CESifo Area Conference on

Employment and Social Protection

27 – 28 May 2005

CESifo Conference Centre, Munich

Workfare, Monitoring, andEfficiency Wages

Christian Holzner, Volker Meierand Martin Werding

CESifoPoschingerstr. 5, 81679 Munich, Germany

Phone: +49 (89) 9224-1410 - Fax: +49 (89) 9224-1409E-mail: office@CESifo.de

Internet: http://www.cesifo.de

Workfare, Monitoring, and Efficiency Wages

Christian Holzner, Volker Meier, and Martin Werding ∗

Ifo Institute for Economic Research,

University of Munich, and CESifoE-mail: holzner@ifo de (Holzner); meier@ifo.de (Meier);

werding@ifo.de (Werding)

January 28, 2005

Abstract

The impacts of increasing effort requirements of welfare recipi-

ents in a workfare program are studied in an efficiency wage model

where a representative firm chooses its level of monitoring activites.

A stricter workfare program raises employment, monitoring activities

and profits, and reduces the tax rate. The impact on the net wage is

ambiguous. Utility levels of employed workers and welfare recipients

may increase even if the net wage declines. The utility differential

between these two groups of workers shrinks.

JEL classification: H53; J41; J60Keywords: Workfare; Welfare; Efficiency wages; Monitoring

∗Corresponding author: Volker Meier, Ifo Institute for Economic Research, Poschinger-

str. 5, D-81679 Munich, Germany, E-mail: meier@ifo.de.

1 Introduction

Throughout the world, welfare recipients are confronted with the obligationto offer work in exchange for the cash benefits they receive. This combinationof paying benefits and asking for a work effort has been coined workfare.Noting that in many cases the output in such programs has been quite low,many countries have refrained from using workfare. However, enforcing thework obligation on a large scale constitutes one of the core elements of theU. S. welfare reform in the last decade (see the surveys by Ellwood, 2000,Haveman and Wolfe, 2000, Blank, 2002, and Moffitt, 2002). Interestingly,the number of welfare recipients in the U. S. has fallen drastically in the lastfew years (Hotz et al., 2002).

Our paper addresses the impacts of introducing workfare institutions onemployment, wages, profits, and utility levels of both employed and unem-ployed workers. A framework of involuntary unemployment is chosen inwhich the impacts of workfare are independent of behavioral changes of wel-fare recipients. If jobs were available, welfare recipients being asked to enter aworkfare program would reduce their reservation wage, causing an increase inemployment. In our setting labor demand reacts to the changes of the incen-tive structure of the employed, who perceive the risk of a dismissal as a moresevere threat. The forward-looking welfare recipients may be compensatedfor the imposed work effort by new job opportunities.

We analyze an efficiency wage model in which workers can choose toshirk. For simplicity, all unemployed are welfare recipients, and benefits arefinanced by a proportional income tax. A representative firm chooses itsmonitoring intensity at the workplace, that is, the share of workers engagedin supervising their co-workers. The workfare program itself may also beassociated with a monitoring cost for ensuring that the participants do notshirk. All individuals are identical with respect to abilities and preferences.

Raising the work requirement for welfare recipients will increase bothemployment and the monitoring intensity, and lower both taxes and wages.As unemployment becomes more uncomfortable, wages can be cut, increas-ing the demand for labor and reducing the opportunity cost of monitoring.Taxes then go down along with a declining welfare caseload. Profits will in-crease while the impact on net wages is ambiguous. Expected lifetime utilitylevels of employed and unemployed may increase even if the net wage falls.The unemployed lose according to their increased disutility of labor, whilethey benefit through improved job opportunities. The chances to reenter

1

employment further increase due to the fact that the higher monitoring in-tensity within firms diminishes the rent of employed workers, thus inducingan additional wage cut. The smaller utility differential reduces losses arisingfrom exogenous splits of employment relationships. The results indicate thatimposing workfare can even lead to a Pareto improvement.

The theoretical literature on workfare programs has extensively dealt withscreening issues. Workfare has been justified with the objective to achievea minimum utility level or a minimum income of social assistance recipientsby Dye and Antle (1986), Blackorby and Donaldson (1988), and Besley andCoate (1992). Further, Chambers (1989), Besley and Coate (1995), Brett(1998) and Cuff (2000) have explored efficiency properties of workfare designsunder voluntary unemployment within optimum income tax frameworks. Thebasic idea is that individuals with high earnings capacities are deterred fromtaking up benefits. In a dynamic perspective, the less comfortable treatmentof the unemployed creates an incentive to acquire human capital.

The consequences of workfare on the participants are subject to disputes.While Coate (1995) stresses that the work requirement may prevent futuredependency on welfare, Peck and Theodore (2000) argue that human capitallosses arise if training is crowded out by the work obligation. A politicaleconomy argument for the widespread acceptance of workfare rules has beenoffered by Moffitt (1999). Combining a high income of the poor with awork requirement may simply reflect the preferences of the political majority.These voters will then be willing to finance unproductive workfare programs.

Surprisingly little has been said about welfare effects in a general equilib-rium context. Solow (1998) stresses that measures reducing the well-beingof welfare recipients will usually lead to an increase in employment. How-ever, he suspects that low skilled workers will be the losers of such a reformdue to falling wages. Schöb (2003) arrives at similar results for a scenariowhere a monopolistic union sets the wage rate. Within a matching model,Fredriksson and Holmlund (2003) confirm the argument of Thustrup Kreinerand Tranæs (2003), stating that workfare allows to achieve a Pareto improve-ment by screening the voluntarily unemployed and increasing the replacementrate in unemployment insurance. In their view, the drawback of workfare incomparison to time limits or monitoring search activities lies in providingrelatively little incentives for job search. Meier (2002) shows for a shirkingmodel with exogenous monitoring that net wages, and lifetime utilities ofemployed and unemployed workers all move into the same direction. Hence,it is not obvious that workfare affects employed and unemployed workers

2

asymmetrically.The remainder of the paper is organized as follows. After introducing

the model in section 2, section 3 analyzes problems of existence and stabilityof equilibria. Comparative static results are derived in section 4. Section 5discusses the results and indicates directions for future research.

2 The Model

The model is based on Shapiro and Stiglitz (1984). We consider N identicalworkers whose preferences are described by the utility function U(ω, ε) = ω−

ε, where ω denotes the monetary compensation and ε is the effort exerted atthe activity. With probability b per unit of time, an employment relationshipbreaks down for exogenous reasons. Workers are infinitely lived andmaximizeW = E

∫∞

0U(ω(s), ε(s)) exp(−rs)ds, where s denotes time, r > 0 is the

discount rate, andE represents the expectations operator. Employed workerscan either shirk (e = 0) or choose the expected effort level (e = 1). Workerswho shirk are detected with probability q per unit of time. Detected shirkersare laid off immediately.

The unemployed have to participate in the workfare program under aworkfare policy in order to receive the benefit w > 0. We neglect the pos-sibility that some of the unemployed may prefer a cut of the benefit even ifit goes down to zero. The net cost of monitoring a participant to excludeshirking accounting for a possible positive value of the output is given bym. The study of Haveman and Wolfe (2000), reporting strong increases inmonthly cost per family upon introducing the workfare program in Wiscon-sin, indicates substantial costs of organizing public jobs and enforcing thework obligation. The disutility of required work is reflected by the effort eu.Recalling that in real world programs the unemployed can choose to opt out,we assume that eu ≤ e holds, that is, the effort requirement in a workfareprogram does never exceed the effort requirement at the workplace. Welfarebenefits are financed by a proportional income tax with t denoting the taxrate. The net wage (1 − t)w of an employed worker will always be set suchthat it exceeds the welfare benefit w.

Let V S

E,V N

E, and Vu denote expected lifetime utility of an employed

shirker, employed non-shirker, and unemployed individual, respectively. Theasset equations for shirkers and non-shirkers are given by

rV S

E= (1 − t)w + (b+ q)(Vu − V S

E) (1)

3

andrV N

E= (1 − t)w − e+ b(Vu − V N

E). (2)

These equations state that the return in each period is equal to sumof the flow benefit and the expected change of the value of the asset. Anemployed worker does not shirk if V S

E≤ V N

E, being equivalent to the no-

shirking condition

(1− t)w ≥ rVu +(r + b+ q)e

q. (3)

The representative firm produces under decreasing returns. Its outputis given by Q = F (LP ) where LP denotes the number of employed pro-duction workers not shirking. The production function satisfies F ′(LP ) >0, F ′′(LP ) < 0 and F ′(N ) > e. The last property implies that full employ-ment would be efficient.

An unemployed worker always takes part in the workfare program. Hegets a new job with probability α per unit of time. The asset equation of anunemployed worker is given by

rVu = w − eu + α(VE − Vu) (4)

with VE = max{V S

E, V N

E

}. If not shirking at the workplace is optimal, (2)

and (4) can be solved. We obtain

VE − Vu =(1− t)w − w − (e− eu)

r + α+ b, (5)

rVu = w − eu + α(1− t)w −w − (e− eu)

r + α+ b, (6)

rVE = (1 − t)w − e− b(1 − t)w −w − (e− eu)

r + α+ b. (7)

Inserting (6) into the no-shirking condition yields

(1 − t)w ≥ w + e− eu +r + α + b

qe. (8)

Inducing workers not to shirk requires a higher wage w if the welfare benefit wis higher, the rate of exogenous splits b increases, the rate of obtaining a newjob α goes up, the tax rate t increases, the rate of time preference r rises or thequality of shirking detection, as measured by q, falls. The inequality indicates

4

that (1 − t)w − e > w − eu must hold to deter shirking. Recalling (6) and(7), employed workers have a higher expected remaining lifetime utility thanthose being unemployed at any given point in time. Thus, unemployment

is involuntary. Employed workers earn the information rent r + α+ bq e due

to the fact that the shirking detection technology is imperfect, that is, q isfinite.

The detection technology is endogenized as follows. Let LS denote thenumber of workers who are engaged in monitoring. The number of productiveworkers is LP = L− LS, and L is total employment. The shirking detection

rate q depends on the share of monitoring labor at the workplace, σ := LSL.

Thus, q = H(σ), where the detection functionH exhibits diminishing returns(H ′ > 0, H ′′ < 0) and satisfies the Inada conditions limσ→0H

′ = ∞ andlimσ→1H

′ = 0. Monitoring workers and productive workers receive the samewage, and monitoring workers control each other. Their shirking decision isidentical to the shirking decision of productive workers. The firm maximizesits net profits subject to the no-shirking condition. The Lagrangian is givenby

Λ = (1 − t) [F ((1 − σ)L)− wL] + λ

[(1− t)w − w − e+ eu −

r + α+ b

H(σ)e

],

with λ denoting the Lagrange multiplier associated with the no-shirking con-straint. Optimizing with respect to total employment L, the monitoringintensity σ, the gross wage w, and λ yields the first-order conditions

∂Λ

∂L= (1 − t) [(1 − σ)F ′

− w] = 0, (9)

∂Λ

∂σ= −(1 − t)LF ′ + λe

r + α+ b

[H(σ)]2H ′ = 0, (10)

∂Λ

∂w= −(1 − t)L+ λ(1− t) = 0, (11)

∂Λ

∂λ= (1 − t)w −w − e+ eu −

r + α+ b

H(σ)e = 0. (12)

The optimality conditions can be interpreted as follows. Equation (9) statesthat additional workers will be hired until the marginal productivity of labor,corrected for losses through unproductive monitoring, is equal to the grosswage. According to (10), a higher monitoring intensity is associated with aprofit reduction of (1− t)LF ′ because more labor becomes unproductive. On

5

the other hand, as indicated by λer + α+ b[H(σ)]2

H ′, the no-shirking constraint

is no longer binding. This may be exploited to increase profits by cuttingthe wage. Last, a higher wage directly increases labor costs. Again, thefirm benefits according to λ(1 − t) because the no-shirking constraint is nolonger binding. This property could be used to raise profits by driving downthe monitoring intensity. It is easy to see that, with employment level andmonitoring intensity being given, the wage will be chosen so as to satisfy theno-shirking condition with equality. From the firm’s perspective, the tax ratet and the reemployment rate α represent parameters. Lemma 1 shows thatthe firm’s optimization problem has a unique solution.

Lemma 1 At given tax rate t and reemployment rate α, an optimum vector

(σ∗, L∗, w∗) > 0, consisting of monitoring intensity σ∗, employment L∗, and

wage w∗, exists and is unique.

Proof. See Appendix A. �

In equilibrium, the number of entries into unemployment is equal to thenumber of exits:

α(N − L) = bL. (13)

Substituting for α in the no-shirking condition leads to

(1 − t)w ≥ w + e− eu +r

qe+

b

qe1

u(14)

with u = N − LN

representing the unemployment rate. At L = 0, the right-

hand side of (14) is equal to w − eu +r + b+ q

q e. It increases in L and goes

to infinity if L→ N .If workers do not shirk, the representative firm will set its labor input to

the point where the marginal product of labor, corrected for losses throughmonitoring, is equal to the gross wage, that is, w = (1 − σ)F ′((1 − σ)L).Utilizing this relationship and the government budget equation

tF ((1− σ)L) = (w +m)(N − L) (15)

shows that feasible allocations require

(1 − t)w = (1− σ)F ′((1 − σ)L)(1 −w +m

F ((1− σ)L)(N − L)). (16)

6

Note that the right-hand side of (16) is equal to F ′((1− σ)N) > 0 if L = N .Moreover, provided that F (0) = 0, an employment level L0 ∈ (0, N) exists

which satisfies (1− w +mF ((1− σ)L)

(N − L)) = 0.

All relevant decisions are taken simultaneously. The government alwaysadjusts the income tax rate instantaneously so as to balance its budget. Thefirm generally takes both the wage and the tax rate as given, and choosesits employment level and its monitoring intensity so as to maximize profits.However, the firm accepts underbidding by unemployed workers if net wagesare higher than necessary to satisfy the no-shirking constraint. Conversely,should the net wage be too low to prevent shirking, the firm will increase thegross wage. Taking as given wages, policy variables, and the unemploymentrate, employed workers choose whether or not to shirk.

3 Equilibria and Stability

The model can be simplified to a system of two equations for σ and L. Com-bining the first-order condition for the optimum monitoring intensity, (10),with the labor market equilibrium condition (13) and the budget equation(15), where the Lagrange multiplier can be derived from (11), yields

f1 = −(1−(w +m)(N − L)

F ((1− σ)L))F ′((1 − σ)L) + e

r + b NN − L

[H(σ)]2H ′(σ) = 0. (17)

Equation f1 defines a relation σ(L) which can exist for some L in the range(Lmin, N) where Lmin is the minimum feasible employment where social aidcan be financed, that is, F (Lmin) = (w + m)(N − Lmin). The first term,

−(1 −(w +m)(N − L)F ((1− σ)L)

)F ′((1 − σ)L), represents the net profit reduction

by the marginal cost per employed worker due to an increase in the mon-

itoring intensity. The second term, er + b N

N − L[H(σ)]2

H ′(σ), is the correspond-

ing marginal benefit in a labor market equilibrium. In general, we havelimσ→0 f1 = ∞ and limσ→σmax

f1 > 0, with σmax denoting the maximum

monitoring intensity being feasible, that is,(w +m)(N − L)F ((1 − σmax)L)

= 1. Ignoring

the possibility of a tangent point, there are usually at least two solutions forσ that satisfy f1(σ,L) = 0. In Figure 1, the equilibrium monitoring intensity

7

σ1 is stable while σ2 is unstable. Considering monitoring intensities close tothe equilibrium values, marginal benefit exceeds marginal cost if σ < σ1, im-plying that there is a tendency to increase the monitoring intensity, and viceversa. For having a stable equilibrium monitoring intensity it is necessarythat f1σ ≤ 0 at any combination (σ,L) that satisfies (17).

Insert Figure 1 about here

Combining the input rule (9) with the no-shirking condition (12), thelabor market equilibrium condition (13), and the budget equation (15) givesus

f2 = (1− σ)F ′((1− σ)L)

[1 −

w +m

F ((1− σ)L)(N − L)

](18)

−w − e+ eu −r + b N

N − L

H(σ)e

= 0.

Equation f2 describes a relation L(σ) which generally exists in some range(0, σmax) where σmax is defined as above. The first term, (1 − σ)F ′((1 −

σ)L)

[1 − w +m

F ((1− σ)L)(N − L)

], represents the marginal net revenue from

increasing employment. In a labor market equilibrium situation, the remain-

ing terms, −w − e + eu −r + b N

N − LH(σ)

e, are equal to −(1 − t)w, the net

marginal cost. Note that limL→Lminf2 < 0 and limL→N f2 = −∞. Neglect-

ing again the possibility of a tangent point, at least two solutions for L existthat satisfy f2(σ,L) = 0. In Figure 2, the low equilibrium employment L1 isunstable, while the high equilibrium employment L2 is stable. For employ-ment levels close to the equilibrium, we see that marginal revenue exceedsmarginal cost if L < L2, implying that there is a tendency to increase em-ployment, and vice versa. For having a stable equilibrium employment, it isnecessary that f2L ≤ 0 at the candidate vector (σ,L) satisfying (18).

Insert Figure 2 about here

8

An equilibrium is a vector (σ,L) that satisfies both (17) and (18). Thedynamics of the monitoring intensity and aggregate employment is describedby

σ̇ = h1(f1(σ,L)), (19)

L̇ = h2(f2(σ,L)), (20)

with h1(0) = h2(0) = 0, h′1> 0, and h′

2> 0. Thus, firms increase the

monitoring intensity or employment, respectively, if the marginal benefit out-weighs the marginal cost.

Taken these pieces together, Figure 3 shows a typical situation that mayemerge. The upper branch of the f1 = 0 curve is related to unstable equilib-rium monitoring intensities and the left branch of the f2 = 0 curve representsunstable equilibrium employment levels. Among the four vectors of moni-toring intensity and employment that satisfy both equations, only (σ1, L1) islocally asymptotically stable.

Insert Figure 3 about here

In the following we confine our attention to the stable branches and, there-fore, treat both σ(L) and L(σ) corresponding to the equilibrium conditionsf1 and f2, as functions. Lemma 2 demonstrates that a third condition has tobe satisfied to ensure local asymptotic stability.

Lemma 2 An equilibrium (σ,L) is locally asymptotically stable only if f1σf2L−f1Lf2σ ≥ 0 holds at (σ,L).

Proof. See Appendix B. �

Insert Figures 4 and 5 about here

9

It should be noted that all equilibria satisfying f1σ < 0, f2L < 0 andf1σf2L− f1Lf2σ ≥ 0 are locally asymptotically stable. Further, if the equilib-rium based on stable branches is unique, it will also be stable. In Figures 4and 5, the equilibria (σ1, L1) and (σ3, L3) are locally stable, whereas this isnot the case for the equilibrium (σ2, L2).

4 Comparative statics

We can now investigate the impacts of a change in the work obligation euon the endogenous variables. In this section we assume that a unique stableequilibrium (σ,L) exists. Our analysis starts by considering the impacts onthe micro level where repercussions on the reemployment rate α and the taxrate t are neglected. The reactions of the firm to a stricter work requirementin the workfare program at given macro variables are described in Proposition1.

Proposition 1 At given tax rate t and reemployment rate α, a higher work

requirement in the workfare program eu induces the representative firm to

increase the monitoring intensity σ, total employment L, and productive em-

ployment (1− σ)L, while the wage w is reduced.

Proof. See Appendix C. �

With a higher work requirement the no-shirking constraint is no longerbinding. This enables the firm to reduce the wage and to hire more labor.At a given marginal productivity of productive labor, the higher employmentlevel raise both the marginal cost and the marginal benefit of an increase inthe monitoring intensity. These two effects just offset each other. At thesame time, the wage cut reflected in the smaller marginal productivity ofproductive labor decreases the opportunity cost of monitoring. Therefore itpays to raise the monitoring intensity. In the new optimum, employment ofproductive labor will be higher because otherwise the incentive to raise themonitoring intensity would not exist.

Noting that the direct reactions create more jobs and decrease the numberof welfare recipients, the reemployment rate will increase and the tax ratewill fall. The reactions of the firm to the changes in these parameters aresummarized in Proposition 2.

10

Proposition 2 At a given tax rate t, the representative firm reacts to an

increasing reemployment rate α by increasing both the control intensity σ

and the gross wage w, and by reducing productive employment (1− σ)L. Ata given job acquisition rate α, a lower tax rate t induces the firm to decrease

the gross wage w and to raise employment L while keeping both the monitoring

intensity σ and the net wage (1− t)w constant.

Proof. See Appendix D. �

A higher job acquisition rate increases the marginal benefit of monitoringdue to a stronger incentive to shirk. This effect works in favor of raisingthe monitoring intensity. In addition, the wage rate will also rise in orderto satisfy the no-shirking condition again. The higher wage rate increasesthe cost of labor while the higher monitoring intensity is associated withproductivity losses. Profit maximization then requires a higher marginalproductivity of productive labor, which can only be achieved by employinga smaller number of productive workers.

A falling tax rate reduces the incentive to shirk. This creates a pressure onthe wage rate and the monitoring intensity. At the same time, the marginalprofit reduction from monitoring becomes larger, again implying a tendencytoward less monitoring. The falling cost of labor leads the firm to hire moreworkers. Recalling the interpretation of Proposition 1, the net effect of ahigher employment level on the monitoring intensity is positive. It turnsout that restoring incentives is achieved by cutting the wage and keeping themonitoring intensity constant. The reduction of the marginal productivity ofproductive labor due to a higher input just offsets the falling tax rate. Sincethe opportunity cost of increasing the monitoring intensity remains constant,the various impacts on the monitoring intensity are neutralized.

We proceed by investigating the impacts of a rising work requirement inthe workfare program in an equilibrium situation.

Proposition 3 A higher disutility of participants in the workfare program,

eu, increases both employment L and the monitoring intensity σ. The life-

time utility differential between employed and unemployed workers, VE − Vu,

shrinks.

Proof. See Appendix E. �

The results can easily be understood by recalling the Propositions 1 and2. Due to the higher work requirement for social assistance recipients, the

11

no-shirking condition is no longer binding. Firms respond by reducing thegross wage and by hiring more labor. Since the net impact on the opportunitycost of monitoring is negative, it turns out to be profitable to increase themonitoring intensity. With a higher labor demand and a smaller number ofwelfare recipients, the tax rate goes down and the chances of obtaining a jobare improved. The falling tax rate yields a further increase in employmentwhile keeping the monitoring intensity unaffected. The falling unemploymentrate again drives up the optimum monitoring intensity. Hence, the impacton the monitoring intensity is unambiguously positive.

In contrast to Meier (2002), there is a distributive impact on workers interms of absolute lifetime utility differentials. The utility differential shrinksdue to the rising monitoring intensity, reducing the information rent of em-ployed workers.

Productive employment evolves according to

∂(1 − σ)L

∂eu= (1− σ)

∂L

∂eu− L

∂σ

∂eu. (21)

Evaluating the derivatives yields

sgn

[∂(1 − σ)L

∂eu

]= sgn [−Lf1L − (1− σ)f1σ] (22)

= sgn

[(w +m)

LF ′

F−

LebNH ′

(N − L)2H2

−(1 − σ)e

[r + b

N

N − L

]HH ′′ − 2(H ′)2

H3

].

Although it proves difficult to demonstrate that productive employment al-ways rises, the opposite direction is clearly implausible. Hence, we assumethat the reduction of productive employment upon a fall in the unemploy-ment rate is never strong enough to compensate for the increases that occur(i) as a direct reaction to a higher work requirement for welfare recipientsand (ii) due to the fall of the tax rate.

As the wage has to satisfy the marginal productivity rule (9), profits canbe written as

π = F ((1− σ)L)− L(1− σ)F ′((1− σ)L). (23)

Profits increase if and only if productive employment goes up:

∂π

∂(1− σ)L= −(1 − σ)LF ′′((1− σ)L) > 0. (24)

12

The change in the tax rate is given by

∂t

∂eu=w +m

F 2

[−F

∂L

∂eu− (N − L)F ′

∂(1− σ)L

∂eu

]. (25)

The tax rate will fall due to both a smaller number of welfare recipients and ahigher total output. The latter is a consequence of an increasing productivelabor input.

Taking into account the equilibrium condition (17), the change of the netwage can be derived as

∂(1 − t)w

∂eu= e

[(1 − σ)bNH ′

H2(N − L)2∂L

∂eu(26)

+

(r + b

N

N − L

)(1 − σ) [H ′′H − 2(H ′)2]−H ′H

H3

∂σ

∂eu

].

The change of the net wage is ambiguous in sign. While both the highermonitoring intensity and the rising level of productive employment contributeto a falling gross wage, the tax cut works in the opposite direction. Thepositive impact of a rising employment on the net wage in equation (26)only captures the consequences of a higher reemployment rate. In general,the ambiguity result arising in the exogenous monitoring scenario in Meier(2002) must carry over because it represents a boundary case of the currentmodel. The basic idea is that the elasticity of the marginal product of laborwith respect to labor input is crucial. If this elasticity is small in absoluteterms, the tax cut tends to be the dominating effect, implying a rise of thenet wage. In contrast, with a high elasticity the falling gross wage will beassociated with small employment gains and a declining net wage.

In terms of per period utility, the welfare impacts on the two groups ofworkers are described by

∂rVE

∂eu=∂(1− t)w

∂eu+

beH ′

[H(σ)]2∂σ

∂eu(27)

and∂rVu

∂eu=∂(1 − t)w

∂eu+

(b+ r)eH ′

[H(σ)]2∂σ

∂eu. (28)

It can easily be seen that both utility measures increase if the net wageremains constant. Lifetime utility of an employed worker can rise even if thenet wage declines because the utility reduction upon the breakdown of theemployment relationship falls.

13

5 Conclusions

It has been shown that introducing a workfare program bears the potential fora Pareto improvement. It increases employment and profits and reduces grosswages. Productive employment increases as well as unproductive monitoring.The latter reduces the rent of employed workers, improving the chances ofthe unemployed to get a job. Interestingly, the unemployed would be thewinners of the reform in terms of absolute utility differentials. Since the taxrate falls due to a higher output and a smaller number of unemployed, the netwage may move in either direction. A constant net wage would already besufficient to achieve a higher expected lifetime utility for everybody. Theseresults may explain why workfare experiments have become more popularduring the last years.

Our model may be too optimistic with respect to the employability of theunemployed. Usually, welfare recipients display less favorable labor marketcharacteristics than short-term unemployed and show a substantially smallerexit rate into employment. It may therefore be the case that workfare forwelfare recipients affects the unemployed in a asymmetric fashion. The im-pact on lifetime utility will then be positive for short-term unemployed andnegative for some groups of long-term unemployed who are made subject tothe workfare rule while the chances of regaining employment remain small.It is therefore typical that introducing workfare without screening the can-didates before imposing the work requirement always harms some of thepoorest individuals.

6 Appendix

A: Proof of Lemma 1

Isolating λ in (11) and F ′ in (9), and inserting the results into (10) yields

−(1− t)Lw

1− σ+ Le

r + α+ b

[H(σ)]2H ′ = 0.

14

Substituting for (1− t)w from (12) then implies

Ψ(σ) = −

w + e− eu +r + α+ bH(σ)

e

1 − σ+ e

r + α + b

[H(σ)]2H ′ (29)

= er + α+ b

H(σ)

[H ′(σ)

H(σ)−

1

1 − σ

]−w + e− eu

1 − σ

= 0.

Note that limσ→0Ψ(σ) =∞, limσ→1Ψ(σ) = −∞, and

Ψ′(σ) = −w + e− eu

(1 − σ)2− e

r + α+ b

H(σ)

H ′(σ)

H(σ)

[H ′(σ)

H(σ)−

1

1 − σ

]

+er + α + b

H(σ)

[H ′′(σ)H(σ)− [H ′(σ)]2

[H(σ)]2−

1

(1 − σ)2

]

=1

1− σ

[er + α+ b

H(σ)

[H ′(σ)

H(σ)−

1

1− σ

]−w + e− eu

1− σ

]

+er + α + b

H(σ)

[H ′′(σ)H(σ)− 2 [H ′(σ)]2

[H(σ)]2

].

Any candidate σ∗ that satisfies Ψ(σ) = 0 then has the property

Ψ′(σ) = er + α + b

H(σ)

[H ′′(σ)H(σ)− 2 [H ′(σ)]2

[H(σ)]2

]< 0.

Thus, an interior solution for the optimum control intensity σ exists, and it isunique. The no-shirking condition (12) then uniquely determines the relatedgross wage w. Last, at given σ and w the input rule (9) uniquely determinesemployment. �

B: Proof of Lemma 2

Recalling that a locally stable equilibrium requires both f1σ ≤ 0 and f2L ≤ 0,the inequality f1σf2L−f1Lf2σ ≥ 0 can be violated only if sgn[f1L] = sgn[f2σ].Suppose that f1σf2L < f1Lf2σ.

If f1L < 0, this would imply f1Lf1σ

>f2Lf2σ

, being equivalent to −f1Lf1σ

<

−f2Lf2σ

. Hence, we would havedσ(L)dL

< 1dL(σ)dσ

< 0. Such a situation corre-

15

sponds to an equilibrium which is only saddle-path stable, but not locallyasymptotically stable (see, for example, the equilibrium (σ2, L2) in Figure 4).

Should f1σf2L < f1Lf2σ hold in combination with f1L > 0 at the equilib-

rium, this would yield f1Lf1σ

<f2Lf2σ

, being equivalent to −f1Lf1σ

> −f2Lf2σ

, im-

plying thatdσ(L)dL

> 1dL(σ)dσ

> 0. Again, the equilibrium is only saddle-path

stable, but not locally asymptotically stable (see, for example, the equilib-rium (σ2,L2) in Figure 5). �

C: Proof of Proposition 1

Totally differentiating (29) and taking into account the proof of Lemma 1yields

∂σ

∂eu= −

∂Ψ∂eu∂Ψ∂σ

= −1

(1− σ)∂Ψ∂σ

> 0.

Note that L can be eliminated from (10) after inserting for λ from (11).Applying the implicit function rule to (10) then yields

∂L

∂σ=

(1− t)LF ′′ + e(r + α+ b)HH ′′ − 2H(H ′)2

H3

(1− t)(1 − σ)F ′′> 0.

Moreover, we arrive at

∂(1− σ)L

∂σ= (1− σ)

∂L

∂σ− L

=e(r + α+ b)

HH ′′ − 2H(H ′)2

H3

(1− t)F ′′> 0.

Last, according to (9) , it follows that

∂w

∂eu= −F ′

∂σ

∂eu+ (1− σ)F ′′

∂(1− σ)L

∂eu< 0.

�

16

D: Proof of Proposition 2

Due to the implicit function theorem, it follows that

∂σ

∂α= −

∂Ψ∂α∂Ψ∂σ

> 0

because ∂Ψ∂α

= eH(σ)

[H ′(σ)H(σ)

−1

1 − σ

]> 0 holds if w + e− eu > 0. Taking

into account (12) then implies

(1 − t)∂w

∂α=

e

H(σ)−r + α + b

H(σ)eH ′(σ)

H(σ)

∂σ

∂α.

Evaluating ∂σ∂α

yields

∂σ

∂α= −

eH(σ)

[H ′(σ)H(σ)

− 11− σ

]

er + α + bH(σ)

[H ′′(σ)H(σ)− 2 [H ′(σ)]2

[H(σ)]2

]

=

H ′(σ)H(σ)

−1

1− σ

(r + α+ b)

[2 [H ′(σ)]2 −H ′′(σ)H(σ)

[H(σ)]2

] .Therefore,

(r + α+ b)H ′(σ)

H(σ)

∂σ

∂α=

(H ′(σ)H(σ)

)2

−H ′(σ)

H(σ)(1− σ)

2 [H ′(σ)]2 −H ′′(σ)H(σ)

[H(σ)]2

< 1

and (1− t)∂w∂α

> 0.With ∂w∂α

> 0 and ∂σ∂α

> 0, the falling level of productive

employment can immediately be seen from the labor demand equation (9).Note that (29), characterizing the optimum monitoring intensity, is unaf-

fected by a variation of the tax rate as long as the job acquisition rate α is

fixed. Hence, we have ∂σ∂t

= 0. Recalling the no-shirking condition (12), this

implies a constant net wage (1 − t)w and an increasing gross wage w. Withan unchanged monitoring intensity σ and a rising wage w, total employmentis reduced according to the input rule (9). �

17

E: Proof of Proposition 3

Applying the implicit function theorem yields

∂L

∂eu= −

f1σ

∆> 0,

∂σ

∂eu=

f1L

∆,

where ∆ = f1σf2L − f2σf1L > 0 is a consequence of the stability conditionfor the equilibrium. Evaluating the numerators leads to

f1σ = (1 − t)LF ′′ + F ′(w +m)(N − L)LF ′

(F )2

+e

[r + b

N

N − L

]HH ′′ − 2(H ′)2

H3,

f1L = −(1− t)(1− σ)F ′′− F ′(w +m)

F + (N − L)(1 − σ)F ′

(F )2

+ebNH ′

(N − L)2H2.

Note that f1σ < 0 is a sufficient stability condition, while f1L can be rewrittenas

f1L = −f2L

(1− σ)+

ebN

H(N − L)2

[H ′

H−

1

1− σ

]with

f2L = (1− σ)2(1 − t)F ′′ + (1− σ)F ′(w +m)F + (N − L)(1− σ)F ′

(F )2

−ebN

H(N − L)2.

The inequality f2L ≤ 0 has to hold in any stable equilibrium. Further,combining the equilibrium conditions (17) and (18) reveals that

sgn

[H ′

H−

1

1− σ

]= sgn [w + e− eu] > 0.

Thus, we arrive at f1L > 0.

18

Noting that the no-shirking condition holds with equality, combining (5)with (8) yields

VE − Vu =e

q,

implying that∂[VE − Vu]

∂eu= − e

q2H ′(σ) ∂σ

∂eu< 0. �

19

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21

�

�

1 σ

f1

0σ1 σ2 σmax

� ��

Fig. 1. Stable and unstable monitoring intensity

22

�

�

N L

f2

0L1 L2Lmin

� ��

Fig. 2. Stable and unstable equilibrium employment

23

�

�

��

�� ����

��

��

�� �

���

N L

σ

0

1

f2 = 0

f1 = 0

L1

σ1

Fig. 3. Stable and unstable branches

24

�

�

��

��

��

��

��

��

N L

σ

0

1

σ(L)

L(σ)

L1 L2 L3

σ1

σ2

σ3

Fig. 4. Stable and unstable equilibria (i)

25

�

�

��

��

�� ��

����

N L

σ

0

1

σ(L)

L(σ)

L1L2 L3

σ1

σ2

σ3

Fig. 5. Stable and unstable equilibria (ii)

26