Marginal Tax Rate Structure Matters for Wage Inequality

Have you ever had a conversation that completely changed your thinking on a subject?  My thinking on inequality and taxes was radically altered about 10 years ago when my office mate and I were studying for an exam in Robin Boadway's PhD course in Public Finance, and we got to discussing tax rate structure.  During that conversation I was convinced that poorly designed marginal tax rates can increase inequality - so much so that I tend to think of progressivity in tax structure as being as a marginal issue as it is a progressive one (to the point where I forget to remember that this is an idiosyncratic view).  Strictly speaking, it isn't - as defined, progressivity is measured in terms of average tax rates.  So ECON 1000 students - don't use term progressive as I have here - you will get the question wrong.

But marginal tax rates matter in terms of inequality beyond their effects on average rates.  And I can prove it, through the use of a simple stylized example:

Consider an economy made up of farmers, operating in a small open economy who produce a good.  The price of this good is determined at the international level, and the number of farmers in our economy is small enough such that our production levels do not alter the world price.

Each farmer has a utility function U = w - l, where w is the after-tax wage they earn from selling their good and l is the disutility they receive from working (measured in dollars).

This economy features two types of workers in equal proportion: Type A and Type B.  Every farmer has the options of work easy or work hard, with the following payoffs:

Work Easy

A: Wage of $20,000 - Disutility of labour $0

B: Wage for $20,000 - Disutility of labour $0

Work Hard

A: Wage of $100,000 - Disutility of labour $25,000

B: Wage of $50,000 - Disutility of labour $25,000

In an economy without taxes, both groups receive a higher level of utility by working hard, so A's earn a labour income of $100,000 (and a utility of 75,000) and B's earn a labour income of $50,000 (and a utility of 25,000).

A 'Progressive' Tax is Introduced

Now suppose the residents of this country introduces a tax that features a progressive average rate structure.  In any analysis such as this we need to consider where the money goes - the simplest version is to assume the money is spent on items that do not have a transfer effect (the money is spent on, non-rival, non-excludable public goods such as roads and national defense.  Or if you prefer, they could be spent on items such as foreign aid or building statues).

Average Tax Rate Structure

The average tax rate for this economy is as follows:

• 0% from $0 to $40,000
• Rising from 0% to 20% from $40,000 to $50,000
• 20% from $50,000 onwards

Thus the average tax rate has two flat portions, and a progressive portion in the $40,000-$50,000.  Nice and progressive, with the tax burden being borne by higher income earners.  This tax structure looks as follows when considering marginal rates.

Marginal Tax Rate Structure

The marginal tax rate for this economy is as follows:

• 0% from $0 to $40,000
• 100% from $40,000 to $50,000
• 20% from $50,000 onwards

Sure, this marginal tax rate structure looks strange, but remember - progressivity, as defined, is an average tax rate story.  The oddness of this marginal tax structure shouldn't matter.  But it does.

Choice for A

If A works easy, he earns $20,000 and has a utility of 20,000.  If A works hard, he earns $50,000 before tax.  However, after tax his income is $40,000 since the portion from $40,000-$50,000 is taxed at 100%.  The disutility from working hard is 25,000, so the overall utility from 'work hard' is 15,000.  Therefore A is better off choosing work easy earning $20,000.

Choice for B

If B works easy, she earns $20,000 and has a utility of 20,000.  If B works hard, she earns $100,000 before tax.  However, after tax her income is $80,000 since the portion from $40,000-$50,000 is taxed at 100% (so $10,000 is lost here to taxes)  and the portion from $50,000-$100,000 is taxed at 20% (so another $10,000 is lost to taxes, making a total of $20,000).  The disutility from working hard is 25,000, so the overall utility from 'work hard' is 55,000 (100,000 - 20,000 - 25,000).  Therefore B is better off choosing work hard an earning $100,000.

Increased Inequality

Prior to the tax being put in place, A earned $50,000 more than B, and the ratio of A's income to B's income was 2-to-1.  A's utility was 50,000 more than B's, and the ratio of their utilities was 3-to-1.

After the tax is put in place, from a pre-tax perspective A earns $80,000 more than B, and the ratio of A's income to B's income is 5-to-1, rather than 2-to-1!

But we should consider inequality in terms of after-tax income, not pre-tax income.  From an after-tax perspective, A earns $60,000 more than B, and the ratio of A's income to B's income is 4-to-1.  Still increased inequality.

Inequality is normally defined in terms of income, but you could argue what really matters is happiness, or utility.  Under the new system, A's utility is 35,000 more than B's, and the ratio of their utilities is 2.75-to-1.  Only if we stretch the definition of inequality, can we suggest that inequality has decreased.

How Did This Happen

Prof. Milligan provided a helpful reminder of why progressivity is defined in terms of average rates:

If I have 100K income, my welfare changes if I pay 20K or 40K taxes on that 100K. This is the average tax burden. if I pay 99% or 1% on hypothetical dollar #100,001, it does not change my current welfare given that I am at $100K.

Note the term given here - when progressivity is measured, the pre-tax labour income is taken as a given.  However, a person's pre-tax income levels are determined in part by marginal tax rates, as these influence a person's decision on how much to work! 

Once the ceteris paribus assumption is removed, marginal tax rates can and do have significant welfare effects beyond how they affect average rates.  Progressive average tax rates, when designed poorly, can have regressive effects due to regressive marginal rates.

Of course, my stylized example and tax rate structure are unrealistic in the real workd.  No argument here - it is used to hilight the importance of marginal rates.  But given that there are many instances in the real world of lower income people facing marginal effective rates of over 100%, it is not that unrealistic.  Poorly designed tax and transfer systems lead to welfare traps than can increase inequality, despite their best intentions.

Comments

[Jason]

Your 'Choice for B' section seems to have two copy and paste errors:
- if B works hard she earns $50,000 before tax (should be $100,000)
- Therefore A is better off choosing work hard... (should be 'B')

[Mike Moffatt]

Thanks Jason - will fix!

[Jay Dixon]

Mike, partly last post, partly this post:
On the last one- I agree with most of the posters on the problems in including the CPP,EI. You say that a lot of people don't believe the CPP will be around when it's their turn, which makes it more akin to income tax when it comes to their labour supply decisions. Perhaps, but isn't that more of an efficiency issue than equity concern?

The rates from your last post make the tax system look regressive over the second portion, which is good from an efficiency stand point (I'm thinking Mirlees). For the discussion above, the point seems to be not that regressive marginal rate structures cause inequality, but discontinuous or bumpy structures do.

[Mike Moffatt]

"Perhaps, but isn't that more of an efficiency issue than equity concern?"

This is where I screwed up big time in my previous post, by not explaining the efficiency/equity tradeoff and my thinking. It made sense in my head, but I didn't write it down, and Apple hasn't come up with a mind-reading app yet.

The idea is as follows - a tax system that is, by nature, inefficient at low-medium levels of income is by it's very nature also inequitable. The reason is that high marginal tax rates deter work effort. If the lower income person is deterred from earning more, while the rich person is not (or at least is to a lesser degree), the gap between rich and poor is, ceteris paribus, going to widen. I completely understand why Prof. Milligan was so confused by my previous post (or just concluded I was an idiot, which would have been a fair assessment).

"the point seems to be not that regressive marginal rate structures cause inequality, but discontinuous or bumpy structures do."

Not necessarily - I could tweak the figures and income levels such that the marginal rate is continuous (say it increases from 0-100% over the range of $30,000 to $40,000) and then back down to 20% from the range $50,000 to $60,000 and still make it all work.

I like to think of marginal tax rates like being similar to hill inclines. Biking up a steeper incline is more difficult, and a tax system does that features regressive *marginal* rates makes the poor-to-middle class face steeper inclines than the rich face.

[Jay Dixon]

Mike: I think by bumpy, I meant to say non-monotonic-it doesn't matter if it's progressive, regressive or flat overall, it's that it shifts.

There is a database consisting of a random sample of tax returns in the US and somewhere in the bowels of Statcan that allows for the calculation of effective marginal tax rates based on a richer set of demographic characteristics. I've seen it used to identify *real* (as opposed to statuatory) changes in marginal effective tax rates. Blanking on the papers/names etc. Do you know of it? I thought it may be of some use, if you weren't aware of it. If you are, could you tell me?