Fairness, freedom, and forgiveness in health care

Abstract

This paper focuses on the optimal allocation between health and lifestyle choices when society is concerned about forgiveness. Based on the idea of fresh starts, we construct a social ordering that permits us to make welfare assessments when it is acceptable to compensate individuals who have mismanaged their initial resources. Our social rule also allows for the inclusion of the fairness and responsibility approach in the model. Grounded on basic ethical principles, we propose the application of the minimax criterion to the existing distance between the individual’s final bundle and her ideal choice.

Appendix: Proofs

1.1 Proof of Proposition 1

Proof

This proof is based on the result obtained by Valletta (2009). For any economy $e\in \mathcal {E}$, let us consider two allocations $z,z'\in Z^{n}$ and two different individuals $j,k\in N$ with $R^t_{j}\ne R^t_{k}$ such that, without loss of generality, $\rho _{j}(z_{j}')>\rho _{j}(z_{k})>\rho _{k}(z_{j})>\rho _{k}(z_{k}')$, and $z_{i}=z_{i}'$ for all $i\ne j,k$. When the relations are different to the one proposed here, the proof is either analogous or immediate.

The proof splits in two steps.

First we need to prove that it must be the case that $z\mathbf P (e)z'$. Contrary to the desired result, let us assume that $z'\mathbf R (e)z$.

$\bullet$ Case i: Let us consider that $\rho _{j}(z_{j}')>0$. Let us introduce two individuals $b,c \in N$ with ex post preferences $R^t_{b}=R^t_{j}$ and $R^t_{c}=R^t_{k}$. Let us also assume that there exist $z''_{j},z'''_{j},z''''_{j},z''_{k},z'''_{k},z_{b},z'_{b},z_{c},z'_{c}\in Z$ with $h''_{j}=h'''_{j}=h''''_{j}=h''_{k}=h'''_{k}=h_{b}=h'_{b}=h_{c}=h'_{c}=h^{*}$ and $\beta ,\varepsilon >0$ such that:

$$\begin{aligned}&c''_{j}=c_{j}^{*}(z'_{j})+\beta ; \ c'''_{j}=c_{j}^{*}(z_{j})+\beta \\&c''_{k}=c_{k}^{*}(z'_{k})+\beta ; \ c'''_{k}=c_{k}^{*}(z_{k})+\beta \\&c_{b}=c'_{b}+\varepsilon ; \ c_{c}=c'_{c}-\varepsilon \\&z'''_{j} P^t_{j}\overline{z}_{j} P^t_{j} z_{b} P^t_{j} z'_{b} P^t_{j} z''''_{j} P^t_{j} z''_{j} P^t_{j} z_{j} P^t_{j} z'_{j} \\&z''_{k} P^t_{k} z'''_{k} P^t_{k} z'_{c} P^t_{k} z_{c} P^t_{k} \overline{z}_{k} P^t_{k} z'_{k} P^t_{k} z_{k}. \end{aligned}$$

Since $\rho _{j}(z_{j}')>\rho _{j}(z_{k})$, and because of continuity, such bundles can always be found. According to the initial assumptions, if we apply Strong Pareto, Separation and Linearity it is straightforward to check that $(z_{-\{j,k\}},z''_{j},z''_{k})\mathbf R (e)(z_{-\{j,k\}},z'''_{j},z'''_{k})$. Because of Separation, we can also add identical agents in both allocations without altering the social ordering, that is, $(z_{-\{j,k\}},z''_{j},z''_{k},z_{b},z_{c})\mathbf R (e)(z_{-\{j,k\}},z'''_{j},z'''_{k},z_{b},z_{c})$. If we apply Equal Preferences Priority we get that $(z_{-\{j,k\}},z''''_{j},z''_{k},z'_{b},z_{c})\mathbf P (e)(z_{-\{j,k\}},z''_{j}, z''_{k},z_{b},z_{c})$, and moreover $(z_{-\{j,k\}},z''''_{j},z'''_{k},z'_{b},z'_{c})\mathbf P (e)(z_{-\{j,k\}},z''''_{j},z''_{k},z'_{b},z_{c})$. Using Strong Pareto we have that $(z_{-\{j,k\}},z'''_{j},z'''_{k},z'_{b},z'_{c})\mathbf P (e)(z_{-\{j,k\}},z''''_{j},z'''_{k},z'_{b},z'_{c})$. Finally, by Transitivity we obtain that $(z_{-\{j,k\}},z'''_{j},z'''_{k},z'_{b},z'_{c})\mathbf P (e)(z_{-\{j,k\}},z'''_{j},z'''_{k}, z_{b},z_{c})$. However, if we apply the Minimum Solidarity axiom it is straightforward to obtain that $(z_{-\{j,k\}},z'''_{j},z'''_{k},z_{b},z_{c})\mathbf R (e)(z_{-\{j,k\}},z'''_{j},z'''_{k},z'_{b},z'_{c})$, which yields the desired contradiction.

$\bullet$ Case ii: Let us consider now that $\rho _{j}(z_{j}')\le 0$. Let us again introduce two individuals $b,c \in N$ with ex post preferences $R^t_{b}=R^t_{j}$ and $R^t_{c}=R^t_{k}$. Let us also assume that there exist $z''_{j},z_{b},z'_{b},z''_{b},z'''_{b},z_{c},z'_{c},z''_{c},z'''_{c}\in Z$ with $h''_{j}=h_{b}=h'_{b}=h''_{b}=h'''_{b}=h_{c}=h'_{c}=h''_{c}=h'''_{c}=h^{*}$ and $\beta >0$ such that:

$$\begin{aligned}&c_{b}=c'_{b}+\varepsilon ; \ c_{c}=c'_{c}-\varepsilon \\&c''_{b}=c_{b}+\beta ; \ c'''_{b}=c'_{b}+\beta \\&c''_{c}=c_{c}+\beta ; \ c'''_{b}=c'_{c}+\beta \\&z_{j} P^t_{j} z''_{b} P^t_{j} z'''_{b} P^t_{j} z''_{j} P^t_{j} z'_{j} P^t_{j} \overline{z}_{j} P^t_{j} z_{b} P^t_{j} z'_{b} \\&z'_{k} P^t_{k} z_{k} P^t_{k} z'''_{c} P^t_{k} z''_{c} P^t_{k} z'_{c} P^t_{k} z_{c} P^t_{k} \overline{z}_{k}. \end{aligned}$$

Again, since $\rho _{j}(z_{j}')>\rho _{j}(z_{k})$, and because of continuity, such bundles can always be found. According to the initial assumptions, if we apply Separation we have that $(z_{-\{j,k\}},z'_{j},z'_{k},z''_{b},z''_{c})\mathbf R (e)(z_{-\{j,k\}},z_{j},z_{k},z''_{b},z''_{c})$. Because of Equal Preferences Priority it is straightforward to see that, $(z_{-\{j,k\}},z''_{j},z'_{k},z'''_{b},z''_{c})\mathbf P (e)(z_{-\{j,k\}},z'_{j},z'_{k},z''_{b},z''_{c})$. Using Strong Pareto we have that $(z_{-\{j,k\}},z_{j},z'_{k},z'''_{b},z''_{c})\mathbf P (e)(z_{-\{j,k\}},z''_{j},z'_{k},z'''_{b},z''_{c})$. Applying Equal Preferences Priority once more we get that $(z_{-\{j,k\}},z_{j},z_{k},z'''_{b},z'''_{c})\mathbf P (e)(z_{-\{j,k\}},z_{j},z'_{k},z'''_{b},z''_{c})$. Finally, by Transitivity we obtain that $(z,z'''_{b},z'''_{c})\mathbf P (e)(z,z''_{b},z''_{c})$. However, if we apply the Minimum Solidarity axiom we have that $(z_{b},z_{c})\mathbf P (e)(z'_{b},z'_{c})$, and because of Linearity we obtain that $(z''_{b},z''_{c})\mathbf P (e)(z'''_{b},z'''_{c})$. Using Separation we get the desired contradiction $(z,z''_{b},z''_{c})\mathbf P (e)(z,z'''_{b},z'''_{c})$.

In the second step of the proof we need to show that whenever there exist $z,z'\in Z^{n}$ such that $\max _{i} \rho _{i}(z_{i}) < \max _{i} \rho _{i}(z'_{i})$, this implies that $z\mathbf P (e)z'$. Let us take now two allocations $z,z' \in Z^{n}$ such that $\max _{i} \rho _{i}(z_{i}) < \max _{i} \rho _{i}(z'_{i})$. By monotonicity of preferences, we can find two allocations $x,x' \in Z^{n}$ in which for all $i\in N, h_{i}=h^{*}, z_{i} P^t_{i} x_{i}$ and $x'_{i} P^t_{i} z'_{i}$. Moreover, there exists $i_{0}$ such that for all $i \ne i_{0}$

$$\begin{aligned} \rho _{i}(x'_{i})<\rho _{i}(x_{i})<\rho _{i_{0}}(x_{i_{0}})< \rho _{i_{0}}(x'_{i_{0}}). \end{aligned}$$

Let $Q=N {\setminus } \{i_{0}\}$ and let us assume a sequence of allocations $(x^{q})_{1 \le q \le |Q|+1}$ such that

$$\begin{aligned}&c^{*}_{i}(x^{q}_{i})=c^{*}_{i}(x'_{i}), \ \ \ \forall i \in Q: \ i \ge q \\&c^{*}_{i}(x^{q}_{i})=c^{*}_{i}(x_{i}), \ \ \ \forall i \in Q: \ i < q, \end{aligned}$$

and for $i_{0}$ let us have

$$\begin{aligned} c^{*}_{i_{0}}(x'_{i_{0}})=c^{*}_{i_{0}}(x^{1}_{i_{0}})<c^{*}_{i_{0}}(x^{2}_{i_{0}}) <\ldots <c^{*}_{i_{0}}(x^{|Q|}_{i_{0}})<c^{*}_{i_{0}}(x^{|Q|+1}_{i_{0}}) =c^{*}_{i_{0}}(x_{i_{0}}). \end{aligned}$$

This implies that $\rho _{i_{0}}(x^{q}_{i_{0}})>\rho _{i_{0}}(x^{q+1}_{i_{0}})>\rho _{q}(x^{q+1}_{q})>\rho _{q}(x^{q}_{q})$, while for all $j \ne q,i_{0}$ we have that $\rho _{j}(x^{q}_{j})=\rho _{j}(x^{q+1}_{j})$. As we have previously proved, for all $q\in Q$ it must be the case that $x^{q+1}\mathbf P (e)x^{q}$. According to the initial assumptions, $z\mathbf P (e)x^{|Q|+1}$ and $x^{1}\mathbf P (e)z'$. Finally, by transitivity we have that $z\mathbf P (e)z'$. $\square$