The curse of the middle-skilled workers Laurent Simula and Alain Trannoy1 EHESS and IDEP-GREQAM2 simul[email protected] [email protected]
May 22, 2005
paper has bene ted greatly from discussions with Professor James Mirrlees, Dr Philippe Choné, Dr Jeremy Edwards, Dr Emanuella Sciubba. The comments of participants at a seminar in Cambridge were also helpful. 2 GREQAM-EHESS, Centre de la Vieille Charité, 2 rue de la Charité, 13236 Marseille cedex 02
Abstract This paper is devoted to optimal income taxation in an open economy. It examines the consequences of the increasing mobility of high skilled workers between developed countries. We focus on a world consisting of two countries, a redistributive country and a laissez-faire country. We rst look into the rst-best optimal tax scheme when emigration of every agent is prevented. We prove that, under the optimal solution, there is a curse of the middle-skilled workers for every social welfare function with a nite aversion to income inequality. Then, we examine if it is always socially optimal to prevent high skilled emigration. Two social criteria are distinguished when emigration actually occurs. In the national criterion, the social objective depends on the welfare of every national. In the resident criterion, it depends on the welfare of every resident. It is proved that, under the national criterion, it is always socially optimal to prevent high-skilled emigration whilst, under the resident criterion, there are cases in which emigration of the highest skilled individuals is socially optimal. Keywords: Optimal Income Taxation, Emigration, Participation Constraints
1 Introduction Following Mirrlees's (1971) seminal article where migration is supposed to be impossible1 , most of the literature on optimal income taxation focuses on closed economies2 . However, the mobility of the individuals is considerably easier than beforehand, nowadays within the EU or the NAFTA for instance (Cremer, Fourgeaud, Leite Monteiro, Marchand, and Pestieau 1996, Cremer and Pestieau 1996, OECD 2002, Iqbal 2000). These models are therefore surely less appropriate to describe the constraints on the design of the income tax schedule. In this paper, we want to examine to what extent the classical results which characterize the optimal tax schedule of a developed redistributive country A whose agents cannot vote with their feet, change with the introduction of a developed free-tax country B, that is a country of laissezfaire, to which the agents can emigrate for a migration cost. The development of high-skilled migration has justi ed further investigation of the theoretical analysis of optimal taxation when economic agents are mobile between the developed countries. A rst class of models describes the optimal tax competition that may occur between countries: Leite-Monteiro (1997) focuses on a 2-country model with two classes of agents and lump-sum taxes and shows that international labour mobility can increase the level of redistribution; Huber (1999) uses the self-selection approach of Stiglitz (1982). Hamilton and Pestieau (2002) examine the effect of mobility on the optimal non-linear tax scheme using a model à la Stiglitz (1982) and put the stress on voting equilibria. Piaser (2003) extends this model to the case where governments are strategic players. This paper examines the effects of the tax schedule on the choice of residence. It is assumed that the individual productivity, equal to the gross wage, does not change with the country of residence. In other words, both countries have the same production function. In consequence, this paper differs from the literature on the brain drain (Bhagwati 1976, Bhagwati 1980, Bhagwati and Hamada 1989, Wilson 1980, Wilson 1989, Mirrlees 1982). The migration of high-skilled individuals between the most developed countries raises indeed speci c issues. It induces a loss in productive capacity and in taxes levied in the "emigration countries" where redistribution of income is usually implemented. A common idea is that governments have no alternative to lowering taxes to prevent emigration of high-skilled individuals. Indeed a government that would not reduce taxes paid by high-skilled individuals would face emigration that would induce a decrease in tax revenue. As a result, both tax revenue and average productivity in the home country would go down. In this case, national income is not altered but redistribution is reduced. Then there is a con ict between the desire to maintain the national income per capita in keeping taxes down and the desire to sustain the redistribution programme. We focus on a world consisting of two countries, a redistributive country A and a laissezfaire country B. The laissez-faire country B does not levy taxes because of material necessity or ideological choice: its tax policy does not depend on the tax policy in the redistributive country. 1 "Migration is supposed to be impossible. Since the threat of migration is a major in uence on the degree of pro-
gression in actual tax systems, at any rate outside the United States, this is an assumption one would rather not make". 2 This point is stressed in Wilson (1992).
Contrary to Piaser (2003), there is no game or strategic competition on taxes between the two countries. It is assumed that every individual is initially living in the redistributive country A. Without this assumption, there could be low-skilled emigration from the laissez-faire country B to the redistributive country B: Such an emigration would not alter the tax policy in the laissez-faire country B if B 0 s government maximizes the average social welfare of its population; however, it would change A0 s tax policy. If low-skilled workers where initially living in the laissez-faire country B; we would have to introduce an additional assumption on the level of the migration costs so as to prevent its occurrence. An assumption we would rather not make. Our model is based on a Mirrleesian economy, there is full employment in both countries and the labour supply is intensive. The agents differ both in productivity and migration costs. However, it is assumed that the migration costs depend on each agent's productivity so that productivity is the only one parameter of heterogeneity. In addition, it is assumed that the migration costs do not increase faster than the laissez-faire utility. Three cases are thus considered: constant, decreasing and increasing migration costs. No assumption is made on the level of the migration costs. In consequence, the agents who will leave the country A to settle down in the laissez-faire country B should be the most skilled in the population. The main theoretical contribution of this paper is to introduce participation constraints into the optimal income tax problem and to examine the way participation constraints and self-selection constraints interact. Participation constraints appear as a natural way of modelling the possibility that the agents vote with their feet3 . However, they are not used in optimal income taxation whilst they are commonly introduced in models based on contract theory (Laffont and Martimort 2002). Participation constraints express the fact that an individual will leave the country where he is living if the net utility he obtains there is less than his reservation utility, which is equal to the net utility he can obtain abroad, taking his migration cost into account4 . The reservation utility is typedependent. The main question addressed in this paper is whether it is always socially optimal to prevent emigration of the high-skilled individuals. We are interested in the social welfare per capita in the home country and do not adopt a world welfare point of view. It is assumed that the government chooses the tax schedule which maximizes an isoelastic social welfare functional depending on its aversion to income inequality. The question of optimality is more complex in an open economy: the government has indeed to choose who are the individuals whose welfare is to count (Mirrlees 1982). Two main social criteria may be distinguished. A rst criterion (national model) is to take the welfare of every national into account, irrespective of his country of residence. A second criterion (resident model) is to take care of every resident irrespective of his nationality. In consequence, the social objective depends on the social criterion that is chosen, on the government's aversion to income inequality, on the distribution of the productivity and on the elasticity of labour supply. 3 In an article devoted to the brain drain, Wilson (1980) focuses on exogenous institutions to prevent emigration.
Participation constraints are an endogenous device the government can use to "close" the economy. 4 This is a static model; so utility and costs are taken in expected values.
We focus on the rst-best analysis where the government can observe each individual's productivity. In a closed economy and in the Mirrlees's model, under the optimal tax scheme and when leisure is a normal good, the indirect utility is strictly decreasing in productivity for every nite government's aversion to income inequality (Mirrlees 1974). It is socially optimal to extract the productive rent on the most productive agents to nance redistribution in favour of the less productive individuals (Roemer 1985). When the government's aversion to income inequality is in nite (Rawlsian case), the indirect utility remains constant irrespective of the ability level. The plan of the paper is as follows. Section 2 presents the model. Section 3 looks into the optimal tax scheme when the government prevents every individual from emigrating. It is established that there is a productivity level from which the participation constraints are binding. As a result, it is no longer possible to require the most talented agents to work as much as without mobility. In other words, the possibility that the high-skilled individuals vote with their feet prevents the government from extracting from them as much as in a closed economy. As a result, the malediction of the middle-skilled workers takes the place of the malediction of the high-skilled workers. We then examine, in section 4, if it is always socially optimal to prevent emigration of the high-skilled individuals. The answer depends basically on the social criterion, on the government's aversion to income inequality and on the distribution of productivity in the population. In the national model, it is always socially optimal to prevent emigration. To have intuitive insight into this result, assume that every emigration is initially prevented in the redistributive country. The agents who would have emigrated in the absence of participation constraints obtain their reservation utility in A: If the government allows them to leave the country, they obtain their reservation utility in B. They then obtain their reservation utility, whether they are living in A or in B: However, their departure induces a decrease in taxes that can be levied in A: As a result, redistribution of income goes down and the social welfare per capita in A is reduced. Eventually, it is not socially optimal to allow emigration of the high-skilled individual because, in this case, the scal rent is not exploited to the maximum in A. Section 5 addresses the same question, but focuses on the resident model. We show that there are cases in which it is socially optimal to allow emigration of the highest skilled workers, that is cases in which a decrease in national income increases social welfare per capita. In fact, the departure of the high-skilled workers has two effects on social welfare per capita. The scal effect of their departure is always negative. However, since the welfare of the highest skilled workers is no longer taken into account by A0 s government when they emigrate, the average social welfare in A can increase when they leave the country. More precisely, the condition for the departure of the highest skilled workers to be socially optimal involves a comparison between the average social welfare and the marginal social welfare. Section 6 concludes the paper and compares the results obtained in the national and resident models. It is shown that the resident model is weakly socially preferable to the national one.
2 The model The world consists of two countries, labelled A and B. A0 s government implements a purely redistributive tax policy. B is committed to being a laissez-faire country. We assume that every individual is initially living in A: We call a national an agent who is initially living in A and a resident an agent who is living in A:
There is only one parameter of heterogeneity which stands for talent and that belongs to a close interval ; of RC : The distribution function of , denoted F; admits a density function f which is assumed to be strictly positive for every of ; : The distribution of the productivity is known to the government.
All individuals have the same preferences over consumption and leisure, represented by a strictly concave utility function U . This function is continuously differentiable, strictly increasing in consumption (x 0) and strictly decreasing in labour (0 l 1; where 1 is the normalized time endowment), U :D U .x; l/ (1) In addition, U tends to 1 as x tends to 0 from above or l tends to 1 from below. The following assumption will be made to establish some properties. Assumption 1 (A1) Leisure is a normal good. Each agent decides about the optimal amount of consumption and labour so as to maximize his utility subject to his budget constraint. The budget constraint in A is: xD l
T . ; l/ C R
where T . ; l/ is the tax function and R stands for nonlabour income . It is assumed that T . ; l/ D T . / from now on. The budget constraint in B is: xD l
An individual who is living in A chooses the consumption and labour supply that maximize (1) subject to (2). Let R be the total nonlabour income an individual receives. The rst-order condition for the individual utility maximization de nes implicitly a Marshallian labour supply l A as a function of and R : l A D l A ; R : Therefore the consumption function depends on and R as well: x A D x A ; R . The indirect utility function in A is de ned as V A :D U .x A ; l A / :
An individual who is living in B maximizes (1) subject to (3). Since there is no tax in this country, his consumption function and labour supply depend on : x B D x B . / ; l B D l B . / : By the envelope theorem, his indirect utility VB :D U .x B ; l B / is strictly increasing in :
Emigration and participation constraints
The agents who decide to leave A sustain a strictly positive migration cost, denoted c. This migration cost is introduced into the model as a loss in utility. This loss corresponds to different costs of moving: material costs, psychological costs, costs of speaking a different language and adapting to another culture, costs of leaving his family and friends, etc. It is assumed that these costs depends on the productivity level, c :D c . / ; where c is twice continuously differentiable. The distribution of the migration costs is known to A0 s government, which cannot observe c . / for every -individual. However, since c depends on ; A0 s government will know c . / if it knows : That is why there is only one parameter of heterogeneity in our model. The reservation utility is de ned as the maximum utility an individual living in A can obtain abroad, that is in B: It is thus equal to the difference between the laissez-faire utility and the migration cost: VB . / c . / : The following assumption is made. Assumption 2 (A2) The reservation utility is strictly increasing in ferentiable. Thus VB0 . / c0 . / > 0:
and twice continuously dif-
Assumption 2 states precisely the relation between migration cost and productivity. Three cases are indeed considered: constant migration costs, decreasing migration costs, increasing migration costs such that c0 . / < VB0 . / : The most natural case is probably that of decreasing costs. A good command of the language spoken in B; a degree or skills that will be easily recognized in the foreign country, should a priori be increasing in : therefore they should reduce the migration costs rather than making them heavier. Assuming that the migration costs are constant irrespective of the skill level amounts to focusing on material costs. Lastly, migration costs that would increase faster than the laissez-faire utility are dif cult to imagine and are ruled out in the following. It is important to stress that Assumption 2 is an assumption on the rate of increase of the migration cost function c . /: no assumption is made on the level of c . / : Since each individual is assumed to maximize his utility, he will leave A if and only if his utility in A is less than his reservation utility, i.e. iff: V A . / < VB . /
Hence, the following constraint has to be introduced into the optimal income-tax programme if the government aims at preventing emigration of the agents whose productivity is : VA . /
VB . /
We refer to this constraint as the participation constraint for the -individuals. 5
Social objective and tax policy
The set of all nationals is ; : The set of all residents is denoted 2 R : Under the assumption made above, every agent has A's nationality. Thus 2 R is a subset of ; : In the presence of emigration, A0 s government has to choose who are the agents whose welfare is to count. That amounts to specifying the set 2. Two models are distinguished. In the national model, the government takes the welfare of each national into account, irrespective of his country of residence. This model rests on the idea that, since the scal system nds its legitimacy in its democratic adoption, the social objective should be de ned on the individuals who have the right to vote5 . In the resident model, the government takes the welfare of every resident. Under the assumption made above, every resident is a national. Then, the resident model is based on the fact that the geographical distance couples with political disinterest6 and on the idea that no taxpayer should be excluded from the objectives of public policies he funds. It should be noted that if A's government introduces participation constraints so as to prevent potential emigration of its nationals, the national model and the resident model are equivalent. The social objective is a social weighted sum of individual utilities, Z 1 (6) .U .x; l// d F . / F .2/ 2 where .U .x; l// :D 1 1 U .x; l/1 for 2 RC n f1g and 1 .U .x; l// :D ln U .x; l/. is an index of the government's aversion to income inequality. The utilitarian case corresponds to D 0, the Rawlsian one to ! 1. In the national model, 2 D ; and F .2/ D 1: In the resident model, 2 is equal to 2 R and F 2 R is the probability measure of the set 2 R : It is important to notice that the social objective (6) is de ned as a social weighted mean of individual welfare. The government implements a redistributive tax policy T which is assumed to be differentiable. It is assumed that taxes can be levied on the resident population 2 R . Therefore the tax revenue constraint is: Z (7) . l x/ d F . / R 2R
where R is a non-negative per capita income that does not enter the individual's utility function U: If R is equal to 0; the tax policy is purely redistributive.
5 In France, the 14th Article of the Declaration of the Rights of Man and of the Citizen, which has constitutional value, provides that: "All citizens have the right to vote, by themselves or through their representatives, for the need for the public contribution, to agree to it voluntarily, to allow implementation of it, and to determine its appropriation, the amount of assessment, its collection and its duration". As an example, twelve senators represent the French nationals who are living abroad. 6 In France, the French nationals who are living abroad are represented by 150 delegates sitting in the High Council of French Nationals Abroad. In 2000, of 2 million French nationals living abroad, only 642 000 were on the electoral register and the abstention amounted to 81%.
Closed economy benchmark
It is useful, as a yardstick, to examine the case in which the population is immobile and is known to the government. Lump-sum taxes are implementable. In this case, T . ; l/ D T . / : The government chooses the tax policy T . / which maximizes (6) subject to (7), with 2 D ; : The following theorem summarizes the solution to the optimal tax problem (Mirrlees 1974). Theorem 1 Assume that leisure in a normal good and that the population is immobile. Then, under the optimal tax scheme, the indirect utility function is decreasing in for every nite 0 whilst it is constant irrespective of when ! 1: So as to refer to this theorem, VC . / denotes the indirect utility function in A when the population is immobile. We also de ne WCR as the social value function when A0 s population is immobile.
The curse of the middle-skilled workers
This section focuses on the optimal tax scheme in A when A0 s government prevents every agent from leaving the country. The participation constraint (5) is thus taken into account for every of ; : 8 2 ; : V A . / VB . / c . / (8) We prove that the malediction of the middle-skilled workers replaces the malediction of the talented for every nite government's aversion to income inequality.
The optimization problem
The government chooses the tax paid by each individual or, equivalently, the consumption-income bundle intended for each individual. The optimal income tax problem is thus as follows. Problem 2 Find x and l to maximize (6) subject to the tax-revenue constraint (7) and to the participation constraint (8), with 2 D 2 R D ; : Let us de ne and . / as the implicit prices of (7) and (8) respectively. The Lagrangean of this problem reads: LD
.U .x; l// C
x/ C . / [U .x; l/
VB . / C c . /] d F . /
The rst-order conditions for a maximum are: 0
0; . /
0I U .x; l/
.U .x; l// C . / Ux .x; l/ D
.U .x; l// C . / Ul .x; l/ D Z [ l x] d F . / x] d F . / R 0; 0
VB . / C c . /
. / [U .x; l/
Since the left-hand side of (10) is strictly positive, we have is always binding. Combining (10) and (11), one obtains: D
(11) R D0
VB . / C c . /] D 0
> 0 : the tax revenue constraint (7)
Ul Ds Ux
which means that the choice of the government coincides with the choice of every agent. Lemma 3 Under Assumptions 1 and 2, if there exists e such that every 2 e; :
e > 0; then
. / > 0 for
Proof. Cf Appendix A. Let us de ne H D 2 ; j . / D 0 and F D 2 ; j . / > 0 and denote the e in mum of F: When exists, Lemma 3 amounts to saying that H and F are two intervals. More precisely, H D ; and F D] ; ]:
Proposition 4 Under Assumptions 1 and 2,
exists if and only if VC
Proof. Cf Appendix B. The previous results are summarized in the following theorem. Theorem 5 Assume VC < VB c and consider any isoelastic social welfare function where is the inequality aversion. Under Assuptions 1 and 2, we have: (1) for a nite 0, the indirect utility is V-shaped: (i) the indirect utility obtained by the agents of H is strictly decreasing in I (ii) the indirect utility obtained by the agents of F is equal to the reservation utility VB . / c . / ; strictly increasing in I (iii) the minimum indirect utility corresponds to D : (2) for ! 1 (Rawls), (i) the indirect utility obtained by the agents of H is reduced compared to the case with immobile population; this utility is constant irrespective of I (ii) the indirect utility obtained by the agents of F is equal to the reservation utility VB . / c . / ; strictly increasing in :
VB - c
V -c B
VC VA VC
Infinite inequality aversion
Finite inequality aversion
Figure 1: The malediction of the middle-skilled workers
Proposition 6 Assume Assumptions 1–2 hold and VC < VB (1) T is strictly increasing in the skill level for every 2 H. (2) The agents of F pay strictly positive taxes.
Proof. Cf Appendix C. The agents at the bottom of the ability distribution pay strictly negative taxes7 , contrary to the high ability workers of F who pay strictly positive taxes. Corollary 7 Assume Assumptions 1–2 hold, VC < VB c and R D 0: Under the optimal tax schedule, the participation constraint cannot be binding for every of 2 R : In consequence, is strictly greater than : Proof. Cf Appendix D. When R > 0; the participation constraint can be binding for every of 2 R : Note also that there are some R that are not possible. The lower the migration costs are, the lower is the maximum per capita incomeR which can be nance in A: 7 This set of agents may be a subset of H or the set H entirely.
Figure 1 illustrates Theorem 5. Recall V A stands for the indirect utility that an -person obtains in A. VC is the utility of the -person when the population is immobile. VB c is the reservation utility. On the left-hand side, the government's aversion to income inequality is nite. In consequence, VC is strictly decreasing in : On the right hand side, the government's aversion to inequality is in nite. This case corresponds to a Rawlsian government: the indirect utility VC remains constant, irrespective of the ability level. Thus Theorem 5 may be interpreted as follows. To begin with, we focus on the implementation of the solution to the optimal tax problem with emigration constraint. First, consider a government that exhibits a nite inequality aversion. When information is imperfect and population is immobile, it is in each agent's interest to hide his skill level so as to disguise himself as the less able individual in the population. All agents behave as if their productivity were equal to : In consequence, redistribution cannot work. In some cases, incentives are partially restored when participation constraints are introduced. Indeed, if there exists 2] ; [ such that .VB c/ . ; all individuals with productivity / VA will reveal their right type. The agents whose productivity is less than behave as if their productivity were : The solution is not optimal; however it is still possible to operate some redistribution of income within the population. Let us now consider a government which exhibits an in nite aversion to income inequality. The solution to the optimal tax problem with emigration constraint is thus incentive feasible, that is that it can be decentralized. So as to understand the malediction of the middle-skilled workers, it is useful to consider an economy in which the population which previously was immobile on account of huge migration costs has now the possibility to emigrate. Faced with that situation, the government introduces participation constraints into the optimal tax programme, constraints that split the population in two intervals. The participation constraint remains inactive for the less able individuals (the agents of H) while it is binding for the abler (those of F) who thus receive in A their reservation utility VB . / c . / : Hence it is no longer possible to require the most talented agents to work as much as without mobility, that is to require that they keep working even though the labour disutility exceeds the gains from the increase in income. The possibility that the abler individuals vote with their feet prevents the government from extracting from them as much as before. In consequence, the situation of the agents of H gets worse. The bene t they received has to be reduced or the tax they paid increased. Eventually, it is from the middle-skilled workers that the productive rent is extracted to the maximum, but within some limits since their utilities have to remain greater than their reservation utilities. The middle-skilled workers, insuf ciently talented to leave the country, appear as the main victims of the globalization. The conditions under which exists depends on the migration costs. When these costs are innite, the participation constraints are never binding. The malediction of the talented agents occurs when the migration costs are suf ciently high so that the reservation utility of these agents remains less than the utility they obtain in A : the migration costs neutralize the scal externality that constitutes the presence of the laissez-faire country B: When the migration costs decrease suf ciently so that exists, the participation constraints become binding for some talented agents. The reduction 10
of the migration costs widens the interval on which these constraints are binding. In consequence, the current process of reduction of the migration costs, whether material (transportation, degree recognition, languages...) or psychological (emergence of a global culture, standardization of consumption patterns...), is apt to make the malediction of the middle-skilled workers more frequent.
It is assumed that (1) is given by: U .x; l/ D ln x C ln .1
and that A0 s social objective is utilitarian. In that case, x A . / D [1 C . /] = and l A . / D max 0I 1 Note that individuals for whom VA . / D
1C . /
do not work at the optimum. The indirect utility is given by:
ln [1 C . /] ln 2 ln [1 C . /] 2 ln
for < 1= ln for
The reservation utility reads: VB . /
c . / D ln
The participation constraint is binding if V A . / D VB . / 8 2 F : 2 ln [1 C . /]
Therefore, . /
We focus on constant migration costs, c . / D c: :D
c. / 2
c . / ; i.e.
ln D ln
which is equivalent to: 8 2F: . /D
c. / 2
is de ned as: c 2
The value of . / can now be substituted into the tax revenue constraint, 0D
[1 C . /] d F . /
1C . /
dF . /
to get: 1C F
dF . / D
dF . /
Since the distribution of is known to the government, (24) is an equation in . When is known, it is possible to determine the values of ; . /, x A . /, l A . / and T . / successively. We focus on a lognormal population with standard deviation D 0:39 and mean 0:408 , which is assumed to be distributed between 0 and 5: The values taken by the distribution are corrected by a uniform distribution with density [1 F .5/] =5 on [0; 5] : We assume that c is equal to 1:3 log 4: Figure 2shows the values of x A ; l A ; TA and V A under the optimal tax scheme. The participation constraint is binding from ' 3:11 corresponding to 2:98% of the population. The average social welfare, equal to 1:31; is greater than the utility of the highest skilled individuals, V A ' 1:58:
This section examines if it is socially optimal, in the national model, to prevent emigration of the high-skilled workers. We focus on cases where the participation constraint is binding for some . It is important to keep in mind that, by Proposition 6, every agent for whom the participation constraint is binding pays strictly positive taxes and that there is no population ow from B to A. By Lemma 3, the resident population corresponds to a close interval 2 R of ; : b is de ned as the maximum of 2 R : Hence, 2 R D ; b : Is the situation where every national is a resident socially optimal? So as to answer this question, we examine if the social objective reaches a maximum for b D along the participation constraints. Proposition 8 Assume potential emigration occurs. In the national model, it is socially optimal to take emigration constraints into account for every of ; :
Proof. Cf Appendix E. The result of Proposition 8 is rather intuitive. In the national model, the welfare of every national enters the government's objective function. The point is that an agent who is constrained to participate in the home economy obtains his reservation utility and pays strictly positive taxes. Hence, his emigration gives rise to a decrease in tax revenue which reduces redistribution within the population and alters the welfare of the individuals remaining in the home country. In addition, 8 These values are used in Mirrlees (1971).
Figure 2: First-best allocations when agents vote with their feet
if A0 s government aims at raising a strictly positive income R; the social cost of the departure of the most skilled individual increases. Eventually, it is not socially optimal not to take emigration constraints into account because, in this case, the scal rent is not extracted to the maximum. The socially optimal resident population is the national population. Figure 3 represents this result in the utilitarian case ( D 0). The hatched area corresponds to the loss in welfare 1 due to a decrease in b D equal to 1 :
This section examines if it is socially optimal, in the resident model, to prevent emigration of the high-skilled workers. By Lemma 3, 2 R is an interval ; b for some b : It is assumed that the b-individuals emigrate in the absence of participation constraint. We examine if the social objective reaches a maximum at b D : The optimal tax problem is the following. Problem 9 Find x and l which maximize: Z 1 F b 2R subject to:
U .x; l/
.U .x; l// d F . /
x/ d F . /
VB . /
c . / 8 2 2R
Condition for optimal emigration
The value function of this optimization problem is denoted W R and depends on b : W R D W R b : is the multiplier of the tax revenue constraint (26). Lemma 10 Let Assumptions 1–2 hold and the participation constraint be binding at b: The social objective W R b is increasing in b if and only if: WR b dW R b R0, T b R db
VB b F b
Proof. Cf Appendix F. F b ; VB b c b and T b are exogenous. The multiplier of the revenue constraint tells us how the social value function W R b changes when there is an increase in taxes levied by the government. T b is the tax paid by an individual of productivity b: Since the participation constraint is assumed to be binding at b; the second point of 6 tells us that T b is strictly positive. Hence, T b corresponds to the positive scal effect of the presence of the b-individuals on social 14
∆θ θ __ θ
Figure 3: National model
welfare per capita. On the right-hand side of (28), the difference between the social welfare per capita when b is the highest productivity level in A; and the marginal social welfare is divided by the number of residents in A: This direct effect of the presence of the b-individuals in A can be negative. It is thus socially optimal to allow emigration of the -individuals, who are initially living in A, when the following condition holds. Proposition 11 Let Assumptions 1–2 hold and the participation constraint be binding at b: Emigration of the -individuals is socially optimal if and only if: T
Two cases have to be distinguished. In the rst case, the marginal social welfare VB c is at least as great as the R average social welfare W . Hence the right-hand side of (29) is negative and condition (29) does not hold: it is always socially optimal to prevent emigration of the -individuals, for any social welfare function which exhibits a nonnegative aversion to income inequality. In the second case, the marginal social welfare VB c is greater than the averR age social welfare W : The right-hand side of (29) is strictly positive and there is a trade off between both effects of the presence of the -individuals. W R VB c is the loss in social welfare when emigration of the -individuals is prevented. This loss 15
is shared out among the residents in A, whose proportion is given by F right-hand side of (29) corresponds to a loss in welfare per capita.
D 1: Thus the
Since condition (29) involves endogenous variables, it should be applied as follows. Assume the optimal tax problem under emigration constraint has been solved. Every variable in (29) is known. If (29) does not hold, it is socially preferable to allow emigration of the highest skilled individuals. Corollary 12 Let Assumptions 1–2 hold and the participation constraint be binding at b: A suf cient condition for emigration of the -individuals not to be socially optimal is that: c
VC . /1
dF . /
2 [0; 1/ [ .1; 1/
Equivalently, a necessary condition for emigration of the -individuals to be socially optimal is that: "Z #11 1 for any 2 [0; 1/ [ .1; 1/ (31) VC . / dF . / 0: First, every agent obtains his reservation utility which is strictly increasing in : Since the social objective is strictly increasing in ; the marginal social welfare is greater than the average social welfare. Secondly, every agent pays strictly positive taxes. Thus the presence of the marginal individuals has two 16
strictly positive effects on the social objective. In consequence, it is socially optimal to preclude the marginal individuals from emigrating and to take emigration constraints into account. When the migration costs tend to zero for every of 2 R ; the participation constraint reads V A . / VB . / : An agent who would pay strictly positive taxes would have a utility that is less than his laissez-faire utility VB . / and would leave the country A: In consequence, the agents living in A cannot pay strictly positive taxes. If R > 0; the tax revenue constraint cannot be satis ed. If R D 0; there is no redistribution of income within the population and every agent receives his reservation utility. In consequence, the participation constraint is binding for every individual and, by Corollary 13, it is socially optimal to take emigration constraints into account. When A0 s social objective is Rawlsian, the marginal social welfare is equal to zero. In consequence, the scal effect of the presence of the b-individuals is the only one that matters. By Proposition 6, this scal effect is positive when the individuals are constrained to participate. In consequence, we nd the same conclusion as in the national model: the b-individuals are socially useful because they pay positive taxes.
It is rst interesting to compare the results obtained in the national and resident models. In the national model, it was established that it is socially optimal to take emigration constraints into account, for every social welfare function which exhibits a nonnegative aversion to income inequality. The social value function is thus to the maximum when every national is a resident. In the resident model, there are cases in which the average social welfare increases when the highest skilled individuals are allowed to leave the country. Since the social value functions obtained under both criteria are equal when emigration is prevented (i.e. b D /; the resident criterion is (weakly) socially preferred to the national criterion9 . The aim of this paper was to build a simple model to examine the effects of high-skilled emigration which occurs between developed countries. The ows of high-skilled emigrants in the North are largely not balanced: some countries appear as "brain drain" losers whilst the United States appears as a magnet. This paper focuses on brain drain losers. A common idea is that these countries have no alternative to lowering taxes , and redistribution, so as to limit high-skilled emigration. The basic question we addressed was whether it is always socially optimal to prevent emigration of high-skilled workers. In order to isolate the effects of emigration, we looked into a world consisting of a redistributive country and a laissez-faire country. The laissez-faire country may be seen as the limit case of a lessredistributive country. Every agent was initially living in the redistributive country. Tax policy was the only difference between both countries. It was assumed that every individual's productivity was known to the government. In consequence, lump-sum taxes were implementable. 9 Recall W N b is the maximum average social welfare obtained in A when the agents whose productivity is greater than b emigrate to B: By Proposition 8, D arg max W N b : At b D ; W N b D W R b : But by Lemma 10,
arg max W R b
: Thus W N b
0 W R b for every b; b
Participation constraints were introduced as a device at the government's disposal for controlling emigration. The government may introduce participation constraints into its optimal tax problem to prevent emigration of some individuals. In a rst step, we examined the optimal solution to the optimal tax problem when the government prevents every individual from emigrating. We proved that, under some conditions on migration costs, the indirect utility was V -shaped for every nite aversion to income inequality, that is decreasing for the low-skilled individuals and increasing for the high-skilled individuals for which the participation constraints are binding. Because the high-skilled workers can vote with their feet, the scal rent is extracted to the maximum from the highest-skilled individuals who are insuf ciently talented to leave the country. This result was called the "malediction of the middle-skilled workers" in reference to the malediction of the high-skilled workers that occurs in closed economy. Then, two social criteria were distinguished. In the national model, it was proved that it was socially optimal to prevent emigration of the high-skilled individuals because they are net taxpayers. In the resident model, it was shown that there is a trade off between a demographic/ethical effect and a scal effect of the departure of the high-skilled individuals. Since the social objective in the redistributive country does not depend on the welfare of the nationals living abroad, it is socially optimal to allow emigration of the individuals whose social welfare is less than the average social welfare when the taxes they pay are insuf cient. This trade off depends basically on the government's aversion to income inequality and on migration costs. In the Rawlsian case, it was shown that it is always socially optimal to prevent emigration of the high-skilled individuals. The model we built has two main limits. It was assumed that both productivity and migration costs were known to the government. The next step in our research agenda is to focus on the effects of high-skilled emigration on the optimal tax scheme when these variables are private information.
References B HAGWATI , J. (1976): The Brain Drain and Taxation: Theory and Empirical Analysis. NorthHolland, Amsterdam. (1980): “Taxation and International Migration: Recent Policy Issues,” mimeo, Conference on U.S. Immigration Issues and Policies, Chicago. B HAGWATI , J., AND K. H AMADA (1989): “Tax Policy in the Presence of Migration,” in International Taxation and International Mobility, ed. by J. Bhagwati, and J. Wilson, chap. 5, pp. 113–140. MIT Press. C REMER , H., V. F OURGEAUD , M. L EITE M ONTEIRO , M. M ARCHAND , AND P. P ESTIEAU (1996): “Mobility and Redistribution: A Survey,” Public Finance/Finances Publiques, 51, 325– 352. C REMER , H., AND P. P ESTIEAU (1996): “Distributive Implications of European Integration,” European Economic Review, 40, 747–757. G UESNERIE , R. (1995): A Contribution to the Pure Theory of Taxation. Cambridge University Press, Cambridge. H AMILTON , J., AND P. P ESTIEAU (2002): “Optimal Income Taxation and the Ability Distribution : Implications for Migration Equilibria,” CORE Discussion Paper 2002/36. H UBER , B. (1999): “Tax Competition and Tax Coordination in an Optimal Income Tax Model,” Journal of Public Economics, 71, 441–458. I QBAL , M. (2000): “The Migration of High-Skilled Workers from Canada to the United States: Empirical Evidence and Economic Reasons,” Center for Comparative Immigration Studies Working Paper n. 20, University of California-San Diego. L AFFONT, J.-J., Press.
D. M ARTIMORT (2002): The Theory of Incentives. Princeton University
L EITE -M ONTEIRO , M. (1997): “Redistributive Policy with Labour Mobility Accross Countries,” Journal of Public Economics, 65, 229–244. M IRRLEES , J. (1971): “An exploration in the theory of optimal taxation,” Review of Economics Studies, 38(2), 175–208.
(1974): “Notes on welfare economics, information and uncertainty,” in Essays in economic behavior under uncertainty, ed. by M. Balch, M. McFadden, and S. Wu, pp. 243–258. North Holland. (1982): “Migration and Optimal Income Taxes,” Journal of Public Economics, 18, 319– 341. (1986): “The Theory of Optimal Taxation,” in Handbook of Mathematical Economics, ed. by K. Arrow, and M. Intriligator, vol. 3, chap. 24, pp. 1197–1249. North Holland. M YLES , G. (1995): Public Economics. Cambridge University Press, 2002 edn., Chapter 5. OECD (2002): International Mobility of Highly Skilled. OECD, Paris. P IASER , G. (2003): “Labor Mobility and Income Tax Competition,” CORE Discussion Paper 2003/6, UCL. ROEMER , J. (1985): “Equality of Talent,” Economics and Philosophy, 1, 151–157. S TIGLITZ , J. (1982): “Self-Selection and Pareto Ef cient Taxation,” Journal of Public Economics, 17(2), 213–240. T IEBOUT, C. (1956): “A Pure Theory of Local Expenditures,” Journal of Political Economy, 64, 416–424. T UOMALA , M. (1990): Optimal Income Tax and Redistribution. Clarendon Press, Oxford. W ILSON , J. D. (1980): “The effect of potential emigration on the optimal linear income tax,” Journal of Public Economics, 14, 339–353. (1989): “Optimal Linear Income Taxation in the Presence of Emigration,” in Income Taxation and International Mobility, ed. by J. Bhagwati, and J. Wilson, chap. 6, pp. 141–158. MIT Press. (1992): “Optimal Income Taxation and International Mobility,” American Economic Review, 82(2), 191–196.
P ROOF OF L EMMA 3
Note that, by assumption, is continuous in since (10) gives . / D =Ux U : 0 First, note that there is no isolated point 0 of ]e; ] such that D 0: Indeed, assume there exist 0 0 e an isolated point > such that > 0. Since . / is continuous, it cannot differ from zero at an isolated point, a contradiction. Then, we prove that there is no interval of ]e; ] on which . / D 0: Since . / is continuous in ; there exists an interval [e; 0 [ with 0 > e on which . / > 0: By (13),
c . / 8 2 [e; 0 [
V A . / D VB . /
Assume . / D 0 for every 2 0 ; 0 C ; for some > 0: On [ 0 ; 0 C ] the rst order conditions of the optimal tax problem are Ux D U and Ul D U : Locally, under Assumption 1, the result shown by Theorem 1 is valid. Thus V A0 . / 0 for every 2 [ 0 ; 0 C ], whilst, by Assumption 2, VB0 . / c0 . / > 0 for every : Hence, on 0 ; 0 C ; V A . / decreases and VB . / c . / strictly increases. By (32),
V A . / < VB . / Since
. / D 0 for every
c. / 8 2
; (13) implies:
V A . / > VB . /
c. / 8 2
a contradiction because of (33).
P ROOF OF P ROPOSITION 4
Two cases have to be distinguished, VC In the rst case,
< VB VC
Assume D 0: By Lemma 3, . / D 0 8 the participation constraints are never binding:
: Using the complementarity slackness condition (13),
V A . / > VB . /
c. / 8
V A . / D VC . / 8
By (36), (34) is equivalent to:
VA a contradiction because of (35). Therefore In the second case,
> 0: By Lemma 3, there exists VB 21
< . (37)
By Theorem 1, VC0 . / < 0: Under Assumption 2, VB0 . / c0 . / > 0. Hence (37) involves that VC . / c 8 : In consequence, V A . / D VC . / 8 and . / D 0 8 : does not VB exist.
P ROOF OF P ROPOSITION 6
2 H; d V A . / =d < 0: Thus d V =d D .d V =dT / .dT =d / < 0: Since d V =dT < 0; dT =d > 0 for 2 H: (2) Since the migration costs c . / are strictly positive, the indirect utilities of the agents of F are strictly
less than their laissez-faire utility. In consequence, they pay strictly positive taxes.
P ROOF OF C OROLLARY 7
The optimal tax schedule maximizes the social value function. Assume the participation constraint is binding for every of 2 R : Then, by Proposition 6, every agent pays strictly positives taxes. Since R D 0; the participation constraint is not binding.
T . /dF . / > 0
Therefore it would be possible to increase the utility level of some individuals (of some low skilled individuals, in fact, since the social objective is concave). These individuals would be better off and the social value function would increase. A contradiction.
P ROOF OF P ROPOSITION 8
When 2 R is given, the optimal tax problem is as follows.
Problem 14 Find x . / and l . / which maximize: Z
.U .x; l// d F . / s.t.
where 2 R D
; b ; with b
x/ d F . /
0 and U .x; l/
VB . /
c . / ; 8 2 2R
The value function of this optimization problem, W N ; depends on b : W N D W N b : The La-
.U .x; l// d F . /
.U .x; l// C
x/ C . / [U .x; l/
VB . / C c . /] d F . /
The complementarity slackness condition corresponding to the participation constraint reads:
0I U .x; l/
VB . / C c . /
. / [U .x; l/
VB . / C c . /] D 0; 8 2 2 R (38)
Applying the envelope theorem, one obtains:
dW N b db
dL b db bl A b
xA b C
Using (38), recalling that T . / :D l A . /
b U xA b ; lA b
VB . / C c . /
x A . /, one nally obtains:
dW N b R0, T b f b R0 db
Proposition 6 says that T b > 0 since the participation constraint is assumed to be binding at b: In addition and f b are strictly positive. In consequence, W N b is increasing in b along the participation constraint.
P ROOF OF L EMMA 10
The Lagrangian of the optimal tax problem (25) is
Z 1 D .U .x; l// d F . / F b 2R Z f . l x/ C . / [U .x; l/ C 2R
VB . / C c . /]g d F . /
where and . / are the multipliers of the tax revenue constraint (26) and of the participation constraint (27) respectively. The complementarity slackness condition corresponding to (27) is
0I U .x; l/
VB . / C c . /
. / [U .x; l/
VB . / C c . /] D 0
for every 2 2 R : Let us now turn to the effect of a marginal change in b on the social value function. Recall that
(39) holds, that l A . / d L b =db; i.e.
dW R b db
x A . / D T . /. Applying the envelope theorem, we have dW R b =db D Z f b
.V A . // d F . / C
VA b C T b F b
Since f . / > 0 for every ; one nally obtains
WR b dW R b R0, T b R db
VA b F b f b F b
VB b F b
C T b
P ROOF OF C OROLLARY 12
Recall that W R is the social value function when participation constraints are taken into account. De ne WCR as the corresponding social value function in the absence of mobility, and thus without participation constraints. We thus have
By Proposition 11, a necessary and suf cient condition for emigration of the -individuals not to be socially optimal is that
Combining (40) and (41), a suf cient condition for emigration -individuals not to be socially optimal is that
Since the left-hand side of (42) is nonnegative, we nally obtain the following suf cient condition for emigration -individuals to be socially optimal:
WCR that is
VC . /
dF . /
2 [0; 1/ [ .1; 1/
Equivalently, a necessary condition for emigration of the -individuals to be socially optimal is that
VC . /
dF . /