The paper I’m presenting this week is focused on modeling tradeoffs between equity and effectiveness in the public sector, and one of the most important parts of an equity model is how it actually measures inequity. In fact, the main reason the author wrote this paper is because many of the equity models that came before it either ignored equity considerations entirely or accounted for them in a “significantly flawed” way (Mandell, 1991). As a result, I thought it would be helpful to use this blog post as an introduction to the equity discussion and to describe what makes a “good” inequity measure.

First, it’s worth noting that there are a lot of different ways to measure inequity. According to Mandell (1991), some of the most common inequity measures used in operations research models include “the minimum or maximum level of service, the range between the minimum and maximum levels of service, and the sum of absolute deviations from the mean level of service”. Additionally, there are also other measures such as the Gini coefficient, the variance of the logarithms, and Theil’s index (Allison, 1978). While there are important differences between all of these measures, one nearly universal characteristic is that they’re zero when service is completely equal and positive when it’s not.

Beyond this basic criterion, the second important characteristic a measure should have is what’s called **scale invariance**. In terms of income inequality, scale invariance means that multiplying everyone’s income by a constant doesn’t change the inequity measure’s value (Allison, 1978). Unfortunately, many measures fail to satisfy this criterion. For example, if everyone’s income is doubled, the range between the minimum and maximum levels of income will also be doubled, implying that the range measure is not scale invariant.

The third criteria, which narrows the list of acceptable inequity measures even further, is the **principle of transfers**. This principle states that the transfer of income from a poorer person to a richer person should increase the value of the inequity measure regardless of the economic status of the two individuals or the amount that is transferred (Allison, 1978). This criteria eliminates the sum of absolute deviations from the mean from consideration because any transfers strictly above or strictly below the mean wouldn’t change the measure’s value. For example, if the mean income in a population is $600 and there is a transfer of $150 from someone with $200 to someone with $300, the inequity measure would still be $700 after the transfer even though the person that originally had $200 is clearly much worse off [1].

Using these criteria to evaluate potential inequity measures, Allison (1978) finds that only three measures satisfy all of them. These three measures are the coefficient of variation, Theil’s index, and the Gini coefficient (Allison, 1978). Although there are a few differences to consider between these three measures, I’ll save that part of the discussion for the presentation on Thursday.

[1] As you can see from the calculation below, transferring income from the person with $200 to the person with $300 does not change the value of this inequity measure.

Before the transfer: Sum of deviations from the mean = (600 – 200) + (600 – 300) = $700

After the transfer: Sum of deviations from the mean = (600 – 50) + (600 – 450) = $700

References

Allison, P. (1978). Measures of inequality. *American Sociological Review,* *43*(6), 865-880.

Mandell, M. B. (1991). Modelling effectiveness-equity trade-offs in public service delivery systems. *Management Science*, *37*(4), 467-482.

I had lots of fun reading this. This blog post makes me really interested to the upcoming presentation of yours.

In regard of principle of transfers I got a quick question; why do we need to assume large population? For me it seems like the mean wouldn’t be changed anyway by a transfer inside of the population.

Thanks!

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Thanks for catching that! You’re actually right. The mean wouldn’t change. I’ll update the blog post so no one else is confused by that.

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