# Income neutrality

## What is income neutrality?

We are only able to rank societies with the same number of people and the same total income using the transfer principle. However, to know whether inequality in a country has been growing over time, our measure must be applicable when comparing societies with different total incomes.

An inequality measure that satisfies **income neutrality** exhibits the same level of inequality when the entire income distribution is scaled up or down by the same number. For example, an inequality measure that satisfies income neutrality would not register any change if every individual’s income increased by 10%.

We can combine income neutrality with the transfer principle to rank societies with different levels of total income.

### Exercise

Is the rich/poor ratio an inequality measure that satisfies income neutrality?

Yes, the rich/poor ratio is an example of an inequality measure that satisfies income neutrality. If the total income of a society is scaled by a constant, the average income of all deciles will also be scaled by that constant. Thus the ratio of tenth decile to the first decile will remain the same.

### Exercise

Let \(X = (10, 30, 60)\) and \(Y = (5, 16, 29)\).

In this example, each society has the same number of people (3) and different total income (100 and 50).

Can we use income neutrality and the transfer principle to determine which society has more income inequality?

The answer is yes.

We can start by multiplying all the income values in \(Y\) by a factor of 2:

\[Y \times 2 = (5, 16, 29) \times 2 \rightarrow Y^{\prime} = (10, 32, 58)\]Now we can apply the transfer principle to move from \(X\) to \(Y^{\prime}\).

Take 2 away from the richest person in \(X\) and give it to the middle person:

\[(10, 30 + 2, 60 − 2) \rightarrow (10, 32, 58) = Y^{\prime}\]We conclude that \(X\) has more income inequality than \(Y^{\prime}\) and thus more income inequality than \(Y\) by the transfer principle.

### Exercise

Let \(X = (32, 63, 70)\) and \(Y = (10, 20, 25)\).

In this example, each society has the same number of people (3) and different total income (165 and 55).

Can we use income neutrality and the transfer principle to determine which society has more income inequality?

We can start by multiplying all the income values in \(Y\) by a factor of 3:

\[Y \times 3 = (10, 20, 25) \times 3 \rightarrow Y^{\prime} = (30, 60, 75)\]Now we can apply the transfer principle to move from \(Y^{\prime}\) to \(X^{\prime}\).

Take 5 away from the richest person in \(Y^{\prime}\) and give it to the middle person:

\[(30, 60 + 5, 75 − 5) \rightarrow (30, 65, 70) = Y^{\prime}\]Next, to reach society \(X\), we can take away 2 from the middle person and give it to the poorest person:

\[(30 + 2, 65 − 2, 70) \rightarrow (32, 63, 70) = Y^{\prime}\]We conclude that \(Y\) has more income inequality than \(X\) because we can move from society \(Y^{\prime}\) to society \(X\) by making equalizing and order-preserving transfers.