf= 0 GdG + tdt + TdT + SdS + fdf= 0 This two-equation system can be expressed using a matricial form as follows (and after solving for dG and dt): dG dt =0 @Gt Gt 1 A 10 @TSf TSf 1 A0 B B @ dT dS df 1 C C A(20) Matricial manipulation on (20) shows that: dG dT =GT=A(tTtT)(21) dG dS =GS=A(tStS)(22) dG df=Gf=A(tftf)(23) dt dT =tT=A(gTgT)(24) dt dS =tS=A(gSgS)(25) dt df=tf=A(gfgf)(26) 5The …rst-order condition (14) can be ignored in this analysis. In a sense, this expression does not admit any in‡uence from federal variables and, consequently, it does not matter at this point. Anyway, expression (14) can be easily inserted in (15) without modifying substantially the analysis below. 8 where Ais 1 GtGt. Turning back to the federal problem, it is clear that its budget constraint can be written as b=NT e+(1)fS (MN). Plugging this into the objective function (17), we obtain the …rst-order conditions for the policy variables of the federal government: FOC(Te) : N(ue)0(1 + tT)+(MN)(ub)0(bT+bTtT+bGGT)(27a) +(uf)0(FNNGGTwNGGT)ubNGGT= 0 FOC(f) : N(ue)0(tf)ubNGGf+ (MN)(ub)0(bf+bttf+bGGf)(27b) +(uf)0(FNNGGfwNGGf)(uf)0= 0 FOC(S) : N(ue)0(tS)+(MN)(ub)0(bS+bttS+bGGS)ubNGGS(27c) +(uf)0(FNNGGSwNGGS) = 0 FOC() : kNTe+k(1 )fk(MN)bkS = 0 (27d) Taking into account that bcan be residually obtained from the above four equation-system, we symplify (27a)-(27d) and the following result is achieved: tT= 0 (28) tf= N(29) tS=1 N;(30) where w=FN(N; G), (21)-(23) and the corresponding partial derivatives of and (according to (15) and (16)) have been used. What is implicitly established in (28)-(30) is the inability of federal government to a¤ect states’behavior. In fact, not only the federal tax rate on employed workers Tehas no e¤ect on the equivalent state tax rate te(equation (28)), but also none of the policy variables of upper level of government has any impact on the state provision of public inputs. Indeed, from expressions (21)-(23), it is clear that GT=Gf=GS= 0, that is, there is no way through which the federal government can modify the provision of public inputs. The unique impact of the federal policy variables (fand Son te) is trivial: an increase (decrease) in some of them reduces (rises) the state tax rate in a magnitude given by the number of employed workers N. Therefore, the highest level of government is not able to replicate not only the …rst-best outcome of (9) but also the optimality rule for the provision of public inputs in an unitary country with unemployment6. 6Anyway, we must be aware that the …rst-best values for Teand teare guaranteed in each scenario as long as they are lump-sum taxes. 9