Majority rule and selfishly optimal nonlinear income tax schedules with discrete skill levels


Röell (Voting over nonlinear income tax schedules, unpublished manuscript, School of International and Public Affairs, Columbia University, New York, 2012) shows that Black’s median voter theorem for majority voting with single-peaked preferences applies to voting over nonlinear income tax schedules that satisfy the constraints of a finite type version of the Mirrlees optimal income tax problem when voting takes place over the tax schedules that are selfishly optimal for some individual and preferences are quasilinear. An alternative way of establishing Röell’s median voter result is provided that offers a different perspective on her findings, drawing on insights obtained by Brett and Weymark (Games Econ Behav 101:172–188, 2017) in their analysis of a version of this problem with a continuum of types. In order to characterize a selfishly optimal schedule, it is determined how to optimally bunch different types of individuals.

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Fig. 1


  1. 1.

    A preference profile can alternatively be described as being single-peaked if on any triple of alternatives there is one of them that is not ranked last by anyone who is not indifferent among all three (Arrow 1951).

  2. 2.

    A related approach to determining optimal bunching regions in a class of quasilinear incentive problems with discrete types has been independently developed by Goeree and Kushnir (2017). Their approach makes use of results from convex analysis and a majorization theorem due to Hardy et al. (1929).

  3. 3.

    There is an extensive literature on majority voting over income tax schedules, some of which restricts the voting to be over selfishly optimal schedules. For a brief introduction to this literature, see Brett and Weymark (2017).

  4. 4.

    It is straightforward to reformulate our analysis in terms of quasilinear-in-consumption preferences. See Sect. 6.

  5. 5.

    In a two-county extension of this model, Dai and Tian (2018) show that the possibility of mobility between the two countries can result in discontinuities at the endpoints of the skill distribution as well.

  6. 6.

    Arrow (1968) was the first to show how to iron a non-monotone schedule in his analysis of optimal capital policy with irreversible investment.

  7. 7.

    We refer to a set of types of this form as being an interval of types.

  8. 8.

    The methodology used by Brett and Weymark (2018) to derive a type’s reduced-form problem is based on a similar methodology to the one used by Weymark (1986b) to reduce the dimensionality of an optimal nonlinear income tax problem with a weighted utilitarian objective when the number of types is finite. Using a different approach, Lollivier and Rochet (1983) had previously shown that such a dimensionality reduction is possible when there is a continuum of types.

  9. 9.

    We say “essentially” because there are some minor differences. Weymark (1986b) allows the marginal disutility of labor \(\gamma \) and the price p of the consumption good to differ from 1, which results in the analogue to the second term in brackets in (15) being multiplied by \(\gamma p\). Here, this product is 1. His objective function does not include an analogue to the scaling factor \(1/Nw_k\). This is of no consequence as only the ordinal properties of the objective function matter.

  10. 10.

    If \(j = 1\), the inequality \( c^{k^*}_{j-1}<c^{k*}_i\) does not apply. Similarly, if \(m = n\), the inequality \(c^{k*}_i <c^{k*}_{m+1}\) does not apply.

  11. 11.

    It follows from (43) that the virtual wages for proposer n are all positive because the skill levels are also all positive.

  12. 12.

    Of course, each type cares about its income, not just its consumption. However, because a type’s selfishly optimal consumption schedule solves its reduced-form problem, incomes have been fully accounted for.

  13. 13.

    With majority rule, if preferences are single-peaked, nobody can obtain a preferred outcome by reporting a single-peaked preference different from the true one. This does not imply that someone cannot manipulate the outcome by reporting a preference that is not single-peaked. Such a false report could be detected and ruled to be inadmissible. Applied to the tax voting problem, if a proposed income tax schedule or corresponding allocation is not selfishly optimal for any type, then this is publicly observable and could be disallowed.

  14. 14.

    This kind of single-crossing property was introduced by Matthews and Moore (1987). See Brett and Weymark (2019) for a discussion. It is also related to the single-crossing property considered by Gans and Smart (1996), which is satisfied if (i) voters are linearly ordered and (ii) if a pair of voters agree on how to rank a pair of alternatives, then all voters in between them concur with this ranking. A median voter according to the linear ordering of the voters is decisive in a pairwise majority vote. Gans and Smart apply this median voter theorem to voting over nonlinear income tax schedules. Bierbrauer and Boyer (2017) prove a median voter theorem for tax reforms from a status quo situation in which the change in the tax liability is a monotonic function of income. They also make use of a single-crossing property.

  15. 15.

    For notational simplicity, we omit the superscript \(^*\) and the dependence of the optimal incomes on the consumptions in this proof. That is, all allocations in this proof are assumed to be optimal for the relevant proposer.


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Correspondence to John A. Weymark.

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We dedicate this article to the memory of Kenneth J. Arrow, whose influence on our own research has been immeasurable. We are grateful to Saumya Deojain, Alessandro Pavan, Laurent Simula, Alain Trannoy, and our referees for their comments. This article has been presented at the Conference in Honour of Alain Trannoy at the Aix-Marseille School of Economics and at the Taxation Theory Conference at Washington University in St. Louis.

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Proof of Lemma 1

Suppose by way of contradiction that (26) does not hold. Let h be the smallest type in \(\{j, \ldots , m\}\) such that \(c^{k*}_h>c^{k*}_{h-1}\). Because \(\mu _h = 0\), summing (39) over types \(h,\ldots , m\) yields

$$\begin{aligned} v^\prime (c^{k*}_i)\left[ \sum _{i=h}^m n_i \beta ^k_i\right] = \sum _{i=h}^m n_i +\mu _{m+1}. \end{aligned}$$

The left-hand side of (A.1) is nonpositive by (25), whereas the right-hand side is positive, a contradiction. \(\square \)

Proof of Lemma 2

If \(j = m\), (27) is vacuous and the lemma is trivially true.

Now, suppose that \(j < m\). By way of contradiction, suppose that there is some type in \(\{j,\ldots ,m\}\) with \(c^{k*}_i>c^{k*} _j\). Let l be the lowest such type. Then, \(\mu _l =0\).

Suppose, first, that type l is bunched with type m (l need not be distinct from m). By (39),

$$\begin{aligned} v^\prime (c^{k*}_j) \sum _{i=j}^{l-1} n_i \beta ^k_i \ge \sum _{i=j}^{l-1} n_i - [\mu _j + \mu _{l-1}], \end{aligned}$$

which implies that

$$\begin{aligned} v^\prime (c^{k*}_j)\left[ \frac{ \sum _{i=j}^{l-1} n_i \beta ^k_i }{\sum _{i=j}^{l-1} n_i}\right] \ge 1 - \frac{[\mu _j + \mu _{l-1}]}{\sum _{i=j}^{l-1} n_i}. \end{aligned}$$

By assumption, the fraction on the left-hand side of (A.3) is positive. Because \(v^\prime (0) = \infty \), if \(c^{k*}_j = 0\), this inequality is violated. Hence, \(c^{k*}_j > 0\) and both (A.2) and (A.3) hold with equality. Also by (39)

$$\begin{aligned} v^\prime (c^{k*}_l)\left[ \frac{ \sum _{i=l}^{m} n_i \beta ^k_i }{\sum _{i=l}^{m} n_i}\right] = 1 + \frac{[\mu _{l+1} + \mu _{m+1}]}{\sum _{i=l}^{m} n_i}. \end{aligned}$$

It follows from the equality version of (A.3), (A.8), and the concavity of v that

$$\begin{aligned} v^\prime (c^{k*}_j)\left[ \frac{ \sum _{i=j}^{l-1} n_i \beta ^k_i }{\sum _{i=j}^{l-1} n_i}\right]<v^\prime (c^{k*}_l)\left[ \frac{ \sum _{i=l}^{m} n_i \beta ^k_i }{\sum _{i=l}^{m} n_i}\right] < v^\prime (c^{k*}_j)\left[ \frac{ \sum _{i=l}^{m} n_i \beta ^k_i }{\sum _{i=l}^{m} n_i}\right] . \end{aligned}$$


$$\begin{aligned} \frac{ \sum _{i=j}^{l-1} n_i \beta ^k_i }{\sum _{i=j}^{l-1} n_i} < \frac{ \sum _{i=l}^{m} n_i \beta ^k_i }{\sum _{i=l}^{m} n_i}, \end{aligned}$$

which contradicts (27).

We have thus shown that there must be a maximal type \(q<m\) that is bunched with l. Note that it may be the case that \(q = l\). By (40),

$$\begin{aligned} v^\prime (c^{k*}_l)\left[ \frac{ \sum _{i=l}^{q} n_i \beta ^k_i }{\sum _{i=l}^{q} n_i}\right] = 1. \end{aligned}$$

Using (39) for types \(q+1\) to m implies

$$\begin{aligned} v^\prime (c^{k*}_l)\left[ \frac{ \sum _{i=q+1}^{m} n_i \beta ^k_i }{\sum _{i=q+1}^{m} n_i}\right] = 1 + \frac{[\mu _{q+2} + \mu _{m+1}]}{\sum _{i=q+1}^{m} n_i}. \end{aligned}$$

The argument leading to (A.6) may be repeated to conclude that

$$\begin{aligned} \frac{ \sum _{i=l}^{l-1} n_i \beta ^k_i }{\sum _{i=j}^{l-1} n_i} < \frac{ \sum _{i=l}^{m} n_i \beta ^k_i }{\sum _{i=l}^{m} n_i}, \end{aligned}$$

which contradicts (27).

Having shown that both possibilities for type l result in a contradiction, no such type is possible and, hence, (28) holds. \(\square \)

Proof of Theorem 2

Because \({\bar{f}}_{\sigma ^k}\) is a convex function and \(v^{\prime }\) is decreasing, the consumption vector \({\mathbf {c}}^{k*}\) defined in (36) satisfies the constraints in proposer \(k\)’s reduced-form problem. Because \({\bar{\beta }}^k\) is nondecreasing in type, the set of all types that share a common value of the adjusted virtual wage is an interval of types. Let \(\{j, \ldots , m\}\) be such an interval. Because \({\bar{f}}_{\sigma ^k}\) is the highest convex function that lies nowhere above \(f_{\sigma ^k}\), (25) holds if \({\bar{\beta }}^k_j \le 0\) and (27) holds if \({\bar{\beta }}^k_j > 0\). Hence, by Lemmas 1 and 2, respectively, all types in \(\{j, \ldots , m\}\) are bunched together. Furthermore, because the types for which the adjusted virtual wages are nonpositive are the first l types for some \(l \in \{0, 1, \ldots , n\}\), Lemma 1 also implies that \(c^{k*}_i\) is zero for any type \(i \le l\).

For any type i for which \({\bar{\beta }}^k_i > 0\), let \(c^{k \circ }_i\) be the common value of the consumption of the types \(\{j, \ldots , m\}\) that are bunched with type i. The optimal value of \(c^{k \circ }_i\) is the solution to

$$\begin{aligned} \max \ \sum _{h=j}^m n_h {\bar{\beta }}^k_i v(c^{k \circ }_i) - \sum _{h=j}^m n_h c^{k \circ }_i \ \ \text {subject to } \ \ c^{k \circ }_i \ge 0. \end{aligned}$$

Because \({\bar{\beta }}^k_i > 0\), v is strictly concave, and \(v^{\prime }(0) = \infty \), this problem has a unique solution and it is positive. This solution is given by the first equality in (36). \(\square \)

Proof of Lemma 3

Suppose, by way of contradiction, that there is a complete bunching interval \(\{j,\ldots ,m\}\) with \(j < m\) that does not include type k. We first consider the case in which \(m < k\). Because \(\mu _j = \mu _{m+1} = 0\), by (39) and (41),

$$\begin{aligned} n_j \beta ^n_j v^\prime (c^{k*}_j) \ge n_j+\mu _{j+1} \quad \text {and} \quad n_m \beta ^n_m v^\prime (c^{k*}_j) \ge n_m-\mu _{m} \end{aligned}$$

or, equivalently,

$$\begin{aligned} \beta ^n_j \ge \frac{1}{v^\prime (c^{k*}_j)}\left[ 1+\frac{\mu _{j+1}}{n_j}\right] \quad \text {and} \quad \beta ^n_m \ge \frac{1}{v^\prime (c^{k*}_j)}\left[ 1-\frac{\mu _m}{n_m}\right] . \end{aligned}$$

The inequalities in (A.11) and (A.12) bind if \(c^{k*}_j > 0\). In this case, (A.12) implies that \(\beta ^n_j \ge \beta ^n_m\). If \(c^{k*}_j = 0\), then \(j = 1\) and (A.12) implies that \(\beta ^n_j \ge 0\). Because there are no types above m with zero consumption, by Theorem 2 and the way that the adjusted virtual wages are determined, it must be the case that \(\beta ^n_m \le 0\). Thus, in this case as well, we have \(\beta ^n_j \ge \beta ^n_m\). However, because type \(n\)’s optimal consumption schedule does not exhibit bunching, by Theorem 1, \(\beta ^n_j < \beta ^n_m\), so we have a contradiction.

A similar argument can be used to show that no complete bunching interval of the form \(\{j,\ldots , m\}\) with \(j < m\) and \(j>k\) exists. In this case, \(c^{k*}_j\) is necessarily positive, so the analogues to (A.11) and (A.12) for \(\beta ^1_j\) and \(\beta ^1_m\) hold with equality. \(\square \)

Proof of Theorem 4

We first suppose that \(i<k\). We show that the utility that type i receives with type \(k\)’s selfishly optimal allocation is greater than or equal to the utility that type i receives with type \((k+1)\)’s selfishly optimal allocation. Formally,

$$\begin{aligned} w_i v(c^{k}_i)-y^{k}_i \ge w_i v(c^{k+1}_i)-y^{k+1}_i. ^{15} \end{aligned}$$

Footnote 15

Substituting (12) into the left-hand side of (A.13) yields

$$\begin{aligned} w_i v(c^{k}_i)-y^{k}_i = w_iv(c^k_i)-\left[ y^k_k + \sum _{j=i}^{k-1} w_j[v(c^k_j)-v(c^k_{j+1})] \right] . \end{aligned}$$

Rearranging terms in (A.14) gives

$$\begin{aligned} \begin{aligned} w_i v(c^{k}_i)-y^{k}_i = {}&w_iv(c^k_i)- y^k_k - w_i v(c^k_i) \\&{} + \sum _{j=i}^{k-2}[w_j-w_{j+1}]v(c^k_{j+1})+w_{k-1}v(c^k_k) . \end{aligned} \end{aligned}$$

Cancelling the terms in \(w_iv(c^k_i)\) and adding and subtracting \(w_k v(c^k_k)\) on the right-hand side of (A.15) yields

$$\begin{aligned} w_i v(c^{k}_i)-y^{k}_i= w_k v(c^k_k) - y^k_k + \sum _{j=i}^{k-1}[w_j-w_{j+1}]v\left( c^k_{j+1}\right) . \end{aligned}$$

Substituting (12) into the right-hand side of (A.13) yields

$$\begin{aligned} w_i v(c^{k+1}_i)-y^{k+1}_i= w_iv(c^{k+1}_i)-\left[ y^{k+1}_{k+1} + \sum _{j=i}^{k} w_j\left[ v\left( c^{k+1}_j\right) -v\left( c^{k+1}_{j+1}\right) \right] \right] . \nonumber \\ \end{aligned}$$

Rearranging terms on the right-hand side of (A.17) gives

$$\begin{aligned} w_i v(c^{k+1}_i)-y^{k+1}_i= w_k v(c^{k+1}_{k+1}) - y^{k+1}_{k+1} + \sum _{j=i}^{k-1}[w_j-w_{j+1}]v\left( c^{k+1}_{j+1}\right) . \end{aligned}$$

Subtracting (A.18) from (A.16), we obtain

$$\begin{aligned} \begin{aligned}&\left[ w_i v(c^{k}_i) - y^{k}_i\right] - \left[ w_i v\left( c^{k+1}_i\right) -y^{k+1}_i \right] = \left[ w_k v(c^k_k) - y^k_k\right] \\&\quad - \left[ w_k v\left( c^{k+1}_{k+1}\right) - y^{k+1}_{k+1}\right] + \sum _{j=i}^{k-1}[w_j-w_{j+1}]\left[ v\left( c^k_{j+1}\right) -v\left( c^{k+1}_{j+1}\right) \right] . \end{aligned} \end{aligned}$$

The first term on the right-hand side of (A.19) is the maximal utility of type k when the proposer is of this type. Because type \(k\)’s adjacent upward self-selection constraint binds at type \((k+1)\)’s selfishly optimal bundle, the second term is the utility that type k receives when the proposer is of type \(k+1\). Because someone of type k can do no better than with its own proposal, the difference between the first and second terms is nonnegative. The final term is also nonnegative because any selfishly optimal consumption schedule is nondecreasing in type. Hence, the entire right-hand side of (A.19) is nonnegative, which establishes (A.13).

If \(i>k\), we can use a similar argument to show that the utility that type i receives with type \(k\)’s selfishly optimal allocation is greater than or equal to the utility that i receives with type \((k-1)\)’s selfishly optimal allocation. This argument uses (13) instead of (12). \(\square \)

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Brett, C., Weymark, J.A. Majority rule and selfishly optimal nonlinear income tax schedules with discrete skill levels. Soc Choice Welf 54, 337–362 (2020).

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