Abstract
We consider a model where agents differ in their preferences about consumption labor and health, in their (healthdependent) earning ability, and in their health disposition. We study the joint taxation of income and health expenditure, under incentivecompatibility constraints, on the basis of efficiency and fairness principles. The fairness principles we consider propose, on one side, to reduce inequalities deriving from factors that do not depend on individuals’ responsibility. On the other side, redistribution should be precluded at least when all agents in the economy have equal physical characteristics. We construct, on the basis of such principles, a particular social welfare function. Then we give the explicit formula for the comparison of tax policies: we prove that a tax reform should always benefit agents with the worst earning ability and the worst health disposition first. Finally, in some cases, at the bottom of the income distribution the optimal tax scheme should exhibit nonuniform tax rates over health expenditure and nonpositive average marginal tax rates over income.
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Notes
Christensen et al. (1999) show that approximately a quarter of the variation in the liability to selfreported health and the number of hospitalizations could be attributed to such factors.
The latter paper in particular explains the positive correlation between marginal tax rates (used as a proxy for social preferences toward redistribution) across different countries and the extent of public provision of health insurance.
Fleurbaey’s paper in fact proposes to maximin the lowest level of full health equivalent consumption occurring in society. The full health equivalent consumption is the sincere answer to the question: how much consumption would you need to be indifferent between your current bundle and being fully healthy? One could in fact think of applying the maximin criterion to the fullhealth leisure equivalent incomes, namely, the amount of income that would be sufficient for an individual if her health problems disappeared and she no longer had to work. As it will be clearer later this might lead to some ambiguous redistributive effects. In particular, redistribution might go, from the hardworking poor to the lazy rich (a skilled agent who decides to work little). Moreover such a way to rank alternatives might lead to redistribution even in a situation where all agents are equally productive and have the same health disposition.
Given a list of objects \(a=(a_i)_{i \in N},\,a_{i}\) will denote the list \((a_1,\ldots ,a_{i1},a_{i+1},\ldots ,a_N)\) and, more in general, given some \(S \subseteq N,\,a_{S}\) will denote the original list excluding all the elements belonging to \(S\).
A typical health policy encompasses the taxation of earned income and the partial reimbursement of health expenditure. The government may also (and it frequently does) provide directly health care. This part of the health policy will not be modeled explicitly here and it is encapsulated in the \(m_i(.)\) functions.
Notice that the dotted part of the indifference curve lies always above the straight line with origin in \((0,m^+)\) and slope \(w^u(1)\).
Arrow’s condition requires social preferences over two allocations to depend only on individual preferences over these two allocations. Fleurbaey and Maniquet (2008), and Flaurbaey et al. (2005), have argued that such an informational requirement is excessively strong, especially in an economic environment like ours.
This axiom was introduced by Fleurbaey and Maniquet and it is reminiscent of the Separability condition, quite familiar in social choice (d’Aspremont and Gevers 1977), and of the Consistency condition, widely used in the theory of fair allocation (see Thomson 2004), except that it does not require to delete the resources consumed by the removed agents from the social endowment.
The idea of giving priority to a fairness requirement at the expense of a robustness requirement is consistent with the hierarchy of norms proposed by Fleurbaey and Maniquet (2008).
In reality such a possibility is often limited by physicians and insurers. We do not consider, for the sake of simplicity, such limitations.
Indeed, \(y'=\alpha w^*_{ub}(0) = \widehat{\alpha }w^*_{us}(0)=\widehat{\alpha }\gamma w^*_{ub}(0)\) which implies \(\widehat{\alpha }=\frac{\alpha }{\gamma }\).
This is quite reasonable for large populations where one could easily imagine that there are agents spread all over the budget set modified by the tax function.
We could consider situations where there are several intervals over which buying health dominates not buying health, this would just make the computations heavier without adding any value to our results.
In particular Choné and Laroque (2010) (in a recent contribution with a utilitarian setting and heterogeneous preferences) have studied the relation between the social weight curve on one side and the specification of the cardinal utility function and the distribution of preferences on the other. They study the condition under which this relation leads to negative marginal tax rates at the bottom of the income distribution. They also point out that as soon as the the distribution of preferences is independent of that of earning abilities then marginal tax rates are always positive.
Interestingly, dropping preferences heterogeneity and keeping social preferences of the leximin type would lead to a considerably different conclusion. For example, Boadway and Jacquet (2008), using a Rawlsian aggregator, show that the optimal tax scheme always exhibits positive marginal tax rates. Their result is mostly due to the fact that they assume agents having identical preferences. In such a framework the maximin aggregator strongly favors agents with the smallest budget set (i.e. unskilled agents) but there is not, like in our setting, a force that limit redistribution among low income earners.
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Acknowledgments
I thank Marc Fleurbaey and François Maniquet for the many helpful discussions and detailed suggestions. I also thank Claude d’Aspremont, Timos Athanasiou, MarieLouise Leroux, Juan D. MorenoTernero, Pierre Pestieau, Erik Schokkaert, Fred Schroyen, the participants to workshops and conferences in Liege, Louvain La Neuve, Montreal, Namur, Padova, Paris and Pittsburgh and two anonymous referees for their comments. A first draft of this paper was prepared while that author was visiting CERSES (Paris) to which he is grateful for hospitality and financial support.
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Appendix
Appendix
The following lemma will be useful for the proof of Theorem 1.
Lemma 1
If a Social Ordering Function satisfies Strong Pareto, Equal Wellbeing for Equal Preferences and Independence then for all \(e \in {\mathcal {D}},\,z,z' \in Z\), if there exist \(i,j \in N\) such that
and \(z_k=z'_k\) for each \(k \ne j \ne i\), one has \(z' \widetilde{P}(e) z\).
Proof of Lemma 1
See proof of Lemma 1 in Fleurbaey and Maniquet (2006). \(\square \)
Proof of Theorem 1
Let \(\widetilde{R}\) be a social ordering function that satisfies the listed axioms. For the sake of clarity the remaining of the proof is divided in three steps. \(\square \)
Step 1. Consider two allocations \(z\) and \(z'\) and two different agents \(j\) and \(k\) such that
and for all \(i \ne j,k, z_i=z'_i\). We want to prove that \(z' \widetilde{P}(e)z\). Suppose, by contradiction, that \(z \widetilde{R}(e)z'\). Introduce two agents \(a,b\) such that \(w_a(.)=w_b(.)=\overline{w}(.),\,m_a(.)=m_b(.)=\overline{m}(.),\,R_a=R_k\) and \(R_b=R_j\). Let \(e'=(R, R_a, R_b,w,w_a(.), w_b(.), m(.), m_a(.), m_b(.))\) and \(e''=(R_a,R_b,w_a(.), w_b(.), m_a(.),m_b(.))\). Finally let \(z_a,z_b,z'_a,z'_b\) be such that:

(1)
\(IT_j(z'_j)>IT_b(z'_b)= IT_a(z_a) > IT_b(z_b)=IT_a(z'_a)> IT_k(z'_k)\),

(2)
\(z_a \in \max _{R_a}B(IT_a(z_a),w_a(.),m_a(.)),\,z'_a \in \max _{R_a}B(IT'_a(z_a),w_a(.),m_a(.))\),

(3)
\(z_b \in \max _{R_b}B(IT_b(z_b),w_b(.),m_b(.)),\,z'_b \in \max _{R_b}B(IT'_b(z_b),w_b(.),m_b(.))\).
By Separation,
By Lemma 1
and
By Transitivity,
By Separation
Which, by Uniform Circumstances Neutrality, yields the desired contradiction.
Step 2. Consider two allocations \(z\) and \(z'\) and two different agents \(j\) and \(k\) such that
and for all \(i \ne j,k, z_i=z'_i\). We want to prove that \(z' \widetilde{I}(e)z\). Suppose, by contradiction, that \(z \widetilde{P}(e)z'\). Introduce two agents \(a,b\) such that \(w_a(.)=w_b(.)=\overline{w}(.),\,m_a(.)=m_b(.)=\overline{m}(.),\,R_a=R_k\) and \(R_b=R_j\). Let \(e'=(R, R_a, R_b, w, w_a(.),w_b(.),m(.),m_a(.),m_b(.))\) and \(e''=(R_a,R_b,w_a(.),w_b(.), m_a(.),m_b(.))\). Finally let \(z_a,z_b,z'_a,z'_b,z''_j,z''_k,z''_a,z''_b\) be such that:

(1)
\(IT_a(z'_a)=IT(z_b)= IT_k(z_k),\,IT_a(z_a)=IT_b(z'_b)= IT_k(z'_k)\)

(2)
\(z''_k I_k z_k, z''_a I_a z_a\) and \(l''_k=l''_a\) \(z''_j I_j z_j, z''_b I_b z_b\) and \(l''_j=l''_b\)

(3)
\(z_a \in \max _{R_a}B(IT_a(z_a),w_a(.),m_a(.)),\,z'_a \in \max _{R_a}B(IT'_a(z_a),w_a(.),m_a(.))\),

(4)
\(z_b \in \max _{R_b}B(IT_b(z_b),w_b(.),m_b(.)),\,z'_b \in \max _{R_b}B(IT'_b(z_b),w_b(.),m_b(.))\).
By Separation,
By Pareto Indifference (which is implied by Strong Pareto) and Equal Wellbeing for Equal Preferences
Again, by Pareto Indifference and Equal Wellbeing for Equal Preferences,
By Pareto Indifference,
By Transitivity,
By Separation,
The desired contradiction.
Step 3.The rest of the proof derives from standard characterizations of the leximin criterion (Hammond 1976).
Proof of Theorem 2
Consider a minimal tax function \(T\) and the incentivecompatible allocation \(z\) supported by it. Since \(T\) is minimal then \(S(T)\) coincides with the envelope curve of the population’s indifference surface in the \((c,y,m)\)space at \(z\). By Assumption 1, over \([0,w^*_{ub}(m)] \times M\), it is the envelope curve of agents from the subpopulation of unskilled agents with a bad health disposition. Let \(N^{ub}=\{i \in N  \ w_i=w^u(.), m_i=m^b(.)\}\). The minimum value of \(IT_i(z_i)\) within this subpopulation is given by
Into the \((c,y,m)\)space the previous expression becomes
We need to prove that this is indeed the wellbeing level of the worstoff agent in society. Let \(N^{ug}=\{i \in N  \ w_i=w^u(.), m_i=m^g(.)\}\). Following the argument we have used for \(IT^{ub}\) we can prove that the minimum value of \(IT_i(z_i)\) within this subpopulation is given by
We will first prove that \(IT^{ub} \le IT^{ug}\). Assume by contradiction that \(IT^{ub} > IT^{ug}\). By definition of \(IT^{ug}\) there exists \((y',m') \in [0, w^*_{ug}(m')]\times M\) such that
By assumption
From the definition of \(IT^{ub}\) it follows that, for all \((y,m) \in [0, w^*_{ub}(m)] \times M]\),
Consider now, \(m'' \in M\) such that
Necessarily, \(m'' \ge m'\) moreover, from 8.2,
so that, by 8.1
which contradicts the fact that \(T\) is minimal.
Let \(N^{sb}=\{i \in N  \ w_i=w^s(.), m_i=m^b(.)\}\). The minimum value of \(IT_i(z_i)\) within this subpopulation is given by
We now want to prove that \(IT^{ub} \le IT^{sb}\). Assume by contradiction that \(IT^{ub} > IT^{sb}\). By definition of \(IT^{sb}\) there exists \((y',m') \in [0, w^*_{sb}(m')]\times M\) such that
By assumption,
From the definition of \(IT^{ub}\) it follows that, for all \((y,m) \in [0, w^*_{ub}(m)] \times M\),
In particular, if we let \(m=m'\) and \(y''=\frac{w^*_{ub}(m')}{w^*_{sb}(m')}y'\) (so that, by construction, \(y'' < y'\) and \(y'' \le w^*_{ub}(m')\)) we obtain
This entails
which contradicts the fact that, for each \(m \in M,\,ymT(y,m)\) is nondecreasing in \(y\). Let finally \(N^{sg}=\{i \in N  \ w_i=w^s(.), m_i=m^g(.)\}\). Using the same reasoning as above it also possible to prove that both
and
So that finally, by transitivity, we also have
which concludes the proof. \(\square \)
Proof of Theorem 3
Let \(ucs((c_i, y_i,m_i), w_i(.), R_i^*)\) denote the closed upper contour set for an unskilled agent with a bad health disposition and with preferences \(R_i^*\) at the bundle \(z_i=(c_i,y_i,m_i)\):
By definition, for the agents belonging to this subpopulation:
Consider now the point \((c_{ub}(\widehat{\alpha };T;0), w^*_{ub}(0),0)\). The value of the \(IT(.)\) at such a point is given by
Since, for each \(i \in N,\,(c_i,y_i,m_i) \in ucs((c_i,y_i,m_i))\) then, necessarily,
On the other hand, from the proof of Theorem 2,
Finally, the fact that, by definition, \(S(T)\) lies nowhere above the envelope curve of the indifference curves in the \((c,y,m)\)space so that
concludes the proof. \(\square \)
Proof of Proposition 1
Let \(z^*\) be an optimal allocation supported by the minimal tax function \(T^*\). By Theorem 2 the worst off agent is an unskilled agent with a bad health disposition and her level of wellbeing is
From the definition of implicit transfer it follows that there is \(i \in N\) such that \(z^*_i I_i x'_i\) for some \(x'_i=(c'_i,y'_i,m'_i)\) satisfying
Using the same reasoning we can also compute the wellbeing level of the worst off agents among the unskilled ones with a good health disposition, \(IT^{ug}\). Let \(N^{ug}=\{i \in N  \ w_i=w^u(.), m_i=m^g(.)\}\),
Again, this means that there is some \(j \in N^{ug}\) such that \(z^*_j I_j x''_j\) for some \(x''_j=(c''_j,y''_j,m''_j)\) such that
Moreover one should notice that, at a minimal tax schedule, an agent who has spent \(m^\) in health care is necessarily an agent with a good health disposition and an agent who has spent \(m^+\) is necessarily an agent with a bad health disposition. That is, at a minimal tax schedule is always possible to infer the actual health disposition of some agents whose health expenditure is strictly positive (it is as if \(h\) was observable). This implies that at an optimal allocation, like \(z^*\), the wellbeing level across unskilled agents with different health disposition has to be equalized. That is \(IT^{ug}=IT^{ub}\).
Let us consider first agents in \(N_{ug}\). The tax raise necessary to obtain \(\widetilde{T}\) must be such that at \(\widetilde{T}\) the wellbeing level of the worst off among them cannot go below \(IT^{ug}\) (we want \(\widetilde{T}\) to be welfare equivalent to \(T^*\)). Moreover none of the agents choosing their health expenditure between \(0\) and \(m^\) should be able to switch to a bundle involving a higher subsidy (or a lower tax) otherwise the new allocation might be unfeasible. In order to properly devise such a tax raise we need to find out who is the agent that, after the raise, will obtain the highest possible subsidy, without affecting the possibility that the wellbeing of the worst off could go below \(IT^{ug}\). In order to do so let us compute,
subject to
The solution to this problem depends on the distribution of the earning ability functions and the health disposition functions. In particular one can easily check that if Assumption 2 holds, namely
then the solution to 8.6 is at the point \((w^*_{ug}(m^),m^)\). That is, the unskilled agent with good health disposition who works full time and has purchased health care is the agent that, when going from \(T^*\) to \(\widetilde{T}\) is allowed to have the highest possible subsidy (or, the smallest possible tax). The size of such subsidy is equal to (it is enough to plug the solution to 8.6 in the objective function)
We can proceed in a similar fashion when it comes to unskilled agents with a bad health disposition. We first solve the problem
subject to
which, if 8.7 holds, has a solution at \((w^*_{ub}(m^+),m^+)\). An unskilled agent with a bad health disposition who works full time and has purchased health care is the agent that when going from \(T^*\) to \(\widetilde{T}\) is allowed to have the highest possible subsidy (or, the smallest possible tax). The size of such subsidy is equal to
We can finally argue that the tax function
is still feasible since it is obtained from \(T^*\) cutting all the subsidies bigger than a certain threshold (or augmenting the taxes below a certain level). Moreover, no agent can switch to a bundle that, given her health disposition, provides her with a higher subsidy (or with a smaller tax).
Let \(\widetilde{z}\) be the allocation decentralized by \(\widetilde{T}(.)\). Using a reasoning similar to that used in theorem 3, one can see that, under \(\widetilde{T}\), the wellbeing level of worst off among the unskilled agents with a bad health disposition is at least \(IT^{ub}\). Indeed,
The right hand side of the previous inequality can be rewritten as
which is equivalent to
This yields
so that finally, by 8.7, one can write \(\min _{i \in N^{ub}} IT_i(\widetilde{z}_i) \ge IT^{ub}\). Moreover, since the point \(x'_i=(c'_i,y'_i,m'_i)\) is still attainable for agent \(i\), one also has \(\min _{i \in N^{ub}} IT_i(\widetilde{z}_i) \le IT^{ub}\) hence \(\min _{i \in N^{ub}} IT_i(\widetilde{z}_i) = IT^{ub}\). Using the same argument one can also prove that
\(\square \)
Proof of Proposition 2
Let \(z^*\) be an optimal (incentivecompatible) allocation for the ACEE social ordering function. From the proof of proposition 1 we know that \(T^*\) and \(\widetilde{T}\) are welfare equivalent. From the way \(\widetilde{T}(.)\) has been constructed it also follows that (a) for \(m=m^+\) and \(y \ge 0,\,\widetilde{T}(y,m^+) \ge IT^{ub}(\overline{w}((m^b)^{1}(m^+))w^*_{ub}(m^+)+m^+\overline{m} ((m^b)^{1}(m^+))),\,y\le w^*_{ub}(m^+), y\widetilde{T}(y,m^+)m^+\frac{\overline{w}((m^b)^{1}(m^+))}{w^*_{ub}(m^+)}y+\overline{m}((m^b)^{1}(m^+)) \ge IT^{ub}\). Hence, for \(y \le w^*_{ub}(m^+)\)
By setting \(y=w^*_{ub}(m^+)\), from 8.11 we obtain that
If we plug 8.12 into the two equations in (a) we obtain (a’) for \(m = m^+\) and \(y \ge 0\), \(\widetilde{T}(y,m^+) \ge \widetilde{T}(w^*_{ub}(m^+),m^+),\,y\le w^*_{ub}(m^+),\,y\widetilde{T}(y,m^+)\ge \frac{\overline{w}((m^b)^{1}(m^+))}{w^*_{ub}(m^+)}y\widetilde{T}(w^*_{ub}(m^+),m^+) (\overline{w}((m^b)^{1}(m^+))w^*_{ub}(m^+))\).
In a similar fashion, it is possible to prove that
and (b’) for \(m = m^\) and \(y \ge 0,\,\widetilde{T}(y,m^) \ge \widetilde{T}(w^*_{ug}(m^),m^),\,y\le w^*_{ug}(m^),\,y\widetilde{T}(y,m^)\ge \frac{\overline{w}((m^g)^{1}(m^))}{w^*_{ug} (m^)}y\widetilde{T}(w^*_{ug}(m^),m^)(\overline{w}((m^g)^{1}(m^))w^*_{ug}(m^))\).
Finally, from 8.12, \(IT^{ub}=w^*_{ub}(m^+)m^+\widetilde{T}(w((m^b)^{1}(m^+)),m^+)\overline{w}((m^b)^{1}(m^+))+\overline{m}((m^b)^{1}(m^+))=c_{ub}(1;T;m^+)\overline{w}((m^b)^{1}(m^+))+\overline{m}((m^b)^{1}(m^+)), \) and from 8.13 \(IT^{ug}=w^*_{ug}(m^)m^\widetilde{T}(w((m^g)^{1}(m^)),m^)\overline{w}((m^g)^{1}(m^))+\overline{m}((m^g)^{1}(m^))=c_{ug}(1;T;m^)\overline{w}((m^g)^{1}(m^))+\overline{m}((m^g)^{1}(m^)),\)
so, since \(\overline{w}((m^g)^{1}(m^))=\overline{w}((m^b)^{1}(m^+))\) and \(\overline{m}((m^g)^{1}(m^)) = \overline{m}((m^b)^{1}(m^+))\), it follows that
and that maximizing \(IT^{ub}\) is equivalent to maximizing \(c_{ub}(1;T;m^+)\). \(\square \)
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Valletta, G. Health, fairness and taxation. Soc Choice Welf 43, 101–140 (2014). https://doi.org/10.1007/s0035501307685
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DOI: https://doi.org/10.1007/s0035501307685