Pigou–Dalton principle

The Pigou–Dalton principle (PDP) is a principle in welfare economics, particularly in cardinal welfarism. Named after Arthur Cecil Pigou and Hugh Dalton, it is a condition on social welfare functions. It says that, all other things being equal, a social welfare function should prefer allocations that are more equitable. In other words, a transfer of utility from the rich to the poor is desired, as long as it does not bring the rich to a poorer situation than the poor.

Formally,^[1]:67–68 let ${\displaystyle u=(u_{1},u_{2},\dots ,u_{n})}$ and ${\displaystyle u'=(u'_{1},u'_{2},\dots ,u'_{n})}$ be two utility profiles. Suppose that at the first profile:

${\displaystyle u_{1}<u_{2}}$

and at the second profile:

${\displaystyle u_{1}'+u_{2}'=u_{1}+u_{2}}$ and

${\displaystyle u'_{3}=u_{3},u'_{4}=u_{4},\dots ,u'_{n}=u_{n}}$ and

${\displaystyle u_{1}<u_{1}'<u_{2}}$ and ${\displaystyle u_{1}<u_{2}'<u_{2}}$

(so ${\displaystyle u_{1}<u_{1}'<u_{2}'<u_{2}}$ or ${\displaystyle u_{1}<u_{1}'=u_{2}'<u_{2}}$ or ${\displaystyle u_{1}<u_{2}'<u_{1}'<u_{2}}$)

Then, the social-welfare ordering should weakly prefer the second profile $u'$, since it reduces the inequality between agent 1 and agent 2 (and may switch which is richer), while keeping unchanged the sum of their utilities and the utilities of all other agents.

PDP was suggested by Arthur Cecil Pigou^[2]:24 and developed by Hugh Dalton^[3]:351 (see, e.g., Amartya Sen, 1973 or Herve Moulin, 2004).

Examples

• The egalitarian function: ${\displaystyle W(u)=\min(u_{1},u_{2})}$ satisfies PDP in a strong sense: when utility is transferred from the rich to the poor, the value of $W$ strictly increases.
• The utilitarian function: ${\displaystyle W(u)=u_{1}+u_{2}}$ satisfies PDP in a weak sense: when utility is transferred from the rich to the poor, the value of $W$ does not increase, but also does not decrease.
• The function ${\displaystyle W(u)=u_{1}^{2}+u_{2}^{2}}$ violates PDP: when utility is transferred from the rich to the poor, the value of $W$ strictly decreases.

References

1. Herve Moulin (2004). Fair Division and Collective Welfare. Cambridge, Massachusetts: MIT Press. ISBN 9780262134231.
2. Pigou, 1912
3. Dalton, H. The measurement of the inequality of incomes, Economic Journal, 30 (1920), pp. 348–461.