Laurent Simula
email: laurent.simula@ens-lyon.fr

Alain Trannoy
email: alain.trannoy@univ-amu.fr

Bunching in Rank-Dependent Optimal Income Tax Schedules *

Keywords: Rank dependence, Gini, Optimal Income Taxation, Bunching, Log-Concavity. JEL Codes: D630, D820, H210

Revised Version

I. INTRODUCTION

Among many other topics, John Weymark has contributed to the literature on the better understanding of inequality indexes of the Gini family (see, e.g., [START_REF] Weymark | Generalized Gini inequality indices[END_REF]) as well as on optimal income taxation (see, e.g., Weymark (1986aWeymark ( ,b, 1987))). In particular, he has studied optimal taxation with a discrete distribution of types and studied the question of "bunching", a situation in which some agents, of different types, choose a same net income/gross income bundle. The present article articulates both literatures, providing an extension of Simula and Trannoy (forthcoming), our first article on optimal non-linear income tax setting for rank-dependent welfare functions à la [START_REF] Menahem | The dual theory of choice under risk[END_REF] in a continuous population setting. It pays full attention to the question of bunching, that was assumed away in the companion paper, and to the similarities and differences between the continuous and discrete settings.

In this framework, individual welfare weights depend on the position of individuals in the distribution of indirect utility. We consider individual weights consistent with second-order stochastic dominance. Social welfare can be written as the egalitarian benchmark, deflated by inequality measured by the Gini, S-Gini or "illfare-ranked single-series Ginis" introduced by [START_REF] Donaldson | A single-parameter generalization of the Gini indices of inequality[END_REF] as well as [START_REF] Bossert | An axiomatization of the single-series Ginis[END_REF]. The illfare-ranked single series Ginis are referred to as the "A-family" below, given that they have extensively been studied by [START_REF] Aaberge | Characterization of Lorenz curves and income distributions[END_REF]. Beyond Lorenz agreement, the S-Gini family favors a reduction of inequality in the lower part of the distribution, while the A-family places more weight on a reduction of inequality in the upper part. [START_REF] Weymark | Comparative static properties of optimal nonlinear income taxes[END_REF] and [START_REF] Simula | Optimal nonlinear income tax and nonlinear pricing: Optimality conditions and comparative static properties[END_REF] already considered optimal income taxation for a social welfare function with exogenous individual weights, diminishing with productivity. The parametric families we consider are specific examples of such decreasing weights.

Following [START_REF] Mirrlees | An exploration in the theory of optimum income taxation[END_REF], the productivity of each agent is private information, and the benevolent policymaker only knows how productivity is distributed within the population. As a result, the optimal allocation maximizes rank-dependent welfare subject to the tax-revenue and incentive-compatibility constraints. The latter are necessary and sufficient to provide each agent with the right incentive for them to truthfully report their private information, i.e., to behave in the way chosen by the policy-designer. Incentive-compatible allocations verify two conditions: on the one hand, indirect utility must increase at a sufficient rate to induce truthtelling; on the other hand, gross income (or equivalently net income) must be non-decreasing with productivity. A situation in which the "monotonicity" condition on gross income is binding is referred to in the literature as "bunching". In other words, "bunching" is said to occur in a social optimum when agents with different productivity levels choose the same grossincome/net-income combination. Absent bunching, ranks in terms of indirect utility, gross income, net income and productivity are all the same. In addition, providing agents behave in a rational way, these ranks corresponds to those in the actual allocation. Ranks then become "invariant" and provide a standard with respect to which tax rates may be expressed, as suggested by [START_REF] Trannoy | Talent, equality of opportunity and optimal non-linear income tax[END_REF]. To cast light on the main mechanisms and intuitions, we consider that individual utility is quasi-linear, linear with respect to net income. Such preferences are widely used in the literature following [START_REF] Piketty | La redistribution fiscale face au ch ômage[END_REF] and [START_REF] Diamond | Optimal income taxation: An example with U-shaped pattern of optimal marginal tax rates[END_REF], even though assuming that there is no income effect on gross income is obviously restrictive. 1For a continuous population and relying on the two families of parameterized rank-dependent welfare functions presented above (with Gini social welfare function as a common element), we pay special attention to situations in which the optimal allocation is not "fully separating" in a continuous setting. Even if there is no specific reason to expect that bunching would be more frequent in the rank-dependent setting, the rank-dependent approach offers a unique opportunity to say more about this question, and provide a set of new exogenous conditions. The latter are derived using an implicit expression for the gross income function and the various parameters of the model. This expression (Proposition 1) is obtained using Fubini's theorem, allowing for mass points in the distribution of welfare weights, which accommodates "limit" cases such as the Rawlsian maximin, and without resorting to differentiability of the gross income function. This is one of the article's technical contributions.

Thanks to rank-dependent welfare weights, we are able to obtain several conditions on the parameters of the model, under which bunching in the social optimum is either present or precluded. These conditions are derived exploiting the implicit expression for the optimal gross income function, coming out from the optimization process. It is particularly simple due to the combined assumptions of quasi-linear individual preferences and rank-dependent social welfare weights. To study bunching, we restrict the class of preferences to those for which the marginal dis-utility of gross income is convex, but less convex the higher the agent's productivity. While the Spence-Mirrlees condition is about the sign of the cross derivative of the utility function with respect to productivity and gross income, our condition is on the slopes of this cross derivative with respect to gross income, on the one hand, and productivity, on the other hand. It thus correspond to additional sorting conditions. When it holds, the discussion about bunching only depends on the shape of the productivity distribution and the class of rank-dependent social welfare functions (S-Gini or A-families).

We in particular resort to log-concavity of the type's survival function or log-convexity of the type's probability density function (pdf). Under log-concavity of survival function of the productivity distribution and for the whole A-family (including Gini), the optimal allocation does not involve any bunching whatsoever; if the focus is on reducing inequality in the top of the income distribution, bunching is precluded. The same assumption of log-concavity does not guarantee the absence of a pooling equilibrium when we pay closer attention to reducing inequalities in the lower part of the distribution, as for the S-Gini family. We then are only able to prove that the optimal gross income function is increasing above a productivity threshold.

We also examine specific distributions as did [START_REF] Boadway | Optimal income taxation with quasi-linear preferences revisited[END_REF] for quasi-linear-in-leisure preferences and weighted utilitarianism and [START_REF] Boadway | Optimal marginal and average income taxation under maximin[END_REF] for separable preferences and the "Rawlsian" maximin. The findings in [START_REF] Boadway | Optimal income taxation with quasi-linear preferences revisited[END_REF] regarding the absence of bunching for any exponential distribution of productivity are extended to our setting. In addition, the result in [START_REF] Boadway | Optimal marginal and average income taxation under maximin[END_REF] regarding the absence of bunching under the maximin, quasi-linear-in-consumption preferences, and a Weibull probability density for types (with scaling parameter above 1), appears as a subcase of a result we obtain for all productivity distributions with a log-concave survival function. Indeed, the maximin appears as a limit case to the S-Gini family, and in this situation, bunching does not occur in the social optimum under the considered assumptions.

Many articles on optimal taxation, among which several of us [START_REF] Simula | Optimal income tax under the threat of migration by topincome earners[END_REF], [START_REF] Lehmann | Tax me if you can! Optimal nonlinear income tax between competing governments[END_REF] or Simula and Trannoy (forthcoming)), adopt the "first-order" approach, ignoring the monotonicity constraints for gross income, and checking it ex post in numerical simulations. [START_REF] Dagobert | Some properties of the optimal income-tax[END_REF] and [START_REF] Lollivier | Bunching and second-order conditions: A note on optimal tax theory[END_REF] were the first to explain how to formally account for this difficulty which already had been noted by [START_REF] Mirrlees | An exploration in the theory of optimum income taxation[END_REF]. For continuous populations, [START_REF] Lollivier | Bunching and second-order conditions: A note on optimal tax theory[END_REF] considered an example in which bunching occured in the social optimum, showing the limitations of the first-order approach. In addition, [START_REF] Ebert | A reexamination of the optimal nonlinear income tax[END_REF] provided an other example of bunching and examined whether the general features of the "optimal" schedule obtained when ignoring the possibility of bunching were preserved in the "full" optimum. For a discrete population, Weymark (1986b) -focusing on quasilinearin-leisure preferences-provided an example in which the solution to the "relaxed problem" (ignoring possible bunching) cannot be the solution to the full problem, before showing how to solve the full problem through "ironing" techniques. Also for a discrete population, [START_REF] Simula | Optimal nonlinear income tax and nonlinear pricing: Optimality conditions and comparative static properties[END_REF] provided conditions for bunching to occur in the social optimum when preferences are quasi-linear, linear with respect to consumption.

Moreover, we obtain tax formulas when the productivity distribution is discrete and compare them with the formulae obtained in the continuous framework (Simula and Trannoy, forthcoming). The mathematical derivations in the continuous and discrete population cases follow the same structure in this article, adapted from the way to proceed in [START_REF] Simula | Optimal nonlinear income tax and nonlinear pricing: Optimality conditions and comparative static properties[END_REF]. The Lagrangian is expressed in terms of gross income thanks to the quasilinearity-in-consumption assumption. We find expressions which are as close as possible to the "ABC" formula for a continuum of types; in spite of their similarities, they remain however essentially distinct. The question of the convergence to the continuous case is still an open question.

The article is organized as follows. Section II presents the setting. Section III establishes an implicit expression for optimal gross income in the "relaxed" optimal income tax problem which ignores the monotonicity condition. Section IV characterizes the bunching pattern, deriving conditions on the primitives of the model. Section V focuses on a discrete population of types to show similarities and differences with regards to the continuous setting. Section VI provides concluding comments.

II. FROM WEIGHTS DEPENDING ON RANKS TO OPTIMAL TAXATION

The set-up is taken from Simula and Trannoy (forthcoming). We reproduce it so as to be able to analyze the bunching issues and develop its discrete counterpart. We start with rankdependent social welfare function and then present how they connect to the literature on inequality measurement. We here assume a continuous population of individuals, differing with a single dimension of heterogeneity. This setting is maintained in Sections III and IV.

II.1. Connecting Rank-Dependent Social Welfare with Inequality Indexes

We consider a population of individuals, heterogeneous with respect to a variable x. For simplicity, we assume that the latter is uni-dimensional and smoothly distributed according to the cumulative distribution function (CDF) denoted F(x), with support X ⊆ R + . We call f (x) the corresponding probability density function (pdf). The average value of x within the population is µ = X x f (x)dx.

We define the quantile function as F -1 (p) = x, where p ∈ [0, 1] stands for the rank or "position", and introduce weights to capture the social planner's aversion to inequality. The marginal weights are denoted λ(p) and the cumulated weights Λ(p) = p 0 λ(ρ)dρ. In the entire article, we require weights to be consistent with the Lorenz criterion, implying that they belong to the set: L = {∀p ∈ (0, 1), λ(p) > 0 and λ (p) < 0; Λ(0) = λ(1) = 0; Λ(1) = 1}.

(1)

Assuming that λ(p) is positive and decreasing implies that the welfare of every individual counts, but to a lower extent the higher the rank. The other assumptions correspond to normalizations. On this basis, rank-dependent social welfare [START_REF] Menahem | The dual theory of choice under risk[END_REF][START_REF] Menahem | A controversial proposal concerning inequality measurement[END_REF] is defined as:

W = 1 0 λ(p)F -1 (p)dp. (2)
If W = µ, it corresponds to the case of an egalitarian distribution. Otherwise, there is a positive gap ∆ ≡ µ -W between the egalitarian benchmark and social welfare W. Dividing this gap by µ, we obtain the mean-invariant inequality index, denoted I, and case rewrite W as:

W = µ(1 -I).
(3)

This "abreviated form" highlights the close connection between rank-dependent social welfare functions and inequality indexes: social welfare W can then be measured as the mean µ, deflated by the inequality index I. The trade-off between economic efficiency and equity is therefore summarized in the social objective function in a very transparent way.

In various places below, we pay special attention to two important families of rank-dependent social welfare functions:

-The "A"-family is obtained when Λ(p) = (δpp δ )/(δ -1) and δ ≥ 2. 2 -The S-Gini family [START_REF] Donaldson | A single-parameter generalization of the Gini indices of inequality[END_REF] is obtained when

Λ(p) = 1 -(1 -p) δ
and δ ≥ 2.

We also consider a "variation" of the A-family obtained when δ → 1. In that case, the cumulated weights are given by Λ(p) = p(1log(p)) and the corresponding inequality index I is the Bonferroni index of inequality. 3 For any δ > 2, the weights λ(p) are concave for the A-family, and convex for the S-Gini family. This has the following implication if we consider a fixed transfer taking place between two agents with equal difference in ranks. For the A-family, the equalizing effect of such transfer is stronger at higher ranks. The priority is thus given to the reduction of inequalities at the top of the distribution. On the contrary, for the S-Gini family, the equalizing effect of the transfer becomes larger the lower the ranks considered. The main focus is thus on reducing inequalities at the bottom of the distribution.

The A and S-Gini families have a single element in common, obtained when δ = 2 and, thus, Λ(p) = p(2p). In that case, it can be shown that the inequality measure I is the Gini 2 [START_REF] Aaberge | Characterization of Lorenz curves and income distributions[END_REF] refers to this class as the Lorenz family for δ = 2, 3, .... It corresponds to the integer subfamily of the "illfare-ranked single-series Ginis" discussed by [START_REF] Donaldson | A single-parameter generalization of the Gini indices of inequality[END_REF] and [START_REF] Bossert | An axiomatization of the single-series Ginis[END_REF]. We refer to the latter as the "A" family. [START_REF] Aaberge | Ranking intersecting Lorenz curves[END_REF] and [START_REF] Aaberge | A theory for ranking distribution functions[END_REF] have indeed shown the usefulness of this family for analyzing inequality when Lorenz curves intersect.

3 Let µ

(x) = x 0 xdF(x) /F(x) and r(x) = (µ -µ(x))/µ. The Bonferroni inequality index is X r(x)dF(x).
coefficient while W corresponds to the Gini social welfare function introduced by [START_REF] Sen | Informational bases of alternative welfare approaches[END_REF]. It is interesting to note that the marginal social weights λ(p) = 2(1p) vary with respect to the rank in a linear way, diminishing from 2 to 0, and being equal to 1 for the median individuals. 4When the parameter δ increases, the S-Gini and A-families tend to two important benchmarks, the Rawlsian maximin on the one hand and pure utilitarianism on the other hand.

II.2. Relaxed and Full Optimal Income Tax Problems

Following Simula and Trannoy (forthcoming), we consider a population of agents differing with respect to productivity θ belonging to a compact subset θ, θ of the positive real line. The parameter of heterogeneity θ is smoothly distributed, with cumulative density function H and probability density function h = H (satisfying h > 0 over its support). Its distribution is common knowledge; but the exact productivity of a given agent is only known to herself. The connection with the distribution of x introduced above will be made clear below.

The starting point is the utility maximization problem of an agent with given productivity θ. The latter chooses consumption c and taxable income z so as to maximize:

u(c, z; θ) = c -v(z; θ) = z -T(z) -v(z; θ), ( 4
)
where T(z) stands for the nonlinear income tax schedule. It should be noted that the dis-utility of gross income v(z; θ) is written in a more general way than the more commonly-used multiplicative form where v(z; θ) = ν(z/θ). 5 In that case, gross income is obtained as the product of the wage rate and working hours. The specification that we use is not just a theoretical refinement. Indeed, empirical findings do not bolster the case for a multiplicative form, except for shift work. In addition, we let the dis-utility of gross income v(z; θ) be increasing and convex in gross income z. Hence, v z (z; θ) > 0 and v zz (z; θ) > 0. In addition, we assume that the Spence-Mirrlees condition is satified.

ASSUMPTION 1 (Spence-Mirrlees Condition). v zθ (z; θ) < 0.

Assumption 1 has the following interpretation. When productivity goes up, it becomes easier for an agent to increase gross income by a small given amount dz. We do not mention it explicitly in Propositions and Lemmas below, but it holds in the rest of the article. The first-order condition of Agent θ's utility maximization program is:

1 -T (z) = v z (z; θ). ( 5
)
We call c(θ) and z(θ) the optimal consumption and taxable income, and

V(θ) = c(θ) -v(z(θ); θ)
the corresponding indirect utility. Because of the taxation principle, a tax function is equivalent to the specification of a (c(θ), z(θ))-allocation subject to incentive-compatibility constraints. The latter ensure that a θ-individual has an incentive to disclose private information, i.e., her productivity level θ. This is the case providing:

V(θ) = max θ c(θ ) -v(z(θ ); θ)for any (θ, θ ) ∈ θ, θ 2 . ( 6
)
As is well-known (see, e.g., [START_REF] Salanié | The Economics of Contracts[END_REF]), this is equivalent to:

V (θ) = -v θ (z(θ); θ), (7)
z(θ) non-decreasing. ( 8) Equation ( 7) is the first-order condition for incentive compatibility. It ensures that indirect utility V(θ) increases at a sufficient rate to prevent agents from downward mimicking. Condition ( 8) is the second-order condition for incentive compatibility. Note that we do not require θ → z(θ) be differentiable in the general case. When it is differentiable, (8) may be rewritten as:

z (θ) ≥ 0. Because T(z(θ)) = z(θ) -c(θ)
, the government's budget constraint can be written as:

θ θ (z(θ) -c(θ))dH(θ) ≥ E, ( 9
)
where E is an exogenous amount of public expenditures. We focus on the case in which the tax policy is purely redistributive. Hence, E = 0 from now on. Moreover, in any social optimum, (9) will be binding. Using (6), we obtain c(θ) = V(θ) + v(z(θ); θ), which can be substituted into the binding government's budget constraint (9) to get:

θ θ (z(θ) -V(θ) -v(z(θ); θ))dH(θ) = 0. ( 10
)
We let x = V. Consequently, p = F(V(θ)) = H(θ), which also implies:

F -1 (p) = V(θ)
. The social objective is to maximize a weighted sum W of indirect utilities V(θ), with weights given by λ

(p) = λ(H(θ)), i.e., W = θ θ λ(H(θ))V(θ)dH(θ). ( 11
)
The optimal income tax problem can therefore be stated as follows:

PROBLEM 1 (Full Problem). Choose z(θ) and c(θ) to maximize W subject to the incentivecompatibility constraints ( 7) and (8) as well as the tax-revenue constraint (10).

We also formulate a variation of Problem 1 from which the second-order condition for incentive compatibility (8) has been removed.

PROBLEM 2 (Relaxed Problem). Choose z * (θ) and c * (θ) to maximize W subject to the firstorder condition for incentive compatibility ( 7) and the tax-revenue constraint (10).

The so-called "first-order approach" consists in solving Problem 1 without explicitly accounting for the second-order condition for incentive compatibility (8). The idea is to solve the Relaxed Problem and verify in a second step whether its solution generates a non-decreasing pattern of taxable income θ → z * (θ). If this is the case, it is also the solution to the Full Problem, and z(θ) = z * (θ) at any productivity θ in θ, θ .

III. OPTIMAL GROSS INCOME ALLOCATION FOR THE

RELAXED PROBLEM

By the taxation principle [START_REF] Peter | Straightforward individual incentive compatibility in large economies[END_REF][START_REF] Guesnerie | A Contribution to the Pure Theory of Taxation[END_REF], characterizing a nonlinear income tax schedule is equivalent to finding an allocation of gross and net incomes satisfying the incentive compatibility constraints ( 7) and ( 8). In Simula and Trannoy (forthcoming), we solved the Relaxed Problem and found "ABC" formulae for optimal marginal tax rates. In this Section, we derive the conditions that the gross income function solution to the Relaxed Problem has to satisfy, to check in Section IV whether it is monotone increasing, and therefore also provides the solution to the Full Problem.

The method of derivation follows the same steps as [START_REF] Brett | Voting over selfishly optimal nonlinear income tax schedules[END_REF] in a politicaleconomy setting. 6 The main advantage of this procedure is to solve the optimal income tax problem in a clear and transparent manner, without resorting to optimal control theory. The proof carries through even in cases when the gross income function θ → z(θ) would be nondifferentiable.

In addition, this derivation accommodates the possibility of mass points in the distribution of social weights. On this basis, the Rawlsian maximin in particular can be obtained as a limit case of the L-class. More generally, cumulated social weights Λ are allowed to be defined as a step function, which is continuous almost everywhere, with a finite number of discontinuity points. This proof complements those provided in Simula and Trannoy (forthcoming), based on optimal control theory and on a small tax reform perturbation respectively, both valid for a continuous distribution of cumulated social weights Λ(p).

We first use the first-order condition for incentive compatibility (7). Integrating between θ and any given θ, we obtain:

V(θ) = V(θ) - θ θ v θ (z(t); t)dt. ( 12
)
Replacing the latter into W given in (11) yields:

W = V(θ) θ θ λ(H(θ))dH(θ) - θ θ λ(H(θ))   θ θ v θ (z(t); t)dt   dH(θ). ( 13
)
Because Λ(1) = 1 for any θ → λ(θ) in L, the first term on the right-hand side of ( 13) is equal to V(θ). Therefore, (13) may be rearranged as:

W = V(θ) - θ≤t≤θ≤θ λ(H(θ))v θ (z(t), t)dH(θ)dt. ( 14
)
which may also be rearranged as follows:

1 -v z (z(θ); θ)) = -v θz (z(θ); θ) Λ(H(θ)) -H(θ) h(θ) for any θ ∈ θ, θ , (22)
providing h(θ) > 0, as assumed in this Section. We obtain the following Lemma.

PROPOSITION 1. Gross income levels z * (θ) solutions to the Relaxed Problem are determined by:

-

1 -v z (z * (θ); θ)) v θz (z * (θ); θ) = Λ(H(θ)) -H(θ) h(θ) for any θ ∈ θ, θ . ( 23
)
The latter equation is of particular interest for the study of bunching because it provides the optimal gross income function implicitly. Its LHS is related to individual preferences, while the RHS is determined by the distributions of both productivity and social weights.

Thanks to the first-order condition of the individual utility maximization program ( 5), 1v z (z * (θ); θ) is equal to the marginal tax rate solution to the Relaxed Problem, that we call T * . If we define p = π and G(p, π) as

G(p(θ), π(θ)) = Λ(p(θ)) -p(θ) π(θ) , (24)
Formula ( 23) implies that the tax function and gross income z * solutions to the Relaxed Problem are related through:

T * (z * (θ)) = -v θz (z * (θ); θ) G(p(θ), π(θ)). ( 25
)
First, -v θz (z * ; θ) in ( 25) is directly related to the Spence-Mirlees single-crossing condition. This term tells us how the slope of the indifference curve in the gross-income/net-income space should vary with productivity. It is is non-negative under the Spencer-Mirrlees condition, implying that the dis-utility of working an extra hour decreases with productivity. The more it does so, the steeper the profile of marginal tax rates will be. Indeed, the potential downward mimicking behavior then becomes less stringent. Everything else being equal, the optimal tax schedule can thus be more marginally progressive. Second, we see that (25) corresponds to a joint determination of the optimal marginal tax rates and the optimal gross income levels. They do not, in particular, incorporate any term related to consumption. This feature allows a sequential determination of the optimal allocation: in a first step, gross incomes and marginal tax rates are jointly obtained. Optimal consumption levels can then be characterized in a second step. This two-step procedure is a byproduct of quasi-linear preferences as well as weights which only depend on ranks. It was emphasized in a discrete population setting by [START_REF] Simula | Optimal nonlinear income tax and nonlinear pricing: Optimality conditions and comparative static properties[END_REF], who develops a sequential procedure to construct the optimal allocation. In other words, (25) captures an interaction between two endogenous variables: the marginal tax rate, on the one hand, and gross income, on the other hand. Applying the implicit function theorem to T * (z) + v θz (z * ; θ)G(p, π) = 0, the effect of marginal tax rates on gross income are given by:

∂z * ∂T * = - 1 v zzθ (z * ; θ) G(p, π) . ( 26
)
Therefore, the following Corollary is obtained:

COROLLARY 1. Everything else being equal, gross income z * decreases with the marginal tax rate if and only iff v zzθ > 0.

This result expresses a relationship between two endogenous variables, computed at the social optimum. This relationship depends on a third derivative of the work disutility v θz . As it turns out, such derivatives will play a key role in the study of bunching.

IV. BUNCHING

This Section explores the possibility of bunching in our framework. More precisely, we examine cases in which θ → z * (θ) is non-decreasing, so that the allocation solution to the Full Problem does not involve any bunching of types; it is then "fully separating". We also examine the opposite situation, in which some types are bunched together. In the following, we place ourselves in situations in which z * (θ) is differentiable.

IV.1. General Properties

We let θ p be the solution in θ to: p = F(θ). The latter increases with p because F is increasing. Given this definition, note that z * (θ p ) being strictly increasing with rank p is a sufficient condition for bunching not to arise, so that z(θ p ) = z * (θ p ) for any rank p. Moreover, a necessary and sufficient condition for bunching to arise is that z * (θ p ) be non-increasing. In that case, z(θ) = z * (θ) on at least a subset of θ, θ with positive measure.

The first strategy we follow is to look at conditions on the primitives of the model for which z * (θ) is increasing over θ, θ and therefore equal to z(θ). Its starting point is to consider the Relaxed Problem's first-order condition (23). We rewrite it as:

1 -v z (z) + v θz (z; θ)Γ(θ) = 0, ( 27
)
where Γ(θ) ≡ G(p(θ), π(θ)) only depends on how types are distributed within the population.

Applying the implicit function theorem, we obtain:

z * (θ) = - -v θz + v zθθ Γ(θ) + v θz Γ (θ) -v zz + v zzθ Γ(θ) = - v zθθ Γ(θ) + v θz (Γ (θ) -1) -v zz + v zzθ Γ(θ) (28)
where some arguments have been omitted. We know that Γ(θ) > 0 , v z (z; θ) > 0, v θz (z; θ) < 0. Consequently, gross income z * (θ) is strictly increasing if we add Γ (θ) ≤ 1 and the following condition:

ASSUMPTION 2. The marginal dis-utility of gross income is convex, but less convex the higher the productivity: v zzθ (z; θ) < 0 and v zθθ (z) > 0.

The assumptions on the third derivatives of v(z; θ) are additional requirements to the Spence-Mirrlees condition, v zθ (z; θ) < 0 (see Assumption 1 above). The first one, v zzθ (z; θ) ≤ 0, already appeared in Corollary 1. The marginal dis-utility of highly productive individuals is then less convex than that of less productive agents. The second one, v zθθ (z) > 0, requires the marginal dis-utility of gross income to decrease with productivity at a decreasing rate. Both conditions are verified when an iso-elastic specification is used for the dis-utility of gross income v(z; θ).

They are also satisfied when the dis-utility of gross income takes the often used multiplicative form, v(z; θ) = φ(z/θ), as soon as we add φ > 0 to the classical conditions φ > 0 and φ > 0. From now on, we assume that this Assumption is verified. The following Proposition summarizes the result. PROPOSITION 2. Let Assumption 2 be verified and Γ (θ) ≤ 1 for any θ. Then, the solution to the Relaxed Problem is the solution to the Full Problem. In particular:

(i) z(θ) = z * (θ) for any θ ∈ θ, θ ;

(ii) The solution is fully separating, i.e., does not involve any bunching.

On this basis, our strategy is to check whether Γ (θ) ≤ 1 for the different scenarios we consider. Starting from the definition of Γ(θ) and differentiating, we obtain:

Γ (θ) -1 = (λ(p(θ)) -2) - (Λ(p(θ)) -p(θ))π (θ) π 2 (θ) . ( 29
)
First, note a sufficient condition for the second term on the right-hand side (RHS) to be non-positive is π (θ) ≥ 0. Second, the first term on the RHS is always negative for the Aaberge Family including Gini because λ(0) = 0 < 2 and λ (p) < 0. For the two other families, S-Gini and Bonferroni respectively, the first term is negative providing λ(p(θ)) < 2. Because λ(•) is monotone decreasing while p is monotone increasing, then (λ • p) (.) is monotone decreasing.

In addition, λ(0) > 2 and λ(θ) = 0, there exists θ * in θ, θ such that (i) λ(p(θ * )) = 2 and (ii)

λ(p(θ)) < 2 if and only if θ ≥ θ * .
Consequently, the first term in ( 29) is negative for all θ > θ * .

The following Proposition follows from these observations. PROPOSITION 3. Let Assumption 2 and π (θ) ≥ 0 be verified. Then:

(i) For any Aaberge social welfare function (including Gini, i.e., δ = 2), the solution to the Relaxed Problem is the solution to the Full Problem, i.e., the optimal solution.

(ii) For the S-Gini and Bonferroni families, the unconstrained solution θ → z(θ) is increasing beyond θ * such that λ(p(θ * )) = 2.

If we instead assume that the pdf of productivity π(θ) is single-peaked, reaching its mode at θ M , we obtain: PROPOSITION 4. Let Assumption 2 hold and π be single-peaked, with mode θ M . Then: (i) For any Aaberge social welfare function (including Gini, i.e., δ = 2), the solution to the Relaxed Problem θ → z(θ) is increasing for all θ < θ M .

(ii) For the S-Gini and Bonferroni families, the unconstrained solution θ → z(θ) is increasing for all θ in θ * , θ M , providing θ * defined by λ(p(θ * )) = 2 is below θ M .

IV.2. Log-Concave and Log-Convex Distributions

In this Subsection, we maintain Assumption 2 and establish properties of the optimal solution using log-concavity or log-convexity. Any given function f is said to be log-concave on an interval if and only if log( f ) is concave on this interval. It turns out that if the probability density function of productivity h(θ) is log-concave on its support, then the corresponding survival function, 1 -H(θ), is also log-concave. To see that, let us consider π > 0. Then pπ <

π 2 ⇒ -π + pπ < π 2 ⇒ -(1 -p)π < π 2 , implying a log-concave survival function. If π < 0 instead, then -(1 -p)π < π 2 ⇒ -π + pπ < π 2 ⇒ pπ < π 2 ,
implying again a log-concave survival function. In addition, many commonly used distributions, such as the normal, exponential, Weibull (with scaling parameter above 1) or power distributions, generate a log-concave h(θ) and, thus, a log-concave 1 -H(θ). See Table 1 in [START_REF] Bagnoli | Log-concave probability and its applications[END_REF]. Because log-concavity of the survival function is less demanding than log-concavity of the pdf, we resort to the former rather than the latter whenever possible. PROPOSITION 5. Let Assumption 2 be verified and consider any Aaberge social welfare function (including Gini, i.e., δ = 2). If the survival function 1 -H(θ) is log-concave, then:

(i) Γ (θ) -1 < 0, so that z * (θ) = z(θ) for any θ in θ, θ ;

(ii) There is no bunching whatsoever.

Proof. For Aaberge social weights, note that:

Γ (θ) -1 = 2 -δ -δp δ-1 δ -1 + p δ -1 -(1 -p δ-1 )π π 2 (30)
where arguments on the RHS have been removed for notational convenience and δ ≥ 2. There are two cases to consider:

-For θ such that π ≥ 0: Γ (θ) ≤ 1 given that δ ≥ 2 and p belongs to [0, 1].

-For θ such that π < 0: Log-concavity of 1 -

H(θ) = 1 -p implies that -(1-p)π π 2 < 1. Therefore, Γ (θ) -1 < 2 -δ -δp δ-1 δ -1 + p δ -1 ≡ Υ(δ). (31)
It turns out that Υ (δ) = -1 -(p δ-1 + δ(δ -1)p δ-2 ) < 0 for δ ≥ 2. Because Υ(2) = -p, we conclude that, for δ ≥ 2, Γ (θ) -1 < 0.

We now turn to the S-Gini family of social welfare functions. We are not able to show that there is no bunching whatsoever for this whole family. Assuming a log-concave survival function 1 -H(θ), we are able to show that the solution θ → z * (θ) is increasing above some threshold denoted θ. PROPOSITION 6. Let Assumption 2 be verified and consider any S-Gini social welfare function (including Gini, i.e., δ = 2). If the survival function 1 -H(θ) is log-concave, then:

-There is a productivity threshold θ defined by p = p( θ) = 1 -[1/(δ -1)] 1/(δ-1) ; -For any θ > θ, z * (θ) is strictly increasing.

Proof. Given S-Gini weights, one obtains:

Γ (θ) -1 = δ(1 -p) δ-1 -2 - (1 -p) 1 -(1 -p) δ-1 ) π (θ) π 2 (θ) , ( 32
) = δ(1 -p) δ-1 -2 + 1 -(1 -p) δ-1 -(1 -p) π (θ) π 2 (θ) . ( 33
)
Because δ ≥ 2, the first square bracket is non-positive. There are two cases to consider:

-At any θ for which π (θ) ≥ 0: the product of the two square brackets on the RHS of ( 33) is negative except for p = 1 where it is zero. Consequently, Γ (θ) < 0 for any θ in (θ, θ). Then θ = θ and z * (θ) is increasing, implying z(θ) = z(θ) for all θ in θ, θ .

-At any θ for which π (θ) < 0: We can rely on log-concavity of the survival function, implying Γ (θ)

-1 < (δ(1 -p) δ-1 -2) + (1 -(1 -p) δ-1 ) = (δ -1)(1 -p) δ-1 -1 ≡ Υ(δ).
In addition:

Υ(δ) < 0 ⇔ (1 -p) δ-1 < 1 δ -1 ⇔ p = p( θ) ≡ 1 - 1 δ -1 1 δ-1 < p. ( 34
)
Consequently, Γ (θ) < 1 for any p ∈ ( p, 1).

For example, when δ = 3, we obtain p = 1 -√ 1/2 ≈ 0.29. When δ tends to infinity, so that the S-Gini social welfare function tends to the Rawlsian maximin, [1/(δ -1)] 1/(δ-1) tends to 1, so that p tends to 1 and θ to θ. Consequently, z * (θ) is increasing for all θ and z(θ) = z * (θ). The following Proposition summarizes this result. PROPOSITION 7. Let Assumption 2 be verified and consider the Rawlsian social welfare function (maximin). If the survival function 1 -H(θ) is log-concave, then the optimal allocation does not involve any bunching of types.

Four points are worth noting. First, it follows from Proposition 7 that, under Assumption 2, adopting the "first-order approach" consisting in solving the Relaxed Problem instead of the Full Problem implies no loss of generality whatsoever to solve for the maximin allocation as soon as the survival function of productivity 1 -H(θ) is log-concave. Second, the findings in [START_REF] Boadway | Optimal income taxation with quasi-linear preferences revisited[END_REF] regarding the absence of bunching for any exponential distribution of productivity and quasi-linear in leisure preferences are extended to quasi-linear in consumption preferences. Third, the result in [START_REF] Boadway | Optimal marginal and average income taxation under maximin[END_REF] regarding the absence of bunching under a Weibull probability density for types (with scaling parameter above 1) and quasi-linear in consumption preferences are a subcase of Proposition 7. Last, it is insightful to compare p to the value of p * introduced in Proposition 3 and for which λ(p * ) = 2. It turns out that: p > p * ⇔ (1/(δ -1)) 1/(δ-1) < (2/δ) 1/(δ-1) ⇔ 2 ≤ δ, which is always satisfied. This is natural because of the positivity of the second term of (29) when the conditions of Proposition 7 are verified.

We were able to provide results for the S-Gini and A families. The following remark examines what would happen for the Bonferroni social welfare function.

REMARK 1 (Bonferroni Social Welfare Function). With Bonferroni social weights, Λ(p)p = -p log p, λ(p) -1 = -1log p and thus:

Γ (θ) -1 = -2 + log p pπ (θ) π 2 (θ) -1 . ( 35
)
If log p is concave, then pπ (θ) π 2 (θ) < 1 and, thus, pπ (θ) π 2 (θ) -1 < 0. Because log p < 0, we then have: log p pπ (θ) π 2 (θ) -1 > 0. We therefore cannot say anything about the sign of Γ (θ) -1.

A second remark focuses on the variation of Γ (θ).

REMARK 2. Computing Γ (θ), we obtain:

Γ = λ π - (λ -1)π π -(Λ -p)( π π -2π 2 π 3
), (36) in which the (θ) arguments were removed for notational convenience. We see that the sign of Γ (θ) depends on the sign of π and π .

-Because λ (θ) < 0 in L, the first term on the RHS of ( 36) is strictly negative.

-There exists θ * * such that λ(p(θ * * )) = 1, λ(p(θ)) > 1 for θ < θ * * and λ(p(θ)) < 1 for θ > θ * * . Then, the second term on the RHS of ( 36) is always strictly negative when the sign of π switches from positive to negative exactly at θ * * , which is the case for any singlepeaked distribution the mode of which coincides with θ * .

-We always have Λ(p)p > 0 for the S-Gini, A-and Bonferroni families. Moreover, a logconcave probability density function satisfies π (θ)π(θ) < [π (θ)] 2 for any θ. This implies that the third term on the RHS of ( 36) is always positive for a log-concave probability density function, and therefore for any concave probability density function (concavity implying log-concavity).

-Therefore, we cannot conclude about the sign of Γ (θ) for log-concave probability density functions. There is indeed a conflict between the sign of the above terms. On the contrary, if the probability density function h(θ) is log-convex. Then, π (θ)π(θ) > [π (θ)] 2 , so that the third term on the RHS of ( 36) is negative, as the first two terms.

The above remark allows us to formulate the following Proposition.

PROPOSITION 8. Let Assumption 2 be satisfied and social weights belong to the A-family (including Gini). Let the probability density function be (i) single-peaked, (ii) log-convex and such that (iii) its mode θ M coincides with the value of productivity for which λ (θ) = 1, i.e., θ M = θ * * . Then, θ → z * (θ) is increasing on it support, so that z(θ) = z * (θ). The optimal allocation is therefore full separating.

Proof. Given the above conditions, the function Γ (θ) is always decreasing. In addition, Γ (0) < 1. Consequently, Γ (θ) -1 < 0 for all θ in θ, θ .

Proposition 8 applies to log-convex probability density functions. According to Table 3 in [START_REF] Bagnoli | Log-concave probability and its applications[END_REF], the Weibull (for scaling parameter below 1), the Gamma (for scaling parameter between 0 and 1) and the Pareto distributions have this property. However, Proposition 8 also requires the mode to be in the interior of the support, a condition that is not satisfied for a Pareto distribution (often used to describe the distribution of productivities at the top). For the log-normal distribution, often used to describe the distribution of productivities (except at the top), we know that the cumulative probability function is log-concave, but the logconcavity/log-convexity of the probability density function is undetermined in general. In that case, we cannot sign Γ (0) -1 and can only rely on Proposition 4 making use of unimodality.

IV.3. Local Properties

In this Subsection, we establish a few local conditions regarding the solution in gross income to the Relaxed Problem. First, note that Γ (0) = λ (0)π(0) -(λ(0)-1)π ( 0)

π(0)
. For all parametric families introduced above, we have λ(0) -1 > 0. Therefore, if we add the following Assumption, we are able to sign z (θ) in the lower part of the distribution. ASSUMPTION 3. Γ (0) < 0 when π (0) > 0.

We already know that Γ (0) -1 < 0. Therefore, if we make Assumption 3 and let π (0) > 0, the following Lemma is obtained: LEMMA 1. Let Assumptions 2 and 3 be satisfied. If π (0) > 0, then there is a θ > θ (with θ possibly arbitrarily large) such that the solution in gross income to the Relaxed Problem is strictly increasing on θ, θ .

For the upper bound of the support, θ, we have:

Γ (θ) = λ (θ)π(θ) + π (θ) π(θ)
. We know that λ (θ -) < 0 for any weights in L (where the notation θ -stands for the limit to the left of θ). In addition, it is likely that π (θ -) ≤ 0. We can therefore formulate the following Lemma.

LEMMA 2. Let Assumption 2 be verified. If π (θ -) ≤ 0, the solution to the Relaxed Problem θ → z(θ) is increasing on an interval with upper bound θ.

IV.4. Results for Specific Distributions

We now consider specific distributions of productivity to see whether we may say more regarding the bunching or separating pattern of the optimal allocation. PROPOSITION 9. Let Assumption 2 be verified, and productivity be described by a Pareto distribution with parameters A > 0 and k > 1. For the Gini welfare function, the optimal allocation is fully separating.

Proof. We have:

H(θ) = (A/θ) k = E > 0, implying p = 1 -(A/θ) k = 1 -E, 1 -p = E, π(θ) = k (A) k /θ k+1 = kE/θ, π 2 (θ) = k 2 E 2 /θ 2 , π (θ) = -k(k + 1)A k /θ k+2 = -k(k + 1)E/θ 2 . Consequently, Γ (θ) = λ(p(θ)) -1 + (Λ(p(θ)) -p(θ))k(k + 1) E θ 2 π 2 (θ) . ( 37
)
Replacing with Gini weights, and using 1p = E,

Γ (θ) = 1 k + E(1 - 1 k ) = 1 k + (1 -p)( k -1 k ) < 1 (38) for k > 1 and p ∈ [0, 1].
PROPOSITION 10. Let Assumption 2 be verified, and productivity be described by a Pareto distribution with parameters A > 0 and k > 1. For social weights belonging to the A-family, the optimal allocation is fully separating.

Proof. Starting from (37), and plugging weights for the A-family,

Γ (θ) = 1 (δ -1)
(1δp δ-1 ) +

Using a majoration argument, we establish that:

Γ (θ) < 1 δ -1 (1 -δp(k) δ-1 ) + 1 δ-1 (p(k)(1 -p(k) δ-1 ) 2 1 -pk (40)
for any k > 1. Moreover, we check that: Γ (θ, k = 1) < 1 for any p and deduce that:

1 δ -1 (1 -δp(k) δ-1 ) + 1 δ-1 (p(k)(1 -p(k) δ-1 ) k+1 k 1 -p(k) < 1, (41)
implying Γ (θ) < 1.

PROPOSITION 11. Let Assumption 2 be verified, and productivity be described by a Pareto distribution with parameters A > 0 and k > 1. For S-Gini (including Gini) welfare weights, then the sign of Γ (θ) -1 follows that of pp(θ)

with p = 1 -( k-1 1+k+kδ ) 1 δ-1 .
Proof. With S-Gini weights:

Λ(p) -p = 1 -(1 -p) δ -p = (1 -p)(1 -(1 -p) δ-1 ) (42)
and

λ(p) -1 = δ(1 -p) δ-1 -1. Using 1 -p = E, Γ (θ) = 1 k + E δ-1 (δ + (k + 1) k ) = 1 k + E δ-1 ( 1 + k + kδ k ). (43)
Consequently,

Γ (θ) ≶ 1 ⇔ E δ-1 ( 1 + k + kδ k ) ≶ k -1 k . ( 44
)
Rearranging, the latter is equivalent t o: p ≶ 1 -( k-1 1+k+kδ )

1 δ-1 .

V. THE DISCRETE POPULATION CASE

The objective of this Section is to highlight the similarities and differences between the solutions to the rank-dependent optimal income tax problem for a continuous population and for a discrete population. The discrete population framework can be seen as an extension to the setting developped in Simula and Trannoy (forthcoming). Optimal income taxation with a finite number of types has been examined by [START_REF] Stiglitz | Self-selection and pareto efficient taxation[END_REF][START_REF] Stiglitz | Pareto efficient and optimal taxation and the new welfare economics[END_REF], providing key insights in a two-type model. This setting was extended to any given number of agents by R öell (1985) under a very weak redistributive assumption, by [START_REF] Weymark | Comparative static properties of optimal nonlinear income taxes[END_REF]Weymark ( , 1986a,b) ,b) when agents have quasilinear-in-consumption preferences, and then by [START_REF] Simula | Optimal nonlinear income tax and nonlinear pricing: Optimality conditions and comparative static properties[END_REF] when preferences are quasilinear, linear in gross income, as in Equation (4). See also [START_REF] Simula | Incentive-Compatible ELIE Transfers[END_REF], in which a discrete-type optimal income taxation model is used to decentralize the so-called "ELIE" transfers proposed by [START_REF] Kolm | Macrojustice: The Political Economy of Fairness[END_REF]. To a large extent, these studies pay attention to the characterization of the socially optimal gross-income/net-income allocation, which contrasts with the literature considering a continuum of types, the main focus of which is the derivation of optimal marginal tax rates (for a survey see, e.g., [START_REF] Piketty | Optimal labor income taxation[END_REF]).

V.1. Continuous Benchmark

As a benchmark, we recall elements obtained in Simula and Trannoy (forthcoming) for a continuous type space. We first note that the elasticity of taxable income with respect to the marginal retention rate is equal to:

1-T (z, θ) = ∂z ∂(1 -T ) 1 -T z = v z (z; θ) zv zz (z; θ) , . (45)
while the elasticity of taxable income with respect to productivity is given by:

θ (z, θ) = ∂z ∂θ θ z = - θv zθ (z; θ) zv zz (z; θ) . ( 46
)
The ratio of both elasticities, denoted e(z, θ), is equal to:

e(z, θ) ≡ 1-T (z, θ) θ (z, θ) = - v z (z; θ) θv zθ (z; θ) . ( 47
)
Given these definitions, Proposition 2 in Simula and Trannoy (forthcoming) establishes that optimal marginal tax rates satisfy:

T * (z * ) 1 -T * (z * ) = 1 e(z * (θ); θ) A 1 -p(θ) θπ(θ) B Λ(p(θ)) -p(θ) 1 -p(θ) C . (48)
In this ABC formula, marginal tax rates are the product of three factors respectively: A captures efficiency, B depends on the type and C summarizes ethical views.

V.2. Discrete Setting

We now turn to the discrete setting. The size of the population is normalized to 1. We define the type distribution by a discrete probability measure {θ k , π k ; k = 1, ..., K}, where the support is a finite set of K points in a compact subset A of the positive real line. The probability of any subset S of A is given by

p(S ) = ∑ θ k ∈S π k δ k (49)
with δ k the Dirac measure at θ k . We let F(x) be the CDF, i.e., the step function corresponding to this discrete probability measure. Namely, F : A → [0, 1] with p = F(θ) gives the cumulative proportion of people for whom θ k ≤ θ for any θ in S. By construction, we have:

p k = k ∑ j=1 π j , (50)
which corresponds to the rank of agents with productivity θ k . For later use, note that p kp k-1 = π k . In addition, it should be highlighted that p = F(θ) is constant between two consecutive steps (θ k , θ k+1 ) of the CDF. We define the quantile function as the right inverse of F, i.e., F -1 (p) = sup F(θ)≤p θ k . The marginal weights λ(p) and cumulative weights Λ(p) are step functions, with an upward jump at each θ k , and a plateau for any θ in [θ k , θ k+1 [. The rank dependent social welfare function is therefore given by:

W = 1 0 λ(p)F -1 (p)dp. (51)
Evaluating the integral for the discrete distribution, we obtain:

W = θ 1 p 1 0 λ(p)dp + θ 2 p 2 p 1 λ(p)dp + ...θ K p K p K-1 λ(p)dp = ∑ k θ k [Λ(p k ) -Λ(p k-1 )] . ( 52
)
If we denote by g(p k ) the per-capita social weight of group k, i.e.,

g(p k ) = Λ(p k ) -Λ(p k-1 ) p k -p k-1 (53)
the social objective may be rewritten as:

W = K ∑ k=1 θ k • g(p k ) • π k . ( 54
)
Applying the mean value theorem, there exists for every k, a cumulative probability p in [0, 1], with p k-1 < p < p k , such that g(p k ) = λ(p). Given that λ(p) is strictly decreasing in L, the per-capita social weight g(p k ) is also strictly decreasing with k.

V.3. Marginal Tax Rate Formula

The way in which the optimal marginal tax rates are derived is formally similar to Simula (2010). The novelty is to show the similarities with the derivation of formulas for optimal marginal tax rates in the continuous case. In this way, we complement the important article by [START_REF] Hellwig | A contribution to the theory of optimal utilitarian income taxation[END_REF], which shows that optimal income taxation in the tradition of [START_REF] Mirrlees | An exploration in the theory of optimum income taxation[END_REF] is built on a single rock, irrespective of whether the population is continuous or discrete. However, [START_REF] Hellwig | A contribution to the theory of optimal utilitarian income taxation[END_REF] does not provide a comparison of optimal income tax formulas for either a continuous or a discrete population.

The proof exploits the fact that, when Assumption 1 is satisfied, only the local and downward incentive-compatibility conditions are binding. This implies that an incentive-compatible allocation must be such that v(z k+1 ; θ k+1 ) = v(z k ; θ k+1 ) for any k = 1, ..., K -1. This corresponds to the first-order condition for incentive compatibility (7) highlighted above. The equivalent to the second-order condition for incentive compatibility ( 8) is now: z = (z 1 , ..., z K ) nondecreasing in the sense that z k+1 ≥ z k for any k = 1, ..., K -1. See [START_REF] Guesnerie | Nonlinear pricing in a finite economy[END_REF] for a detailed presentation. On this basis, and following the same steps as in [START_REF] Simula | Optimal nonlinear income tax and nonlinear pricing: Optimality conditions and comparative static properties[END_REF], it is straightforward to establish that the Relaxed Problem formulated above is now equivalent to: PROBLEM 3 (Relaxed Discrete-Population Problem). Find gross incomes z 1 , ..., z K maximizing:

W = K ∑ k=1 θ k • g(p k ) • π k - K ∑ k=1 (Λ k -p k ) [v(z k ; θ k ) -v(z k ; θ k+1 )] . ( 55
)
It is worth noting that ( 55) is analogue, in the discrete setting, to (20) in the continuous case. The Full Discrete-Population Problem is similar, with the extra requirement that the solution vector z is non-decreasing. From now on, we focus on the "first-order approach" and assume that the monotonicity condition on z is verified. When W as defined in ( 55) is maximized:

1 -v z (z k ; θ k ) π k -(Λ(p k ) -p k ) v z (z k ; θ k ) -v z (z k ; θ k+1 ) = 0, ∀k = 1, ..., K -1. ( 56
)
In the continuous-population case, the above condition corresponds to (21). We let:

T (z j ; θ k ) = 1 -v z (z j ; θ k ). ( 57
)
This corresponds to the "implicit marginal tax rate" faced by the θ k -individuals at gross income z j . In particular, T (z k ; θ k+1 ) corresponds to the marginal tax rate an agent with productivity θ k+1 would face when choosing the gross-income level z k designed for the θ k -individual. Note that T (z j ; θ k ) only depends on gross income because, in the gross-income/net-income space, any given θ k -individual's indifference curves are obtained from one another through vertical displacements, due to quasilinear-in-consumption preferences. Using the above definition, Equation ( 56) may be rewritten as:

T (z k ; θ k ) = Λ(p k ) -p k π k v (z k ; θ k ) -v (z k ; θ k+1 ) , = 1 -p k π k Λ(p k ) -p k 1 -p k v (z k ; θ k ) -v (z k ; θ k+1 ) . (58)
Following [START_REF] Simula | Optimal nonlinear income tax and nonlinear pricing: Optimality conditions and comparative static properties[END_REF], we define the Spence-Mirrlees wedge as:

SM(z k ; θ k ; θ k+1 ) = T (z k ; θ k+1 ) -T (z k ; θ k ) = MRS(z k ; θ k ) -MRS(z k ; θ k+1 ), = v z (z k ; θ k ) -v z (z k ; θ k+1 ), (59)
where MRS(z j ; θ k ) stands for the marginal rate of substitution of a θ k -agent at gross income z j . Under Assumption 1, the Spence-Mirrlees wage SM(z k ; θ k ; θ k+1 ) is positive. Using this definition, (59) becomes:

T (z k ; θ k ) = 1 -p k π k Λ(p k ) -p k 1 -p k SM(z k ; θ k ; θ k+1 ). (60)
In this form, the optimal marginal tax rate appears as the product of three factors: the "demographic" factor (1p k )/π k depends on the distribution of the population across ranks; the "ethical" factor (Λ(p k )p k )/(1p k ) depends on the society's preference for redistributing incomes across ranks, as captured by p → Λ(p); and the last factor SM(z k ; θ k ; θ k+1 ) ensures that marginal tax rates are adjusted in such a way that indirect utility increases at a sufficient rate to prevent downward mimicking of θ k -indivuals by θ k+1 -individuals. Formula (60) shows in a transparent way the role played by the Spence-Mirrlees condition (here Assumption 1), which was key for the solution to the optimal income tax problem as emphasized in Mirrlees's Nobel 20

Lecture [START_REF] Mirrlees | Information and incentives: The economics of carrots and sticks[END_REF]. Dividing (60) by 1 -T (z k ; θ k ) = v (z k ; θ k ), we obtain:

T (z k ; θ k ) 1 -T (z k ; θ k ) = 1 -p k π k Λ(p k ) -p k 1 -p k SM(z k ; θ k ; θ k+1 ) v z (z k ; θ k ) . ( 61
)
For a discrete population, gross income is not a function, but a vector taking K values, one for each productivity level. Each of these k values implicitly depends on the marginal tax rate faced by the θ k -individuals, and thus on the retention tax rate, as well as on productivity θ k . Using the differentiability of indifference curves, the elasticities of gross income with respect to the retention rate (assuming a constant marginal tax rate) and productivity, defined in ( 45) and ( 46) respectively, directly translate to the discrete population setting. To make use of these definitions, we may rearrange the last factor in the right-hand side of Formula (61) as follows:

SM(z k ; θ k ; θ k+1 ) v z (z k ; θ k ) = 1 z k v zz (z k ; θ k ) MRS(z k ; θ k ) -MRS(z k ; θ k+1 ) v z (z k ; θ k )/(z k v zz (z k ; θ k )) = θ k -θ k+1 z k v zz (z k ; θ k ) 1 1-T (z k ; θ k ) MRS(z k ; θ k ) -MRS(z k ; θ k+1 ) θ k -θ k+1 ≈ - θ k+1 -θ k z k v zz (z k ; θ k ) 1 1-T (z k ; θ k ) MRS θ (z k ; θ k ), (62)
where the approximation is valid for θ k+1 sufficiently close to θ k . Because MRS θ (z k ; θ k ) = v zθ (z k ; θ k ), we obtain:

T (z k ; θ k ) 1 -T (z k ; θ k ) ≈ 1 -p k π k Λ(p k ) -p k 1 -p k 1 1-T (z k ; θ k ) -θ k v zθ (z k ; θ k ) z k v zz (z k ; θ k ) θ k+1 -θ k θ k , ≈ 1 -p k π k Λ(p k ) -p k 1 -p k θ (z k ; θ k ) 1-T (z k ; θ k ) θ k+1 -θ k θ k , ≈ 1 e(z k ; θ k ) 1 -p k θ k π k Λ(p k ) -p k 1 -p k (θ k+1 -θ k ) , (63)
where the behavioral term e(z k ; θ k ) is equal to the ratio 1-T (z k ; θ k )/ θ (z k ; θ k ). In this form, the first-order approximation of the optimal marginal tax rates obtained for a discrete population is given by the same ABC terms as in the continuous-population case (see Equation ( 48)), multiplied by θ k+1θ k . When θ k+1 tends to θ k , the RHS of (63) tends to zero. This implies that this approximation does not converge to the continuous-population formula when θ k+1 and θ k become arbitrarily close. Indeed, by (60), two consecutive marginal tax rates do not correspond to the same discrete change, with:

T (z k ; θ k ) = 1 -p k π k Λ(p k ) -p k 1 -p k SM(z k ; θ k ; θ k+1 ), (64)
T (z k+1 ; θ k+1 ) = 1p k+1 π k+1

Λ(p k+1 )p k+1 1p k+1 SM(z k+1 ; θ k+1 ; θ k+2 ). (65)

Taking the limit of (63) does not provide the adequate limit. [START_REF] Bastani | Using the discrete model to derive optimal income tax rates[END_REF] compares graphically the solutions of the optimal income tax model for a continuous population, approximated by a discrete distribution, and for a discrete population. The considered social objective is weighted utilitarianism. He concludes that "the discrete model should be considered a viable alternative to the continuous model when using numerical sim-ulations to derive optimal income tax" based on a comparison between curves obtained for a 2000 point approximation. In Simula and Trannoy (forthcoming), we provide simulations of the continuous rank-dependent model, using a discrete approximation of the type distribution. The question of the numerical convergence remains open in our setting.

VI. CONCLUSION

This article contributes to the exploration of bunching by considering the rank-dependent optimal income tax setting (Simula and Trannoy, forthcoming). Paying special attention to two families, generating welfare weights depending on a single-dimension parameter, we derive conditions on the primitives of the model precluding or implying bunching in the social optimum. In particular, we obtain general results for bunching not to occur for either log-concave type's survival functions or log-convex type's pdf, providing the dis-utility of gross income verifies an assumption on its third cross derivatives. This assumption can be seen as an additional sorting condition, linking types and preferences, which complements Spence-Mirrlees condition. An avenue for further research would be to explore whether this new condition could help find conditions for a fully separating optimum allocation in the usual welfarist setting.

This article also exhibits "ABC"-like formulas for optimal marginal tax rates in a discrete population setting. If the formulas look like the ones obtained for a continuous population in Simula and Trannoy (forthcoming), we show that the former do not converge to the latter when the distance between two consecutive productivity levels tend to zero.

Because each population setting has some irreductible specificities, it would be of interest to explore how comparative statics techniques and properties for discrete-population optimal income-tax models introduced in [START_REF] Weymark | Comparative static properties of optimal nonlinear income taxes[END_REF] seminal article should be adjusted when a continuum of agents is considered. This would necessitate a conceptual adjustment of what is meant by a change in productivity such as those considered for discrete populations by [START_REF] Brett | The impact of changing skill levels on optimal nonlinear income taxes[END_REF] and [START_REF] Simula | Optimal nonlinear income tax and nonlinear pricing: Optimality conditions and comparative static properties[END_REF].

In the above-mentioned companion paper(Simula and Trannoy, forthcoming), we relax this assumption but focusing on cases in which bunching is assumed away.

For every interior p, the cumulated weights are larger in the Bonferroni case than in the Gini one, implying that the former stresses poverty more than the latter.

Note that z is not bounded from above. This implies that agents always have a possibility to earn sufficiently to pay tax liabilities required from them, and that we do not have to account for a nonnegativity on z when solving the optimal income tax problem. This precludes bunching due to the violation of this nonnegativity constraint, a phenomenon highlighted in[START_REF] Boone | The simple economics of bunching: Optimal taxation with quasilinear preferences[END_REF].

[START_REF] Brett | Voting over selfishly optimal nonlinear income tax schedules[END_REF] characterize the income tax schedule maximizing the utility of a given agent subject to incentive-compatibility constraints and budget balancedness.[START_REF] Brett | Voting over selfishly optimal nonlinear income tax schedules[END_REF] themselves rely on[START_REF] Lollivier | Bunching and second-order conditions: A note on optimal tax theory[END_REF] who present a simple non-linear taxation model in which the agents, with quasi-linear in labor preferences, are indexed by a one-dimensional parameter.

v z (z(θ); θ))h(θ) = -v θz (z(θ); θ) [Λ(H(θ)) -H(θ)] for any θ ∈ θ, θ ,(21)

(δ-1) (p(1p δ-1 ) (k+1) k E . (39)


Audrey De Jong

Ashish Khanna

Airway management in the critically ill patient with COVID-19