On Measuring Welfare ‘Behind a Veil of Ignorance’

Abstract

How should we rank different income distributions? Should we adopt the Rawlsian criterion that focuses on the minimal income in the distribution? Or should we rather maximize the geometric mean, a criterion advocated by gamblers and welfarists alike? This paper microfounds these two criteria by showing that each of them can be obtained by granting veto rights to the members of society ‘behind a veil of ignorance’, where society is represented by the set of regular utilities (Hart 2011). The Rawlsian maximin criterion is obtained by granting each member of society a right to veto acceptance of any candidate income distribution, while the geometric-mean criterion is obtained by granting instead a right to veto rejection of income distributions. This way, the proposed method circumvents the need to arbitrarily choose a representative agent when ranking income distributions. The Rawlsian maximin criterion is further shown to be robust to extending the set of utilities that constitutes “society” to all risk-averse utilities, while the geometric-mean criterion is not as robust.

Appendix: proofs

Proof of part 1 of Proposition 1

For any income distribution (random variable) z, let $L(z)\equiv \min z$ and let $M(z)\equiv \max z$. Notice that for any income distribution z, and every $u\in U^{*}$, u accepts z at any $w\le L(z)$. Now let $w_{0}=L(z)+\epsilon$ for some arbitrary small and positive $\epsilon$, and let $p_{L}$ denote the non-zero probability of $L(z)\in$ supp$\left( z\right)$. Now let $\hat{u}_{\alpha }\left( w\right) :=\left( \log \left( \alpha w\right) -1\right) /e$ for $w\le \frac{1}{\alpha }$ and $\hat{u}_{\alpha }\left( w\right) :=-\exp \left( -\alpha w\right)$ for $w\ge \frac{1}{\alpha }$; then $\gamma _{\hat{u}_{\alpha }}(w)=1$ for $w\le \frac{1}{\alpha }$ and $\gamma _{\hat{u}_{\alpha }}(w)=\alpha w$ for $w\ge \frac{1}{\alpha }$ and so $\hat{u}_{\alpha }\left( w\right) \in U^{*}$ for each $\alpha >0$.¹⁴ For every $\alpha >\frac{1}{L(z)}$, we get that $\hat{u}_{\alpha }\left( w\right)$ rejects distribution z at $w_{0}$ if and only if $\hat{u}_{\alpha }(w_{0})\ge E[\hat{u}_{\alpha }(z)]$. A sufficient condition for a rejection of z by $\hat{u}_{\alpha }\left( w\right)$ at $w_{0}$ is then

$$\begin{aligned} -\exp \left( -\alpha w_{0}\right)= & {} -\exp \left( -\alpha \left( L(z)+\epsilon \right) \right) \\\ge & {} p_{L}\left( -\exp \left( -\alpha L(z)\right) \right) +\left( 1-p_{L}\right) \left( -\exp \left( -\alpha M(z)\right) \right) . \end{aligned}$$

Dividing by $-\exp \left( -\alpha L(z)\right)$ and setting $\alpha \left( \epsilon \right) =-\frac{\log \left( p_{L}\right) }{\epsilon }$, we get that $\hat{u}_{\alpha \left( \epsilon \right) }\left( w\right)$ rejects distribution z at $w_{0}$, because $p_{L}\le p_{L}+\left( 1-p_{L}\right) p_{L}^{\frac{M(z)-L(z)}{\epsilon }}$. Letting $\epsilon \rightarrow 0$, we get that $\forall \epsilon >0$, $\exists \alpha \left( \epsilon \right)$ such that every $\hat{u}_{\alpha }\left( w\right) \in U^{*}$ with $\alpha >\alpha \left( \epsilon \right)$ rejects z at wealth level $L(z)+\epsilon$. Consequentially, income distribution x is unanimously accepted at w if and only if $w\le L(x)$, and income distribution y is unanimously accepted at w if and only if $w\le L(y)$, and so x unanimously-acceptance dominates y if and only if $L(x)\ge L(y)$. $\square$

Lemma 1

Let $u_{1},u_{2}\in U_{2}$ be two utility functions with absolute risk aversion coefficients $\rho _{1}$ and $\rho _{2}$ respectively, where $\rho _{1}\left( w\right) \ge$ $\rho _{2}\left( w\right)$ for every $w>0$. Then for every income distribution z with finite support in $\mathfrak {R}^{+}$ and every $w>0$, if $u_{2}$ rejects z at w then $u_{1}$ rejects z at w too.¹⁵

Proof

Let $\psi$ be such that $u_{1}=\psi \circ u_{2}$; then $\psi$ is strictly increasing (since $u_{1}$ and $u_{2}$ are such), and concave (since for every $w>0$ we have $\psi \prime \left( u_{2}\left( w\right) \right) =u_{1}^{\prime }\left( w\right) /u_{2}^{\prime }\left( w\right)$, hence $\left( \log \psi \prime \left( u_{2}\left( w\right) \right) \right) ^{\prime }=\left( \log u_{1}^{\prime }\left( w\right) \right) ^{\prime }-\left( \log u_{2}^{\prime }\left( w\right) \right) ^{\prime }=-\rho _{1}\left( w\right) +$ $\rho _{2}\left( w\right) \le 0$, and so $\psi ^{\prime \prime }\le 0$). Therefore $E[u_{2}(z)]\le u_{2}\left( w\right)$ implies by Jensen inequality and by the monotonicity of $\psi$ that $E[u_{1}(z)]=E[\psi \left( u_{2}(z)\right) ]\le \psi \left( E[u_{2}(z)]\right) \le \psi \left( u_{2}\left( w\right) \right) =u_{1}\left( w\right)$. $\square$

Proof of part 2 of Proposition 1

For any income distribution z and any wealth level $w>0$, I will show that z is unanimously rejected at w if and only if $\log (w)\ge E[\log (z)]$. This means that unanimous rejection by all $u\in U^{*}$ implies and is implied by rejection by the log utility, hence rejection-based domination boils down to the preferences of the log utility (i.e., x rejection-based dominates y if and only if $E[\log (x)]>E[\log (y)]$). The “only if” direction is immediate: if $\log (w)<E[\log (z)]$, then $u_{l}\equiv \log (w)\in U^{*}$ accepts z at w, thus z is not unanimously rejected at w. The proof for the “if” direction is less straightforward. Lemma 8 in Hart (2011) implies that if u is increasing and strictly concave, and $\lim \limits _{w\rightarrow 0^{+}}u(w)=-\infty$ (as implied by property (R4) of $U^{*}$), then $\lim \limits _{w\rightarrow 0^{+}}\gamma _{u}(w)\ge 1$. From property (R3) of $U^{*}$ (weakly increasing $\gamma _{u}(w)$), it follows that $\forall u\in U^{*},$ $\gamma _{u}(w)\ge 1$ for all $w>0$. Thus, Lemma 1 implies that $\forall u\in U^{*}\subset U_{2}$, if $u_{2}=u_{l}$ rejects z at w then $u_{1}=u$ rejects z at w too (because $\gamma _{u}(w)\ge 1=\gamma _{u_{l}}(w)\Rightarrow \rho _{u}(w)\ge \rho _{u_{l}}(w)$), i.e., $\log (w)\ge E[\log (z)]$ implies that z is unanimously rejected at w. $\square$

Proof of part 1 of Proposition 2

The proof of part (1) of Proposition 1 applies here too, for both statements (a) and (b), because $\hat{u}_{\alpha }\left( w\right) \in U^{*}\subset U_{3}\subset U_{2}$, hence the fact that every $\hat{u}_{\alpha }\left( w\right)$ with $\alpha >\alpha \left( \epsilon \right)$ rejects z at wealth level $L(z)+\epsilon$ (see the proof of part (1) of Proposition 1) is relevant here too, implying that income distribution x is not unanimously accepted at w whenever $w>L(x)$; and at the same time it still holds that for any income distribution z, and every $u\in U_{2}$ (hence also every $u\in U_{3}\subset U_{2}$), u accepts z at any $w\le L(z)$.$\square$

Proof of part 2 of Proposition 2

• (a) For any income distribution z and any wealth level $w>0$, I will show that income distribution z is unanimously rejected at w if and only if $w\ge E[z]$ (this would then imply that $x\ge _{RD_{2}}y$  if and only if $E[x]\ge E[y]$). Let $u_{neut}\in U_{2}$ denote the risk-neutral utility function. The “only if” direction: if $w<E[z]$, then $u_{neut}$ accepts z at w, thus z is not unanimously rejected at w. The “if” direction: Lemma 1 implies that $\forall u\in U_{2}$, if $u_{2}=u_{neut}$ rejects z at w then $u_{1}=u$ rejects z at w too (because $\gamma _{u}(w)\ge 0=\gamma _{u_{neut}}(w)\Rightarrow \rho _{u}(w)\ge \rho _{u_{neut}}(w)$), i.e., $w\ge E[z]$ implies not only that $u_{neut}$ rejects z at w, but also that z is unanimously rejected at w.

• (b) The same proof of (a) holds, substituting $U_{2}$ with $U_{3}$ and $\ge _{RD_{2}}$ with $\ge _{RD_{3}}$. $\square$