BUSINESS( NO PLAGARISM A+ WORK, ON TIME)
Soc Choice Welf (2010) 35:199–220 DOI 10.1007/s00355-009-0437-x
Optimal nonlinear income tax and nonlinear pricing: optimality conditions and comparative static properties
Laurent Simula
Received: 15 August 2008 / Accepted: 3 December 2009 / Published online: 19 December 2009 © Springer-Verlag 2009
Abstract Using the Mirrlees optimal income tax model, with no income effects on labour supply, this article shows that the discrete population approach provides new insights into the characterization of the optimal tax system, which complements the previous findings. The analysis is based on the “Spence-Mirrlees wedge” which corre- sponds, at each observed gross income level, to a ratio between the marginal tax rate of the individual for whom the bundle is designed and that of his nearest more productive neighbour if he chooses to mimic. Using this wedge, a necessary and sufficient con- dition for bunching to be optimal is obtained in terms of the primitives of the model, separating optima are characterized geometrically, and comparative statics properties derived, notably with respect to skill levels and individual social weights. It is then shown that the analysis extends to adverse-selection problems where participation constraints replace the tax revenue constraint.
Keywords Optimal tax · Income tax · Nonlinear pricing · Adverse-selection · Comparative statics
JEL Classification D82 · D86 · H21 · H31
1. Introduction
This article analyses the nonlinear income tax problem à la Mirrlees (1971) with a finite distribution of workers as in Guesnerie and Seade (1982). In this setting, the optimal tax structure is the product of different sorts of interacting influences:
L. Simula (B) Uppsala Center for Fiscal Studies, Department of Economics, Uppsala University, P.O. Box 513, SE-75120 Uppsala, Sweden e-mail: laurent.simula@nek.uu.se
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individual skill levels, government’s aversion to income inequality and responsive- ness of labour supply. Moreover, the way in which these influences interact is affected both by the incentive-compatibility constraints and the tax revenue constraint. Merely assuming that individual preferences are concave and increasing both in consumption and leisure does not yield many general results. Consequently, investigations often resort to numerical simulations (Tuomala 1990). However, if simulations are very useful to quantify the optimal tax rates, they are not ideally suited for shedding light on the economic intuitions behind the results. The present article follows an alterna- tive path: it concentrates on the benchmark case where there are no income effects on labour supply to derive new qualitative features of the solution to the optimal income tax schedule.
Following Lollivier and Rochet (1983), many articles have focused on the polar case in which all income effects are absorbed by labour supply, i.e. on preferences over consumption and leisure which are separable and linear in leisure. In the finite population framework, it was used to characterize the optimal allocation (Weymark 1986a,b), provide comparative static results with respect to a preference parame- ter, individual productivities and social weights (Brett and Weymark 2008; Weymark 1987), and examine the impact of changing individual productivity in an economy with two classes of agents where the government adopts a maximin or maximax objective function (Hamilton and Pestieau 2005). It was also used in the continuous population framework to provide an example in which different types of individuals are bunched together when the tax schedule is set optimally (Ebert 1992) and to characterize the optimum allocation with a particular focus on bunching (Boadway et al. 2000; Boone and Bovenberg 2007). There are however limitations in the linear-in-leisure model. In particular, the tax schedule is then independent of the labour response because the dis- utility of labour is constant, a fact which seems to be incompatible with the empirical findings.
Because most of the empirical studies give credence to small income effects rel- ative to substitution effects as regards labour supply (Blundell 1992; Blundell and MaCurdy 1999), assuming no income effects on labour supply, i.e. focusing on sep- arable preferences in consumption and leisure, linear in consumption (referred to as quasilinear-in-consumption), provides a relevant benchmark for the policymaker. This has been acknowledged since the work by Diamond (1998) who examines con- ditions for marginal tax rates to be U-shapped when the population is continuous. The present article shows that the discrete population approach provides new insights into the characterization of the optimal tax system, which complements the previous findings. Most of the analysis is based on a wedge corresponding, at each observed gross income level, to a ratio between the marginal tax rate of the individual for whom the bundle is designed and that of his nearest more productive neighbour if he chooses to mimic. This wedge is referred to as the “Spence-Mirrlees” wedge and cannot be defined when there is a continuum of agents because the space between two productivity levels has then measure zero. Thanks to the Spence-Mirrlees wedge, a necessary and sufficient condition under which bunching is not optimal is obtained in terms of the primitives of the model. When the optimal allocation is fully sepa- rating, the optimum is characterized by the equality, at each observed income level, between the Spence-Mirrlees wedge and the cumulative social weight of the individ-
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Optimal nonlinear income tax and nonlinear pricing 201
uals with earnings up to this income level. This feature is first used to construct the optimal allocation geometrically in a series of very simple steps. It is then employed to examine the impact of changing the marginal utility of money, skill levels, and social weights. Skill levels are the fundamental source of the self-selection problem in Mirrlees model since they basically condition the effort a given individual has to undertake when he chooses to mimic everyone else. It is shown that a slight alter- ation in the skill level of any individual only affects the implicit marginal tax rates and pre-tax incomes of himself and his nearest less productive neighbour, which is in sharp contrast with Brett and Weymark (2008): for quasilinear-in-leisure prefer- ences, all marginal tax rates and incomes are usually affected, except for the top individual. Changes in skill levels are then combined with adjustments in social weights since a policymaker might be less concerned by the well-being of some- one who becomes more productive. The analysis is then extended to adverse-selec- tion problems where participation constraints replace the tax revenue constraint. The Spence-Mirrlees wedge appears as a key determinant of the optimum menu of con- tracts and its bunching properties, while hazard rates play the part of cumulative social weights.
The article is organized as follows. Section 2 sets up the model. Section 3 derives the reduced form of the optimal nonlinear income tax problem and characterizes the optimal allocation. Section 4 is devoted to the comparative static analysis. Section 5 extends the analysis to nonlinear pricing. Section 6 concludes. Most proofs are rele- gated to the Appendix.
2. The model
The population consists of I ≥ 2 individuals, indexed by i ∈ I := {1, . . . , I }. There are two goods, consumption and leisure. Units of the consumption goods are cho- sen so that one unit costs one euro. Person i’s consumption and labour supply are denoted xi and �i , respectively. The economy is competitive, with constant-returns- to-scale technology; so person i’s wage rate is fixed and equal to his productivity θi . For convenience, only one person has a given productivity level. Individuals are thus indexed in terms of productivity. This simplification is not particularly restric- tive as the distance between two productivity levels is free to vary. Without loss of generality, the vector of productivities θ := (θ1, . . . , θI ) is taken to be monotonically increasing,
0 < θ1 < · · · < θI . (1)
An individual with productivity θi working �i units of time has gross income zi := θi�i .
All individuals have the same preferences over consumption and leisure, represented by the utility function U : R × R+ → R,
U (xi , �i ) := γ xi − v (�i ), i ∈ I, (2)
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where γ ∈ R++ is the marginal utility of money.1 It is assumed that the disutility of labour v : R+ → R+ is a C3-function which satisfies v′ > 0, v′′ > 0, v′′′ > 0,
v (0) = 0 and lim�i →∞ v′ (�i ) → ∞. The assumption v′′′ > 0 means that the increase by which additional labour supply becomes unattractive becomes larger with �i .2 Because U (xi , �i ) = U (xi , zi/θi ), individuals have person- alized utility functions u : R × R+ → R in the gross-income/consumption space,
u (xi , zi ; θi ) := γ xi − v (zi/θi ), i ∈ I. (3)
The marginal rate of substitution s (zi ; θi ) of person i at the (xi , zi )-bundle is inde- pendent of his consumption level xi , with
s (zi ; θi ) := −u′ zi
(xi , zi ; θi )
u′ xi
(xi , zi ; θi ) = v′ (zi/θi )
γ θi , i ∈ I. (4)
So, in the gross-income/consumption space, i’s indifference curves are parallel ver- tical displacements of each other. Moreover, the Spence-Mirrlees condition is met: for a given gross income level, more productive individuals have flatter indifference curves.
A social allocation specifies a consumption and gross income level for each indi- vidual. It is represented by a vector a = (x, z) ∈ R
I × R I+, with x = (x1, . . . , xI )
and z = (z1, . . . , zI ). The tax policymaker knows the functional form of the utility function and the distribution of wages in the population. He is however unable to observe each individual’s productivity. As a result, he is restricted to setting taxes as a function of gross income zi . By the taxation principle, a nonlinear income tax schedule is equivalent to a mapping θ −→ a which satisfies the incentive compatibility constraints
I Ci j : u (xi , zi ; θi ) ≥ u ( x j , z j ; θi
) , ∀ (i, j) ∈ I2, (5)
and the tax revenue constraint
I∑
i=1
zi ≥ I∑
i=1
xi . (6)
An allocation a is budget-balanced if the tax revenue constraint (6) is binding. It is a simple monotonic chain to the left (“SMCL” hereafter) if and only if all
1 The assumption xi ∈ R is introduced for convenience; it can be checked ex post that xi is nonnegative. Similarly, it is typically assumed that �i ∈ R in the quasilinear-in-leisure version of Mirrlees’s model (Lollivier and Rochet 1983; Weymark 1986b, 1987). 2 This assumption plays an important role in the comparative statics. The role of third derivatives in such exercises has been emphasized by the literature devoted to risk and uncertainty. For instance, see Kimball (1990), Laffont (1989), Menegatti (2001). It also yields uniqueness of the solution to the optimal income tax model, as established below. In this respect, it could be relaxed to “weakly decreasing consumption-specific risk aversion” introduced by Hellwig (2007), which is sufficient for uniqueness.
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Optimal nonlinear income tax and nonlinear pricing 203
adjacent downward incentive-compatibility constraints in (5) are binding, i.e. iff u (xi+1, zi+1; θi+1) = u (xi , zi ; θi+1) for i = 1, . . . , I − 1. Because of (3), these equalities are equivalent to
u (xi+1, zi+1; θi+1) − u (xi , zi ; θi ) = R (zi ; θi , θi+1), i = 1, . . . , I − 1 (7)
where R (zi ; θi , θi+1) is defined, for later reference, as
R (zi ; θi , θi+1) := v (zi/θi ) − v (zi/θi+1), i = 1, . . . , I − 1. (8)
The social welfare function W : R I × R
I+ → R is a weighted sum of individual utilities,
W (a) := I∑
i=1
λi u (xi , zi ; θi ), (9)
in which λ := (λ1, . . . , λI ) are individual social weights. The policymaker’s taste for redistribution from the high to the low productive individuals is captured through the requirement that the higher the individual productivity the less the weight in the social objective, i.e.
0 < λI < · · · < λ1. (10)
The social objective function (9) can thus be regarded as a linear approximation of∑I i=1 φ (u (xi , zi ; θi )) , where the function φ would be strictly concave.3 If I = 2,
assumption (10) amounts to considering the “normal” case studied by Stiglitz (1982) in which only the incentive compatibility constraint of the high type is binding.
As W (a) is homogeneous of degree one in λ, the sum of the social weights can be normalized without loss of generality. It is convenient to define �i as the cumulative social weight of the i less productive individuals, i.e. �i = ∑ j=i
j=1 λi for i ∈ I, and to set
�I = I. (11)
Consequently, admissible parameters (θ, γ, λ) belong to the set
P := {θ |(1) is satisfied} × R++ × {λ|(10) and (11) are satisfied}. (12)
The optimal nonlinear income tax problem can thus be formulated as follows.
Problem 1 (Optimal Nonlinear Income Tax Problem) For (θ, γ, λ) ∈ P, choose an allocation a ∈ R
I × R I+ to maximize W (a) under the incentive-compatibility con-
straints (5) and the tax revenue constraint (6).
3 So, pure utilitarianism in which all individual social weights λi are equal is not considered. This is jus- tified since, in this case, the solution to the second-best optimal income tax problem is the laissez-faire, whose properties are already well-known.
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3. The optimal allocation
The optimal nonlinear income tax problem involves two sets of control variables, gross income z and net income x . It can however be transformed into a reduced-form problem in which the policymaker chooses only one of these sets. The reduced-form problem makes it easier to interpret the social value function as well as the optimality conditions and to derive comparative static results. For this purpose, Problem 1 is separated into two subproblems. In the first one, gross income is arbitrarily chosen within the set of incentive-feasible gross income levels,
Z := {
z ∈ R I |0 ≤ z1 ≤ · · · ≤ zI
} . (13)
Subproblem 1 Given a gross income vector z ∈ Z and the parameters (θ, γ, λ) ∈ P,
choose the consumption vector x ∈ R I to maximize the social welfare function
W (a) subject to the incentive-compatibility constraints (5) and the tax revenue constraint (6).
Let X ∗ (z; θ, γ, λ) be the set of maximizers. Then, if X ∗ (z; θ, γ, λ) contains only one consumption vector x∗ (z; θ, γ, λ), the reduced-form problem can be stated as follows.
Subproblem 2 Given the parameters (θ, γ, λ) ∈ P, choose z ∈ Z to maximize the social welfare function W (x∗ (z; θ, γ, λ) , z).
The incentive-compatibility constraints (5) place structure on the solutions to theses problems.
Proposition 1 Given z ∈ Z and (θ, γ, λ) ∈ P,
(i) any allocation a = (x∗, z) where x∗ ∈ X ∗ (z; θ, γ, λ), is a SMCL which is budget-balanced;
(ii) there is a unique solution to Subproblem 1, which is independent of λ, twice continuously differentiable and defined by x∗ : Z × R
I++ × R++ → R I+ with
x∗ 1 (z; θ, γ ) = 1
I
⎧ ⎨
⎩
I∑
j=1
z j − 1
γ
I∑
j=2
(I + 1 − j)
[ v
( z j
θ j
) − v
( z j−1
θ j
)] ⎫ ⎬
⎭ ,
(14)
x∗ i (z; θ, γ ) = x∗
1 (z; θ, γ ) + 1
γ
i∑
j=2
[ v
( z j
θ j
) − v
( z j−1
θ j
)] , i = 2, . . . , I.
(15)
An implication of (i) is that (7) holds at any optimum allocation. Hence, R (zi ; θi , θi+1) indicates at which rate utility must be increased for the tax sched- ule to induce individual truthtelling. So, it may be regarded as the marginal rent the policymaker has to leave to the more productive i + 1-individual to prevent him from mimicking the i-individual.
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Optimal nonlinear income tax and nonlinear pricing 205
3.1. The reduced form
We can now take stock of the previous results to give a more compact formulation of Subproblem 2. For this purpose, it is convenient to introduce the new vector of social parameters β = (β1, . . . , βI ), with
βi := �i − i, i ∈ I. (16)
Because of (10) and (11), the graph of i → �i is hump-shaped and above the 45◦-line. Hence, βi > 0 for i = 1, . . . , I − 1, whilst βI = 0. Skill levels do not appear directly in (16), unlike in the analogous expression in Weymark (1986b).
The parameters βi summarize in a transparent way the redistributive taste of the government. First, they would all be equal to zero if the government adopted pure utilitarianism as a social objective. They thus express the policymaker’s strict aver- sion to income inequality. Second, to get further insight into βi , it is instructive to consider the effects of the government’s decision to give each of the i less productive individuals one extra euro of consumption in the absence of incentive effects. On the one hand, the utility of each of them is increased by γ, the marginal utility of money. Hence, the gross social benefit amounts to γ�i . On the other hand, the consump- tion of every individual must be decreased by i/I in order to satisfy the tax revenue constraint (6). This reduces individual welfare by γ i/I and thus social welfare by �I ×γ i/I = γ i. Summing both effects, it appears that γβi is the net social benefit of marginally increasing the consumption of the i less skilled individuals whilst ignor- ing informational externalities. So, βi is this net social benefit expressed in monetary units. Third, the parameters βi can alternatively be defined as
βi := I − i − I∑
j=i+1
λ j , i = 1, . . . , I − 1. (17)
They thus also corresponds to the net social cost, expressed in euros, of marginally increasing the consumption of the I − i most productive individuals. That is why they are henceforth referred to as net cumulative social weights. Thanks to them Subprob- lem 2 can be rewritten as follows.
Problem 2 (Reduced Form) For (θ, γ, λ) ∈ P, choose z ∈ Z so as to maximize the social objective function W∗ (z; θ, γ, λ) : Z × P → R with4
W∗ (z; θ, γ, λ) := I∑
i=1
[ γ zi − v (zi/θi )
]− I∑
i=1
βi R (zi ; θi , θi+1). (18)
W∗ is strictly concave over the convex set Z because v′′ is positive and increas- ing. Hence, there is a unique gross income vector which maximizes W∗ for every
4 Since βI = 0, R ( zI ; θI , θI+1
) is defined arbitrarily without loss of generality. Note that the expression
of W∗ (z; θ, γ, λ) is justified in the proof of Proposition 2.
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(θ, γ, λ) ∈ P. It remains to substitute it in x∗ i (gz (θ, γ, λ) ; θ, γ ) to get the optimal
allocation.
Proposition 2 For (θ, γ, λ) ∈ P, there is a unique allocation a = (gx (θ, γ, λ) , gz
(θ, γ, λ)) solution to Problem 2, with
{ gz
i (θ, γ, λ) = arg maxzi ∈Z W∗ (z; θ, γ, λ)
gx i (θ, γ, λ) = x∗
i (gz (θ, γ, λ) ; θ, γ ) , i ∈ I. (19)
A first observation is that both gx (θ, γ, λ) and gz (θ, γ, λ) are functions. More- over, Proposition 2 implies that the social allocation solution to the optimal nonlinear income tax problem is a monotonic chain to the left. As a consequence, the optimal tax schedule is not differentiable at each observed gross income level zi . It is nevertheless possible to use the differentiability of the indifference curves in order to define implicit marginal tax rates as
T ′ (zi ; θ j ) := 1 − s
( zi ; θ j
) = 1 − v′ (zi/θ j )
γ θ j , (i, j) ∈ I2.5 (20)
Every T ′ (zi ; θ j )
is less than one because v′ > 0 and decreases in zi since v′′ > 0.
Since at the optimum only the adjacent downward incentive-compatibility constraints are binding, two implicit marginal tax rates are of particular interest at each observed gross income level zi : the implicit marginal tax rate T ′ (zi ; θi ) faced by i for whom the( x∗
i (zi ; θ, γ ) , zi ) -bundle is designed and the implicit marginal tax rate T ′ (zi ; θi+1)
the nearest more productive i + 1-individual would face if he were mimicking the i-individual.
The implicit marginal tax rates allow us to get further understanding of the reduced- form objective function W∗ (z; θ, γ, λ) . Indeed, let z be a fixed gross income vector and consider that the gross income zi of the i-individual is increased at the margin. As
dW∗ (z; θ, γ, λ)
dzi = γ T ′ (zi ; θi ) − βi R′ (zi ; θi , θi+1), i ∈ I, (21)
by (18) and (20), the impact on social welfare may be thought of as proceeding in two steps. In the first step, the i-individual pays T ′ (zi ; θi ) additional euros in taxes, which relaxes the tax revenue constraint (6). As γ is the marginal utility of money, the positive effect on social welfare amounts to γ T ′ (zi ; θi ) . In the second step, the effect on incentives is taken into account. Person i receives 1 − T ′ (zi ; θi ) extra euros of consumption. As a result, the I − i more productive individuals have to sacrifice less consumption when they decide to mimic i . So, cheating becomes more attrac- tive to them. For i + 1, the marginal rent is R′ (zi ; θi , θi+1) . This person’s binding incentive-compatibility constraint I Ci+1,i is restored by adjusting consumption by
5 Note that implicit marginal tax rates are also defined at bundles not chosen. This construction is useful in the subsequent analysis.
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Optimal nonlinear income tax and nonlinear pricing 207
R′ (zi ; θi , θi+1) /γ. Increasing the consumption of everybody of higher ability by the same amount preserves the monotonic chain to the left. Because γβi is the net social cost of marginally increasing consumption of the I − i most productive individuals, social welfare is reduced by βi R′ (zi ; θi , θi+1) . If the social optimum is interior, it is therefore obtained when the positive effect on social welfare due to the relaxation of the tax revenue constraint offsets the negative one stemming from private information, i.e.
dW∗ (z; θ, γ, λ)
dzi = 0 ⇔ T ′ (zi ; θi ) = βi
γ R′ (zi ; θi , θi+1), i ∈ I. (22)
3.2. Optimal allocations and bunching
In order to characterize the optimal allocation, it is useful to consider the relaxed form of Problem 2, obtained when all monotonicity conditions on z are removed (but not the non-negativity constraint z � 0).
Problem 3 (Relaxed Form) For (θ, γ, λ) ∈ P, choose z � 0 so as to maximize the social objective function Ŵ (z; θ, γ, λ) : R
I+ × P → R with
Ŵ (z; θ, γ, λ) := I∑
i=1
[ γ zi − v (zi/θi )
]− I∑
i=1
βi R (zi ; θi , θi+1). (23)
This problem is closely related to Problem 2. Its first-order conditions are:
∂Ŵ (z; θ, γ, λ) /∂zi ≤ 0 (= 0 if zi > 0) , i = 1, . . . , I. (24)
Note that Ŵ is strictly concave over the convex set R I+ because v′′ is positive and
increasing. Therefore, the first-order conditions (24) are both necessary and sufficient and yield a unique gross income vector, solution to Problem 3. This solution is denoted ẑ (θ, γ, p) = (̂z1 (θ, γ, p) , . . . , ẑ I (θ, γ, p))and satisfies
T ′ ( ẑi ; θi ) ≤ βi
γ R′ ( ẑi ; θi , θi+1) (= 0 if ẑi > 0), i ∈ I. (25)
If ẑ (θ, γ, p) is non-decreasing, then Problems 2 and 3 have both the same solution; in other words, ẑ (θ, γ, p) = gz (θ, γ, λ). Otherwise, the optimal allocation involves bunching. If that case, let μ1, . . . , μI be the Kuhn-Tucker multipliers of the different constraints in Z and consider the Lagrangian of Problem 2,
L = W∗ (z; θ, γ, λ) + μ1z1 + μ2 (z2 − z1) + · · · + μI (zI − zI−1). (26)
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208 L. Simula
The corresponding first-order conditions are:
∂L/∂zi = ∂W∗ (z; θ, γ, λ) /∂zi + μi − μi+1 = 0, i = 1, . . . , I − 1, (27)
∂L/∂zI = ∂W∗ (z; θ, γ, λ) /∂zI + μI = 0, (28)
μ1 ≥ 0 (= 0 if z1 > 0), (29)
μi ≥ 0 (= 0 if zi > zi−1), i = 2, . . . , I. (30)
Now assume that person i is not bunched together with anyone else at the social opti- mum, in the sense that 0 ≤ zi−1 < zi < zi+1. Then, (30) implies that μi = μi+1 = 0.
Therefore, by (27), zi satisfies (22). Because zi > 0, it is clear that (25) yields the same solution as (22). The following Lemma is obtained.
Lemma 1 For (θ, γ, λ) ∈ P, if person i is not bunched together with anyone else at the social optimum, then gz
i (θ, γ, λ) = ẑi .
Two types of bunching must be distinguished. (a) Bunching due to the violation of the non-negativity constraints, called z = 0 bunching by Boone and Bovenberg (2007), can only occur at the bottom, in which case the individuals bunched together have gross income equal to 0. (b) The other kind of bunching stems from the violation of the monotonicity constraints and can happen either at the bottom or in the interior of the skill distribution.
To gain further insight, let us assume that persons n = i, . . . , i + κ − 1, are all bunched together but not with anyone else, and denote by z their gross income level. If there is z = 0 bunching, then: i = 1, μ1 > 0 and μ1+κ = 0. Summing (27) for n = 1, . . . , κ and using (30), one gets:
1
κ
κ∑
n=1
∂W∗ (z; θ, γ, λ) /∂zn ∣∣ zn=0 < 0. (31)
If there is bunching due to the violation of the monotonicity constraints, then μi = μi+κ = 0. Summing again (27) for n = i, . . . , i + κ − 1, one obtains:
1
κ
i+κ−1∑
n=i
∂W∗ (z; θ, γ, λ)
∂zn
∣ ∣∣∣ zn=z
= 0. (32)
Consequently, the optimal gross income vector is solution to the relaxed Problem 2, except on a number of bunching sets where it is characterized by (31) and (32). Loosely speaking, the first-order conditions of Problem 2 are “decoupled” in the absence of bunching and only re-coupled inside an interval of “bunch”. The following proposition summarizes these results.
Proposition 3 For (θ, γ, λ) ∈ P, the optimal allocation is such that:
(i) gz i (θ, γ, λ) = ẑi (θ, γ, p), except on a number K of disjoint sets Bk :={ i k, . . . , j k
} , k = 1, . . . , K , i k increasing with k, where gz
n (θ, γ, λ) = zk for every n ∈ Bk;
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Optimal nonlinear income tax and nonlinear pricing 209
(ii) 0 < T ′ (zi ; θi ) < 1 for i = 1, . . . , I − 1, T ′ (zI ; θI ) = 0 and bunching at the top is ruled out;
(iii) if i k = 1,
1
#Bk
∑
n∈Bk
∂W∗ (z; θ, γ, λ)
∂zn
∣∣∣ ∣ zn=zk
≤ 0 ( = 0 if zk > 0
) ; (33)
(iv) for each interior Bk,
1
#Bk
∑
n∈Bk
∂W∗ (z; θ, γ, λ)
∂zn
∣∣∣ ∣ zn=zk
= 0. (34)
The fact that there is no bunching and no distortion at the top follows from a well- known efficiency argument (cf. Seade 1977; Guesnerie and Seade 1982). Note that, when I = 2, Proposition 3 implies that the optimal allocation is fully separating.
To gain further insights into the optimal allocation, it is useful to define the function αi : [0, θiv
′−1 (γ θi ) [→ R+ as
αi (zi ; θi , θi+1, γ ) := R′ (zi ; θi , θi+1)
γ T ′ (zi ; θi ) = T ′ (zi ; θi+1) − T ′ (zi ; θi )
T ′ (zi ; θi ) ,
i = 1, . . . , I − 1. (35)
For i < I, the domain of αi corresponds to the gross incomes for which T ′ (zi ; θi ) is strictly positive; hence, by Proposition 3, gz
i (θ, γ, λ) ∈ [0, θiv ′−1 (γ θi ) [. Moreover,
αi is continuous and strictly increasing over its domain.6 Using (22) and (35), ẑ must satisfy
αi ( ẑi ; θi , θi+1, γ ) ≥ 1
βi (= if ẑi > 0), i = 1, . . . , I − 1. (36)
So, if ẑi > 0, ẑi = α−1 i (1/βi ; θi , θi+1, γ ). Since the optimal gross income vector
gz (θ, γ, λ) lies in the interior of Z if and only if 0 < ẑ1 < · · · < ẑ I , the following characterization is obtained.
Proposition 4 For (θ, γ, λ) ∈ P, the optimum is fully separating if and only if βi ∈ αi ([0, θiv
′−1 (γ θi ) [) for i = 1, . . . , I − 1 with
0 < α−1 1 (1/β1; θ1, θ2, γ ) < . . . < α−1
I−1 (1/βI−1; θI−1, θI , γ ) < θI v ′−1 (γ θI ) .
(37)
This proposition provides a necessary and sufficient condition under which bunch- ing is either or not optimal in terms of the primitives of the model β, θ and γ. There
6 For i < I, αi is strictly increasing since dT ′ (zi ; θi ) /dzi < 0 and R′′ (zi ; θi , θi+1 )
> 0.
123
210 L. Simula
x
xi
zi
z
i+1
1
45°
i
B
A
Fig. 1 αi ( zi ; θi , θi+1, γ
) = A/B
is no analogue condition in literature considering a continuous population. The set of parameters for which the optimum is fully separating is denoted P0.
3.3. Separating optimum allocations
When the primitives of the model imply that bunching is suboptimal, the optimality conditions—obtained from (36)—are strikingly simple.
Proposition 5 For (θ, γ, λ) ∈ P0, z is socially optimal if and only if
αi (zi ; θi , θi+1, γ ) = 1/βi , i = 1, . . . , I − 1, (38)
and T ′ (zI ; θI ) = 0.
For a gross income zi , the wedge αi (zi ; θi , θi+1, γ ) tells us to which extent person i + 1’s implicit marginal tax rate is higher than person i’s one. Geometrically, it thus corresponds to the tangent of the angle between i’s and i + 1’s indifference curves divided by T ′ (zi ; θi ), as shown in Fig. 1. It is closely related to the single-crossing condition and thus henceforth referred to as the Spence-Mirrlees wedge. Indeed, the single-crossing condition corresponds to a restriction on its sign, which must be strictly positive. The conditions for social optimality (38) introduce an additional restriction on its magnitude: an allocation is socially optimal only if, at each observed gross income level zi , the Spence-Mirrlees wedge is equal to the inverse of the cumulative social weight βi .
Proposition 5 allows a simple two-step geometric construction of the optimal allo- cation, as illustrated in Fig. 2. In the first step, the tax revenue constraint is ignored.
123
Optimal nonlinear income tax and nonlinear pricing 211
Fig. 2 Construction of the optimal allocation
Start at (0, x̂1), where x̂1 is arbitrarily chosen. Move along θ1’s indifference curve until α1 (z1; θ1, θ2, γ ) = 1/β1. This point is (x1, z1). Then, move along θ2’s indifference curve through (x1, z1) until α2 (z2; θ2, θ3, γ ) = 1/β2. This point is (x2, z2) . And so on until i = I − 1. The determination of zI exploits the no-distortion-at-the-top result: starting from (x I−1, zI−1), gross income is increased along the indifference curve of the most productive individual until the tangent to this line has slope one. By construction, the allocation (x, z) is a SMCL and is, thus, incentive compatible. However, it is not necessarily budget-balanced. So, in the second step, each xi is varied by a same amount ε = 1
I
∑I i=1 (zi − xi ) so as to get a binding tax revenue constraint.
The resulting allocation (x, z) = (x + ε, z) is socially optimal.
4. Comparative static properties
The Spence-Mirrlees wedge allows us to examine the impact of changing the parame- ters of the model at the margin. For expositional reasons, the analysis will concentrate on the comparative statics of separating allocations. Yet, using Proposition 3, it can be shown that the results derived below remain valid when there is bunching at the opti- mum, provided changes in (θ, γ, λ) do not modify the sets Bk of individuals bunched together. The next property is particular useful for the following analysis.
Proposition 6 The functions gx and gz are C1 at every (θ, γ, λ)in P0.
4.1. Changing the marginal utility of money
The marginal utility of money γ measures the intensity of the individual preference for private consumption. When it goes up, the marginal utility of leisure expressed in
123
212 L. Simula
consumption good is reduced. So, the indifference curves of every individual become flatter in the (z, x)-space. Everyone is thus willing to sacrifice more leisure to obtain a certain amount of additional consumption. As a consequence, changing γ at the margin casts light on how taxes should be adjusted in a country where all individuals would like to work more to consume more.
Proposition 7 For (θ, γ, λ) ∈ P0 and i ∈ I, ∂gz i (θ, γ, λ) /∂γ > 0.
Hence, the adjustment in the tax system should not discourage more hard-working individuals to earn more. National income is increased and thus, since production efficiency is preserved by Lemma 1, total consumption goes up.
4.2. Changing individual productivities
In Mirrlees’s model, individuals are born with different abilities to turn effort into output. These fixed skill levels are the sole source of heterogeneity within the popu- lation. They constitute the most basic ingredients of the model because they give rise to the adverse selection problem: indeed, the effort level z j/θi a given i-individual must provide to earn the gross income of everyone else directly depends on his pro- ductivity. In practice, skill levels are subject to changes. For instance, a highly skilled individual can catch an illness which impairs his productivity while a low skilled can benefit from on-the-job training. Another argument rests on the technological side of the economy: since a person’s productivity depends on the capitalistic intensity in his branch of activity, a new investment can make him more productive. The impact of changing a person’s skill level on his own choices, but also on the policymaker and the whole population, is as follows.7
Proposition 8 For (θ, γ, λ) ∈ P0 and (i, j) ∈ {1, . . . , I − 1} × I,
∂zi/∂θi+1 < 0, (39)
∂zi+1/∂θi+1 > 0, (40)
∂z j/∂θi+1 = 0 for j /∈ {i, i + 1}, (41)
and
dT ′ (zi ; θi ) /dθi+1 > 0, (42)
dT ′ (zi+1; θi+1) /dθi+1 < 0 when i �= I − 1 (= 0 when i = I − 1), (43)
dT ′ (z j ; θ j ) /dθi+1 = 0 for j /∈ {i, i + 1} , (44)
Increasing person i + 1’s productivity only alters his gross income and that of his nearest less productive neighbour. Indeed, by Proposition 5, only the wedges αi (zi ; θi , θi+1, γ ) and αi+1 (zi+1; θi+1, θi+2, γ ) depend on θi+1. Accordingly, the gross income levels of all other individuals, with j /∈ {i, i + 1}, remain unaltered. As regards persons i and i + 1, the adjustment process combines three effects.
7 The fact that the productivity vector θ is strictly monotonically increasing ensures that (1) remains satisfied once a given individual productivity is changed at the margin.
123
Optimal nonlinear income tax and nonlinear pricing 213
First, the variation in θi+1 gives rise to a local substitution effect. The increase in person i + 1’s productivity results in a rise in his net-of-tax wage rate, which leads him to increase his labour supply in efficiency units, zi+1.
Second, changing θi+1 has an incentive effect. As he becomes more efficient, per- son i + 1 has to provide less effort if he wants to imitate person i . Consequently, his indifference curve through person i’s gross-income/consumption bundle flattens. By (20), this corresponds to an increase in the implicit marginal tax rate T ′ (xi , zi ; θi+1)
he would face if he were cheating. Third, person i incurs an informational externality induced by the incentive effect.
Since the cumulative social weight βi is unaltered, the wedge αi (zi ; θi , θi+1, γ ) must stay constant (Proposition 5). Consequently, the increase in T ′ (xi , zi ; θi+1) must be associated with an increase in the implicit marginal tax rate T ′ (xi , zi ; θi ) and thus with a reduction in person i’s net-of-tax wage rate. Finally, the substitution effect leads person i to work less.
The changes in gross income ensure that a new monotonic chain to the left a′ is obtained. However, this incentive-compatible allocation is not necessarily budget- balanced. Therefore, in a second step, all consumption levels are adjusted by a same amount in order to obtain a production-efficient allocation. This corresponds to a ver- tical displacement of all indifference curves through the bundles of a′. However, the direction of this displacement cannot be signed in the general case, which explains why changes in consumption are ambiguous.
4.3. Changing social weights
Social and political changes often promote greater social concern for some groups of the population. We examine the impact of increasing person i’s social weight while all other welfare weights λ j , with j �= i, are scaled down proportionally. This is equiv- alent to increasing one welfare weight, holding the others constant, in the absence of the normalization rule (11).
These changes in the individual social weights result in variations in the net cumu- lative social weights β: for every j < i, the decrease in λ j implies a reduction in the cumulative social weight � j and thus in β j ; in contrast, since the magnitude of the raise in λi is equivalent to the reduction in all other social weights in the population, � j
and β j go up for every j such that i ≤ j < I. By Proposition 5, (i) gz j (θ, γ, λ) is inde-
pendent of βi for i �= j; (ii) a rise in βi reduces gz i (θ, γ, λ) because αi (.; θi , θi+1, γ )is
strictly increasing around the optimum as previously noted. Increasing person i’s social weight thus results in the following changes.
Proposition 9 Let (θ, γ, λ) ∈ P0, z ≡ gz (θ, γ, λ)and i ∈ I. Consider that person i’s social weight is increased at the margin while all other welfare weights λ j , with j �= i, are scaled down proportionally. Then,
(i) z j is increased and T ′ (z j ; θ j )
goes down for every j < i ; (ii) if i �= I, z j is decreased and T ′ (z j ; θ j
) goes up for i ≤ j ≤ I − 1;
(iii) zI and T ′ (zI ; θI ) are unaltered.
123
214 L. Simula
4.4. Productivity change accompanied by a social weight adjustment
Up to now, we have considered changes in productivity without any impact on indi- vidual social weights and changes in social weights independent of variations in skill levels. However, since the policymaker wants to redistribute incomes from the more to the less productive, he might be less concerned for the well-being of an individual who becomes more productive and reduce the social weight of the latter accordingly. For concreteness, let us consider that the policymaker responds to a marginal increase in person i + 1’s productivity by a slight decrease in λi+1. The overall effect on the optimal gross incomes and implicit marginal tax rates follow from the combination of Propositions 8 and 9.
Proposition 10 Let (θ, γ, λ) ∈ P0 and (i, j) ∈ {1, . . . , I − 1} × I. Consider a mar- ginal increase in θi+1 accompanied by a slight decrease in λi+1, whilst all other social weights are scaled up proportionally. Then,
(i) for every j ≤ i, z j is reduced and T ′ (z j ; θ j )
is increased; (ii) for every j > i, z j is increased;
(iii) for every i < j < I, T ′ (z j ; θ j )
is reduced; (iv) T ′ (zI ; θI ) is unaltered.
5. Extension to nonlinear pricing
The Spence-Mirrlees wedge appears as a key determinant of the properties of the opti- mal solution to the optimal income tax in the above quasilinear economy. This sec- tion shows how the analysis extends to generic adverse-selection problems (see, e.g. Laffont and Martimort 2002; Salanié 2005). One considers a generic situation where a ‘principal’ contracts with an ‘agent’ to produce a certain amount of some good zi and compensates the agent with a monetary transfer xi . The agent knows his type θi before signing the contract. The cost he incurs when producing zi is given by v (θi�i ) ≡ v (zi ), where �i represents his effort level. His utility is given by u (xi , zi ; θi ) := xi − v (zi ) .
The principal only knows that type θi occurs with probability λi/I. He designs a menu of contracts which maximizes his expected utility in the exchange with the agent.
Problem 4 (Optimal Nonlinear Pricing) For (θ, γ, λ) ∈ P, choose a menu of con- tracts a ∈ R
I × R I+ to maximize
∑I i=1 λi [h (zi ) − xi ], with h′ > 0 and h′′ < 0,
subject to the incentive compatibility constraints (5) and the participation constraints
γ xi − v (zi/θi ) ≥ 0, i = 1, . . . , I. (45)
The marginal profit T ′ (zi ; θ j )
is now defined as
T ′ (zi ; θ j ) := h′ (zi ) − s
( zi ; θ j
) = h′ (zi ) − v′ (zi/θ j )
γ θ j . (i, j) ∈ I2. (46)
Because any optimum menu of contracts is a SMCL, (15) applies. Moreover, the agent’s participation constraint in state 1 is the only one to be binding; it plays the
123
Optimal nonlinear income tax and nonlinear pricing 215
same part in determining x∗ 1 (z; θ, γ ) as the tax revenue constraint, in the optimal
income tax problem. So, x∗ 1 (z; θ, γ ) = v (z1/θ1) /γ. Using these observations, the
principal’s reduced-form objective function is
W∗ (z; θ, γ, λ) = − I
γ v
( z1
θ1
) +
I∑
i=1
λi h (zi )
− 1
γ
I∑
i=2
⎛
⎝ I∑
j=i
λ j
⎞
⎠ [ v
( zi
θi
) − v
( zi−1
θi
)] .
Because W∗ is strictly concave, the necessary and sufficient conditions for an interior maximum are
∂W∗ (z; θ, γ, λ)
∂zi = 0 ⇔ λi
( h′ (zi ) − v′ (zi/θi )
γ θi
) = 1
γ
⎛
⎝ I∑
j=i+1
λ j
⎞
⎠ [ v′ (zi/θi )
θi
−v′ (zi/θi+1)
θi+1
]
⇔ R′ (zi ; θi , θi+1)
γ T ′ (zi ; θi ) = λi
I − �i ⇔ αi (zi ; θi , θi+1, γ ) = 1
βi , i = 1, . . . , I − 1,
(47)
where βi is defined as
βi := I − �i
λi , i = 1, . . . , I − 1, (48)
and T ′ (zI ; θI ) = 0. The analysis of the previous sections thus extends as follows.
Proposition 11 The menu of contracts solution to Problem 4 is characterized by Propositions 3, 4 and 5 with T ′ (zi ; θ j
) and βi defined by (46) and (48).
Note that the inverse of the hazard rate (I − �i ) /λi plays the same part, in the char- acterization of the optimum menu of contracts, as the net cumulative social weight in the characterization of the optimum income tax. Because the utility of the agent is not linear, the monotonicity of the hazard rate is not sufficient to prevent bunching. Thanks to the Spence-Mirrlees wedge an alternative condition is obtained: bunching is optimal if and only if the hazard rates do not satisfy (37). When the optimal menu of contracts is separating, the optimal contract is characterized by the equality between the Spence-Mirrlees wedge and the inverse of the hazard rate for all i = 1, . . . , I − 1.
The comparative static properties of the optimum menu of contracts are the follow- ing provided (θ, γ, λ) ∈ P0. Proposition 11 implies that the impact of increasing in γ and θi is given by Propositions 7 and 8. Proposition 9 must be adapted because an increase in λi has not the same effects on the hazard rates as on the cumulative social weights. Let us consider a rise in the probability of state i accompanied by propor- tional adjustments in all other probabilities: β j increases for all j < i and decreases
123
216 L. Simula
otherwise. By Proposition 11, z j is independent of βi for i �= j while a rise in β j
reduces z j around the optimum. Therefore, in the optimum, the principal decides to ask the agent to produce more output in all states of nature with a lower β j and less output in all other states of nature.
6. Conclusion
This article introduces the notion of Spence-Mirrlees wedge and shows its usefulness in the analysis of two important classes of incentives problems. In the optimal income tax problem, this wedge αi (zi ; θi , θi+1, γ ) indicates to which extent person i + 1’s marginal tax rate is higher than person i’s one when both earn gross income zi . In the standard adverse-selection problem, it tells us to which extent, for a given output level zi , the marginal profit is higher in state i + 1 than in state i. In both models, the Spence-Mirrlees wedge is a key determinant of the optimum menu of contracts. It allows the derivation of a necessary and sufficient condition, expressed in terms of the primitives of the models, under which bunching is optimal. It also allows a simple geometric construction of the separating menus of contracts as well as the derivation of their comparative static properties. These observations suggest that the Spence- Mirrlees wedge might be a useful tool to cast light on the structure of other screening problems in which incentive compatibility constraints are binding.
Acknowledgements I would like to thank two anonymous reviewers, the editor, Craig Brett, Antoine d’Autume, Julia Cagé, Guy Gilbert, Laurence Jacquet, Guy Laroque, Jacques Le Cacheux, Etienne Lehmann, Pierre Pestieau, Alain Trannoy, Gwenola Trotin and John A. Weymark for their very helpful comments and suggestions. The usual caveat applies.
Appendix
Proof of Proposition 1 (i.a) a is a SMCL. Assume a is not a SMCL, i.e.
u (
x∗ j , z j ; θ j
) �= u
( x∗
j−1, z j−1; θ j
) for some j ≥ 2. This is equivalent to
considering that there exists j ≥ 2 for which
γ x∗ j − v
( z j/θ j
) > γ x∗
j−1 − v ( z j−1/θ j
) . (A.1)
For (A.1) to be satisfied, (
x∗ j , z∗
j
) >>
( x j−1, z j−1
) . Hence,
(i) is established if z j ≤ z j−1. If z j > z j−1, let xi = x∗ i + ε for i =
1, . . . , j − 1, and xi = x∗ i − j−1
I− j+1ε for i = j, . . . , I, where ε >
0 is arbitrarily chosen. The new allocation (x, z) satisfies all incentive compatibility constraints for sufficiently small ε and is feasible because∑
i x i = ∑ i x∗
i . In addition,
W (x, z) − W ( x∗, z
) = γ
I∑
i=1
λi ( xi − x∗
i
)
123
Optimal nonlinear income tax and nonlinear pricing 217
= γ
⎡
⎣ j−1∑
i=1
λi − j − 1
I − j + 1
I∑
i= j
λi
⎤
⎦ ε, (A.2)
which can be minored thanks to (10) to get W (x, z) − W (x∗, z) ≥ γ ( j − 1) ε
[ λ j−1 − λ j
] > 0, contradicting x∗ ∈ X ∗ (z; θ, γ, λ) .
(i.b) ais budget-balanced. Fix z in Z . The constraints (5) are satisfied, with zi = zi and xi = x∗
i . Assume (6) is not binding. So, (6) is still satisfied if every x∗
i is increased by a sufficiently small ε > 0. This Pareto-improving increase is incentive compatible since a same amount γ ε is added to both sides of I Ci, j for every (i, j) ∈ I2, contradicting x∗ ∈ X ∗ (z; θ, γ, λ).
(ii) Because a is a SMCL by (i.a),
xi = xi−1 + 1
γ
[ v (zi/θi ) − v (zi−1/θi )
] , i = 2, . . . , I, (A.3)
and, thus,
xi = x1 + 1
γ
i∑
j=2
[ v ( z j/θ j
)− v ( z j−1/θ j
)] , i = 2, . . . , I.
(A.4)
By (i.a), the equality form of (6) can be substituted in ∑I
i=1 xi obtained from (A.4):
I∑
i=1
zi = I x1 + 1
γ
I∑
i=2
i∑
j=2
[ v
( z j
θ j
) − v
( z j−1
θ j
)] = I x1
+ 1
γ
I∑
i=2
(I + 1 − i)
[ v
( zi
θi
) − v
( zi−1
θi
)] . (A.5)
This equation has a unique solution in x1, that is substituted in (A.4). �� Proof of Proposition 2 It is sufficient to establish that substitution of x∗(z; θ, γ ) into W yields W∗ for every z ∈ Z. By (7),
u ( x∗
i , zi ; θi ) = u
( x∗
1 , z1; θ1 )+
i−1∑
j=1
R ( z j ; θ j , θ j+1
) , i = 2, . . . , I, (A.6)
from which
I∑
i=1
u ( x∗
i , zi ; θi ) = I u
( x∗
1 , z1; θ1 )+
I−1∑
i=1
(I − i) R (zi ; θi , θi+1). (A.7)
123
218 L. Simula
In addition, summing (3) over i on I and employing the equality form of (6),
I∑
i=1
u ( x∗
i , zi ; θi ) = γ
I∑
i=1
zi − I∑
i=1
v (zi/θi ). (A.8)
Plugging (A.8) in (A.7) and solving for u ( x∗
1 , z1; θ1 ) ,
u ( x∗
1 , z1; θ1 ) = 1
I
[
γ
I∑
i=1
zi − I∑
i=1
v
( zi
θi
) −
I−1∑
i=1
(I − i) R (zi ; θi , θi+1)
]
.
(A.9)
Using (A.6) and (A.9),
W = u ( x∗
1 , z1; θ1 ) I∑
i=1
λi + I∑
i=2
i−1∑
j=1
λi R ( z j ; θ j , θ j+1
)
= u ( x∗
1 , z1; θ1 )
I + I−1∑
i=1
⎛
⎝ I∑
j=i+1
λ j
⎞
⎠ R (zi ; θi , θi+1)
= I∑
i=1
⎡
⎣γ zi − v
( zi
θi
) + ⎛
⎝i − i∑
j=1
λ j
⎞
⎠ R (zi ; θi , θi+1)
⎤
⎦, (A.10)
in which R (zI ; θI , θI+1) is an arbitrary number. �� Proof of Proposition 3 (i) If i /∈Bk∀k, zi−1 < zi < zi+1. Then, by (30), μi = μi+1 = 0. Therefore, by (27), zi satisfies ∂W∗ (z; θ, γ, λ) /∂zi = 0, whose unique solution is ẑi . Hence, zi = ẑi . (ii) See Guesnerie and Seade (1982, Proposition 7). (iii)–(iv) Assume
{ i k, . . . , j k
} are bunched together at the optimum. Case (a): bunching at
the bottom due to the constraint � 0. Bk = { 1, . . . , j1
} , μ1 > 0 and μ j k+1 = 0.
Summing (27) for n = 1, . . . , j k,
∑
n∈Bk
∂W∗ (z; θ, γ, λ) /∂zn ∣∣ zn=zk = −μ1 ≤ 0, (A.11)
that is divided by #Bk > 0. Case (b): bunching due to the violation of the monotonicity constraints. Because μi k = μ j k+1 = 0, summing (27) for n = i k, . . . , j k, yields the equality form of (A.11). �� Proof of Proposition 6 By Proposition 5, gz
i (θ, γ, λ) = α−1 i (1/βi ; θi , θi+1, γ )for
i < I and gz I (θ, γ, λ) = θI v
′−1 (γ θI ) . Since α′ i > 0 and v′′ > 0, gz (θ, γ, λ) ∈ C1.
Thus, by Proposition 1, gx (θ, γ, λ) ∈ C1. ��
123
Optimal nonlinear income tax and nonlinear pricing 219
Proof of Proposition 7 Let z ≡ gz (θ, γ, λ) . For i = I, zI = θI v ′−1 (γ θI ) because
T ′ (zI ; θI ) = 0 by Proposition 3. Since v′′ > 0, ∂zI /∂γ > 0. For i ≤ i − 1, let φi := (1/βi ) − αi (zi ; θi , θi+1, γ ) and note that ∂φi/∂γ > 0. Since v′′′ > 0 and 0 < θi < θi+1,
∂φi
∂zi = βi
θ2 i+1
v′′ (
zi
θi+1
) − 1 + βi
θ2 i
v′′ (
zi
θi
) < 0. (A.12)
By the implicit function theorem, φi ≡ 0 defines zi as a C1-function of γ, zi = ϕ γ
i (γ )
with
∂zi
∂γ ≡ dϕ
γ
i (γ )
dγ = − ∂φi/∂γ
∂φi/∂zi > 0. (A.13)
�� Proof of Proposition 8 Let z ≡ gz (θ, γ, λ). It is clear from Proposition 5 that z j does not depend on θi+1 except for j = i, i + 1. Hence, ∂z j/∂θi+1 = 0 for j /∈ {i, i + 1}, implying (44).
It remains to examine the effect of a change in θi+1 on zi and zi+1. If θi+1 <
θI , φi ≡ 0 implicitly defines zi and zi+1 as C1-functions of θi+1, zi = ϕθ i (θi+1) and
zi+1 = ϕθ i+1 (θi+1), respectively, with derivatives
dϕθ i (θi+1)
dθi+1 = −∂φi/∂θi+1
∂φi/∂zi and
dϕθ i+1 (θi+1)
dθi+1 = −∂φi+1/∂θi+1
∂φi+1/∂zi+1 . (A.14)
(39) and (40) hold because ∂φi/∂zi < 0 and ∂φi+1/∂zi+1 < 0 by (A.12) while
∂φi
∂θi+1 = − βi
θ2 i+1
[ v′ (
zi
θi+1
) + zi
θi+1 v′′ (
zi
θi+1
)] < 0, (A.15)
∂φi+1
∂θi+1 = (
1 + βi+1
θ2 i+1
)[ v′ (
zi+1
θi+1
) + zi+1
θi+1 v′′ (
zi+1
θi+1
)] > 0, (A.16)
If θi+1 = θI , the change in zI−1 is obtained as previously and that in zI comes from the observation that zI = θI v
′−1 (γ θI ).
Two cases must be distinguished as regards marginal tax rates. (i) i < I − 1 : as ∂zi/∂θi+1 < 0, (20) and v′′ > 0 imply an increase in T ′ (zi ; θi ); similarly, ∂zi+1/∂θi+1 > 0 implies a reduction in T ′ (zi+1; θi+2) . (ii) i = I −1 : T ′ (zi+1; θi+1)
is unaltered because of Proposition 3. T ′ (zi ; θi ) changes as in (i). ��
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