# Transfer principle

## Why move away from simple measures?

The rich/poor ratio and similar measures are crude and insensitive to changes within and across most deciles. It might be preferable to use all the information available to us—the entire income distribution. To do this, we can produce a set of principles that might lead us to a sensible inequality measure. We will consider three such principles: the transfer principle, the income neutrality principle, and the population neutrality principle.

## What is the transfer principle?

A transfer of income is **equalizing** and **order-preserving** if it shifts income from a richer individual to a poorer individual without changing anyone’s rank in the income distribution.

An inequality measure satisfies the **transfer principle** if equalizing and order-preserving transfers result in a decline in measured inequality.

We can use the transfer principle to compare inequality across societies: if you can move from society \(X\) to society \(Y\) through a series of equalizing, order-preserving transfers, then society \(Y\) has less inequality than society \(X\).

The rich/poor ratio is an example of an inequality measure that does not satisfy the transfer principle. If income is moved from someone in the second highest decile to someone in the third highest decile, it will not change the value of the rich/poor ratio.

### Exercise

Let \(X=(15, 16, 69)\) and \(Y=(17, 34, 49)\).

In this example, each society has the same number of people (3) and the same total income (100).

Can we move from society \(X\) to society \(Y\), or from \(Y\) to \(X\), using only equalizing, order-preserving transfers?

The answer is yes.

Applying the transfer principle:

Take 18 away from the richest person in \(X\) and give it to the middle person:

\[(15, 16 + 18, 69 − 18) \rightarrow (15,34,51)\]Then, take 2 away from the richest person in \(X\) and give it to the poorest person:

\[(15 + 2, 34, 51 − 2) \rightarrow (17, 34, 49) = Y\]We were able to move from society \(X\) to the more equal society \(Y\) by making equalizing and order-preserving transfers. Thus, we can conclude that \(X\) has more income inequality than \(Y\) by the transfer principle.

The transfer princple can only be used to compare two societies if they have the same number of people and the same total income. Even in this case, it may not be possible to rank societies, as shown in the following exercise.

### Exercise

Let \(X = (15, 40, 45)\) and \(Y = (17, 34, 49)\).

In this example, each society has the same number of people (3) and same total income (100).

Can we move from society \(X\) to society \(Y\), or from \(Y\) to \(X\), using only equalizing, order-preserving transfers?

The answer is no.

Moving from \(Y\) to \(X\), we can start by transfering 4 units from the richest person in \(Y\) to the middle person.

\[(17, 34 + 4, 49 − 4) \rightarrow (17, 38, 45)\]This was an equalizing, order-preserving transfer. However, to reach society \(X\), we must now transfer 2 units from the poorest person to the middle person.

\[(17 − 2, 38 + 2, 45) \rightarrow (15, 40, 45) = X\]Moving income from lower to higher earners is not equalizing. Therefore, we cannot move from society \(Y\) to society \(X\) through equalizing, order-preserving transfers. We also cannot go from \(X\) to \(Y\) with a sequence of equalizing, order-preserving transfers, as this move would require us to add 4 units of income to the highest earner in \(X\).

We conclude that we are unable to say whether \(X\) or \(Y\) has more inequality using the transfer principle.