STIGMA, OPTIMAL INCOME TAXATION, AND THE OPTIMAL WELFARE PROGRAM: A THEORETICAL ANALYSIS ∗

ZHIYONG AN China Economics and Management Academy Central University of Finance and Economics Beijing, China 100081 Cell: (86)15010653641 Email: [email protected]

ABSTRACT This paper integrates the theory of welfare stigma (Moffitt, 1983) and the theory of optimal income taxation. We study optimal income taxation and an optimal welfare program within a unified framework while taking welfare stigma into account. In our framework, the government is assumed to have two policy instruments: (1) general income taxation; and (2) a welfare program. Individuals are assumed to be heterogeneous along two dimensions: (1) wages (skill); and (2) welfare stigma. Each individual is assumed to take the income tax schedule and the parameters of the welfare program as given and make labor supply decisions to maximize his utility. The government is assumed to choose both an optimal income tax schedule and an optimal welfare program so as to maximize a social welfare function subject to its own revenue budget constraint and individuals’ behavioral response to the income tax schedule and the parameters of the welfare program. Our analysis shows that: (1) it can be optimal for the government to offer both general income taxation and a welfare program; and (2) the more intensely people suffer from welfare stigma, the higher the welfare benefit should be.

Key words: Stigma; Optimal Income Taxation; Optimal Welfare Program

∗

The author thanks George A. Akerlof, Robert M. Anderson, Alan J. Auerbach, Raj Chetty, John M. Quigley, Emmanuel Saez and Brian D. Wright for valuable discussions and advice. The author also thanks all the Public Finance seminar participants at UC Berkeley. Comments from Kim M. Bloomquist at the Internal Revenue Service (IRS) are highly appreciated.

I.

INTRODUCTION AND LITERATURE REVIEW

Mirrlees (1971) and Fair (1971) broke ground with their research on optimal income taxation. In their models, the government is assumed to have only one instrument at hand, namely, general income taxation. Individuals are assumed to be heterogeneous only along one dimension, namely, wages (skill). Each individual is assumed to take the income tax schedule as given and choose his labor supply to maximize his own utility. The government is assumed to choose the optimal income tax schedule by maximizing a social welfare function subject to its own revenue budget constraint and individuals’ behavioral response to the chosen income tax schedule. The basic trade-off captured in their models is between equity and efficiency. Intuitively, due to progressive income taxation, the government can redistribute income from rich people to poor people. The redistribution of income can improve distributional equity because poor people have higher marginal utility than rich people. However, labor supply is elastic. The progressive income taxation will change people’s labor supply and thus result in deadweight loss (efficiency cost). Because Mirrlees (1971) assumed a continuum of unbounded skills in his model, it is very difficult to reach general conclusions. In contrast, Fair (1971) assumed a limited discrete number of individuals in his model, thus laying a good foundation for his numerical simulations. His simulations show that the average tax rate should increase with income (skills). In other words, the optimal tax schedule should be progressive. Since then, the theory of optimal income taxation based on the original Mirrlees-Fair framework has been considerably developed (e.g., Sadka (1976); Seade (1977); Akerlof (1978); Nichols and Zeckhauser (1982); Diamond (1998); Saez (2002)). Several recent papers attempt to introduce psychological concepts into the study of optimal income taxation (e.g., Ireland (1998, 2001); Corneo (2002)).

1

However, to the best of our knowledge, there is no published research on optimal income taxation that has taken into account the stigma factor (Moffitt, 1983). In order to account for the low take-up rate of welfare benefits 1 , Moffitt proposed the idea of stigma, “disutility arising from participation in a welfare program per se.” He said that there might be both a “flat” stigma that arises from the participation in welfare programs itself and a “variable” stigma that varies with the size of the benefit. The “flat” stigma and the “variable” stigma have different implications for people’s participation decisions. If only a “flat” stigma exists, then an individual will participate in welfare programs if the utility gain from the welfare benefit is greater than the utility loss due to the “flat” stigma. If only a “variable” stigma exists, then the individual will only participate if his utility increases from the welfare benefit. Moffitt’s empirical estimation shows that stigma appears to arise mainly from the flat component, i.e., a fixed cost associated with participation in the program. Clearly, Moffitt’s work shows that stigma plays an important role in people’s utility functions and thus distorts people’s labor supply decisions. Because stigma plays an important role in people’s utility functions and their labor supply decisions, it must also have important implications for the optimal income tax schedule. Therefore, in this paper we integrate the theory of welfare stigma and the theory of optimal income taxation by studying optimal income taxation and an optimal welfare program within a unified framework while taking welfare stigma into account. To the best of our knowledge, our work is the first touch on this important topic. In order to study optimal income taxation and an optimal welfare program within a unified framework, we build a theoretical model that is a variant of the model in Akerlof’s “tagging” paper (1978). In our model, we assume that the government has two policy 1

According to Moffitt (1983), the participation rate in the Aid to Families with Dependent Children (AFDC) was estimated to be only about 69 percent in 1970 and the participation rate in the Food Stamp Program was only 38 percent.

2

instruments: (1) general income taxation; and (2) a welfare program. We assume that individuals are heterogeneous along two dimensions: (1) wages (skill); and (2) welfare stigma. Each individual is assumed to take the income tax schedule and the parameters of the welfare program as given and make two decisions to maximize his utility: (1) whether to self-select into the general income taxation or the welfare program; and (2) conditional on self-selecting into the general income taxation, how much labor to supply. If an individual self-selects into the welfare program, then in addition to the utility derived from income and leisure, he will also incur a utility loss due to welfare stigma (Moffitt, 1983). However, if an individual self-selects into the general income taxation, there is no extra utility loss, because it is generally believed that people do not attach stigma to general income taxation, whether it is positive or negative 2 . Intuitively, other things being equal, individuals with high welfare stigma tend to self-select into the general income taxation, while individuals with low welfare stigma tend to self-select into the welfare program. The government is assumed to choose both an optimal income tax schedule and an optimal welfare program so as to maximize a social welfare function subject to its own budget constraint and individuals’ behavioral response to the income tax schedule and the parameters of the welfare program. Our theoretical analysis shows that: (1) it can be optimal for the government to offer both general income taxation and a welfare program; and (2) the more intensely people suffer from welfare stigma, the higher the welfare benefit should be. Our first result intuitively makes sense. Because the government needs to address two concerns, namely, income redistribution and welfare stigma, the government should use two policy instruments. In other words, because individuals are heterogeneous along two dimensions

2

This is common assumption in the literature on optimal income taxation. It is implicit in Mirrlees and Fair’s modeling (Mirrlees, 1971; Fair, 1971).

3

(namely, wages (skill) and welfare stigma), the government should use two instruments. This result might be related with the famous Atkinson-Stiglitz result (Atkinson and Stiglitz, 1976). Atkinson and Stiglitz showed that if individuals’ utility function is separable between consumption and leisure, optimal non-linear income taxation makes commodity taxation useless. Saez (2002) reinterpreted the Atkinson-Stiglitz result. He showed that the separability between consumption and leisure is misleading for the Atkionson-Stiglitz result. He said that due to the separability, individuals are essentially heterogeneous along one dimension, namely, wages. If individuals are heterogeneous along two dimensions, namely, wages and tastes for consumption goods, then both income taxation and commodity taxation are desirable. Similarly, in our model individuals are heterogeneous along two dimensions, namely, wages and welfare stigma, the government therefore should use two policy instruments. Our second result also intuitively makes sense. The more intensely people suffer from welfare stigma, the government should use higher welfare benefit to compensate larger utility loss due to welfare stigma. The remainder of this paper is organized as follows. Section II presents our theoretical model and theoretical analysis. Section III concludes.

II.

THEORETICAL MODEL AND THEORETICAL ANALYSIS

Our model is a variant of the model in Akerlof’s “tagging” paper (1978). In his model, Akerlof assumed that there are two types of labor: skilled labor and unskilled labor. Skilled labor can take both difficult and easy jobs, whereas unskilled labor can only take easy jobs. With only two types of labor, Akerlof could derive explicit solutions. In addition, although there are only

4

two types of labor, the key insight and the general framework of Mirrlees-Fair are carried over so that his conclusions hold in a richer economic context. Hence, his model is a useful starting point for our research purpose. We expand Akerlof’s model as follows. First, we assume that one-half of the population is made up of skilled people and the other one-half of the population is made up of unskilled people. We assume that all the skilled people have welfare stigma φ . We assume that (1 − α ) of the unskilled people have welfare stigma φ while α of the unskilled people have no welfare stigma at all. In summary, we assume that there are three types of people: (1) skilled people with stigma (

1 1−α of the whole of the whole population); (2) unskilled people with stigma ( 2 2

population); and (3) unskilled people without stigma (

α 2

of the whole population). These data

are summarized in Table 1. Therefore, individuals are heterogeneous along two dimensions in our model: (1) wages (skill); and (2) welfare stigma. Skilled people have three options: (1) take a difficult job; (2) take an easy job; or (3) be on

welfare.

These

people’s

utility

function

is

given

by

max {u (q D − t D ) − δ D , u (q E + t E ) − δ E , u (b ) − φ } . We assume that u (0 ) = 0 , u ' (.) > 0 , and u" (.) < 0 . If a skilled person takes a difficult job, his output is q D and his disutility of doing the

job is δ D . As the economy is assumed to be competitive, his pre-tax income is also q D . His utility depends on his after-tax income and his disutility of doing the difficult job and is given by u (q D − t D ) − δ D . We assume that u (q D ) − δ D > 0 . We assume that if an individual takes an easy

job, whether he is a skilled person or an unskilled person, his output is q E and thus his pre-tax income is q E . His disutility of doing the easy job is δ E whether he is skilled or unskilled. Thus

5

his pre-tax utility is given by u (q E ) − δ E and his after-tax utility is given by u (q E + t E ) − δ E . We assume that u (q E ) − δ E < 0 . Finally, if a skilled person chooses to be on welfare, his income is the welfare benefit b and he does not supply any labor. However, because he attaches stigma φ to being on welfare by assumption, his utility is given by u (b ) − φ . We assume that if a skilled person is indifferent between taking a difficult job, taking an easy job, and being on welfare, then he chooses to take the difficult job. We assume that if a skilled person is indifferent between taking an easy job and being on welfare, then he chooses to take the easy job. Unskilled people with stigma have two options: (1) take an easy job or (2) be on welfare. These people’s utility function is given by max {u (q E + t E ) − δ E , u (b ) − φ } . If an unskilled person with stigma takes an easy job, his utility is given by u (q E + t E ) − δ E . If an unskilled person with stigma chooses to be on welfare, his utility is given by u (b ) − φ . We assume that if an unskilled individual with stigma is indifferent between taking an easy job and being on welfare, then he chooses to take the easy job. In our model, we follow Akerlof (1978) and assume that unskilled people cannot work at difficult jobs. An alternative but equivalent assumption is that they can work at difficult jobs but that the disutility of doing so is so high that in equilibrium they will not choose to. Unskilled people without stigma have two options: (1) take an easy job or (2) be on welfare. These people’s utility function is given by max { u (q E + t E ) − δ E , u (b )} . If an unskilled person without stigma takes easy jobs, his utility is given by u (q E + t E ) − δ E . If an unskilled person without stigma chooses to be on welfare, his utility is given by u (b ) because he does not attach stigma to being on welfare. We assume that if an unskilled individual without stigma is indifferent between taking an easy job and being on welfare, then he chooses to be on welfare.

6

The proceeding data are summarized in Table 2. The social welfare function is given by: ⎧max {u (q D − t D ) − δ D , u (q E + t E ) − δ E , u (b ) − φ }, ⎪ SW = min ⎨max { u (q E + t E ) − δ E , u (b ) − φ }, ⎪max {u (q + t ) − δ , u (b )} E E E ⎩

⎫ ⎪3 ⎬ . ⎪ ⎭

Each individual is assumed to take (t D , t E , b ) as given and make optimal labor supply decisions to maximize his utility. The government is assumed to choose (t D , t E , b ) to maximize SW subject to its own revenue budget constraint and individuals’ behavioral response to

(t D , t E , b ) , the parameters of the general income taxation and the welfare program. Throughout the theoretical analysis, we assume that under equilibrium it is optimal for an

(

)

individual to stay in the “system”. In other words, we assume u q D − t D* − δ D > 0 ,

(

)

( )

u q E + t E* − δ E > 0 , and u b * − φ > 0 . Zero is the utility outside the “system”.

Because u (q D ) − δ D > 0 and u (q E ) − δ E < 0 by assumption, there are two possible cases for

this

problem:

u (q D ) − δ D > 0 > u (q E ) − δ E > −φ ;

(1)

and

(2)

u (q D ) − δ D > 0 > −φ > u (q E ) − δ E .

Proposition 1: If u (q D ) − δ D > 0 > u (q E ) − δ E > −φ , and if α is close to one, then the

equilibrium is a separating equilibrium, and the optimal conditions are given by:

(

)

( )

(

)

(

(1) u q E + t E* − δ E = u b * ,

)

(2) u q D − t D* − δ D = u q E + t E* − δ E , and (3) t D* = (1 − α )t E* + αb * .

3

The standard social welfare function assumed in the optimal income taxation literature is a “sum” weighted by population. The “min” social welfare function is chosen so that it is feasible to derive the optimal solutions to the model. One justification for the “min” social welfare function is that it turns out that the intuition underlying the key conclusions does not depend on this specific assumption.

7

Proof: See Appendix 2.

Proposition 2: If u (q D ) − δ D > 0 > u (q E ) − δ E > −φ , and if α is close to zero, then the

equilibrium is either a separating equilibrium or a pooling equilibrium, and the optimal conditions are given either by for the separating equilibrium:

(

)

( )

(

)

(

(4) u q E + t E* − δ E = u b * ,

)

(5) u q D − t D* − δ D = u q E + t E* − δ E , and (6) t D* = (1 − α )t E* + αb * , or by for the pooling equilibrium:

(

)

( )

(

)

( )

(7) u q E + t E* − δ E < u b * − φ , (8) u q D − t D* − δ D = u b * − φ , and (9) t D* = b * . Proof: See Appendix 2.

Proposition 3: If u (q D ) − δ D > 0 > −φ > u (q E ) − δ E , and if α is close to one, then the

equilibrium is a separating equilibrium, and the optimal conditions are given by:

(

)

( )

(

)

(

(10) u q E + t E* − δ E = u b * ,

)

(11) u q D − t D* − δ D = u q E + t E* − δ E , and (12) t D* = (1 − α )t E* + αb * . Proof: See Appendix 2.

8

Proposition 4: If u (q D ) − δ D > 0 > −φ > u (q E ) − δ E , and if α is close to zero, then the

equilibrium is a pooling equilibrium, and the optimal conditions are given by:

(

)

( )

(

)

( )

(13) u q E + t E* − δ E < u b * − φ , (14) u q D − t D* − δ D = u b * − φ , and (15) t D* = b * . Proof: See Appendix 2.

Proposition 5: For both cases, whether α is close to one or close to zero, it can be optimal for

the government to provide both general income taxation and a welfare program. Proof: Proposition 5 is obvious from Propositions 1, 2, 3, and 4.

Proposition 5 intuitively makes sense. As the government has two concerns (namely, income redistribution and welfare stigma) to address, the government should use two instruments. In other words, because individuals are heterogeneous along two dimensions (namely, wages (skill) and welfare stigma), the government should use two instruments. This result might be related with the famous Atkinson-Stiglitz result (Atkinson and Stiglitz, 1976). Atkinson and Stiglitz showed that if individuals’ utility function is separable between consumption and leisure, optimal non-linear income taxation makes commodity taxation useless. Saez (2002) reinterpreted the Atkinson-Stiglitz result. He showed that the separability between consumption and leisure is misleading for the Atkionson-Stiglitz result. He said that due to the separability, individuals are essentially heterogeneous along one dimension, namely, wages. If individuals are heterogeneous along two dimensions, namely, wages and tastes for consumption goods, then both income taxation and commodity taxation are desirable. Similarly, in our model

9

individuals are heterogeneous along two dimensions, namely, wages and welfare stigma, the government should use two policy instruments.

Proposition 6: For both cases, if α is close to zero, then the equilibrium is likely to be a pooling

equilibrium so that all the unskilled people are on welfare, and the optimal conditions are given by:

(

)

( )

(

)

( )

(16) u q E + t E* − δ E < u b * − φ , (17) u q D − t D* − δ D = u b * − φ , and (18) t D* = b * . Proof: Proposition 6 is obvious from Propositions 2 and 4.

Proposition 6 also intuitively makes sense. As α decreases, the unskilled people’s capability of “producing” utility is decreasing on average, which implies that the burden on the skilled people will increase. Therefore, conditional on a separating equilibrium, as α decreases, t D* will increase. When α decreases enough, t D* will increase enough so that skilled people will

switch from difficult jobs to easy jobs. In order to prevent skilled people from switching from difficult jobs to easy jobs, when α is small enough, the optimal condition is changed from

(

)

( )

(

)

( )

u q D − t D* − δ D = u b * under a separating equilibrium to u q D − t D* − δ D = u b * − φ under a

pooling equilibrium so that the government can extract more from the skilled people while still keeping the skilled people taking difficult jobs.

Proposition 7: Conditional on a pooling equilibrium so that all the unskilled people are on

welfare, then

db * > 0. dφ

10

Proof: If the equilibrium is a pooling equilibrium so that all the unskilled people are on welfare,

(

)

( )

(

)

( )

then the optimal conditions are given by u q E + t E* − δ E < u b * − φ , u q D − t D* − δ D = u b * − φ ,

(

)

( )

and t D* = b * . The optimal conditions imply that u q D − b * − δ D = u b * − φ , which further db * > 0. implies that dφ

In summary, Proposition 5 says that it can be optimal for the government to offer both general income taxation and a welfare program. Proposition 6 says that if most of the unskilled people suffer from welfare stigma (i.e., if α is close to zero), then it is likely to be optimal for the government to put all the unskilled people on the welfare program (i.e., the equilibrium is likely to be a pooling equilibrium so that all the unskilled people are on welfare). In addition, conditional on a pooling equilibrium so that all the unskilled people are on welfare, Proposition 7 says that the more intensely people suffer from welfare stigma (i.e., the higher φ ), the higher the welfare benefit should be (i.e., the higher b * ).

III.

CONCLUSION

Moffitt (1983) shows that stigma plays an important role in people’s utility functions and thus distorts people’s labor supply decisions. Because stigma plays an important role in people’s utility functions and their labor supply decisions, it must also have important implications for the optimal income tax schedule. However, the traditional literature on optimal income taxation does not take this important factor into account.

11

In this research, we integrate the theory of welfare stigma and the theory of optimal income taxation. In order to do so, we study optimal income taxation and an optimal welfare program within a unified framework while taking welfare stigma into account. In our framework, the government has two instruments at hand: (1) general income taxation; and (2) a welfare program. Individuals are heterogeneous along two dimensions: (1) wages; and (2) welfare stigma. Each individual is assumed to take the income tax schedule and the parameters of the welfare program as given and choose his labor supply to maximize his utility. The government is assumed to choose both an optimal income tax schedule and an optimal welfare program so as to maximize a social welfare function subject to its own budget constraint and individuals’ behavioral response to the income tax schedule and the welfare program parameters. The key conclusions that our theoretical analysis reaches are: (1) it can be optimal for the government to offer both general income taxation and a welfare program; and (2) the more intensely people suffer from welfare stigma, the higher the welfare benefit should be.

12

APPENDIX 1: TABLES

Table 1: Output of Worker by Type of Worker by Type of Job

Type of Worker (Percent of Workforce)

Difficult Jobs

1 ) qD 2 1−α Unskilled with Stigma ( ) Not Applicable 2 Skilled with Stigma (

Unskilled without Stigma (

α

2

)

Not Applicable

Easy Jobs

On Welfare

qE

0

qE

0

qE

0

Note: q D > q E

13

Table 2: Utility of Workers by Type of Workers by Type of Job (After Tax and Transfer)

Type of Worker (Percent of Workforce)

Difficult Jobs

1 ) u (q D − t D ) − δ D 2 1−α Unskilled with Stigma ( ) Not Applicable 2 Skilled with Stigma (

Unskilled without Stigma (

α

)

Not Applicable

2 Note: u (q D ) − δ D > 0 > u (q E ) −δ E

Easy Jobs

On Welfare

u (q E + t E ) − δ E

u (b ) − φ

u (q E + t E ) − δ E

u (b ) − φ

u (q E + t E ) − δ E

u (b )

14

APPENDIX 2: PROOF OF PROPOSITIONS 1, 2, 3, AND 4

In order to prove Propositions 1, 2, 3, and 4, we need to prove eight lemmas first.

Lemma 1: Conditional on that under equilibrium skilled people choose to take difficult jobs and all unskilled people choose to take easy jobs, the optimal conditions are given by:

(

)

( )

(

)

(

(1) u q E + t E* 1 − δ E > u b1* ,

)

(2) u q D − t D* 1 − δ D = u q E + t E* 1 − δ E , and (3) t D* 1 = t E* 1 .

(

)

The maximum value of the social welfare function is SW1 = u q D − t D* 1 − δ D , where t D* 1 satisfies

u (q D − t D* 1 ) − δ D = u (q E + t D* 1 ) − δ E , i.e., t D* 1 = u −1 (u (q D − t D* 1 ) − δ D + δ E ) − q E . Proof: In order to induce skilled people to take difficult jobs, the government should set parameters such that u (q D − t D1 ) − δ D ≥ u (q E + t E1 ) − δ E and u (q D − t D1 ) − δ D ≥ u (b1 ) − φ . In order to induce unskilled people with stigma to take easy jobs, the government should set parameters such that u (q E + t E1 ) − δ E ≥ u (b1 ) − φ . In order to induce unskilled people without stigma

to

take

easy

jobs,

the

government

should

set

parameters

such

that

u (q E + t E1 ) − δ E > u (b1 ) . Taken together, in order to induce skilled people to take difficult jobs

and all unskilled people to take easy jobs, the government should set parameters such that u (q E + t E1 ) − δ E > u (b1 ) and u (q D − t D1 ) − δ D ≥ u (q E + t E1 ) − δ E . The government’s budget

constraint implies that t D1 = t E1 . Conditional on that u (q E + t E1 ) − δ E > u (b1 ) , u (q D − t D1 ) − δ D ≥ u (q E + t E1 ) − δ E , and t D1 = t E1 , in order to maximize the social welfare function, the government should extract as

much as possible from the skilled people without inducing them to switch from difficult jobs to

15

easy

jobs.

Therefore,

the

government

should

set

parameters

such

that

u (q D − t D* 1 ) − δ D = u (q E + t E* 1 ) − δ E . We also have u (q E + t E* 1 ) − δ E > u (b1* ) so that all unskilled people choose to take easy jobs under equilibrium. The budget constraint implies that t D* 1 = t E* 1 . Thus, we have proved the optimal conditions.

(

)

( )

(

)

(

)

With u q E + t E* 1 − δ E > u b1* , u q D − t D* 1 − δ D = u q E + t E* 1 − δ E , and t D* 1 = t E* 1 , the

(

)

utility of skilled people is u q D − t D* 1 − δ D , the utility of unskilled people with stigma is

u (q E + t E* 1 ) − δ E , and the utility of unskilled people without stigma is u (q E + t E* 1 ) − δ E . Because u (q D − t D* 1 ) − δ D = u (q E + t E* 1 ) − δ E , we have u (q D − t D* 1 ) − δ D = u (q E + t D* 1 ) − δ E

by

SW1 = u (q D − t D* 1 ) − δ D , where

u (q D − t D* 1 ) − δ D = u (q E + t E* 1 ) − δ E

t D* 1

and

satisfies

t D* 1 = t E* 1 .

u (q D − t D* 1 ) − δ D = u (q E + t D* 1 ) − δ E is equivalent to t D* 1 = u −1 (u (q D − t D* 1 ) − δ D + δ E ) − q E . Thus, we have proved Lemma 1.

Lemma 2: Conditional on that under equilibrium skilled people choose to take difficult jobs and all unskilled people choose to be on welfare, the optimal conditions are given by:

(

)

( )

(

)

( )

(4) u q E + t E* 2 − δ E < u b2* − φ , (5) u q D − t D* 2 − δ D = u b2* − φ , and (6) t D* 2 = b2* .

(

)

The maximum value of the social welfare function is SW2 = u q D − t D* 2 − δ D , where t D* 2 satisfies

u (q D − t D* 2 ) − δ D = u (t D* 2 ) − φ , i.e., t D* 2 = u −1 (u (q D − t D* 2 ) − δ D + φ ). Proof: In order to induce skilled people to take difficult jobs, the government should set parameters such that u (q D − t D 2 ) − δ D ≥ u (q E + t E 2 ) − δ E and u (q D − t D 2 ) − δ D ≥ u (b2 ) − φ . In order to induce unskilled people with stigma to be on welfare, the government should set

16

parameters such that u (q E + t E 2 ) − δ E < u (b2 ) − φ . In order to induce unskilled people without stigma

to

be

on

welfare,

the

government

should

set

parameters

such

that

u (q E + t E 2 ) − δ E ≤ u (b2 ) . Taken together, in order to induce skilled people to take difficult jobs

and all unskilled people to be on welfare, the government should set parameters such that u (q E + t E 2 ) − δ E < u (b2 ) − φ

and

u (q D − t D 2 ) − δ D ≥ u (b2 ) − φ . The government’s budget

constraint implies that t D 2 = b2 . Conditional on u (q E + t E 2 ) − δ E < u (b2 ) − φ , u (q D − t D 2 ) − δ D ≥ u (b2 ) − φ , and t D 2 = b2 , in order to maximize the social welfare function, the government should extract as much as possible from the skilled people without inducing them to switch from difficult jobs to welfare.

(

)

( )

Therefore, the government should set parameters such that u q D − t D* 2 − δ D = u b2* − φ . We also

(

)

( )

have u q E + t E* 2 − δ E < u b2* − φ so that all unskilled people choose to be on welfare under equilibrium. The budget constraint implies that t D* 2 = b2* . Thus, we have proved the optimal conditions.

(

)

( )

(

)

( )

With u q E + t E* 2 − δ E < u b2* − φ , u q D − t D* 2 − δ D = u b2* − φ , and t D* 2 = b2* , the utility

(

)

( )

of skilled people is u q D − t D* 2 − δ D , the utility of unskilled people with stigma is u b2* − φ , and the

utility

of

unskilled

people

without

stigma

is

u (b2* ).

Because

u (q D − t D* 2 ) − δ D = u (b2* ) − φ < u (b2* ) , we have SW2 = u (q D − t D* 2 ) − δ D , where t D* 2 satisfies u (q D − t D* 2 ) − δ D = u (t D* 2 ) − φ

by

u (q D − t D* 2 ) − δ D = u (b2* ) − φ

and

t D* 2 = b2* .

u (q D − t D* 2 ) − δ D = u (t D* 2 ) − φ is equivalent to t D* 2 = u −1 (u (q D − t D* 2 ) − δ D + φ ). Thus, we have proved Lemma 2.

17

Lemma 3: Conditional on that under equilibrium skilled people choose to take difficult jobs, unskilled people with stigma choose to take easy jobs, and unskilled people without stigma choose to be on welfare, then the optimal conditions are given by:

(

)

( )

(

)

(

(7) u q E + t E* 3 − δ E = u b3* ,

)

(8) u q D − t D* 3 − δ D = u q E + t E* 3 − δ E , and (9) t D* 3 = (1 − α )t E* 3 + αb3* .

(

)

The maximum value of the social welfare function is SW3 = u q D − t D* 3 − δ D , where t D* 3 satisfies

( ((

)

)

)

((

)

)

t D* 3 = (1 − α ) u −1 u q D − t D* 3 − δ D + δ E − q E + αu −1 u q D − t D* 3 − δ D . Proof: In order to induce skilled people to take difficult jobs, the government should set parameters such that u (q D − t D 3 ) − δ D ≥ u (q E + t E 3 ) − δ E and u (q D − t D 3 ) − δ D ≥ u (b3 ) − φ . In order to induce unskilled people with stigma to take easy jobs, the government should set parameters such that u (q E + t E 3 ) − δ E ≥ u (b3 ) − φ . In order to induce unskilled people without stigma

to

be

on

welfare,

the

government

should

set

parameters

such

that

u (q E + t E 3 ) − δ E ≤ u (b3 ) . Taken together, in order to induce skilled people to take difficult jobs, unskilled people with stigma to take easy jobs, and unskilled people without stigma to be on welfare, the government should set parameters such that u (b3 ) − φ ≤ u (q E + t E 3 ) − δ E ≤ u (b3 ) and u (q D − t D 3 ) − δ D ≥ u (q E + t E 3 ) − δ E .

The

government’s

budget

constraint

implies

that

t D 3 = (1 − α )t E 3 + αb3 . Conditional

on

that

u (q D − t D 3 ) − δ D ≥ u (q E + t E 3 ) − δ E ,

u (b3 ) − φ ≤ u (q E + t E 3 ) − δ E ≤ u (b3 ) , and t D 3 = (1 − α )t E 3 + αb3 , in order to maximize the social welfare function, the government should extract as much as possible from the skilled people

18

without inducing them to switch from difficult jobs to easy jobs. Therefore, the government

(

)

(

)

should set parameters such that u q D − t D* 3 − δ D = u q E + t E* 3 − δ E . It can be proved by

(

)

( )

contradiction that u q E + t E* 3 − δ E = u b3* . Otherwise, if the government reduces b3* by a certain amount, increases t E* 3 by a certain amount, and reduces t D* 3 by a certain amount, the social welfare can be improved. The budget constraint implies that t D* 3 = (1 − α )t E* 3 + αb3* . Thus, we have proved the optimal conditions.

(

)

( )

(

u q E + t E* 3 − δ E = u b3* ,

With

)

(

)

u q D − t D* 3 − δ D = u q E + t E* 3 − δ E ,

(

and

)

t D* 3 = (1 − α )t E* 3 + αb3* , the utility of the skilled people is u q D − t D* 3 − δ D , the utility of unskilled

(

)

people with stigma is u q E + t E* 3 − δ E , and the utility of unskilled people without stigma is

( )

(

)

(

)

( )

(

)

u b3* . Because u q D − t D* 3 − δ D = u q E + t E* 3 − δ E = u b3* , we have SW3 = u q D − t D* 3 − δ D .

(

)

(

((

)

)

((

)

)

)

( )

((

)

Because u q D − t D* 3 − δ D = u q E + t E* 3 − δ E = u b3* , we have b3* = u −1 u q D − t D* 3 − δ D t E* 3 = u −1 u q D − t D* 3 − δ D + δ E − q E .

Substitute

t E* 3 = u −1 u q D − t D* 3 − δ D + δ E − q E

( ((

)

into

)

)

((

)

and

and

)

and

we

obtain

b3* = u −1 u q D − t D* 3 − δ D

t D* 3 = (1 − α )t E* 3 + αb3* ,

((

)

)

)

t D* 3 = (1 − α ) u −1 u q D − t D* 3 − δ D + δ E − q E + αu −1 u q D − t D* 3 − δ D . Thus, we have proved Lemma 3. Lemma 4: It is impossible for the government to set parameters such that under equilibrium unskilled people with stigma choose to be on welfare and unskilled people without stigma choose to take easy jobs simultaneously. Proof: In order to induce unskilled people with stigma to be on welfare, the government should set parameters such that u (q E + t E 4 ) − δ E < u (b4 ) − φ . In order to induce unskilled people without stigma

to

take

easy

jobs,

the

government

should

set

parameters

such

that

19

u (q E + t E 4 ) − δ E > u (b4 ) . As u (q E + t E 4 ) − δ E < u (b4 ) − φ and u (q E + t E 4 ) − δ E > u (b4 ) cannot be

satisfied simultaneously, it is impossible for the government to set parameters such that under equilibrium unskilled people with stigma choose to be on welfare and unskilled people without stigma choose to take easy jobs simultaneously. Thus, we have proved Lemma 4. Lemma 5: If u (q D ) − δ D > 0 > u (q E ) − δ E > −φ , and if α is close to one, then we have t D* 3 < t D* 1 and t D* 3 < t D* 2 , which further imply that SW3 is the largest among SW1 , SW2 , and SW3 .

( ((

)

)

)

((

)

)

Proof: We have t D* 3 = (1 − α ) u −1 u q D − t D* 3 − δ D + δ E − q E + αu −1 u q D − t D* 3 − δ D by Lemma 3. Because u (0 ) = 0 and u (.) is concave by assumption so that u −1 (0) = 0 and u −1 (.) is convex, we

have

((

)

)

((

)

)

((

)

u −1 u q D − t D* 3 − δ D + δ E − q E > u −1 u q D − t D* 3 − δ D + u −1 (δ E ) − q E > u −1 u q D − t D* 3 − δ D u (q E ) < δ E

because

((

so

)

that

u −1 (δ E ) − q E > 0 .

)

((

Therefore,

)

we

) have

)

t D* 3 < u −1 u q D − t D* 3 − δ D + δ E − q E . Because t D* 1 = u −1 u q D − t D* 1 − δ D + δ E − q E by Lemma 1, it can be proved by contradiction that t D* 3 < t D* 1 .

((

)

)

((

)

)

If α is close to one, we have t D* 3 ≈ u −1 u q D − t D* 3 − δ D < u −1 u q D − t D* 3 − δ D + φ .

((

)

)

Because t D* 2 = u −1 u q D − t D* 2 − δ D + φ by Lemma 2, it can be proved by contradiction that t D* 3 < t D* 2 . Thus, we have shown that if u (q D ) − δ D > 0 > u (q E ) − δ E > −φ , and if α is close to one, then we have t D* 3 < t D* 1 and t D* 3 < t D* 2 .

20

(

)

(

)

Because SW1 = u q D − t D* 1 − δ D by Lemma 1, SW2 = u q D − t D* 2 − δ D by Lemma 2, and

(

)

SW3 = u q D − t D* 3 − δ D by Lemma 3, t D* 3 < t D* 1 and t D* 3 < t D* 2 imply that SW3 is the largest among SW1 , SW2 , and SW3 . Thus, we have proved Lemma 5. Lemma 6: If u (q D ) − δ D > 0 > u (q E ) − δ E > −φ , and if α is close to zero, then we have t D* 3 < t D* 1 , * * * but we cannot compare t D* 1 and t D 2 , and we cannot compare t D3 and t D 2 , which further imply

that either SW3 or SW2 is the largest among SW1 , SW2 , and SW3 . Proof: In the proof of Lemma 5, we have shown that t D* 3 < t D* 1 .

(

)

(

)

By Lemma 1, we have u q D − t D* 1 − δ D = u q E + t D* 1 − δ E . By Lemma 2, we have

(

)

( )

u q D − t D* 2 − δ D = u t D* 2 − φ . Although u (q E ) − δ E > −φ , the utility contribution of q E * decreases with t D* 1 . Therefore, we cannot compare t D* 1 and t D 2 .

( ((

)

)

)

((

)

)

We have t D* 3 = (1 − α ) u −1 u q D − t D* 3 − δ D + δ E − q E + αu −1 u q D − t D* 3 − δ D by Lemma

((

(

)

)

is close to zero, t D* 3 ≈ u −1 u q D − t D* 3 − δ D + δ E − q E , which is equivalent to

3. If α

)

(

)

* u q D − t D* 3 − δ D = u q E + t D* 3 − δ E . Therefore, when α is close to zero, we cannot compare t D3 * and t D 2 , either.

(

)

(

)

Because SW1 = u q D − t D* 1 − δ D by Lemma 1, SW2 = u q D − t D* 2 − δ D by Lemma 2, and

(

)

SW3 = u q D − t D* 3 − δ D by Lemma 3, therefore t D* 3 < t D* 1 and the facts that we cannot compare

t D* 3 and t D* 2 imply that either SW3 or SW2 is the largest among SW1 , SW2 , and SW3 . Thus, we have proved Lemma 6.

21

Lemma 7: If u (q D ) − δ D > 0 > −φ > u (q E ) − δ E , and if α is close to one, then we have t D* 3 < t D* 2 < t D* 1 , which further implies that SW3 > SW2 > SW1 .

(

)

(

)

Proof: We have u q D − t D* 1 − δ D = u q E + t D* 1 − δ E by Lemma 1. Because u (0 ) = 0 and u (.) is concave

by

assumption,

and

because

− φ > u (q E ) − δ E

by

assumption,

(

we

)

have

( )

u (q E + t D* 1 ) − δ E < u (t D* 1 ) + u (q E ) − δ E < u (t D* 1 ) − φ . Thus, we have u q D − t D* 1 − δ D < u t D* 1 − φ .

(

)

( )

Because u q D − t D* 2 − δ D = u t D* 2 − φ by Lemma 2, it can be proved by contradiction that t D* 2 < t D* 1 .

( ((

)

)

)

((

)

)

We have t D* 3 = (1 − α ) u −1 u q D − t D* 3 − δ D + δ E − q E + αu −1 u q D − t D* 3 − δ D by Lemma 3. If α is close to one, then t D* 3 ≈ u −1 (u (q D − t D* 3 ) − δ D ) < u −1 (u (q D − t D* 3 ) − δ D + φ ) . Because

((

)

)

t D* 2 = u −1 u q D − t D* 2 − δ D + φ , it can be proved by contradiction that t D* 3 < t D* 2 . Thus, we have shown that if u (q D ) − δ D > 0 > −φ > u (q E ) − δ E , and if α is close to one, then we have t D* 3 < t D* 2 < t D* 1 .

(

)

(

)

Because SW1 = u q D − t D* 1 − δ D by Lemma 1, SW2 = u q D − t D* 2 − δ D by Lemma 2, and SW3 = u (q D − t D* 3 ) − δ D by Lemma 3, t D* 3 < t D* 2 < t D* 1 implies that SW3 > SW2 > SW1 . Thus, we have proved Lemma 7. Lemma 8: If u (q D ) − δ D > 0 > −φ > u (q E ) − δ E , and if α is close to zero, then we have t D* 2 < t D* 1 and t D* 2 < t D* 3 , which further imply that SW2 is the largest among SW1 , SW2 , and SW3 . Proof: In the proof of Lemma 7, we have shown that t D* 2 < t D* 1 .

( ((

)

)

)

((

)

)

We have t D* 3 = (1 − α ) u −1 u q D − t D* 3 − δ D + δ E − q E + αu −1 u q D − t D* 3 − δ D by Lemma

((

)

)

3. If α is close to zero, we have t D* 3 ≈ u −1 u q D − t D* 3 − δ D + δ E − q E , which is equivalent to

22

(

)

(

)

(

)

( )

u q D − t D* 3 − δ D = u q E + t D* 3 − δ E . Because u q D − t D* 2 − δ D = u t D* 2 − φ by Lemma 2, we can show that t D* 2 < t D* 3 following the same argument in the proof of Lemma 7. Thus, we have shown that if u (q D ) − δ D > 0 > −φ > u (q E ) − δ E , and if α is close to zero, then we have t D* 2 < t D* 1 and t D* 2 < t D* 3 .

(

)

(

)

Because SW1 = u q D − t D* 1 − δ D by Lemma 1, SW2 = u q D − t D* 2 − δ D by Lemma 2, and SW3 = u (q D − t D* 3 ) − δ D by Lemma 3, t D* 2 < t D* 1 and t D* 2 < t D* 3 imply that SW2 is the largest among SW1 , SW2 , and SW3 . Thus, we have proved Lemma 8. Proposition 1: If u (q D ) − δ D > 0 > u (q E ) − δ E > −φ , and if α is close to one, then the equilibrium is a separating equilibrium, and the optimal conditions are given by:

(

)

( )

(

)

(

(10) u q E + t E* − δ E = u b * ,

)

(11) u q D − t D* − δ D = u q E + t E* − δ E , and (12) t D* = (1 − α )t E* + αb * . Proof: Because skilled people have three options (take a difficult job, take an easy job, or be on welfare), unskilled people with stigma have two options (take an easy job, or be on welfare), and unskilled people without stigma have two options (take an easy job, or be on welfare), there are C 31 * C 21 * C 21 = 12 possible equilibrium outcomes in principle. However, it is impossible to be optimal so that under equilibrium skilled people choose to take easy jobs or be on welfare because the government will then have no resource to subsidize unskilled people. Thus, we only need to consider C11 * C 21 * C 21 = 4 possible equilibrium outcomes.

23

By Lemma 4, it is impossible for the government to set parameters such that under equilibrium unskilled people with stigma choose to be on welfare and unskilled people without stigma choose to take easy jobs simultaneously. Thus, we only need to consider three possible equilibrium outcomes: (1) skilled people choose to take difficult jobs, and all unskilled people choose to take easy jobs; (2) skilled people choose to take difficult jobs, and all unskilled people choose to be on welfare; and (3) skilled people choose to take difficult jobs, unskilled people with stigma choose to take easy jobs, and unskilled people without stigma choose to be on welfare. Conditional on the first possible equilibrium outcome, the maximum social welfare is SW1 by Lemma 1. Conditional on the second possible equilibrium outcome, the maximum social

welfare is SW2 by Lemma 2. Conditional on the third possible equilibrium outcome, the maximum social welfare is SW3 by Lemma 3. If u (q D ) − δ D > 0 > u (q E ) − δ E > −φ , and if α is close to one, then by Lemma 5, SW3 is the largest among SW1 , SW2 , and SW3 . The optimal conditions in Lemma 3 are therefore the overall optimal conditions. Thus, we have shown that if u (q D ) − δ D > 0 > u (q E ) − δ E > −φ , and if

α

(

is close to one, the optimal conditions are given by

)

(

(

)

( )

u q E + t E* − δ E = u b * ,

)

u q D − t D* − δ D = u q E + t E* − δ E , and t D* = (1 − α )t E* + αb * . Under the optimal parameters, skilled people choose to take difficult jobs, unskilled people with stigma choose to take easy jobs, and unskilled people without stigma choose to be on welfare. The equilibrium is a separating equilibrium. Thus, we have proved Proposition 1.

24

Proposition 2: If u (q D ) − δ D > 0 > u (q E ) − δ E > −φ , and if α is close to zero, then the equilibrium is either a separating equilibrium or a pooling equilibrium, and the optimal conditions are given either by for the separating equilibrium:

(

)

( )

(

)

(

(13) u q E + t E* − δ E = u b * ,

)

(14) u q D − t D* − δ D = u q E + t E* − δ E , and (15) t D* = (1 − α )t E* + αb * . or by for the pooling equilibrium:

(

)

( )

(16) u q E + t E* − δ E < u b * − φ , (17) u (q D − t D* ) − δ D = u (b * ) − φ , and (18) t D* = b * . Proof: Following the same argument as in the proof of Proposition 1, we only need to consider three possible equilibrium outcomes: (1) skilled people choose to take difficult jobs, and all unskilled people choose to take easy jobs; (2) skilled people choose to take difficult jobs, and all unskilled people choose to be on welfare; and (3) skilled people choose to take difficult jobs, unskilled people with stigma choose to take easy jobs, and unskilled people without stigma choose to be on welfare. Conditional on the first possible equilibrium outcome, the maximum social welfare is SW1 by Lemma 1. Conditional on the second possible equilibrium outcome, the maximum social

welfare is SW2 by Lemma 2. Conditional on the third possible equilibrium outcome, the maximum social welfare is SW3 by Lemma 3. If u (q D ) − δ D > 0 > u (q E ) − δ E > −φ , and if α is close to zero, then by Lemma 6, either SW3 or SW2 is the largest among SW1 , SW2 , and SW3 . Therefore, either the optimal conditions

25

in Lemma 3 or the optimal conditions in Lemma 2 are the overall optimal conditions. If SW3 > SW2 ,

(

)

then

the

(

optimal

conditions

are

given

by

(

)

( )

u q E + t E* − δ E = u b * ,

)

u q D − t D* − δ D = u q E + t E* − δ E , and t D* = (1 − α )t E* + αb * . If SW2 > SW3 , then the optimal

(

)

( )

conditions are given by u q E + t E* − δ E < u b * − φ , u (q D − t D* ) − δ D = u (b * ) − φ , and t D* = b * . Thus, we have proved Proposition 2. Proposition 3: If u (q D ) − δ D > 0 > −φ > u (q E ) − δ E , and if α is close to one, then the equilibrium is a separating equilibrium, and the optimal conditions are given by:

(

)

( )

(

)

(

(19) u q E + t E* − δ E = u b * ,

)

(20) u q D − t D* − δ D = u q E + t E* − δ E , and (21) t D* = (1 − α )t E* + αb * . Proof: Following the same argument as in the proof of Proposition 1, we only need to consider three possible equilibrium outcomes: (1) skilled people choose to take difficult jobs, and all unskilled people choose to take easy jobs; (2) skilled people choose to take difficult jobs, and all unskilled people choose to be on welfare; and (3) skilled people choose to take difficult jobs, unskilled people with stigma choose to take easy jobs, and unskilled people without stigma choose to be on welfare. Conditional on the first possible equilibrium outcome, the maximum social welfare is SW1 by Lemma 1. Conditional on the second possible equilibrium outcome, the maximum social

welfare is SW2 by Lemma 2. Conditional on the third possible equilibrium outcome, the maximum social welfare is SW3 by Lemma 3. If u (q D ) − δ D > 0 > −φ > u (q E ) − δ E , and if α

is close to one, then we have

SW3 > SW2 > SW1 by Lemma 7. Therefore, the optimal conditions in Lemma 3 are the overall

26

optimal conditions. Thus, we have shown that if u (q D ) − δ D > 0 > −φ > u (q E ) − δ E , and if α is close

(

to

one,

)

then

(

the

optimal

conditions

are

given

by

(

)

( )

u q E + t E* − δ E = u b * ,

)

u q D − t D* − δ D = u q E + t E* − δ E , and t D* = (1 − α )t E* + αb * . Under the optimal parameters, skilled people choose to take difficult jobs, unskilled people with stigma choose to take easy jobs, and unskilled people without stigma choose to be on welfare. The equilibrium is a separating equilibrium. Thus, we have proved Proposition 3. Proposition 4: If u (q D ) − δ D > 0 > −φ > u (q E ) − δ E , and if α is close to zero, then the equilibrium is a pooling equilibrium, and the optimal conditions are given by:

(

)

( )

(22) u q E + t E* − δ E < u b * − φ , (23) u (q D − t D* ) − δ D = u (b * ) − φ , and (24) t D* = b * . Proof: Following the same argument as in the proof of Proposition 1, we only need to consider three possible equilibrium outcomes: (1) skilled people choose to take difficult jobs, and all unskilled people choose to take easy jobs; (2) skilled people choose to take difficult jobs, and all unskilled people choose to be on welfare; and (3) skilled people choose to take difficult jobs, unskilled people with stigma choose to take easy jobs, and unskilled people without stigma choose to be on welfare. Conditional on the first possible equilibrium outcome, the maximum social welfare is SW1 by Lemma 1. Conditional on the second possible equilibrium outcome, the maximum social

welfare is SW2 by Lemma 2. Conditional on the third possible equilibrium outcome, the maximum social welfare is SW3 by Lemma 3.

27

If u (q D ) − δ D > 0 > −φ > u (q E ) − δ E , and if α is close to zero, then SW2 is the largest among SW1 , SW2 , and SW3 by Lemma 8. Therefore, the optimal conditions in Lemma 2 are the overall optimal conditions. Thus, we have shown that if u (q D ) − δ D > 0 > −φ > u (q E ) − δ E , and if

α

(

)

( )

is close to zero, the optimal conditions are given by u q E + t E* − δ E < u b * − φ ,

u (q D − t D* ) − δ D = u (b * ) − φ , and t D* = b * . Under the optimal parameters, skilled people choose to take difficult jobs, and all unskilled people choose to be on welfare. The equilibrium is a pooling equilibrium. Thus, we have proved Proposition 4.

28

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29

Sadka, Efraim, “On Income Distribution, Incentive Effects and Optimal Income Taxation,” Review of Economic Studies, 1976, 43(2), 261-267. Saez, Emmanuel, “The Desirability of Commodity Taxation under Non-linear Income Taxation and Heterogeneous Tastes,” Journal of Public Economics, 2002, 83, 217-230. Saez, Emmanuel, “Optimal Income Transfer Programs: Intensive versus Extensive Labor Supply Responses,” Quarterly Journal of Economics, August 2002, 117, 1039-1073. Seade, Jesus K., “On the Shape of Optimal Tax Schedules,” Journal of Public Economics, 1977, 7(2), 203-235.

30