On Redistributive Taxation under the Threat of High-Skill Emigration

The increasing international mobility of high-skill individuals is often seen as posing a threat to domestic social welfare, by limiting the ability of governments to tax these individuals and redistribute to the poor. In this note, we examine a simple dynamic nonlinear income tax model without commitment. In this setting, it is shown that the threat of emigration by high-skill individuals facilitates redistribution and increases social welfare in the short run, and has no e¤ect on social welfare over the long run. Keywords: nonlinear taxation; migration; commitment. JEL Classications: H21; H24; F22. Department of Economics and Related Studies, University of York, Heslington, York, YO10 5DD, U.K. E-mail: alan.krause@york.ac.uk.


Introduction
Optimal tax analyses typically assume that individuals cannot emigrate to avoid domestic taxation. Such an assumption is, however, increasingly viewed as being unrealistic, especially as it relates to high-skill individuals. When high-skill individuals are immobile, redistributive taxation must take into consideration that these individuals may change their labour supply along the intensive margin. That is, high-skill individuals may work less, thus reducing the amount of income available for redistribution. When high-skill individuals are internationally mobile, they have the additional option of changing their labour supply along the extensive margin, i.e., they may emigrate. An often-employed method to capture the threat of emigration is to introduce type-dependent participation constraints into the optimal tax problem; see, e.g., Osmundsen (1999), Krause (2009a), and Simula andTrannoy (2010, 2012). Naturally, these additional constraints reduce the level of social welfare attainable.
In this note, we introduce the threat of emigration by high-skill individuals into a dynamic nonlinear income tax model without commitment. As in the related literature, this threat is captured by introducing a participation constraint for high-skill individuals.
However, we show that the introduction of the high-skill type's participation constraint actually facilitates redistribution and increases social welfare in the short run, and has no e¤ect on social welfare over the long run. The intuition is as follows. In period 1 the government does not know each individual's skill type, and therefore implements standard (incentive-compatible) nonlinear income taxation which induces individuals to reveal their types. However, high-skill individuals know that if they reveal their type in period 1, they lose their information advantage and will be subjected to …rst-best taxation from period 2 onwards. This means that high-skill individuals must be o¤ered a very attractive tax treatment in period 1 to reveal their type, to compensate them for having to face …rst-best taxation thereafter. Accordingly, the ability of the government to redistribute in period 1 is severely limited. However, the threat of emigration reduces the extent of redistribution possible under …rst-best taxation, meaning that high-skill individuals require less compensation in period 1 to reveal their type. This enables the government to implement more redistribution in period 1, which correspondingly increases …rst-period social welfare. The threat of emigration limits redistribution and reduces social welfare from period 2 onwards, but optimal taxation balances the shortrun bene…ts against the long-run costs. Thus the threat of emigration has no e¤ect on the level of social welfare summed over all periods.
In terms of previous results, the paper most closely related to ours is that by Leite-Monteiro (1997). He also …nds that increased international mobility may enhance redistribution, but his model is entirely di¤erent to ours. In Leite-Monterio's model, the government can always implement …rst-best personalised lump-sum taxes, and changes in the country's skill composition after migration takes place is what makes enhanced redistribution a possibility. By contrast, in our model the participation constraint ensures that no one migrates, so the skill composition remains unchanged. Our note is also related to the literature on dynamic Mirrlees (1971) nonlinear income taxation without commitment, e.g., Roberts (1984) can be when the commitment assumption is relaxed. The present note provides another example of a result that, at …rst glance, appears quite counter-intuitive.
The remainder of the note is organised as follows. Section 2 outlines the model and the structure of optimal taxation. Section 3 presents and discusses our result, while Section 4 concludes. The proof of our result is contained in an appendix.

A Simple Model
We consider an in…nite-horizon model with a unit measure of individuals and with the following timing. In period 1 the government knows there are 2 (0; 1) highskill individuals and (1 ) low-skill individuals in the economy, but it does not know any individual's skill type. The government therefore implements standard secondbest (incentive-compatible) nonlinear income taxation, under which each individual is willing to reveal their type. Then, from period 2 onwards, the government knows each individual's skill type, and is tempted to use this information to implement …rst-best redistributive taxation. However, high-skill individuals have the option of emigrating, so redistribution is limited by the high-skill type's participation constraint. For simplicity we assume that individuals do not save or borrow, so the only link between periods is the revelation and use of skill-type information.
Speci…cally, the government in period t (where t 2) solves the following problem.
Choose tax treatments hc t L ; y t L i and hc t H ; y t H i for the low-skill and high-skill individuals, respectively, to maximise: where c t i is type i's consumption (or post-tax income) in period t, y t i = w i l t i is type i's pre-tax income in period t, with w i denoting type i's wage rate and l t i denoting type i's labour supply in period t. It is assumed that w H > w L > 0 and that wages remain of V H by emigrating, which is their reservation utility and is assumed to remain constant 1 The literature on the comparative statics of optimal nonlinear income taxes (e.g., Weymark (1987), Weymark (2008b, 2011), and Simula (2010)) has shown that results are generally obtainable only when the utility function is quasi-linear. As we make use of comparative statics methods, we also assume that the utility function is quasi-linear. 2 We assume that the government cannot save or borrow, and that its revenue requirement is zero. Thus taxation is implemented only for redistributive purposes. In period 1 the government does not know each individual's skill type, and therefore implements second-best (incentive-compatible) nonlinear income taxation. It chooses tax treatments hc 1 L ; y 1 L i and hc 1 H ; y 1 H i for the low-skill and high-skill individuals, respectively, to maximise: subject to: where equation (2.4) is the …rst-period utilitarian social welfare function, and equation (2.5) is the government's …rst-period budget constraint. Equation (2.6) is the highskill type's incentive-compatibility constraint, with 2 (0; 1) denoting the individuals' discount factor. If a high-skill individual chooses hc 1 H ; y 1 H i in period 1, they are revealing their type to the government, and will then receive the high-skill type's reservation utility, V H , in each period t 2. Therefore, a high-skill individual will be willing to choose hc 1 H ; y 1 H i in period 1 if the utility they obtain from this tax treatment, plus the utility they then receive from period 2 onwards, is greater than or equal to the utility they could obtain by pretending to be a low-skill individual. That is, if a high-skill individual chooses hc 1 L ; y 1 L i in period 1, they are announcing to the government that they are low skill, and will therefore receive the low-skill type's tax treatment, hc t L ; y t L i, in each period t 2. We omit the low-skill type's incentive-compatibility constraint, because we make the standard assumption that the redistributive goals of the government create an incentive for high-skill individuals to mimic low-skill individuals, but not vice versa. 3 We also omit the high-skill type's …rst-period participation constraint, as we assume that it is their incentive-compatibility constraint that binds. 4 The solution to programme (2:4) (2:6) yields the value function W 1 ( ; w L ; w H ; V H ; ), which represents the level of social welfare attainable in the …rst period.

The Threat of Emigration and Social Welfare
It is shown in the appendix that: Proposition In our dynamic nonlinear income tax model without commitment, the threat of high-skill emigration increases social welfare in period 1 (@W 1 ( )=@V H > 0), decreases social welfare in period t 2 (@W t ( )=@V H < 0), and has no e¤ect on social welfare summed over the model's horizon (@W 1 ( ) The proposition implies that the threat of high-skill emigration facilitates redistribution and increases social welfare in the short run, and has no e¤ect on the level of social welfare attainable over the long run. Social welfare is increased in period 1 because the threat of emigration relaxes the incentive-compatibility constraint, i.e., it becomes cheaper for the government to obtain skill-type information. In general, high-skill individuals must be o¤ered an attractive tax treatment in period 1 to reveal their type, as compensation for having to face …rst-best taxation from period 2 onwards. Therefore, the ability of the government to redistribute in period 1 is severely limited. However, the threat of emigration also restricts the ability of the government to redistribute after high-skill individuals have revealed their type. This means that the government in period 1 can o¤er high-skill individuals a less attractive tax treatment, which in turn means more redistribution and a higher level of social welfare. In each period from period 2 onwards, the threat of emigration reduces social welfare, because the government cannot implement unrestricted …rst-best redistribution. Over the long run, however, the threat of emigration has no e¤ect on social welfare, because optimal taxation balances the short-run bene…ts vis-a-vis relaxation of the incentive-compatibility constraint against the long-run costs of the binding participation constraints.

Concluding Comments
In this note we have shown, using a simple dynamic nonlinear income tax model without commitment, that the threat of high-skill emigration increases social welfare in the short run and has no e¤ect on social welfare over the long run. While this result may be viewed as being primarily of theoretical interest, it also provides a new example of how restrictions on policy instruments that are clearly welfare-reducing when the government can commit, may in fact be bene…cial when the government cannot commit.

Appendix
Proof of the Proposition The Lagrangian corresponding to programme (2:1) (2:3) is: where t > 0 and t H > 0 are Lagrange multipliers. The …rst-order conditions are: By the Envelope Theorem: where the last equality follows from (A.3) and (A.5). This shows that @W t ( )=@V H < 0 for all t 2.
The Lagrangian corresponding to programme (2:4) (2:6) is: where 1 > 0 and 1 H > 0 are Lagrange multipliers, and use has been made of: The …rst-order conditions on y 1 L and y 1 H are, respectively: By the Envelope Theorem: Equation (A.13) can then be simpli…ed to: where use has been made of (A.11) and (A.12). This shows that @W 1 ( )=@V H > 0.