Does the Shadow of the St. Petersburg Paradox Fall Upon Poverty Indices?

Abstract

The lessons I (Gana 2014) have learned from the vivid contrast between the mean and median fair values (Hayden and Platt 2009) of the game (Bernoulli 1738) underlying the St. Petersburg Paradox (SPP) are applied, by implicating the median, to modify two unidimensional poverty indices: the Sen (1976, 1979) index and the Palma (2011) ratio. By implicating the median in economic analysis when necessary, the SPP urges us to consider the economic and ethical consequences of our interpretation of the complex idea called “probability” (Ergodos 2014) in the pursuit of the political end which Aristotle (circa 350 BC), at the very beginning of his Nicomachean Ethics, defined as the “good for man”. The seminal Sen index, as modified by Shorrocks (1995) and decomposed by Osberg and Xu (2000), is S_P = H_poor × |P| × (1 plus G_poor), where H_poor is the proportion of the population (the “poor”) with incomes below a poverty line, P is the average income gap ratio ( 0) of the rich relative to the wealth line, and G_rich is the Gini coefficient of the censored wealth gap ratios for the population. S_P and S_R are modified, to S_P* and S_R*, respectively, by replacing |P| and R by P* ≡ |min {P, P_med}| and R* ≡ min {R, R_med}, respectively; where P_med and R_med are the median poverty and wealth gap ratios, respectively. The modified indices are used to modify the Palma ratio to S_R* / S_P*. Economic policies targeting these modified indices may benefit the “missing poor” living in the shadows of society incompletely touched by indices that only take mean values as inputs. That is, only accounting for mean poverty gaps may well benefit the “richer” among the poor at the expense of the “poorer” among the poor. Furthermore, because it is better to be rich than poor (S_R* / S_P* > 1), alleviation of poverty may well dependent on also targeting the contrast between the rich and the poor. Using the modified indices to alleviate poverty will be a bit more in line with the critically important moral heuristics, and demands, of Rawls (1971, 2001) and Sen (1992), because the modifications give greater weight to the poorest of the poor.