Vertical and horizontal equity: What are they?

This post is dry.  But if we want to talk about policy and fairness we gotta do some of the hard work making sure we understand how our ethical principles can be measured.  See it as part of trying to build more measures to help us understand arguments around policy given what Sen raises here.  So with that in mind lets define equity:

Equity.  Is the word economists unjustifiably confuse with fairness in order to pay lip service to distributional concerns

Ok I’m being a bit of a dork – in all fairness equity is a good start in asking these questions, but we have to see these measures as only a start!

At the most basic level, when we think about output/income and its distribution in society we consider the average of the income distribution (the mean) and its dispersion (the variance).  If incomes are rising over time as they have been for 200 years, then the variance also rises so we normalise such measures.  This is where inequality measures like the Gini coefficient come from.

The idea of (income) equity goes a step further than just describing the general distribution of income – it considers what happens when we impose an external policy that changes that distribution.  It measures a couple of principles that we may – or may not – value when applying a policy that changes the distribution of income:

  1. Vertical equity:  Captures the proportionality of the system applied – if we introduce taxes are people with higher initial incomes paying proportionally more, if we introduce transfer payments are people with lower initial incomes receiving proportionally more?
  2. Horizontal equity:  When we have two individuals we see as “equals” does this policy system treat them the same way?

With taxes and transfers these measures involve comparing the way people are treated by the tax-transfer system based on a view on what constitutes “equals”.  Specifically, these two concept can only fit together without conflict when looking at income if equals are defined as people with the same income.

Now in this post I will concentrate only on Vertical Equity – we can do Horizontal Equity another time!  And in line with my desire to be a bit more useful I want to focus on how we might measure these concepts, and what we are assuming when we do.

Wait, what is vertical equity?

Vertical Equity (VE) is the idea that those with greater access to resources SHOULD pay PROPORTIONALLY more – not just more, but more as a percentage of their income.  You may personally agree or disagree, that is fine, that is just what it is.

Given that idea, there is a certain level of proportionality that is desired – so there is some optimal level of VE, which could be higher or lower than it is.  Again, this is a value statement so I just leave that as is.

The key take away is that VE is a measure of the proportionality of the tax-transfer system, and we don’t necessarily want more or less VE – we need to come up with a set of values that articulates how much VE we believe is appropriate.

Now we need to figure out how to measure this concept assuming that we can boil everything down to income!

Stop.  You can calculate a Gini coefficient for pre-tax and transfer income and post-tax and transfer income and just look at the difference.  Why complicate matters.

Look I feel you random comment – but things are never that simple.

Let us put Horizontal Equity (HE) to the side for now, as it is an epic can of worms.  Instead, let’s think about Vertical Equity (VE) and fairness.

At face value we are taking proportionally more from those with more and giving proportionally more to those with less, it sounds perfectly fair.  But lets do a thought experiment regarding this idea to see how – even before asking about family structure and differences in need (what income for a couple is equivalent to an income for a single person for meeting needs?) – this concept alone can twist into unfairness.  Imagine a parade of dwarves – not real ones, this income one here:

All along this parade we have people organised by their incomes – from lowest income to highest income.

A Gini coefficient gives us a summary measure of this parade, by looking at the absolute distance each of these people are from all the other individuals in the parade.

So say we walked up to this parade and took the person at the back (the poorest person) and swapped their income with the person at the front (the richest person).

What happens to the Gini coefficient after this change?  Nothing – everyone is still the same absolute distance from other people.

But what about VE?  The wealthiest person has paid proportionally more than the poorest person (sacrificing nearly all their income and giving it to the poorest person) and so there is a disproportional transfer which meets our definition of vertical equity!

So even though the Gini coefficient is unchanged, this policy change refers to a progressive transfer and so produces vertical equity.

But there is an issue.  If we have no reason for valuing one person above another (anonymity) then we can see that this transfer has just swapped the position of two individuals – specifically there has been no reduction in income inequality, instead the policy has just changed who is in what position.

If we were to reorder the parade of dwarves into the same ordering by after policy income, the person at the top who lost all their income would move to the bottom while the person at the bottom who gained all the riches of the top person would move to the top.  As a result, their relative position in the income parade changed!  This is termed reranking.

As a result, we can have a progressive transfer that takes from those with higher incomes and gives to those on lower incomes – but if that transfer changes the relative incomes of individuals (thereby leading to reranking) it leads to a smaller reduction in overall income inequality than this progressive transfer alone would suggest.

Reranking is often termed Horizontal Inequity (although as we get into in the future, it is only one type of HI) as given the assumption of anonymity a transfer that is so large it swaps the relative position of a higher income individual and lower income individual appears inequitable.  Given VE simply states that those on higher incomes should pay proportionally more it does not rule out circumstances where they pay so much more that they end up worse off than people who were below them on the income parade.  Reranking captures that difference.

I thought you were going to measure these things?

As stated, reranking involves allowing the income parade to be reordered after a tax-transfer policy is introduced.  So reranking compares the income parade before this reordering, with the new post tax-transfer incomes to the income parade after this reordering.

When measuring VE and reranking we take advantage of the fact that these orderings correspond to different income parades, and calculate summary statistics that represent these parades.

The initial income parade can be thought of in terms the share of cumulative income as we move from the poorest to the richest person – the Lorenz curve.  The Gini coefficient is then a summary measure of this Lorenz curve.

When a progressive transfer is introduced we can keep everyone in the same order, but the density of income will rise towards the bottom of this curve.  The summary measure of this is the concentration coefficient from this curve (Note:  The Gini is just a concentration coefficient too but ordered from lowest to highest income in whatever income we are discussing).

Once reordering occurs based on the new incomes we get a new Lorenz curve for post tax and transfer income.  The Gini coefficient for this can be thought of as our summary measure of that distribution.

Vertical equity is then shown with regards to the difference between the Lorenz curve for market income and the concentration curve for disposable income – these are simply cumulative income of each measure for the same ordering of the population (by market income).  The difference between these curves tells us if people who had higher market income were paying proportionally more!

Reranking is then the difference between the concentration curve for disposable income and the Lorenz curve for disposable income – or in other words the same disposable income numbers, but ordered in different ways.  If no individuals changed position these would be the same thing!

You’ll notice something.  VE takes us from the original Gini to another number, then Reranking takes us from that number to the new Gini.  As a result, the change in the Gini coefficient after we introduce policy (commonly called the redistributive effect) can be nicely split into a change due to VE (transfer from richest to poorest) and reranking.

How can we think about this?  Say we have some target redistribution and we want to to achieve for some reason.  A system that has less reranking can do that with a less progressive tax-transfer system than a system that has more reranking.

A super clear description of this can be found here.  And I’ve discussed it here.  For clarity I’ve also been giving visual examples in a presentation of my results -> Nolan VE-HI presentation.

Cool, so the transformation of market income to disposable income – given the same ordering – is VE.  I have an idea that ….

Wait.  Urg things get a bit ugly here.

This is an estimate of VE, but eagle eyed readers may noticed something is missing.  Why doesn’t the imposition of policy change MARKET income as well.

What are our “counterfactuals” here.  One scenario with policy and one without.  A world without tax and transfer policies will have a different market distribution of income as well … which implies that VE is wrong.

Now people who write this literature (eg people like me) make the claim that we are looking at policy changes and considering the CHANGE in VE (not the level).  As a result, we assume that the distribution of market income does not change due to this.  This implies that we have to be a little bit careful with the measure.  I like the way it is described here.

The fixed-income approach proposed by Kasten et al. (1994) provides a straightforward framework to isolate these effects. Widely used in the literature on income redistribution and tax policy, this method provides a baseline for the identification of policy effects by keeping the distribution of market incomes fixed and by applying the tax and benefit schemes of different periods to this distribution of reference.

It is important to recognise, however, that this approach only isolates what we could call the immediate policy effects as it does not account in any way for behavioural responses to these policy reforms.

Well, what is the bias then?

Great question.

So for this we need to figure out how the introduction of the tax-transfer system would change market income.  Or if we focus solely on changes, how the change in the system would change market income.  Here we need to answer questions such as:

  • Who faces the incidence of tax due to these changes
  • How do patterns of employment and capital expenditures change following these changes
  • How will the characteristics of households change following these changes

This is more than a question of “will wages rise”.  It is a question of “how will the distribution of wages be influenced by the change in the tax-transfer system”.  And these will depend on the type of policy that was used to introduce a change in progressivity – not just the fact that the tax-transfer system has become more progressive!

For some hypothetical examples where we are keeping the average tax rate fixed but changing progressivity we can say:

  1. If redistributing income does not change any behaviour and the full payment of tax-transfers occurs by households then nothing happens … this is the “fixed income”.
  2. If redistributing income does not change any behaviour, the tax-transfer system is levied on individuals, the tax-transfer system is linear in income, and the actual tax payment is equally split between households and firms then a more PROGRESSIVE tax system will INCREASE pre-tax income inequality … and thereby exaggerate the amount of VE as the “true” inequality in pre-tax incomes without a progressive tax system would be lower.
  3. If by redistributing income we reduced investment in the capital used by low income earners then the amount of VE caused by the tax system would be overstated.
  4. However, if by redistributing income we increased the incentive for low income earners to enter the labour market the amount of VE caused by the tax system change would be understated.

Man, those examples hurt my head – and it is not very evident which way the bias goes in a very complex system (both the economy and tax system) such as the ones we observe in reality.  I have guesses I would hazard, but I’m not going to.

I haven’t seen this type of work undertaken for VE (Kasten et al (1994) discuss it with reference to EMTRs) – but that doesn’t mean it isn’t important 😉

Ok, what about HE – I think that is very important to me

Nice question.  I have no doubt Horizontal Equity (HE) is important to you, real important.  Constantly we hear about discrimination by sex, race, and even age.  On top of this we may view treating those with the same initial income as violating horizontal equity.

But this post is already long, so I’ll leave this discussion for another time – I tell you what though, it will be a doozy … or at least long again!