In-Work Benefits and the Nordic Model

In-Work Bene…ts and the Nordic Model
Ann-So…e Kolmyand Mirco Toninz December 14, 2012

Abstract Welfare bene…ts in the Nordic countries are often tied to employment. We argue that this is one of the factors behind the success of the Nordic model, where a comprehensive welfare state is associated with high employment. In a general equilibrium setting, the underlining mechanism works through wage moderation and job creation. The bene…ts make it more important to hold a job, thus lower wages will be accepted, and more jobs created. Moreover, we show that the incentive to acquire higher education improves, further boosting employment in the long run. These positive e¤ects help counteracting the negative impact of taxation. JEL codes: H24, J21, J24 Keywords: Nordic model, in-work bene…ts, wage adjustment, unemployment, education, skill formation, earnings

1

Introduction

A prominent feature of the so-called Nordic model is a comprehensive welfare state …nanced by taxes on labor. In fact, the public sector in many
We want to thank Torben Andersen, Martin Flodén, Richard Freeman, Mathias Herzing, Eddie Lazear, Ethienne Lehman, Bruno van Linden, and participants at the Conference on the Economics of the Nordic Model. y Department of Economics, Stockholm University, S-106 91 Stockholm, Ph. [phone]. Fax [phone], E-mail address: ann-so…[e-mail] z Economics Division, University of Southampton, UK; Economics Department, UniCredit & Universities Fellow, Central European University, Budapest; and IZA, Bonn. Email address: [e-mail]

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of the Nordic countries is responsible for the distribution and allocation of resources amounting to more than half of their country’ GDP (Eurostat, s 2012). With an emphasis on redistributional transfers and service provision …nanced by taxes on labor, a concern with the model is, of course, that it induces weak incentives to work. In a more long term perspective, such a system may also reduce incentives to acquire skills, with a negative impact on future productivity and labor market outcomes. However, external observers are often surprised that the Nordic countries manage to combine low unemployment and high labor force participation with high taxes and generous welfare arrangements. So, how is this possible? One answer to this question is that many of the welfare arrangements in the Nordic countries are closely tied to market work. The generosity of the bene…ts are, in general, related to earnings. In addition, eligibility to a number of bene…ts and social services is conditional on employment. Subsidized childcare, for example, is, in principle, only available to employed workers. Also, some generous elements of the paid parental leave schemes are only accessible to employed workers. In addition, the more recently introduced earned income tax credit is by de…nition exclusively targeted to employed workers. The idea is that these bene…ts, by increasing the returns from working, increase the supply of labor. The observation that the Nordic countries have sustained high economic activity because bene…ts are closely tied to market work is not new. In fact this was noted as a contributing factor to the high participation rate observed in Sweden when a group of NBER economists studied the Swedish welfare state in the mid 1990s (see Freeman et al., 1997). This was also an important message in the discussion on the prospects and challenges of the Scandinavian model in Andersen (2008). The starting point for this paper is that entitlement to many of the bene…ts available in the Nordic countries is conditional on employment. As discussed above, this tends to increase the gains from working, which encourages labor supply. However, we argue that this is not the end of the story. To investigate the full impact of welfare state arrangements of this type, one needs to account for the general equilibrium e¤ects. This is particularly relevant because many bene…ts have been available to the whole population for a long period of time. Clearly, to investigate the e¤ects of these bene…ts on employment, which is an equilibrium outcome, both supply-side and demand-side factors must be included in the analysis. Moreover, beside considering the equilibrium outcome for the existing workforce, it is important to account 2

for the impact of these bene…ts on incentives to acquire skills. The equilibrium composition of the workforce in terms of educational attainment is a crucial variable for the sustainability of the Nordic model, both in terms of its growth potential and international competitiveness (Andersen, 2008) and in terms of the political support for the welfare state (Hassler et al., 2003). To carry out such an analysis, we develop a simple model of a non-clearing labor market featuring involuntary unemployment as an equilibrium outcome. Labor force participation is also endogenously determined. Moreover, individuals di¤er in their ability to acquire education and choose educational attainments based on a cost-bene…t analysis. In particular, we focus on the choice between proceeding to higher, i.e. tertiary, education or not. The aim is to investigate the implications of bene…ts that are conditional on work on unemployment and labor force participation, accounting for their long term impact on educational attainments. We show that bene…ts available only to employed workers moderate wages, reduce unemployment rates, and increase labor force participation and employment. Moreover, one could expect that welfare bene…ts, even if conditional on work, could induce an outright reduction in education as they represent an important subsidy for low skilled workers. What we …nd instead is that the incentives to proceed to higher education are actually strengthened. This is a consequence of the relatively stronger increase in labor market opportunities for highly educated workers that follow when wages are moderated. Wages, in turn, fall because workers are more willing to accept lower wages when bene…ts are conditional on work and thus the value of having a job is higher. Lower wages increase job creation and lower the unemployment rate. Thus, total employment increases for three sets of reasons. First, the bene…ts reduce the unemployment rate for workers at all educational levels. Second, more workers choose to proceed to higher education where expected unemployment spells are shorter. Third, as labor force participation increases with the bene…ts, a larger share of the population will be employed. We also look at the impact of bene…ts when they are …nanced through a proportional tax on wages. Taxation actually reinforces wage moderation and, as such, does not overrule that bene…ts reduce wages, increase job creation, and reduce unemployment rates. However, it weakens the incentives to acquire higher education and participate in the labor force, thus inducing a counteracting e¤ect on educational attainment and labor force participation. The element of the Nordic model that this paper underlines is the wage moderation stemming from bene…ts conditional on work. Also, we …nd this 3

mechanism to be very robust to the choice of model. Moreover, looking at bene…ts through this channel highlights how they have a positive impact on educational attainment and participation, thus counteracting, at least partly, the negative e¤ect that taxation has on skill acquisition and labor force participation. The analytical results are followed up with a numerical example illustrating the e¤ects of the bene…ts on labor market performance and educational attainment. The simulations indicate that bene…ts can have an important impact on unemployment for both low- and high- skilled. Without distortinary taxation, bene…ts also have a positive impact on skill acquisition, thus further reducing overall unemployment in the long run. When …nancing through proportional taxation on wages is included in the model, the negative e¤ect of taxation on educational attainment dominates the positive e¤ect of bene…ts, thus resulting in a decrease in the share of the workforce acquiring tertiary education. Nonetheless, bene…ts still have a positive overall impact on unemployment. Considering the previous literature, there are a number of studies that have tried to explain why the Nordic countries have performed so well despite high taxes and generous welfare arrangements. As mentioned, some of these studies have emphasized the importance of that bene…ts are tied to market work for the successful outcome in terms of employment and participation (see Aronsson and Walker, 1997). A related view is provided by Rogerson (2007). He argues that the governments’spending pattern in the Scandinavian countries, compared to other high tax countries, can potentially explain the large number of aggregate work hours observed in these countries. He shows, holding tax rates constant, that it matters if the revenue is spent on disability payments which may only be received when an individual does not work or subsidies for day care for working mothers. The reason is that childcare subsidies create jobs. Our study also …nds that how the government choose to spend tax revenues matters for labor market performance, although for a di¤erent reason. In contrast to Rogerson (2007), our results materialize through general equilibrium e¤ects working through wage moderation. There is also a large number of studies focusing on particular features of the welfare state in the Nordic countries, looking for instance at the impact of childcare subsidies and paid parental leave schemes on labor supply and a number of other outcome variables.1 In contrast to our study, this literature
1

See Datta Gupta et al (2008) and Kolm and Lazear (2010). For a recent empirical

4

has, in general, only been concerned with the short run impact of these policies on labor supply, thus disregarding the impact on wages and job creation in the long run. This study is also related to the large literature on earned income tax credits (EITCs) as such a tax credit is available only to workers with income from work.2 This literature, however, has only recently been concerned with the implications of EITC policies when wages respond to the policy. Rothstein (2010) investigates the impact of the US EITC in a model featuring a perfectly competitive labor market, accounting for the behavioral responses in labor force participation and work hours. He …nds that the increased labor supply following the EITC leads to lower wages in equilibrium. This, in turn, dampens the equilibrium impact on labor supply. Kolm and Tonin (2011) contrast the impact of an EITC when wages are …xed and when equilibrium wage adjustments are accounted for using a search and matching model. They also …nd that wages fall with the tax credit in equilibrium, but this actually ampli…es the positive impact of the EITC on search intensity, participation, employment, and unemployment. The theoretical predictions of wage restraints of the EITC are supported by Leigh (2010) who uses variation in state EITC supplements in the US to investigate how hourly wages are a¤ected by the tax credit. He …nds that there is a signi…cant reduction in the wages following the tax credits at the lower end of the income distribution.3 Research on the educational impact of an EITC is rather limited. While not looking at education, the paper by Heckman, Lochner and Cossa (2003) is related as it studies the impact of wage subsidies on on-the-job skill formation, distinguishing between a model with learning-by-doing and a model with training on the job. They show that on-the-job training models predict that wage subsidies reduce skill formation, while learning-by-doing models predict the opposite. A recent paper by Malul and Luski (2009) contrasts the e¤ects of a minimum wage and an EITC on incentives to acquire human capstudy on the impact of childcare subsidies on female labor supply in Norway see Havnes and Mogstad (2011). 2 Theoretical papers, usually based on standard neoclassical labor supply models, investigate the e¤ects of the EITC on work hours (Eissa and Hoynes, 2006) or on the extensive margin (Saez, 2002). For empirical papers, see Eissa and Liebman, 1996, Meyer and Rosenbaum, 2001, Chetty, Friedman and Saez, 2012. 3 Using microdata from Sweden, Bennmarker et al. (2011) investigate the combined impact of an earned income tax credit, reduced generosity in the unemployment bene…ts, and reduce payroll taxes, on wages. They show that wages were restrained, although they are not able to disentangle the e¤ects from the di¤erent policy changes.

5

ital. They …nd that a minimum wage policy increases the professional level, as individuals need to "defend" themselves against unemployment, while the EITC reduces the incentive to invest in human capital because of the implicit tax created by the "phase out" of the EITC subsidy. In contrast to the existing literature, our paper highlight the impact of in-work bene…ts on educational attainment going through the general equilibrium e¤ects, in particular through the impact on wages and job creation. The rest of the paper is organized as follows. Section 2 presents a simple model of the labor market and analyses the implications of bene…ts conditional on work on labor market outcomes and educational choice. Section 3 introduces proportional taxation to …nance bene…ts. Section 4 provides a numerical example to illustrate the mechanisms at work. In section 5 we show that the impact of conditioning bene…ts to work on labor market outcomes is robust to alternative models of wage settings. The last section concludes.

2

The Model

This section develops a simple model of a non-clearing labor market with unemployment featuring as an equilibrium outcome. More speci…cally, the labor market is characterized by trading frictions due to the costly and timeconsuming matching of workers and …rms.4 The policy in consideration is bene…ts conditional on work. As mentioned, a crucial feature of many welfare policies in the Nordic countries is that bene…ts, in di¤erent ways, are conditional on employment. For example, highly subsidized childcare has, until recently, only been available to employed workers.5 Moreover, some of the generous features of the paid parental leave schemes are only accessible to employed workers (Kolm and Lazear, 2010). For example, employed workers have the legal right to return to their job after a more than 12 month period of parental leave in Sweden. In addition, we have the more recently introduced earned income tax credits, which by de…nition are exclusively targeted to employed workers.6 In the
The Nordic model is, in addition to its comprehensive welfare state and high taxes, associated with strong unions. This suggests that a union-…rm wage bargaining model is very relevant. In section 5.1 we show that the results derived in this paper are qualitatively the same using a model where unions that represent workers bargain with …rms over wages. 5 In 2002 unemployed workers got limited access to subsidized childcare in Sweden. 6 Also bene…ts that are accessible when not in employment, like unemployment bene…ts,
4

6

model, we will let one parameter, denoted IWB , capture the in-work bene…ts. To highlight the impact of these bene…ts, we will initially abstract from their …nancing (or equivalently, consider …nancing through lump-sum taxes). The population is heterogenous in terms of ability to acquire education. Abilities, a, are for simplicity distributed according to a standard uniform distribution, and individuals decide on the educational level they wish to pursue based on their individual ability. For simplicity the educational choice is between acquiring higher education, such as a college education, or not. Allowing for more educational levels in this setting would produce the same results. Also labor force participation is endogenously determined. The population is heterogenous in terms of how leisure when out of the labor force, l, is valued, which for simplicity is also distributed according to a standard uniform distribution and, for analytical tractability, is assumed to be independent from the distribution of ability.

2.1

Matching

Unemployed workers with a higher level of education will only search for jobs targeted to workers with higher educational level, and vice versa for workers with a lower level of education. Along the lines of Mortensen and Pissarides (1999), we can allow workers to look for jobs where they are ’ over quali…ed’ and thus allow …rms to employ workers with an educational level , above what is required for the job. In equilibrium, however, workers will not …nd it optimal to search for jobs where they are over quali…ed, and …rms will not …nd it optimal to hire overquali…ed workers, leading to the endogenous outcome of a segmented equilibrium as modelled here. The matching process of vacancies and unemployed job searchers within an educational category is captured by a concave and constant-returns-to1 scale matching function of the Cobb-Douglas form, Xj = vj uj , where Xj is the matching rate, vj is the vacancy rate, and uj is the unemployment rate. Index j = L; H refers to the educational categories: low educated (L), and high educated (H). The matching, unemployment, and vacancy rates are de…ned relative to the labor force of the educational category. are strongly tied to market work due to the fact that their generosity, as well as the entitlement to them, often is based on earnings in previous periods. Although these bene…ts may weaken the incentives to quickly …nd a new job, the fact that entitlement and generosity are tied to earnings increases the returns to work.

7

The transition rate into employment for a worker with a given level of 1 education is Xj =uj = ( j ) = j , where j = vj =uj denotes labor market tightness. Firms …ll vacancies at the rate Xj =vj = q ( j ) = j . Higher labor market tightness, j , increases workers’ probability of …nding a job, but reduces the probability of a …rm …nding a worker, i.e., 0 ( j ) > 0 and q0 ( ) q 0 ( j ) < 0, where = q( jj) j is the elasticity of the expected duration of a vacancy with respect to tightness.

2.2

Workers and Firms

Let Ej and Uj denote the expected present values of employment and unemployment of workers with a given educational level. The ‡ value functions ow for a worker i with education j can then be written: rEji = wji + IW B s (Eji Uji ) Cj (ai ) ; j = L; H; rUji = ( j ) (Ej Uji ) Cj (ai ) ; j = L; H; (1) (2)

where r and s are the exogenous discount and separation rates and w is the wage. The term IW B represents the in-work bene…t which, by de…nition, is a bene…t accessible only when employed. To acquire higher education is costly in terms of e¤ort to the individual, and potentially also in terms of pecuniary means.7 The cost of acquiring the low level of education is, for simplicity, normalized to zero, whereas the cost of attaining higher education is Ci = c (ai ), where c0 (ai ) < 0 captures that workers with high ability face lower e¤ort costs of education. There is a large number of small …rms searching for workers with a particular education. Each …rm employs one worker only. Let Jj and Vj denote the expected present values of an occupied and vacant job for a given level of educational requirements. The asset equations of a speci…c occupied job and a vacant job can then be written as: rJji = yj wji s (Jji Vj ) ; j = L; H; rVj = k + q ( j ) (Jj Vj ) ; j = L; H;
7

(3) (4)

We model the educational cost as a cost to acquire and maintain skill. This is a simplifying assumption and is not important for the results. The assumption enables us to use a model without having workers continuously being born and dying. Such a model would, however, generate the same qualitative expressions.

8

where k denotes vacancy costs and yj denotes productivity. Firms that search for highly educated workers adopt a more advanced technology, which implies that the productivity will be higher in those …rms once production starts. For the same reason will …rms that search for less educated workers adopt a less advanced technology with the implication that productivity is lower once production gets started. Thus, we have yH > yL .

2.3

Wage Formation and Tightness

Matching frictions create quasi-rents for any matched pair providing a scope for Nash bargaining. In symmetric equilibrium with free entry, i.e. with Vj = 0, the bargaining solution satis…es Jj = (1 ) (Ej Uj ) ; where is the worker’ bargaining power. This condition and the ‡ value functions s ow in (1)-(4) yield the wage rule: wj = (yj + k j ) (1 ) IW B; j = L; H: (5)

From the free entry assumption facing …rms, Vj = 0, and equations (3)(4), tightness in equilibrium is determined by: k (r + s) = (1 q ( j) ) (yj + IW B) k j ; j = L; H; (6)

where the equilibrium wage follows recursively from (5) once tightness is pinned down by (6). In equilibrium, the ‡ into unemployment equals the ‡ out of unow ow 8 employment for each category of educated workers. The equilibrium unemployment rate facing workers with a given level of education is: uj = s s+
1 j

; j = L; H;

(7)

which depends positively on the separation rate and negatively on tightness. We can now derive the following results:
Thus, s (1 uL ) LF PL = ( L ) uL LF PL , and s (1 uH ) LF PH = ( H ) uH LF PH , where LF Pj , j = L; H; denotes the labor force for each educational category. The size of the labor force for each educational level is endogenous and will be determined in the next section. However, as the unemployment rates are independent of the size of the labor force it is of no importance how we note them here.
8

9

Proposition 1 An IW B will reduce wages, increase tightness, and reduce the unemployment rate for workers in all educational categories. Proof. See appendix. The in-work bene…t, which by de…nition is conditional on work, simply increases the attractiveness of having a job. When holding a job becomes more attractive, wage demands will be moderated. This makes it more profitable for …rms to open vacancies, which in turn, induces tightness to increase and the equilibrium rate of unemployment to fall. Considering the e¤ect on take-home pay, the following proposition summarizes the result: Proposition 2 An IW B will increase the take-home pay, wj + IW B, j = L; H, although not by the full amount of the IW B. Proof. See appendix. The IW B will thus restrain wage demands leading to a smaller increase in take-home pay compared to the value of the bene…t. As wage restraint stimulates job creation which, in turn, reduces the expected unemployment spells, more workers will transit from unemployment into jobs, thus leading to higher expected life time earnings.

2.4

Education, Labor Force Participation, and Employment

We assume that educational attainment only gives a payo¤ to workers in jobs.9 Thus, only workers that will participate in the labor market will consider whether they should acquire higher education or not. As workers enter the labor market into the state of unemployment, in their decision they compare the value of unemployment at di¤erent educational attainments. This comparison reveals that the educational gain in terms of a higher expected income needs to exceed the individual cost of acquiring education, in order for the individual to attain additional education.10 Thus, workers with very
Education could, of course, also have some consumption value. Accounting for this would not change the results and one could consider the cost of education as modelled here to be net of any bene…t enjoyed regardless of labor market status. 10 By use of (1) and (2), the value of unemployment is written as rUj = (1 (uj )) [wj + IW Bj ] Cj (ai ), where 1 (uj ) can be interpreted as the expected ( ) time in employment. The weight, 1 (u) = r+s+ j( j ) , reduces down to the employment rate, 1 uj ; when the discount rate approaches zero.
9

10

low ability will not …nd it worthwhile to proceed to higher education, whereas very high ability workers will …nd it more than worthwhile to do so. Using (1) and (2), we can write the condition determining the ability level of the marginal worker as: rUL = rUH (^) ; a (8)

where a is the ability level of the worker which is indi¤erent between acquiring ^ higher education or not. Thus, all workers that participate in the labor market and have an ability level equal to or higher than a will proceed to ^ higher education, whereas workers with an ability level below a (and high ^ ability workers who will not participate in the labor market) will not. The equation in (8) can be rewritten using (1), (2), together with the …rst order conditions for wages, and the equations in (4) under the assumption of free entry. This yields: ( H (9) L) : (1 ) The right hand side of equation (9) is the expected income gain of getting a college education. In order to guarantee that at least some workers acquire additional education, expected income must increase with education. Ignoring the IW B, this can be shown to hold formally by use of the equations in (6) where H > L if yH > yL . The IW B may a¤ect the individual incentives to acquire education by a¤ecting tightness, and thus the expected income, in a di¤erent way at the two levels of education. This is the particular issue up for investigation here. By assuming that the cost function ful…lls lima!1 c(a) = 0 and lima!0 c(a) = +1, we can focus on the non-trivial case where at least some workers …nd it worthwhile to acquire higher education while others don’ Although equat. tion (9) is used to pin down whom in the labor force will proceed to higher education and whom will not, to get an expression for the number of workers in the population with higher education, we also need to know whom will participate in the labor market. A worker enters the labor force into the state of unemployment by becoming available to the labor market. It will be worthwhile to enter the labor market if the returns of entering exceed the returns from not entering. Let N denote the expected present value of non-participation. The ‡ value of ow not participating in the labor force is given by the per period real value of leisure, l, which di¤ers across workers. c (^) = a 11 k

The ‡ value function for non-participation, rNi = li , is then added to ow the ‡ value functions for employment and unemployment in (1)-(2). The ow assumption is that it is not important if the worker has a higher education or not for the workers evaluation of leisure when out of the labor force, and thus subindex j is absent. The function determining the valuation of leisure which makes the worker indi¤erent between participating and not participating in the labor market is given by the following continuous function: ^ = rUL if a < a; l ^ ^ = rUH (a) if a a: l ^ This function can be rewritten by use of the ‡ equation in (2) in symmetric ow equilibrium, the Nash bargaining solutions, Jj = (1 ) (Ej Uj ), and the free entry condition, Vj = 0, together with (4), j = L; H, as11 : ^ = l ^ = l
L

k ) k )

(1
H

if a < a; ^ c (a) if a a: ^

(1

A worker that would not proceed to higher education when participating, i.e. a worker with a < a, will not …nd it worthwhile to participate in the ^ labor market if his or her valuation of leisure exceeds ^ = (1L k) . Workers l with very high ability, on the other hand, may choose to participate in the labor market even if they have a high valuation of leisure. This follows as their pay-o¤ on the labor market is very high accounting for that they fairly e¤ortless can acquire higher education and reap a higher expected income. Figure 1 illustrates the choice of participation and education in the population. Areas A, B and C in the left panel give the stock of workers participating in the labor market. Areas B and C give the stock of workers that will acquire higher education. Area D captures workers that would, in case of participation, acquire higher education. However, as they will not participate in the labor market due to their high valuation of leisure, they will
As we use the standard uniform distribution for ^ the value of the function should l, not exceed unity. For simplicity we assume that this is not binding, that is, this threshold level is lower than unity. In what follows we will assume to be in an interior solution.
11

12

Figure 1: Labor force participation and educational attainment

not acquire higher education as education is costly and only gives a payo¤ when working. This implies that the labor force participation rate is larger for workers with high ability. More speci…cally, the labor force participation rate is given by the area (B + C) = (B + C + D) for workers with high ability, i.e. a a, and by the area A=(A + E) for workers with lower ability, i.e. ^ a < a. The labor force participation rate for workers with low education is ^ A= (A + D + E), while by assumption workers acquire high education only if they intend to participate in the labor market. It is then straightforward to derive labor force participation, LF P , the stock of educated workers, Edu, and total employment, Emp, in the economy as:

13

LF P =

Z1
0

^ ^ ^ da = a k L + (1 a) k l (1 ) (1 ) a) k ^ (1 ) uL )
H

H

Z1 a ^

c (a) da;

(10)

Edu =

(1

Z1 a ^

c (a) da; 2 (1 a) k ^ (1 ) Z1 a ^

(11) 3

Emp = (1

a k L ^ + (1 (1 )

The e¤ect of in-work bene…ts on labor force participation, education, and employment are summarized in the following proposition. Proposition 3 An IWB will increase labor force participation, the stock of workers with higher education, and aggregate employment. Proof. See appendix. Labor force participation increases when bene…ts are conditional on work because job creation is stimulated. This tends to increase the expected income of participation both by reducing the expected unemployment spells and by increasing the take home pay for workers at all educational levels. Perhaps more surprisingly is that bene…ts conditional on work increase the incentives for workers to acquire higher education. There are two reasons for this. First, an IW B will increase the relative demand for workers with a higher level of education.12 The improved labor market opportunities for
Labor market opportunities improves for all workers. However, the increased demand for workers will be dampened by the fact that it will take a longer time to …ll a vacancy when the competition for workers becomes higher. More …erce competition for workers will hit …rms hiring low educated more severely, leading to that labor market conditions particularly improve for workers with higher education. This is because tightness is lower for low educated workers and, thus, one more …rm entering with a vacancy will increase the expected time to …ll a vacancy relatively more for these workers compared to workers with higher education. To put it in another way, there is more competition for workers with a college education relative to workers with no higher education. Then, an additional vacancy has a relatively smaller negative externality on other …rms with an open vacancy for college educated. This increases the relative demand for college educated workers, and therefore increases the incentives for workers to proceed to higher education.
12

uH ) 4

H

c (a) da5 :(12)

14

highly educated workers relative to less educated workers increase the attractiveness of acquiring higher education. Second, as an IW B increases the return to participation, some workers with very high ability will …nd it worthwhile to leave their state of non-participation and join the labor force. As these workers have a very high ability, they will fairly e¤ortless acquire higher education …rst. The right panel in Figure 1 illustrates how labor force participation and the stock of workers with higher education are a¤ected by an IW B. What can be noticed is that there is a leftward shift in the line denoting the ability level where workers are indi¤erent between acquiring higher education or not. This corresponds to the …rst of the two reasons provided above for the increase in higher education. The second reason is represented by the upward shift in the line denoting, for each ability level, the value of leisure leaving a worker indi¤erent between participating or not in the labor market. Aggregate employment increases for three sets of reasons. First, employment increases because bene…ts conditional on work increase job creation which reduces the unemployment rate for all educational categories. Second, as the bene…ts improve the incentives to acquire education, employment increases as the expected unemployment spells are shorter among highly educated workers. Third, as labor force participation increases, employment increases because some of the workers entering the labor market will become employed.

3

IWB Financed by Proportional Tax

In this section, we study the e¤ects of an IWB when it is …nanced through distortionary income taxation.13 There are a number of ways to …nance the bene…t. Below we formally consider the case when taxation is proportional. The ‡ value function for employment in (1) is now written: ow rEji = wji (1 t) + IW Bji s (Eji Uji ) Cj (ai ) ; j = L; H; (13)

where t is a proportional income tax rate. The rest of the equations in (1)-(4) and the ‡ value of participation remain unchanged. To derive ow the equilibrium equations determining wages and tightness, we follow the
13

The IW B being …nanced by payroll taxation would yield the same results.

15

same procedure as in the basic setting, taking into account that the Nash bargaining solution now satis…es (1 t) Jj = (1 ) (Ej Uj ) for j = L; H. This yields the following equations: wj = (yj + k j ) ) yj + (1 IW B 1 t ) IW B ; j = L; H; 1 t k j ; j = L; H: (14) (15)

k (r + s) = (1 q ( j)

The tax rate will now have a direct e¤ect on the wage, wj , and tightness, j . In fact, as the tax rate, t, increases, the wage demands are reduced and tightness will increase.14 This follows as a higher tax rate reduces the value of the earned income, but it will not reduce the value of the IW B. Thus, an increase in the tax rate increases the importance of the IW B as a source of income when employed and will thus work in a similar way as an increase in the IW B. Taxes will also have a direct impact on the incentives to acquire education. The ability level of a worker on the labor market that is indi¤erent between acquiring or not acquiring higher education when participating in the labor force is given by equation c (a) = k (1 (1 t) ( )
H L) :

(16)

The equation determining the valuation of leisure which makes the worker indi¤erent between participating and not participating in the labor market is now given by: ^ = l ^ = l
L

k (1 t) if a < a; (1 ) t) H k (1 c (a) if a (1 )

(17) a:

It is then straightforward to derive labor force participation, LF P , the stock of educated workers, Edu, and total employment, Emp, in the economy as:
This is a standard result in models of the equilibrium rate of unemployment. The tax rates will have an impact on producer costs, tightness and unemployment if there is a …xed compensation or cost on the employed or unemployed workers’side. See Pissarides (1998).
14

16

a k L (1 t) (1 + LF P = (1 ) Edu = (1 a) k (1
H

a) k (1 Z1 a H

(1 )

t)

Z1 a c (a) da;

(18)

(1 )

t)

c (a) da;

(19) (20)

The total wage bill in the economy is given by W ageBill = wL (1 +wH (1 uL ) a k L (1 t) + (1 ) 2 (1 a) k H (1 uH ) 4 (1 )

a k L (1 t) Emp = (1 uL ) + (1 ) 2 (1 a) k H (1 +(1 uH ) 4 (1 )

t)

Z1 a c (a) da5 :

3

(21)

(22) t) Z1 a where the expressions for wages are given by (14). Finally, the government budget constraint for an IWB …nanced by proportional taxation is given by t W ageBill = IW B Emp: We can show the following: Proposition 4 An IWB …nanced by proportional taxes on wages will reduce wages, increase tightness, and reduce the unemployment rate for workers at all educational levels provided a higher tax rate implies higher …scal revenues. Proof. See appendix. An IWB …nanced by proportional taxation on wages will again reduce wages and the unemployment rate for workers at all educational levels. There is an ambiguous e¤ect on the incentives to proceed to higher education. As the demand, and thus the employment probabilities, for highly educated 17 (24)

c (a) da5 ; (23)

3

workers increases with the IWB , more workers tend to proceed to higher education. However, the fact that the taxation directly reduces the payo¤ from education will reduce the incentives for workers to attain higher education. Taxes will also have a direct negative e¤ect on labor force participation and employment. Next we turn to numerical simulations to illustrate these e¤ects.

4

A Numerical Example

In this section we calibrate the model and simulate the impact of an in-work bene…t on the main variables of interest. Given the simplicity of the model we consider these calculations as illustrative of how the mechanisms highlighted in the previous sections work, without aiming to provide speci…c guidance in terms of the empirical impact of having bene…ts conditioned on work.

4.1

Calibration

The month is the basic time unit. To ensure that the labor force participation rate for low skilled workers is always less than 1, productivity for low skilled workers, yL , is …xed at 0:75. Worker bargaining power, , is set to 0:6, while the real interest rate r is 0:0025. equals 0:5, while parameters k, s, and yH are set to replicate an average duration of unemployment for the low educated of 6 months and an unemployment rate of 0:08 for low skilled, and 0:06 for high skilled in absence of an in-work bene…t. For analytical convenience, we assume that the cost of acquiring high school education is given by c (ai ) = 2 (1 Then, the expression Z1 a ^

ai ) :

(25)

c (a) da is given by Z1 a ^

c (a) da = (1

a)2 : ^

(26)

Using (9) we get

18

a=1 ^

k( H 2 (1

L)

)

:

(27)

The share of people with high education in the population, Edu, is given by (11). The parameter is set to replicate a skill distribution in absence of an in-work bene…t with Edu = 0:35. The table below summarizes the parametrization, with further details about the calibration provided in the Appendix. yL yH 0:75 1:34 0:6 k r s 15:4 0:0025 0:014 0:5 0:718

4.2

Numerical Results

We …rst analyze the e¤ect of bene…ts conditional on work without distortionary taxation. Figure 2 plots the main variables of interest as a function of an in-work bene…t going from 0 to 0:25, equivalent to one third of the productivity of people with low education. As expected, tightness increases as wages fall. The share of the population with higher education increases, going from 35% of the population with no bene…ts to 44% when bene…ts are at 0:25. Unemployment rates also fall, both for the highly educated (from 6% to 5:5%) and for people with low education (from 8% to 6:9%). Beside the unemployment rate for each of the two educational categories, the overall unemployment rate is also in‡ uenced by the composition of the workforce, and it decreases from 7% to 6:3%. Also as expected, the labor force participation rate increases and, as a result of a higher participation and lower unemployment, the total employment rate in the population increases. When we consider in-work bene…ts …nanced by proportional taxation, the picture is rather di¤erent (see Figure 3). The tax rate corresponding to the maximum in-work bene…t of 0:25 is 28%. The most striking di¤erence is that higher bene…ts now reduce the incentives to acquire higher education, as they imply a higher …scal pressure. As showed by Proposition 4, the unemployment rate falls, both for the highly educated (from 6% to 5:3%) and for people with low education (from 8% to 6:6%). Overall, unemployment falls from 7% to 6:2%. The overall labor force participation rate and employment rate are rather stable. This is due to the combination of a positive e¤ect for low-educated and a negative composition e¤ect as less people acquire higher education. 19

Figure 2: IWB - No Distortionary Tax
Y:T ightness X:IW B 0.06 0.05 0.04 0.03 0 0.05 0.1 0.15 0.2 0.25 0.08 1.2 1 0.8 0 0.05 0.1 0.15 0.2 0.25 0.075 0.07 0.065 0.06 0 0.05 0.1 0.15 0.2 0.25 0.44 0.42 0.4 0.38 0.36 0 0.05 0.1 0.15 0.2 0.25 Y:Education X:IW B

Y:W age X:IW B

Y:Unemployment X:IW B

Y:LF P rate X:IW B 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0 0.05 0.1 0.15 0.2 0.25 High Low 0

Y:Employment X:IW B

0.05 All

0.1

0.15

0.2

0.25

Figure 3: IWB - Distortionary Tax
Y:T ightness X:IW B 0.07 0.06 0.05 0.04 0.03 0 0.05 0.1 0.15 0.2 0.25 0.08 1.2 0.07 1 0.06 0.8 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 Y:W age X:IW B Y:Unemployment X:IW B 0.25 0.35 0.3 Y:Education X:IW B

Y:LF P rate X:IW B 0.85 0.8 0.75 0.7 0.65 0 0.05 0.1 0.15 0.2 0.25 High Low 0.8 0.7 0.6 0

Y:Employment X:IW B

0.05 All

0.1

0.15

0.2

0.25

20

To appreciate the di¤erence made by bene…ts conditional on work even when they are …nanced by distortionary taxation, in Figure 4 we show the impact when distortionary taxes are instead used to …nance a lump-sum transfer available to all. To facilitate the comparison, the x-axis in each graph is still IW B. This should be interpreted as the tax rate resulting in the corresponding level of in-work bene…ts when taxes are used to …nance them, so that IW B = 0:25 corresponds to a tax rate of 28%. As expected from (14) and (15), when taxes are used to …nance a lump-sum transfer, they have no e¤ect on tightness, wages, and unemployment rates. They do have a very strong negative e¤ect on the incentive to acquire education. At t = 28%, the share of the population with higher education falls from 35% to 18% when bene…ts are not conditional on work, as opposed to 24:7% when bene…ts are conditional on work. Because of this composition e¤ect, the overall unemployment rate increases with taxation, even if the unemployment rate for the two educational categories is una¤ected. Labor force participation decreases, both because of the composition e¤ect and because of the negative incentive to participate for the low-educated. As a result, employment also falls. Instead, with bene…ts conditional on work, both participation and employment remained substantially stable, while unemployment declined.

5

Alternative Models of Wage Setting

In this paper we underline the impact of bene…ts conditional on work stemming from wage moderation. This mechanism is very robust to various possible assumptions about the way wages are set. The induced wage moderation, increased hires, and the lower unemployment rate follow in all basic standard models featuring unemployment as an equilibrium outcome. In an e¢ ciency wage model, for instance, bene…ts conditional on work would provide …rms with an instrument to discipline workers. Thus, the …rm would not need to pay workers as high wages in order to prevent them from shirking, since the threat of loosing the bene…t when …red will do the job. Also in a static or dynamic union-…rm wage bargaining model of the Right-to-manage type, or a Monopoly union model, would the same result materialize.15 As a unionIn fact, also a model with a perfectly competitive labor market will do the job of explaining the fall in wages and the job creation. However, such model can not explain the observed low unemployment rate in the Nordic countries as involuntary unemployment is absent.
15

21

Figure 4: Lump-sum Transfer - Distortionary Tax
Y:T ightness X:IW B 0.05 0.045 0.04 0.035 0.03 0 0.05 0.1 0.15 0.2 0.25 0.08 1.2 1 0.8 0 0.05 0.1 0.15 0.2 0.25 0.075 0.07 0.065 0.06 0 0.05 0.1 0.15 0.2 0.25 0.3 0.25 0.2 0.15 0 0.05 0.1 0.15 0.2 0.25 Y:Education X:IW B

Y:W age X:IW B

Y:Unemployment X:IW B

Y:LF P rate X:IW B 0.8 0.7 0.6 0.5 0.4 0 0.05 0.1 0.15 0.2 0.25 High 0 Low 0.5 0.4 0.6

Y:Employment X:IW B

0.05 All

0.1

0.15

0.2

0.25

…rm wage bargaining model is particularly relevant for the Nordic countries, in this section we set up such a model to demonstrate the e¤ects of in-work bene…ts on the main labor market outcomes.

5.1

Union-Firm Wage Bargaining

One feature of the Nordic countries is the strong presence of unions. Here we assume that unions, representing workers at the …rm level, bargain with …rms over the wage. However, once the wage is set, the …rms will decide on how many workers to hire. The problem is solved through backward induction. Thus, at the second stage, …rms decide on the number of workers to hire, N , so as to maximize their pro…t, , taking the wage, w, as predetermined. Firms then solve M axN =N wN , where, for simplicity, the production technology is captured by a Cobb-Douglas function, and the individual index is dropped. The …rm’ demand for labor is then given by s N= w
1 1

; with " =

dN w 1 = dw N 1

:

(28)

22

Wage are determined through decentralized union-…rm Nash bargaining. The union’ utilitarian objective function is captured by = N [w + IW B]+ s N N B, where N is the number of members, and B capture the unemployment bene…t which is received by all unemployed workers. The union face a trade-o¤ in that a higher wage improves the well-being of their employed members, but a higher wage will, at the same time, render more members unemployed which reduces the well-being of those members. As all workers receive unemployment bene…ts in case the bargain breaks apart, N B captures the union fallback position, leaving N [w + IW B B] as the union "rent" contribution in the bargain. As the …rm makes no pro…t in case the bargain breaks apart, the Nash product is given by = [N (w + IW B B)] [N wN ]1 ; where is the relative bargaining strength of the union compared to the …rm, 2 (0; 1]. The Nash product is maximized by choosing w, accounting for that N = N (w) through (28). From the …rst order condition, the following wage setting curve can be derived: w= + (1 ) [B IW B] ;

where it is clear that also in a union-…rm wage bargaining model will wage moderation follow due to in-work bene…ts. The intuition is analogous to the one found in the basic matching model. As an in-work bene…t increases the value of having a job, the union wage demand is restrained because they want more of their member to be in jobs. Employment increases as more workers are hired when wages are lower (see (28) which determines employment in the economy with the number of …rms normalized to unity). The model collapses to a Monopoly union model when = 1 is imposed, i.e. all bargaining power is given to the union. Note that using ‡ value functions in this union-…rm wage bargaining ow set-up would yield the same steady state result as this simple static model of decentralized bargains or union wage setting.

6

Conclusions

As mentioned in the introduction, the Nordic countries are characterized by a comprehensive welfare state …nanced through taxes and social security contribution, with public expenditures amounting in 2011 to more than half of 23

GDP. However, one could argue that high social spending …nanced through high taxes characterizes many other countries, especially in continental Europe. Indeed, in 2011, general government total expenditure amounted to 49.1% of GDP in the EU-27 (the average for the period 2002-2011 is 47.7%), with the largest share, 43.4%, of this expenditure devoted to the redistribution of income through social transfers in cash or in kind (Eurostat, 2012). What this paper has emphasized is that one feature of the Nordic model, namely the fact that many of the welfare arrangements are strongly tied to work, makes a di¤erence. In particular, we have underlined how bene…ts structured in such a way induce job creation and lower the unemployment rate through their wage moderating e¤ect. Moreover, they do provide incentives to pursue further education, and increase labor force participation and employment. For a given amount of tax revenue, the labor market outcome would be very di¤erent if spending would instead be directed towards programs that are not conditional on work, or, even more, towards programs conditional on not working. We have also emphasized one crucial aspect behind the long-term sustainability of the Nordic model, namely its e¤ect on incentives to pursue higher education. As Andersen (2008) noticed in his discussion on the prospects and challenges of the Nordic model, "a compressed wage structure and high taxation have a negative e¤ect on the return to education". This paper shows how bene…ts conditional on work mitigate this negative incentive and may contribute, together with other policies like the public …nancing of education, to maintain the educational attainment in the Nordic countries at high levels.16

References
[1] Andersen T. M., 2008, The Scandinavian Model - Prospects and Challenges, International Tax and Public Finance 15, p. 45-66. [2] Aronsson T., and Walker, J., 1997, in: Freeman, R., B. Swedenborg, and R. Topel (Eds.), The Welfare State in Transition: Reforming the Swedish Model , National Bureau of Economic Research, Inc.
According to OECD data (OECD, 2011), the percentage of the population with at least upper secondary education or tertiary education in the Nordic countries is above the OECD and the EU-21 averages both for the working age population (25-64) and for youth (25-34).
16

24

[3] Bennmarker, H., Calmfors, L., and A. Larsson, 2011, Wage Formation and the Swedish Labour Market Reforms 2007-2009. Report to the Swedish Fiscal Policy Council 2011/1. [4] Datta Gupta, N., Smith, N., and Verner, M., 2008. The Impact of Nordic countries’ Family Friendly Policies on Employment, Wages, and Children, Review of Economics of the Household, Springer, vol. 6(1), 65-89. [5] Chetty, R., Friedman, J., and Saez, E., 2012. Using Di¤erences in Knowledge Across Neighborhoods to Uncover the Impacts of the EITC on Earnings, mimeo. [6] Eissa, N., and Hoynes, H., 2006. Behavioral responses to taxes; Lessons from the EITC and Labour Supply. Tax Policy and the Economy 20, 73-110. [7] Eissa, N., and Liebman, J., 1996. Labor Supply Responses to the Earned Income Tax Credit. Quarterly Journal of Economics 111, 605-637. [8] Eurostat, 2012. Statistics in focus N. 27. [9] Freeman, R., B. Swedenborg, and R. Topel (Eds.), 1997, The Welfare State in Transition: Reforming the Swedish Model, National Bureau of Economic Research, Inc. [10] Havnes T. and Mogstad M., 2011, Money for Nothing? Universal Child Care and Maternal Employment. Journal of Public Economics 95, 14551465. [11] Hassler, J., Storesletten, K., and Zilibotti, A. 2003. The Survival of the Welfare State. The American Economic Review, 93(1), 87-112. [12] Heckman, J., Lochner, L., and Cossa, R. 2003. Learning-by-doing vs. On-the-job Training: Using Variation Induced by the EITC to Distinguish Between Models of Skill Formation. In: Phelps, E. S. (Ed.). Designing inclusion: tools to raise low-end pay and employment in private enterprise. Cambridge university press. [13] Kolm, A-S., and Lazear, E., 2010, Policies A¤ecting Work Patterns and Labor Income for Women, in Reforming the Welfare State - Recovery and Beyond in Sweden, Eds., R. Freeman, B. Swedenborg, and R. Topel, NBER, The University of Chicago Press, Chicago, 57-82. 25

[14] Kolm, A-S., and Tonin, M., 2011, In-Work Bene…ts and Unemployment. International Tax and Public Finance, 18 (1), 74-92. [15] Leigh, A. 2010. Who Bene…ts from the Earned Income Tax Credit? Incidence Among Recipients, Coworkers and Firms. The BE Journal of Economic Analysis & Policy, 10(1). [16] Malul, M., & Luski, I. 2009. The Optimal Policy Combination of the Minimum Wage and the Earned Income Tax Credit. The BE Journal of Economic Analysis & Policy, 9(1). [17] Meyer, B., and Rosenbaum, D., 2001. Welfare, the earned income tax credit and the labour supply of single mothers. Quarterly Journal of Economics 116, 1063-1114. [18] Mortensen, D., and Pissarides, C., 1999. Unemployment Responses to ’ Skill-Biased’ Technology Shocks: The Role of Labour Market Policy. The Economic Journal 109, 242-265. [19] OECD, 2011, Education at a Glance 2011: OECD Indicators, OECD Publishing. [20] 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(1), 155-183. [21] Pissarides, C., 2000. Equilibrium unemployment theory, MIT Press, Boston, MA. [22] Rogerson, R., 2007. Taxation and Market Work: is Scandinavia an Outlier? Economic Theory, 32(1), 59-85. [23] Rothstein, J., 2010, Is the EITC as Good as an NIT? Conditional Cash Transfers and Tax Incidence, American Economic Journal: Economic Policy 2 (1), 177-208. [24] Saez, E., 2002. Optimal Income Transfer Programs: Intensive Versus Extensive Labor Supply Responses. Quarterly Journal of Economics 117, 1039-1073.

26

7

Appendix d j dIW B duj dIW B

A1 Proofs of propositions
Propositions 1-4. Di¤erentiation of the equations in (6) yields:
1 (1 k (1 )

= =

1 + (r + s)
)s
1 j j

1 j

> 0: Then di¤erentiation of (7) yields

(s+ d j dIW B

)

2

d j dIW B

yields

dwj dIW B

< 0: Di¤erentiation of (5) making use of the expressions for ! 1 = (1 ) 1+
1 (r+s)
1 j

< 0. This proves proposi1

tion 1. The impact of an IWB on the take-home pay, concluded in proposition ! ! 2, is: d(wj +IW Bj ) dIW B

=

dw 1+ dIWjB

=

+

1 (r+s)

1 j

1+

1 (r+s)

1 j

2 (0; 1):

The impact of an IWB on labor force participation, the stock of education, and employment is considered through di¤erentiation of (9)-(12) and using a ^ d L dLF the previous proposition. Di¤erentiation of (10) yields dIW P = (1 k ) dIW B + B
(1 a) k d H ^ (1 ) dIW B

> 0 as changes in a will have no impact on LF P . Di¤erenti^ d j dIW B

ation of (9) using the expression for
H

and the facts that c0 ( ) < 0 and
(r+s)
1 H

>

L

yields

d^ a dIW B

=

1 c0 (^) a

(1 k

)

1

1+ dEdu dIW B a) k ^ (1 )

1+ k d^ a dIW B (1 Z1 a ^
L

(r+s)
1 L

1

< 0.

Di¤erentiation of (11) and (12) yield 2 and dEmp dIW B

=
H

=

^ duL a k L dIW B (1 )

duH dIW B

^ a ^ a ^ d L d H uH ) + (1 uL ) (1 k ) dIW B + (1 uH ) (1 a) )k dIW B > 0 using that (1 (1 ) k) H (1 Z1 c (a) da > 0 as that is the labor force participation of highly educated a ^

4 (1

c (a) da5

) 3

^ d H + (1 a) )k dIW B > 0 (1 k d^ a dIW B (1
L

)

(uL

workers, and that

d^ a dIW B

< 0 and

Di¤erentiation of (15) yields

d H > 0. This proves dIW B [1 t+IW B dt ] d j (1 = (1 t)) k(r+s) dIW B > 2 1 dIW B + k] [ duj dIW B

proposition 3. 0 if dt dIW B

> 0.

From (7) it then follows that

dt < 0 if dIW B > 0. (s+ ) d Also, di¤erentiation of (14) and by use of the expression for dIWjB we have

=

(1

)s

j

1 j

2

d j dIW B

27

dwj dIW B

=

> 0 from the direct e¤ect in (24) (ignoring the indirect clear that e¤ects working through the tax bases of employment and the wage bill). Accounting for the dynamic e¤ects implies that the government revenue can dt both increase and fall with higher taxes. By assuming dIW B > 0, we assume that the dynamic e¤ects working through the tax bases are not dominating the direct e¤ects. This proves proposition 4.

(1 ) (1 t)2 dt dIW B

k(r+s) k+k(r+s)

1 1

1

dt t + IW B dIW B

< 0 if

dt dIW B

> 0. It is

A2 Calibration - Derivation
The parameters k and s are set to replicate, in absence of an IW B, an unemployment rate for the low skilled of uL and an average duration of unemployment for the low skilled of duL months. Given (7) and the fact that duL = (1L ) = 11 , we get
L

s= and
L

uL ; duL (1 uL ) s 1 uL uL j 1 1

(29)

=

:

(30) , we can get the following

Then, using (6) and the fact that q ( j ) = expression for k (1 ) yL k= (r + s) L +

:
L

(31)

Given uH and using the expression for H corresponding to (30), it is possible to derive the implied H . Using (6) again and the value for H derived above, it is then possible to calculate yH yH = k (r + s) 1
H

+ k

H

:

(32)

Finally, using (27) and the fact that the share of people with higher education, Edu, is given by (11), we get, after using (26), 2 = ( k) ( H )L )( H + L ) : 2 4(1 Edu

28

A3 Simulations with Financing
Using equations (15), (16), and (24), when …nancing is taken into account and for a given level of IW B, we have four unknowns ( H ; L ; a; t) solving the system of equations IW B + k L=0 1 t IW B (1 ) yH + + k H =0 1 t k (1 t) ( H a L) = 0 2 (1 ) IW B IW B (yH + k H ) (1 ) + 1 t t (1 ) yL +
L)

k (r + s) k (r + s) 1 (1 +a a)
1 L 1 H 1 H

L

(33) (34) (35)

H

s+
1 L

s+

(yL + k

(1

)

IW B 1 t

IW B =0 t
H

(36) and t, (37)

We can use equation (33) to solve for IW B as a function of IW B = k (r + s)
H

+ k H (1 )

yH (1

)

(1

t) :

We can then use this expression to replace IW B in (34) to get (1 ) (yH yL ) k (r + s) (
H L)

k(

H

L)

= 0:

(38)

We can use expression (35) to replace a in (36), expression (37) to replace IW B in the …rst line of (36) and a similar expression using (34) instead of (33) to replace IW B in the second line of (36). This gives the following equation
1 L

s+ 1 L [(1 ) yL + [(1
1 H

L

[2 (1

)

k (1
L

t) ( (1

H

L )] L] +

(39) (40)

(1

t) k (r + s) 1 k (1 2 t) (
H

t) k +

s+ ) yH

1 H

L) ( H

L) H]

(1

t) k (r + s) 29

H

(1

t) k

=0

(41)

Using (38) and (39), we have a system of two equations in two unknowns ( H ; L ) that we solve numerically. We can then get a from (35) and IW B from (37).

30