The Unemployment Bene…t System: Wage Taxes versus Payroll Taxes Daniel Cardonay Universitat de les Illes Balears
Fernando Sánchez-Losada Universitat de Barcelona
This version: January 2007
Abstract We analyze the relationship between the labor tax composition (wage and payroll taxes) and the unemployment rate in an unionized economy with capital accumulation and a self-…nanced unemployment bene…t system. We show that when the unemployment bene…t system is budget-balanced then multiple equilibria can arise. This possibility depends on both the particular unemployment bene…t scheme considered and the …scal instrument used to balance the budget. Also, the e¤ects on employment of the labor tax composition strongly depend on the speci…c structure of the unemployment bene…ts. JEL Classi…cation Code: E24, E62, H53, J50, J65. Keywords: unemployment bene…t system, payroll tax, wage tax, multiple equilibria.
We are grateful to Salvador Bal le, Xavier Peris, Patricio García-Mínguez, Xavier Raurich, and specially to Jöerg Lingens for their valuable comments. Fernando Sánchez-Losada acknowledges …nancial support from the Generalitat de Catalunya through grant SGR2005-00984 and Ministerio de Ciencia y Tecnología through grant SEJ2006-05441. Daniel Cardona acknowledges …nancial support from the Ministerio de Ciencia y Tecnología through grant SEJ2006-05441. y Corresponding author: Departament d’Economia Aplicada; Universitat de les Illes Balears; Ctra. Valldemossa km. 7,5; 07071 Palma de Mallorca; Spain; e-mail:
[email protected]; Ph.: (+ 34) 971172845; Fax: (+ 34) 971173089.
1
1
Introduction
The provision of unemployment bene…ts for those who eventually become unemployed is a widely extended social program in most developed countries. These programs may di¤er because of a di¤erent concept of the bene…ts that they provide. In a Beveridgean system, unemployment bene…ts consist of a …xed amount that guarantees a minimal income to any person. This system is used in Australia, United Kingdom, Iceland, Ireland, New Zealand or Poland. Alternatively, in a Bismarckian system the unemployment bene…ts are determined as a proportion of an earnings base when employed. This system is observed in Belgium, Canada, France, Italy, Netherlands, Spain, Sweden or the United States when the earnings base is the gross wage, and in Austria or Germany when the earnings base is the net wage.1 Hence, we can also di¤erentiate between a gross and a net Bismarckian system, respectively. At the same time, while Beveridgean systems are not …nanced through speci…c taxes, Bismarckian systems are usually …nanced with speci…c labor taxes consisting on both a wage tax levied on workers and a payroll tax levied on …rms.2 Whereas in a neoclassical framework only the overall tax payments or tax burden distort the economy, but not its composition, this is not necessarily true when the wage setting process is non-competitive. In this paper, we analyze the relationship between the labor tax composition and the unemployment rate in an unionized economy with capital accumulation and a self-…nanced unemployment bene…t system. In particular, we ask the following questions: does a unique equilibrium exist? Does the composition of the tax burden (the sum of tax rates) or the tax wedge (the ratio between the net wage received by the worker and the wage paid by …rms) a¤ect employment? If the answer to the last question is positive, what speci…c labor tax structure maximizes employment? Finally, has the structure of the unemployment bene…ts any in‡uence on the previous questions/answers? Previous literature analyzing the relationship between the labor tax structure and unemployment in economies with a non-competitive labor market has followed three main lines of research. The …rst one focuses on the progresssivity of the tax system. Koskela and Vilmunen (1996) analyze income taxation under alternative wage setting processes to show that increasing the tax progression for workers a¤ects employment positively. In contrast, Pissarides (1998) shows that increasing the progressivity of the labor taxes paid by …rms also reduces unemploy1
The institutional details of the unemployment bene…t system for the di¤erent countries can be found in OECD (2004). 2 For instance, in 1998 the wage and payroll taxes were about 3.3% and 5.2% in France, 1.6% and 6.2% in Spain, both taxes were 3.25% in Germany, and in the U.S. only a payroll tax of 6.2% was collected; see OECD (2000).
2
ment. The second line of research considers the bene…t side of the unemployment bene…t system. Heer and Morgenstern (2005) study how restructuring the “progressivity”of the bene…ts a¤ects employment. They show that indexing unemployment bene…ts to wages (while maintaining the total employment bene…ts constant) is good for employment in a partial equilibrium framework. Lingens and Wälde (2005) show that a reduction of both the wage tax and the replacement ratio may be compatible with a self-…nanced net Bismarckian system and a lower unemployment.3 Egger (2002) concludes that unemployment may be lower if both unions bargain over wages and employment and the unemployment bene…ts contain a …xed part. Beissinger and Egger (2004), in a dynamic setting, show that a reduction of the replacement ratio increases unemployment. However, both Egger (2002) and Beissinger and Egger (2004) lack of a government (unemployment bene…ts) budget constraint. Goerke and Madsen (2003) show that an increase in the replacement ratio makes unemployment to decrease if unions maximize the rent from unionization. Nonetheless, in the last three papers the unions worry about the gross wage (and not the net wage) received by the workers. The third line of research considers some restriction relating taxes. Goerke (2000) imposes a constant tax wedge and …nds that shifting taxes from …rms to workers reduces unemployment when the alternative income (outside the …rm) of the worker is also taxed. Contrary, when the alternative income of the workers is not taxed, restructuring taxes has no e¤ects on employment. However, we think that a natural restriction to analyze changes in the labor tax composition is to consider a government budget constraint. Moreover, while a constant tax wedge implies a balanced government budget constraint under a Beveridgean unemployment bene…t system with proportional wage taxes (see Koskela and Schöb, 2005), this is not the case under alternative speci…cations of labor taxes or unemployment bene…ts. In fact, under a Beveridge system and a budget-balanced rule, restructuring taxes from …rms to workers a¤ects employment positively either when there is a tax exemption for workers (see Koskela and Schöb, 1999) or when the unemployment bene…ts are taxed at the same proportional tax rate than wages (see Picard and Toulemonde, 2001). In contrast with these papers, we show that when the government has an unemployment bene…t budget constraint then multiple equilibria can arise. Moreover, the e¤ects on employment of the labor tax composition depend on the speci…c structure of the unemployment bene…ts. In particular, under a gross Bismarckian unemployment bene…t system a reduction in the wage tax enhances employment, regardless of the equilibrium attained by the economy. 3
The replacement ratio is the proportion of the wage earned when working that is paid to the unemployed.
3
The …rst part of the paper analyzes the most widely extended system: the gross Bismarckian unemployment bene…t system. Such a system is …nanced exclusively through both a wage and a payroll tax. Since the government uses a budget-balanced rule, then equilibrium indeterminacy may arise, as it has been put forward by Schmith-Grohé and Uribe (1997) in a real business cycle competitive model, or by Rocheteau (1999a, 1999b) in a matching model with either a Bismarckian or a Beveridgean unemployment bene…t system …nanced only through payroll taxes.4 Moreover, Guo and Harrison (2004) …nd that such a possibility strongly depends on the speci…c instrument used to balance the budget.5 Our model corroborates these results. We show that when both the payroll and the wage tax are …xed and the replacement ratio is endogenous (in order to balance the unemployment bene…t budget), then the economy converges to a unique equilibrium. However, when the replacement ratio is …xed and both the payroll and the wage tax are endogenously determined, then a continuum of self-ful…lling equilibria arise. Nonetheless, this is a direct consequence of having two endogenous variables to balance the budget. In fact, as we reduce the instruments to balance the budget either a unique or (generically) two di¤erent unemployment rates may be attained. In particular, if the wage tax is the endogenous …scal instrument then two stable self-ful…lling equilibria arise. Expectations are crucial: if agents expect a higher wage tax rate, then they demand higher wages. Firms respond with a lower labor demand and, therefore, unemployment rises. Since the …nancial necessities of the unemployment system rise, the wage tax increases, which con…rms agents’ expectations. Thus, a wage demand spillover is created and unemployment rises. Alternatively, if agents expect a lower wage tax rate, they demand lower wages, too. Since …rms respond with a higher labor demand, the unemployment level and the wage tax decrease. Thus, two di¤erent equilibria are possible, one with a high level of employment (namely, the optimistic equilibrium) and the other with a high level of unemployment (the pessimistic equilibrium). In contrast, when the payroll tax is the endogenous …scal instrument then indeterminacy disappears and a unique equilibrium is attained. When the wage tax is endogenous, the level of unemployment determines the level of the wage tax via the budget-balanced requirement, but the wage tax in turn determines the level of unemployment via the wage bargaining. Instead, when the payroll tax is endogenous, this wage bargaining channel transmission disappears and, thus, the source for multiplicity vanishes. This is in contrast with Rocheteau (1999a, 1999b), who has multiplicity of equilibria when the payroll tax is endogenous. We show that in a gross Bismarckian system the relevant variable explaining unemployment is the tax wedge. When labor taxes are exogenous and the replacement ratio adjusts in order to 4 5
Hence, the Bismarckian system is at the same time gross and net. Schmitt-Grohé and Uribe (1997) suggest the same results, too.
4
satisfy the budget-balanced rule, the crucial role of the tax wedge explains why the composition of the tax burden (and not the total tax burden) in‡uences unemployment. In particular, an increase in the payroll tax accompanied by a decrease in the wage tax such that the tax burden remains constant causes the unemployment rate to fall. Moreover, both the replacement ratio and the net wage that workers receive increase. When the replacement ratio is kept …xed, we derive our main result: the lower the wage tax, the lower the unemployment rate, regardless of both the labor tax used to balance the budget and the equilibrium reached by the economy. Hence, a change in the labor tax composition that satis…es the budget-balanced rule a¤ects the tax wedge.6 Moreover, a lower wage tax is consistent with a lower payroll tax either when the wage tax is endogenous and the economy reaches the pessimistic equilibrium or under an exogenous relatively high wage tax rate. In such cases, employment increases because the tax distortion in the economy is reduced. Otherwise, a lower wage tax is associated with a higher payroll tax so that there is a shift of the tax incidence from wage to payroll taxes. The rationale in these cases is that a lower wage tax implies a lower wage demand, which a¤ects negatively the unemployment bene…ts. Hence, since the tax base is the same, the increase in the payroll tax needed to satisfy the budget-balanced rule may be smaller than the reduction in the wage tax, which makes the bargained wage to decrease. The rest of the paper is devoted to analyze alternative speci…cations of unemployment bene…t systems and bargaining patterns. While the tax wedge determines unemployment in the gross Bismarckian system, this is not the case when considering other situations. In particular, the unemployment rate does not depend on how the unemployment bene…t system is …nanced when the unemployment bene…t system is net Bismarckian. Moreover, in this case the equilibrium unemployment rate is unique and depends on the replacement ratio instead of the tax wedge or the tax burden. An usual feature of the Beveridgean systems is that there are no speci…c taxes to …nance the unemployment bene…ts. However, most of the papers considering constant unemployment bene…ts introduce speci…c taxes to …nance them. We also discuss this possibility by describing a scenario where an income tax for unemployed, a payroll tax, a wage tax and a tax exemption for the workers are present. In contrast with the gross Bismarckian system, two steady state equilibria are attained when the instrument used to adjust the budget-balanced rule is the payroll tax. We show that the labor tax composition does not a¤ect the unemployment rates of the economy if the unemployed do not pay taxes and the employed pay a constant marginal tax rate. Otherwise, an increase in the wage tax always makes the unemployment rate to decrease 6
Goerke (2000) considers changes in taxes that maintain the tax wedge constant, which in a gross Bismarckian system would imply that the unemployment bene…t budget is not necessarily balanced.
5
in the optimistic equilibrium and to increase in the pessimistic equilibrium. Moreover, as in Koskela and Vilmunen (1996), we have that an increase in the tax exemption yields a lower unemployment rate in the optimistic equilibrium. We also analyze a mixed bene…t system with a …xed and a variable part.7 We show that when the payroll tax is exogenous two steady state equilibria arise, but now the pessimistic equilibrium may be unstable. Moreover, unemployment is reduced in the optimistic equilibrium and increased in the pessimistic when the payroll tax rate is increased. While an increase in the payroll tax rate implies a reduction in the wage tax rate in the optimistic equilibrium, this is not necessarily true in the pessimistic one. In this case, an increase in the payroll tax rate is compatible with either an increase or a reduction in the wage tax, depending on the parameter con…guration. The negotiation between unions and …rms is a right-to-manage one, where unions focus on the (net) wage mass but bargain only on wages. We show that the qualitative results remain unchanged when employment is also bargained on, and in a seniority model. Thus, we obtain the same result than Creedy and McDonald (1991), who show that the qualitative e¤ects of taxes on employment do not depend on the type of negotiation, but a di¤erent one to Cardona and Sánchez-Losada (2006), who analyze a segmented labor market. The paper is also related to the literature concerning …scal increasing returns, …rstly analyzed by Blanchard and Summers (1987). They show that increases in employment and output allow for a decrease in taxes such that the tax cuts become self-…nancing. Therefore, the increase in the tax base compensates the decrease in the tax rate. Our paper shows that the existence of …scal increasing returns: (1) depends on the considered unemployment bene…t system; (2) is compatible with an endogenous government expenditure; (3) is compatible with a tax increase if there is an appropriate decrease of another tax; (4) does not depend on the considered decentralized wage bargaining. The paper is organized as follows. In the next section we present the economy. In section 3, we analyze the gross Bismarckian unemployment bene…t system. In section 4, we discuss alternative unemployment bene…t systems and bargaining patterns. Finally, section 5 concludes. 7
Curiously, such system has been only considered when studying how a reallocation of the unemployed bene…ts between the indexed part and the constant part a¤ects employment while maintaining the total unemployed bene…ts constant (see Vijlbrief and Wijngaert, 1995, Schluter, 1997, Pissarides, 2000, Goerke and Madsen, 2003, Heer and Morgenstern, 2005, or Lingens and Wälde, 2006).
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2
The economy
We construct a very simple economy with constant population, whose mass is normalized to one. Individuals are endowed with one unit of labor which they o¤er inelastically at each period t. The market of goods is competitive. Individuals. We assume Solow individuals: each one saves a constant fraction of her income. Therefore, savings for the individual i are (1)
sit = s Iit ; where Iit is the individual income, and s 2 (0; 1) is the constant propensity to save.8
Firms. Each …rm j produces according to a constant returns to scale Cobb-Douglas technology given by 1 (2) Yjt = AKjt Ljt ; where Kjt and Ljt denote the capital and labor used by …rm j at period t, A > 0; and We assume that capital fully depreciates in one period.
2 (0; 1).
Union bargaining: The negotiation between (risk-neutral) unions and …rms is a rightto-manage one, where unions focus on the (net) wage mass but bargain only on wages. This bargaining takes place at a decentralized level, so that neither the …rm nor the union perceive the e¤ects of their actions on the economy via the unemployment bene…t budget constraint.9 The timing of the bargaining is as follows: once the …rm j has selected the level of capital, the …rm and the union bargain over the wage wjt .10 Then, the …rm chooses the employment level. Hence, given wjt and Kjt , the employment level chosen by any …rm solves 1 max AKjt Ljt Ljt
wjt Ljt (1 +
t)
(1 + rt )Kjt ;
where rt is the interest rate and t is a proportional tax on the wage paid by the …rm, referred as payroll tax. The optimal condition implies (1 +
t )wjt
1 = A Kjt Ljt 1 .
8
(3)
We obtain the same savings using a Cobb-Douglas utility function and an overlapping generations economy à la Diamond. If instead we assume an in…nite horizon economy, we would recover the same savings only in steady state. 9 In the fourth section, we show that the qualitative results remain unchanged when employment is also bargained on, and in a seniority model. The decentralized bargaining considered in the paper is adapted from Layard and Nickell (1990). 10 The …rm bargains with the union since it looks for greater income for the capital owners.
7
The objective function of the union in the bargaining process is given by the net wage mass, while the …rm’s objective are pro…ts. The disagreement point of the union is given by wt while the …rm’s fall-back position is (1 + rt )Kt ; since at this point its level of capital has been already selected. Thus, according to the Nash solution and denoting the bargaining power of the union and the …rm by and (1 ) ; respectively, with 2 [0; 1], the wage solves max [(wjt (1 wjt
t)
wt ) Ljt ]
1 AKjt Ljt
wjt Ljt (1 +
t)
1
subject to the labor demand given by equation (3), where t is a proportional tax on the wage paid by the worker, referred as wage tax. The optimal wage is (implicitly) de…ned by wjt (1
t)
(4)
= Qwt ;
where Q = [ + (1 )] = > 1. Thus, the net wage is set as a mark-up of the union’s disagreement point. Therefore, as long as the payroll tax does not in‡uence the union’s disagreement point, it does not a¤ect the wage setting curve. Since the …rm anticipates both the level of employment and the wage, it chooses the level of capital that solves max Kjt
1 AKjt Ljt
wjt Ljt (1 +
t)
(1 + rt )Kjt
subject to equations (3) and (4). Solving this problem yields the capital demand.11 Equilibrium: The wage paid by any …rm wjt coincides with the average wage of the economy wt , because all the …rms have the same bargaining power. Therefore, equation (3) also holds in aggregate; i.e., (5) (1 + t )wt = A Kt1 Lt 1 ; where Lt is the aggregate employed labor and Kt is the aggregate capital of the economy. Following Layard and Nickell (1990), in equilibrium the disagreement point is given by wt = Lt wt (1
t)
+ (1
Lt ) Bt ;
(6)
where the …rst part is the probability of …nding a job elsewhere times the net wage and the second part is the probability of being unemployed times the unemployment bene…ts Bt . Note that because the population is normalized to one the employment and the unemployment rates are Lt and (1 Lt ), respectively. 11
Since the propensity to save is constant, the interest rate is not needed to …nd the equilibrium.
8
An important feature to be emphasized is the dimension of the unions’disagreement alternative. It is a convex combination of the unemployment bene…ts and the wage that a worker can get by returning to the market. Hence, in terms of Layard and Nickell (1990), by including such a disagreement outcome (outside opportunities) we “capture the notion that individuals who do not gain employment in the particular …rm under consideration have potential opportunities elsewhere”.
3
Payroll or wage taxes?
In this section, we focus on a self-…nanced gross Bismarckian unemployment bene…t system. Even though unemployment bene…ts are a proportion of the wage received by the worker when employed, in order to make the analysis more tractable we consider that such bene…ts are related to the average wage of the economy wt , as in Layard and Nickell (1990), Manning (1993) or Altenburg and Straub (1998).12 Therefore, the unemployment bene…ts are Bt = t wt ; where t 2 (0; 1) is usually known as the bene…t replacement ratio. Thus, equation (6) becomes wt = Lt wt (1
t)
+ (1
Lt ) t wt :
(7)
The system is …nanced by the wage and the payroll taxes. Accordingly, the unemployment bene…t budget constraint is given by wt Lt ( t +
t)
= (1
Lt ) t wt :
(8)
Note that only …rms and employed agents are taxed in order to …nance unemployment bene…ts. An important aspect of this budget-balanced rule is that revenues are also endogenous, since both employment and wages are endogenous. Hence, and in contrast with many models that only consider revenue-neutral rules, …nancial necessities may vary when unemployment changes.13 12
Beissinger and Egger (2004) consider that unions are aware on the possibility that workers currently employed could be …red in the next period and, therefore, unions realize that the bargained wage of today a¤ects the unemployment bene…ts of tomorrow. However, they disregard that …rms should also realize that the bargained wage of today a¤ects the bargained wage of tomorrow. Nonetheless, if unions only worry about the current wage of the workers of the …rm, as in Oswald (1985), then the steady state equilibria would remain unaltered. 13 There exists some confusion in some papers between a budget-balanced and a revenue neutral rule when the unemployment bene…t system is analyzed. Since we consider a self-…nanced program, tax revenues are required in order to (exactly) compensate the unemployment bene…t spending. Hence, the reasonable assumption should be to require a budget-balanced rule.
9
We begin the analysis in a static framework where the …rms’ capital remains constant. Later, we show that in a dynamic economy where capital accumulation is considered, the main conclusions remain unchanged.
3.1
The static framework: short-run equilibria
First, we analyze the indeterminacy problem. The unemployment bene…t budget constraint (8) can be written as t Lt = . (9) + t t+ t Using equations (4) and (7) yields Lt =
1 Q [1
Q
t
t t]
t
(10)
:
Then, from equations (9) and (10) we have that any pair ( t ; 1 Q [1
Q
t t
t t]
t
= t
+
t
+
t)
satisfying (11)
t
constitutes an equilibrium, which shows that multiple self-ful…lling equilibria may arise. Moreover, combining equations (9) and (10) yields the employment as a function of the tax wedge,14 Lt =
1 1+
t
Q 1.
(12)
t
As we can observe from this equation, when unemployment bene…ts are …nanced through …xed labor tax rates (and hence the replacement ratio becomes endogenous), there exists only one equilibrium, which is determined by the tax wedge. Moreover, the same tax burden is compatible with di¤erent tax wedges and, therefore, with di¤erent unemployment rates. In particular, shifting the tax incidence from wage to payroll taxes such that the tax burden is kept constant reduces unemployment. Proposition 1 When the replacement ratio is used to balance the unemployment bene…t budget constraint, then a reduction in the wage tax such that the tax burden remains unaltered increases employment. 14
We call tax wedge to the ratio between the net wage received by the worker and the wage paid by the …rm, i.e. (1 t ) = (1 + t ) ; as in Blanchard and Summers (1987) and Goerke (2000). In other papers it is the sum of the tax rates (what we call the tax burden) what is called the tax wedge; see for instance Nickell et al. (2005).
10
Proof. Di¤erentiating equation (12) and using d sign(dLt ) = sign (d[(1 = sign ( which is positive when d
t
t
+d
t
= 0, we have that
t
t )= (1
+
t )])
=
2 t)
+ t ) d t = (1 +
,
(13)
> 0:
When the replacement ratio is …xed and both the payroll and the wage tax are endogenous, then equation (11) shows that there exists a continuum of pairs ( t ; t ) compatible with the replacement ratio. Moreover, such a continuum of tax pairs induces a continuum of di¤erent tax wedges and, therefore, a continuum of equilibria arise. Instead, when the replacement ratio and one of the two labor taxes are …xed, the possibility of having a continuum of selfful…lling equilibria disappears, but the possibility of having indeterminacy remains unaltered. In particular, when the wage tax is …xed (and hence the payroll tax becomes endogenous), it turns out from equation (10) that for any t the economy reaches a unique equilibrium. Likewise, when the payroll tax is …xed (and hence the wage tax becomes endogenous), from equation (9) we have that ( t + t ) Lt t , (14) t = Lt and substituting for t from equation (14) into equation (10) we get Q (1 +
2 t ) Lt
(1 +
t
+
t ) Lt
+
t
(15)
= 0;
which has the following two solutions, Lt =
(1 +
t
+
t)
(1 + t + t )2 2Q (1 + t )
4 t Q (1 +
t)
1=2
:
(16)
Hence, two di¤erent and consistent with the same payroll tax equilibria (generically) arise.15 Note that the government can …x a speci…c replacement ratio and avoid indeterminacy by …xing the wage tax and adjusting the payroll tax to balance the unemployment bene…t budget constraint. Proposition 2 When the wage tax is used to balance the unemployment bene…t budget constraint, two unemployment rate equilibria arise. However, there exists a unique equilibrium when the payroll tax is used to balance the unemployment bene…t budget constraint. 2
15
For an equilibrium to exist, the condition (1 + t + t ) 4 t Q (1 + 2 when (1 + t + t ) = 4 t Q (1 + t ), a unique equilibrium is attained.
11
t)
has to be imposed. Moreover,
Di¤erent expectations make the same …xed payroll tax to be compatible with two di¤erent equilibria, one with a low unemployment rate and the other with a high one. When unions expect a low unemployment rate, they also expect a low wage tax and, thus, they can reduce their wage demand at the same time that the net wage increases, what makes unemployment to stay low. We call this equilibrium the “optimistic equilibrium”. However, when unions expect a high unemployment rate, then they also expect a high wage tax. Therefore, and in view of an expected low net wage, their wage demand increases, what makes unemployment to rise. We call this equilibrium the “pessimistic equilibrium”. Moreover, since in the short-run capital per …rm is constant, from the right-to-manage condition we can observe that the gross wage paid by …rms in the pessimistic equilibrium is higher than that of the optimistic one. Therefore, the unemployment bene…ts are also higher in the pessimistic equilibrium. Figures 1 to 3 illustrate these statements for the case where = 2=3; = 0:1 and = 0:65. In particular, Figure 1 represents the pairs ( t ; t ) compatible with the budget-balanced rule (11); in Figure 2 employment is related to the wage tax rates using equation (10); and Figure 3 relates the payroll tax to the employment level using equations (10) and (11). The U-shaped relationship between both taxes shows when the budget-balanced rule causes multiplicity. In particular, given a wage tax, there is only one payroll tax that satis…es the budget constraint. Instead, given a payroll tax, there are two wage taxes satisfying the budget constraint and, therefore, two employment levels arise.
Figure 1: Compatible tax rates.
12
Figure 2: Wage taxes and employment.
Figure 3: Payroll taxes and employment.
We show next how a tax reallocation between workers and …rms a¤ects the unemployment rate when the replacement ratio is …xed. Proposition 3 Given a replacement ratio, a decrease in the wage tax reduces the unemployment rate of the economy. Proof. When the wage tax is exogenous, it is direct to show from equation (10) that dLt t (1 = d t Q [1
Q) t
2 t]
(17)
< 0;
since Q > 1. Thus, a decrease in the wage tax implies an increase in the employment rate. When the payroll tax is exogenous, from equation (15) we have dLt = d t
Lt (QLt 1) 2Q (1 + t ) Lt (1 +
t
+
t)
Lt
=
(1 +
t
(Q t
+
1) = (1 2
t)
t
4 t Q (1 +
t) t)
1=2
;
(18)
where we have used equation (10) in the numerator and equation (16) in the denominator. Thus, in the optimistic equilibrium we have that dLt =d t > 0 and in the pessimistic equilibrium dLt =d t < 0. From equation (10) we have that QLt < 1. Moreover, from this equation we get Q [1
t
t ] Lt
(1 13
t
Q t ) = 0;
(19)
where Lt is a function of gives
t
only (since
d d
t
is endogenous). Implicit di¤erentiation of this equation Q [1
t
=
t
QLt
t
t]
@Lt @ t
1
;
(20)
which is negative at the optimistic equilibrium and positive at the pessimistic one. Note that this result always holds regardless of the …scal instrument that the government uses in order to balance the unemployment bene…t budget constraint. Then, given a replacement ratio, unemployment is minimized if unemployment bene…ts are paid exclusively by …rms. When the payroll tax is exogenous, the reduction in the wage tax is induced by an increase in the payroll tax in the optimistic equilibrium. However, in the pessimistic equilibrium the wage tax decreases as far as the payroll tax is reduced, too. In both cases the unemployment rate decreases. A reduction of the wage tax decreases the gross wage paid by …rms and increases the net wage received by the employees in both equilibria, while unemployment bene…ts are reduced in the optimistic equilibrium and may decrease or increase depending on the speci…c parameter values in the pessimistic one.
3.2
The dynamics: long-run equilibria
Next, we analyze whether capital accumulation alters the conclusions obtained in the static model. In fact, similar results are obtained, but in the dynamic setup low unemployment rates imply high capital stocks, which reinforces employment. The introduction of capital implies the existence of a capital market clearing condition: the amount saved at period t equals the stock of physical capital at t + 1, i.e.,16 Z 1 Z 1 sit di = s Iit di = sF (Kt ; Lt ) = sAKt1 Lt ; (21) Kt+1 = 0
0
from where we can isolate the employment level, Lt =
Kt+1 sAKt1
16
1
:
(22)
Note that the entire income which is generated in the economy is distributed across workers and capital owners. Since individuals can have di¤erent income source weights, the expected income of all union members is not maximal if the economy’s output is maximal. Thus, unions will pursue to raise wages.
14
Note that this equation implies that in steady state the capital-labor proportion K=L is …xed regardless of the unemployment bene…t system. However, it must be noted that capital per capita could be a¤ected by the unemployment bene…t system. Using the unemployment bene…t budget constraint (8) and the de…nition of the disagreement point (7), we get wt = wt Lt (1 + t ); (23) which combined with equations (4) and (22) yields (1
t)
= Q (1 +
t)
And from equations (8), (5) and (22), we have " 1
Kt+1 ( t +
t)
= (sA)
1
t
1
Kt+1 sAKt1
1
Kt+1 sAKt1
1
(24)
:
#
1
Kt
.
(25)
Equations (24) and (25) recover the dynamics of the economy. As in the static case, equation (24) indicates that when unemployment bene…ts are …nanced through …xed tax rates and hence the replacement ratio is endogenous, there exists only one steady state equilibrium. Likewise, when the replacement ratio is …xed and both the payroll and the wage tax are endogenous, there exists a continuum of steady state equilibria which di¤er in the capital stock and the unemployment rate. Similar conclusions are derived when the replacement ratio is kept …xed and only one labor tax is endogenous. When both the replacement ratio and the payroll tax are …xed, and hence the wage tax adjusts to balance the unemployment bene…t budget constraint, substituting for t from equation (24) into equation (25) yields the capital dynamics as 1
(1 +
t
+
t ) Kt+1
Q (sA)
1
1
Kt
2
Kt+1 (1 +
t)
(sA)
1
1
t Kt
= 0,
(26)
from where we have two positive steady state equilibria implicitly de…ned by17 Q (sA)
1
K 2 (1 + )
(1 +
+ ) K + (sA)
1
(27)
= 0:
Solving this equation yields h (sA) (1 + K= 2Q (1 + ) 1
17
+ )
(1 +
We do not consider the zero capital equilibrium.
15
+ )2
4Q (1 + )
1=2
i
;
(28)
from where we have a pessimistic equilibrium K1 and an optimistic one K2 , with K1 < K2 . Since from equation (22) in steady state the capital-labor proportion K=L is …xed, then the higher the capital is, the higher the employment rate is. Moreover, di¤erentiating equation (26), evaluating the resulting expression in steady state, and using equation (27), we have dKt+1 dKt
=1
(29)
;
Kt+1 =Kt
which indicates that both steady state equilibria are stable without cycles. Although we observe from equations (5) and (22) that in any steady state the gross wage paid by the …rm is constant, labor demand is not. It depends on the capital level of the economy, which may di¤er depending on the agents’ expectations about wage taxes. Hence, given a payroll tax rate, the equilibrium negotiated wage is the same in both equilibria, which implies that whereas the unemployment bene…ts are the same in both equilibria, the net wage is higher in the optimistic equilibrium than in the pessimistic one. When the replacement ratio and the wage tax are …xed, and hence the payroll tax adjusts to balance the unemployment bene…t budget constraint, substituting for t from equation (24) into equation (25) yields the capital dynamics as 1
(1
t
t ) Kt+1
(1
t) Q
1
1
t
1
(sA) Kt
= 0,
(30)
which leads to a unique steady state equilibrium, K=
(1
1
Q ) (sA) . Q (1 )
(31)
Di¤erentiating equation (30), evaluating the resulting expression in steady state, and using equation (31), we have dKt+1 =1 ; (32) dKt Kt+1 =Kt which indicates that the steady state equilibrium is stable without cycles. The next proposition summarizes these results. Proposition 4 When the wage tax is used to balance the unemployment bene…t budget constraint, two stable without cycles steady state equilibria arise. However, when the payroll tax is used to balance the unemployment bene…t budget constraint, there exists a unique stable without cycles steady state equilibrium. 16
In this economy, the steady state capital-labor proportion is …xed regardless of the unemployment bene…t system, which implies a constant gross wage paid by …rms. This is a crucial di¤erence with respect to the static model. In this dynamic setup, the capital adjusts in order to maintain the labor productivity in the long-run. Hence, if employment is a¤ected by the …scal policy, then capital per capita is a¤ected accordingly. In fact, in steady state the e¤ects on employment from a change in the composition of the labor taxes are identical to those in the static model: a reduction of the wage tax has an unambiguously positive e¤ect on employment regardless of the two scenarios we consider, either an exogenous or an endogenous wage tax.18 Obviously, the dynamics of wages can be di¤erent although the steady state gross wage paid by …rms remains constant. Next, we state and prove this result. Proposition 5 Given a replacement ratio, a decrease in the wage tax reduces the unemployment rate of the economy. Proof. When the wage tax is exogenous, it is direct to show from equation (31) that 1
dK (sA) = d Q [1
(1
Q) < 0: ]2
(33)
When the payroll tax is exogenous, implicit di¤erentiation of equation (27) yields 1
K 1 Q (sA) K dK = 1 d 2Q (sA) K (1 + ) (1 +
;
(34)
+ )
which numerator is positive, because equation (24) implies 1
Q (sA)
1
K =
1
1 1+
> 0;
(35)
and, from equation (28), the denominator is positive at the optimistic equilibrium and negative at the pessimistic one. Therefore, an increase in the payroll tax increases (reduces) capital and therefore employment in the optimistic (pessimistic) equilibrium. Di¤erentiating equation (24) evaluated in steady state yields d = d
Q (sA)
1
K + (1 +
18
t)
dK d
:
(36)
Proposition 1 can also be applied in the long-run. Hence, when the replacement ratio is used to balance the unemployment bene…t budget constraint, a decrease in the wage tax accompanied by an increase in the payroll tax such that the tax burden remains constant implies a decrease in the unemployment rate.
17
Substituting equation (34) into equation (36), and using equation (24) we have 1
d Q (sA) K (1 = 1 d 2Q (sA) K (1 + )
) (1 +
;
(37)
+ )
which is negative (positive) in the optimistic (pessimistic) equilibrium since the denominator is positive (negative). Therefore, a decrease in the payroll tax implies an increase in the capital level (and thus in the employment rate) and a decrease in the wage tax in the pessimistic equilibrium, whereas in the optimistic equilibrium an increase in the payroll tax implies an increase in both the capital level and the employment rate and a decrease in the wage tax. The introduction of capital does not change the results: the unemployment rate is minimized when bene…ts are paid exclusively by …rms. Moreover, depending on the equilibrium attained, a reduction in the wage tax is consistent with either an increase or a decrease in the payroll tax. Hence, the evolution of the wage earnings depends on the speci…c equilibrium the economy is converging to. In the pessimistic equilibria, characterized by a high wage tax, a decrease in the payroll tax implies a reduction in the wage tax. This causes the wage to increase and, thus, both the net wage and the unemployment bene…ts increase. However, in the optimistic equilibria, wages are reduced by shifting taxes from workers to …rms. Hence, unemployment bene…ts are reduced while net earnings increase and, hence, there are more workers with higher net earnings and less unemployed with lower bene…ts.19
4
Alternative unemployment bene…t systems and bargaining patterns
In this section, we consider alternative unemployment bene…t systems. First, we analyze the case where unemployment bene…ts are tied to net earnings. We show that a budget-balanced rule induces a unique steady state unemployment rate which is independent of the labor tax composition. Second, we consider a Beveridgean system where unemployment spending is …nanced through both an income and a payroll tax. Finally, we analyze alternative bargaining patterns. 19
A reduction of unemployment increases inequality. However, previously unemployed agents become employed and get higher earnings. Hence, a welfare analysis should contemplate also such shift from unemployment to employment.
18
4.1
Net Bismarckian system
The net Bismarckian system is characterized by unemployment bene…ts being a proportion of 20 the net wage; i.e. Bt = t wt (1 t ). In this case, the budget-balanced rule is given by wt Lt ( t +
t)
= (1
Lt ) t wt (1
(38)
t ):
In equilibrium, the disagreement point of the unions becomes wt = wt Lt (1
t)
+ (1
Lt ) t wt (1
t) .
(39)
And combining it with equation (4), we obtain Lt =
Q 1 1
t
.
(40)
t
Hence, there exists an unique equilibrium, which is independent on how spending is …nanced. Unemployment is exclusively explained by the technology, the unions’ bargaining power and the replacement ratio. Moreover, combining equations (38) and (40) we get t
=
t
(1
t)
Q 1 1 Q t
t
which shows that the equilibrium is consistent with di¤erent labor tax combinations.21 Proposition 6 Given a replacement ratio, in a net Bismarckian unemployment bene…t system the labor tax composition does not a¤ect the unique unemployment rate of the economy.
4.2
Beveridgean system
Unemployment bene…ts are …xed in a Beveridgean unemployment bene…t system; i.e., Bt = bt . As in the analysis of the previous systems, a budget-balanced rule is considered. In general, we assume that the unemployment bene…ts bt are taxed at a di¤erent rate bt than wages.22 For simplicity, we …x bt = t with 2 [0; 1]. We assume that the net wage is wt
t
(wt
abt )
abbt = wt
20
t
[wt
abt (1
)] ;
(41)
We consider, as in the gross Bismarckian system, that the unemployment bene…ts are related to the average wage of the economy. See footnote 12. 21 Note that this equation also indicates that the tax wedge must be constant. 22 It would be more appropriate to refer to these taxes as income taxes.
19
where a 0 is an arbitrary constant. This formalization allows us to consider alternative scenarios: (1) if a = = 0 we have a system where the unemployed do not pay taxes while workers pay the same marginal tax rate for all their income; (2) if a = 1 then we have an income tax where workers face two tax brackets: they pay the same marginal tax rate than the unemployed for the same income but they pay a higher marginal tax rate for the rest of their income; (3) if a = 0 and 6= 0 then both the unemployed and the employed pay a di¤erent marginal tax rate for all their respective income; (4) if = 1 then all individuals pay the same marginal tax rate for all their income; and (5) if = 0 and 0 < a 6= 1 then employed have a …scal exemption (namely, a0 = ab): This last case coincides with Koskela and Vilmunen (1996). Now, equation (4) is given by wt (1
t)
t
= Qwt ;
)] = (1
Lt ) (1
+ Qabt (1
)
(42)
and the budget-balanced rule is wt Lt
t
+ Lt t [wt
abt (1
(43)
t ) bt :
The disagreement point in this case can be written as wt = Lt (wt
t
[wt
abt (1
)]) + (1
Lt ) (1
t ) bt .
(44)
From equations (43) and (44) we have wt = Lt wt (1 +
(45)
t) :
And from equations (5), (42) and (45) we get AKt1
Lt
1
(1 (1 +
t) t)
+ Qabt (1
)
t
= Q AKt1
Lt :
) + bt (1
Lt ) (1
(46)
Combining equations (5) and (43) we obtain AKt1
Lt = AKt1
Lt
(1 (1 +
t) t)
+ abt Lt t (1
t) .
(47)
Equations (46), (47) and (21) yield the dynamics of the economy. We consider the case where the instrument used to adjust the unemployment bene…t budget is the payroll tax. Substituting the payroll tax from (46) into (47), and evaluating in steady state, gives AK 1
L
+1
Q
AK 1
L + b (1
L) (1
) + abL (1 20
)
Qab (1
) L = 0; (48)
which can be written as G (L; ) = A (K=L)1 A (K=L)1
+ (Q
1) ab (1
QL2
v) + b (1
) L + b (1
) = 0:
(49)
From equation (22) we have that K=L is constant in steady state. Hence, and in contrast with the gross Bismarckian system, equation (49) yields (generically) two steady state employment rates, one pessimistic L1 and the other optimistic L2 , with L1 < L2 .23 Moreover, since equation (49) is quadratic, we have @G (L; ) @L
< 0 and L=L1
@G (L; ) @L
> 0.
(50)
L=L2
Di¤erentiating equation (49) yields @L (1 = @
L) b + (Q 1) ab (1 @G (L; ) =@L
)L
,
(51)
and hence, @L @L 0 and 0. (52) @ L=L1 @ L=L2 Thus, the e¤ects of restructuring taxes are completely opposed to those obtained in the gross Bismarckian system. Except in the case that a = = 0; an increase in the wage tax always reduces (increases) the steady state unemployment rate in the optimistic (pessimistic) case. Therefore, having either two tax brackets (a = 1) or an exemption ( = 0 and 0 < a 6= 1) are qualitatively equivalent. Moreover, we have that an increase in the tax exemption yields a lower unemployment rate in the optimistic equilibrium. This result coincides with Koskela and Vilmunen (1996). When the unemployed do not pay taxes and the employed pay the same marginal tax rate for all their income, a = = 0, then @L=@ = 0, as in the net Bismarckian system. In the Appendix we show, as a subcase of the mixed unemployment bene…t system, that in this case the optimistic equilibrium is always stable whereas the pessimistic equilibrium may now be unstable. Proposition 7 Given a level of bene…ts, in a Beveridgean unemployment bene…t system the labor tax composition does not a¤ect the unemployment rates of the economy if and only if the unemployed do not pay taxes and the employed pay a constant marginal tax rate. Otherwise, an increase in the wage tax causes the unemployment rate to decrease in the optimistic equilibrium and to increase in the pessimistic equilibrium. 23
Two steady state equilibria are also attained when the wage tax is endogenous. We do not consider this case and analyze the case where the payroll tax is endogenous because the results di¤er from those of the gross Bismarckian system in the last case.
21
4.3
Mixed unemployment bene…t system
Given the assumptions that marginal taxes are constant and exclusively levied on workers and …rms, the implications of a change in the labor tax composition have been shown to be di¤erent under a gross Bismarckian from those under a Beveridgean system. What are, then, the e¤ects of a change in the labor tax composition when the unemployment bene…ts consist of both a …xed part bt and a variable part wt ? Since when the unemployed do not pay taxes the labor tax composition does not a¤ect employment in a Beveridgean system, intuition would suggest that the positive e¤ect on employment of a reduction in the wage tax under a gross Bismarckian system should prevail. But this is not always the case. In the Appendix it is shown that when both the replacement ratio and the payroll tax are exogenous, two steady state equilibria arise. Unemployment is reduced in the (stable) optimistic equilibrium if there is a shift of the tax incidence from wage taxes to payroll taxes. In the pessimistic equilibrium, unemployment is reduced if the payroll tax is decreased, which may be compatible with either an increase or a decrease in the wage tax. Moreover, the pessimistic equilibrium may be either stable (with or without cycles) or unstable (see the numerical simulations in the Appendix).
4.4
Alternative bargaining patterns in the gross Bismarckian system
Bargaining over wage and employment. We have considered that the negotiation is a right-to-manage one. A di¤erent situation arises when the unions bargain on both the wage and the employment level.24 In this case, after the capital level has been selected by the …rm, the Nash solution implies that the wage and the employment level solve max [(wt (1
fLt ;wt g
t)
w t ) Lt ]
AKt1
Lt
wt Lt (1 +
t)
1
;
from where the following optimal conditions are satis…ed: wt (1 + wt Lt (1 +
t)
t)
= AKt1
= (1
t)
Lt + (1
AKt1
Lt
)wt (1 +
1
(53)
;
t ) Lt = (1
t) :
(54)
Substituting for the payroll tax from equation (53) into equation (54) we obtain wt (1
t)
24
= Qwt :
(55)
This model is usually called the e¢ cient bargaining model. However, since in our model there is a previous decision on capital that is not considered in the bargaining, we do not use this term in order to avoid any confusion about e¢ ciency.
22
And using equations (7) and (55) we get the employment as a function of the wage tax, Lt =
1 Q (1
tQ
t t
t)
:
(56)
Note that this is the same condition (10) than that in the right-to-manage model and, thus, the same results apply. Seniority. In this case, employment is not considered by unions, who only focus on wages (see, for instance, Oswald, 1985). Then, …rms and unions bargain over wages and the …rm has the right-to-manage. While both the unemployment bene…t budget constraint and the disagreement point are the same as in section 3, the optimal condition from the Nash problem is now slightly di¤erent. Combining such condition with the right-to-manage condition (5) yields b wt (1 (57) t ) = Qw t ;
b = (1 where Q )=( ) is assumed to be greater than 1, indicating that the net wage received by workers is higher than the disagreement alternative.25 Hence, all the results of b section 3 remain qualitatively unchanged (just replace Q by Q).
Corporatist institutions. The analysis of this paper assumes a decentralized negotiation. Thus, both unions and …rms bargain at the …rm level without taking into account the indirect e¤ects that their decisions have on the unemployment bene…t system. In contrast, in a corporatist economy or an economy with a centralized negotiation, unions and …rms would take into account this indirect e¤ect and, as a result, unemployment would be lower.26 Moreover, multiplicity would disappear, because expectations would be coordinated among the agents.
5
Conclusions
In the context of an unionized economy with capital accumulation and a self-…nanced unemployment bene…t system, the existence of an unemployment bene…t budget-balanced rule can generate multiple equilibria. We show that the speci…cation of the unemployment bene…ts scheme determines the existence (or not) of multiplicity. In particular, a unique equilibrium is 25
Note that from (57) must be higher than since otherwise the net wage received by the workers would b > 1. be negative. Moreover, > implies that Q 26 A similar situation is presented in Goerke and Madsen (2003) to show that the e¤ects of restructuring the unemployment bene…ts are di¤erent than those of a model where unions do not consider the aggregate e¤ects of their decisions.
23
always attained in case that the unemployment bene…ts consist of a …xed proportion of the net wage (net Bismarckian system). Otherwise, when the unemployment bene…ts are either a …xed proportion of the gross wage (gross Bismarckian system) or a constant payment (Beveridgean system) multiple equilibria may arise. Moreover, in the gross Bismarckian system such a possibility depends crucially on the …scal instrument used to balance the budget. Hence, both the unemployment bene…t system and the speci…c tax used to equilibrate expenditures a¤ect the possibility of multiple equilibria. The analysis reveals the particular relevance of the labor tax composition on unemployment. In particular, the paper emphasizes the role of the labor tax composition in determining unemployment. Moreover, the institutional setup regulating the unemployment bene…t system reveals to be crucial. While the tax wedge is directly related to unemployment in countries with a gross Bismarckian system, what is important when the unemployment system is net Bismarckian is the replacement ratio. Likewise, when the unemployment system is Beveridgean, identifying the taxes paid by the unemployed might be useful to explain unemployment. All qualitative results hold if alternative types of negotiation between unions and …rms are considered.
24
References Altenburg, L. and M. Straub, 1998, E¢ ciency Wages, Trade Unions, and Employment, Oxford Economic Papers 50, 726-46. Beissinger, T. and H. Egger, 2004, Dynamic Wage Bargaining if Bene…ts Are Tied to Individual Wages, Oxford Economic Papers 56, 437-60. Blanchard, O.J. and L.H. Summers, 1987, Fiscal Increasing Returns, Hysteresis, Real Wages and Unemployment, European Economic Review 31, 543-66. Cardona, D. and F. Sánchez-Losada, 2006, Unions, Quali…cation Choice, and Output, Oxford Economic Papers 58, 50-76. Creedy, J. and I.M. McDonald, 1991, Models of Trade Union Behaviour: A Synthesis, Economic Record 67, 346-59. Egger, H., 2002, Unemployment May Be Lower if Unions Bargain over Wages and Employment, Labour 16, 103-33. Goerke, L., 2000, The Wedge, Manchester School 68, 608-23. Goerke, L. and J.B. Madsen, 2003, Earnings-related Unemployment Ben…ts and Unemployment, Economic Systems 27, 41-62. Guo, J.T. and S.G. Harrison, 2004, Balanced-Budget Rules and Macroeconomic (In)stability, Journal of Economic Theory 119, 357-63. Heer, B. and A. Morgenstern, 2005, The Labor market E¤ect of Indexing Unemployment Bene…ts to Previous Earnings, Public Finance Review 33, 385-402. Koskela, E. and R. Schöb, 1999, Does the Composition of Wage and Payroll Taxes matter under Nash Bargaining?, Economics Letters 64, 343-49. Koskela, E. and R. Schöb, 2005, The Structure of Labour Taxation and Unemployment in E¢ ciency Wage Models, Helsinky Center of Economic Research, Discussion Paper 59. Koskela, E. and J. Vilmunen, 1996, Tax Progression Is Good for Employment in Popular Models of Trade Union Behaviour, Labour Economics 3, 65-80.
25
Layard, R. and S. Nickell, 1990, Is Unemployment Lower if Unions Bargain over Employment?, Quarterly Journal of Economics 105, 773-87. Lingens, J. and K. Wälde, 2005, Pareto-Improving Unemployment Policies, Université catholique de Louvain, Département des Sciences Economiques, Discussion Paper 2005-033. Manning, A., 1993, Wage Bargaining and the Phillips Curve: The Identi…cation and Speci…cation of Aggregate Wage Equations, The Economic Journal 103, 98-118. Nickell, S., Nunziata, L. and W. Ochel, 2005, Unemployment in the OECD Since the 1960s: What Do We Know?, The Economic Journal 115, 1-27. OECD, 2000, Taxing Wages. Taxes on Wages and Salaries, Social Security Contributions for Employees and their Employers, Child Bene…ts. 1998-1999. Paris. OECD, 2004, Bene…ts and Wages: OECD Indicators. Paris. Oswald, A.J., 1985, The Economic Theory of Trade Unions: An Introductory Survey, Scandinavian Journal of Economics 87, 160-93. Picard, P.M. and E. Toulemonde, 2001, On the Equivalence of Taxes Paid by Employers and Employees, Scottish Journal of Political Economy 48, 461-70. Pissarides, C.A., 1998, The Impact of Employment Tax Cuts on Unemployment and Wages; the Role of Unemployment Bene…ts and Tax Structure, European Economic Review 42, 155-83. Pissarides, C.A., 2000, Equilibrium Unemployment Theory, The MIT Press, Cambridge, MA. Rocheteau, G., 1999a, Can an Unemployment Insurance System Generate Multiple Natural Rates?, International Tax an Public Finance 6, 379-87. Rocheteau, G., 1999b, Balanced-Budget Rules and Indeterminacy of the Equilibrium Unemployment Rate, Oxford Economic Papers 51, 399-409. Schluter, C., 1997, On the Performance of Social Bene…t Systems, The Economic Journal 107, 489-502. Schmitt-Grohé, S. and M. Uribe, 1997, Balanced-Budget Rules, Distortionary Taxes, and Aggregate Instability, Journal of Political Economy 105, 976-1000.
26
Vijlbrief, H. and R. Wijngaert, 1995, Unemployment Insurance Policy and Union wage Formation, Labour 9, 233-51.
27
Appendix: The mixed unemployment bene…t system The right-to-manage and the wage conditions arising from the Nash bargaining are identical to those of section 2, given by equations (5) and (4), respectively. But the unemployment bene…t budget constraint and the unions’disagreement point di¤er. When bene…ts consist both of a …xed part bt and a variable part, which is a proportion of the wage t wt , the unemployment bene…t budget constraint becomes27 wt Lt ( t +
t)
= (1
(A.1)
Lt ) ( t wt + bt ) ;
while the disagreement point of the unions is wt = wt Lt (1
t)
+ (1
(A.2)
Lt ) ( t wt + bt ) :
Combining equations (4), (A.1) and (A.2), and using equation (22) yields (1
t)
= Q (1 +
Kt+1 sAKt1
t)
1
.
(A.3)
On the other hand, equations (5), (A.1) and (22) yield h 1 1 1 1 1 1 1 1 Kt+1 ( t + t ) = s s A Kt+1 Kt A Kt+1 Kt t s
i + bt (1 + t ) :
(A.4)
We consider the case where the instrument used to adjust the unemployment bene…t budget is the wage tax. Hence, substituting for t from equation (A.3) into equation (A.4), we have the dynamics of the economy, Kt+1 1 +
t
1+
Qs
1
A
1
t
+s
1
A
1
1+
1
1
t
Kt+1 Kt
1+
t
1
1
1
s A Kt+1 Kt
1
1
Kt+1 Kt
bt
(A.5)
sbt = 0;
from where the steady state equilibrium solves Qs
1
A
1
K2
1+
1+
+s
1
A
1
b K+
1
1+
1
s A + sb
= 0:
(A.6)
Note that in a Beveridgean system, = 0, taxes have no in‡uence on capital and, thus, on unemployment. Instead, in a mixed unemployment bene…t system there are always two 27
Note that the unemployment bene…ts are not taxed.
28
equilibria, one pessimistic K1 and the other optimistic K2 , with K1 < K2 . Using equations (A.1) and (5) we get A (sA)
t
Lt =
A (sA)
1
1
+ bt (1 +
( t+
t+
t)
(A.7)
;
t ) + bt (1 +
t)
which combined with equations (22) and (A.3) yield a continuum of equilibrium pairs ( t ; which are implicitly de…ned by A (sA)
t
A (sA)
1
1
+ bt (1 +
( t+
t
+
t)
t)
=
+ bt (1 +
t)
1 t : Q (1 + t )
t)
(A.8)
In contrast with the gross Bismarckian unemployment bene…t system, for each wage tax two di¤erent values of the payroll tax solve the previous equation because bt > 0. Hence, two equilibria arise regardless of the tax used to adjust the unemployment bene…t budget. Next, we analyze the stability properties of the equilibria. Di¤erentiating equation (A.5) and evaluating in steady state yields dKt+1 dKt
= Kt+1 =Kt
H H
H ; M
(A.9)
where H = Qs
1
A
1
K2
1
s
A
1
1
bK
1
(A.10)
s A ;
1+
and M=
Qs
1
A
1
K2 +
1+
K
1+ 1
1
1
s A :
1+
(A.11)
1
From equation (A.6) we have that M = sb s A bK, which is positive.28 Su¢ cient conditions for the equilibria to be stable without cycles are that either H 0; or H > 0 and H > M: A su¢ cient condition for the equilibria to be stable with cycles is that H > 0 and H < M =(2 ). Using equations (A.6), (A.10) and (A.11) we have that H
M = 2 Qs
1
A
1
K2
s
1
A
1
b+
1+
K;
1+
(A.12)
which is positive at the optimistic equilibrium and negative at the pessimistic one.29 Hence, the optimistic equilibrium is always stable, since H > M > 0. In the pessimistic equilibrium we 28
1
1
1
1
1
1
Note that sb s A bK = sb 1 s A K > 0 since in steady state s A K = L < 1. 29 As equation (A.6) is quadratic, di¤erentiating equation (A.6) and multiplying the resulting expression by K gives directly the sign of (A.12) for each equilibrium.
29
have that H M < 0. Thus, in order to have stability it must be the case that either H 0 or H > 0 and (2 )H < M . These last two conditions can be reduced to (2 )H M < 0: Note that if b = 0 then this condition is satis…ed, since M = 0 and H < M . However, for high values of b the stability properties of the pessimistic equilibria change. There is a subspace of the parameter space 0. From equation (A.3) we also have d = d
Q (sA)
1
K + (1 +
t)
dK d
;
(A.15)
indicating that in the optimistic equilibrium d =d < 0. Regarding the pessimistic equilibrium, Table 1 shows that this derivative may be either positive or negative. 30
We have chosen the following values for the parameters: A = 4; s = 0:1; = 0:02 and = 0:66: In this numerical example we have included some cases where the pessimistic equilibrium is unstable (i.e, (2 )H1 M1 > 0) or stable with cycles ( i.e., H1 > 0 and (2 )H1 M1 < 0) in a larger proportion than the observed ones in the simulations.
30
b
(%)
L1
L2
w
1 (%)
2 (%)
H1
0:4 0:4 0:4 0:4
1 2 3 4
0:55 0:55 0:55 0:55
0:829 0:820 0:813 0:806
0:940 0:943 0:945 0:947
1:63 1:61 1:60 1:58
15:4 15:2 15:3 15:2
4:0 2:8 1:6 0:4
0:45 0:45 0:45 0:45 0:45
1:5 2 3 4 5
0:55 0:55 0:55 0:55 0:55
0:886 0:876 0:863 0:853 0:844
0:910 0:917 0:925 0:930 0:934
1:62 1:61 1:60 1:58 1:57
9:1 9:6 10:1 10:3 10:4
6:6 5:4 3:7 2:2 0:8
+
0:55 0:55 0:55
4 5 6
0:5 0:5 0:5
0:884 0:870 0:860
0:911 0:921 0:927
1:58 1:57 1:55
7:0 7:6 7:8
4:2 2:2 0:7
+
0:77 0:77 0:77 0:77 0:77
1 2 3 4 5
0:35 0:35 0:35 0:35 0:35
0:882 0:871 0:863 0:857 0:851
0:913 0:920 0:925 0:928 0:931
1:63 1:61 1:60 1:58 1:57
9:9 10:1 10:1 9:9 9:7
6:7 5:1 3:7 2:4 1:1
+ + + + +
0:85 0:85 0:85 0:85
1 2 4 6
0:3 0:3 0:3 0:3
0:879 0:870 0:858 0:848
0:915 0:921 0:928 0:932
1:63 1:61 1:58 1:55
10:3 10:2 9:8 9:1
6:5 5:0 2:4 0:08
+ + + +
1 1 1
1 3 5
0:2 0:2 0:2
0:859 0:851 0:845
0:927 0:931 0:934
1:63 1:60 1:57
12:3 11:3 10:3
5:3 3:1 0:9
+ + +
1:3 1:3
1 3
0 0
0:832 0:832
0:939 0:939
1:63 1:60
15:1 13:4
4:1 2:2
+ +
1:33 1:33
1 3
0 0
0:864 0:864
0:924 0:924
1:63 1:60
11:8 10:0
5:6 3:7
+ +
Table 1. Stability properties of the equilibria and taxes.
31
(2
) H1
+
+ +
M1
