Many people advocate progressive taxation as the most "fair" tax because it falls most heavily on those best able to afford it. But "fairness" is a merely a subjective platitude. The real question is, does progressive taxation work? All taxes have a negative impact on economic activity, but are necessary to generate government revenue. I define the tax's "efficiency ratio" as the ratio of the funds generated to its negative economic impact. Thus asking whether progressive taxation "works" means comparing its efficiency ratio with a flat tax.

First it is worth observing that even a flat tax is progressive in absolute terms. Those who make N times more money, pay N times as much tax. The rich pay more, the poor pay less. But progressive taxation refers not to absolute dollars but to progressive percentage rates. Thus those who make N times more money, pay much more than N tines as much tax. The question is, which tax has a better efficiency ratio?

The issue hinges on the notion of marginal utility. This is the simple concept that people satisfy their most important needs first. For example, when you work every hour you get paid the same amount. But the first few dollars you earn go toward rent, food and other necessities. Each subsequent dollar you earn is used to satisfy less important needs. Thus even though each dollar you earn is the same, the utility they provide is not the same. The first dollars you earn provide high utility, and each subsequent dollar provides less and less utility. This means that the subjective value of each subsequent dollar you earn decreases.

The work you do is itself a disutility. For simplicity, the disutility you incur from each hour of work can be viewed as roughly the same, though it may increase slightly with each subsequent hour. Either way, marginal utility explains why people eventually stop working. This serves as a theoretical explanation for the 8 hour work day or 40 hour work week:

As you continue to work, the disutility of each additional hour of work is about the same, while the utility or value that hour of work provides to you is decreasing. Eventually the utility decreases to the point where it is less than the disutility, at which point you stop working because it is not worth it.

Of course, the concept of marginal utility applies to all goods, not just money.

Now let us analyze taxation from the perspective of marginal utility. Proponents of progressive taxation use a seemingly common sense argument that goes like this: Because of marginal utility, every extra dollar a rich person earns is providing a need less urgent than that same dollar would provide if given to a poor person. Thus progressive taxation increases the overall utility of the wealth in our society by putting dollars in the hands of people who derive the greatest utility from them.

Sounds reasonable and based on sound economic theory, doesn't it? Perhaps it was true in the middle ages, when wealth could only be hoarded because the ability to invest it was risky, limited or non existent. But that is not true today. Today, dollars not immediately spent are saved in a bank or other financial institution. Those dollars are loaned to individuals and businesses. If spent by individuals they increase demand for consumer products. If spent by business to meet that increased demand, they are used to hire people, to purchase machinery which increases the productivity of labor, to expand facilities, etc.

Some key observations from the above:
None of these pursuits can occur until somebody saves money.
Nobody saves money until he has already fulfilled his most urgent needs.
The rich save more, both in absolute and in percentage terms, than the poor.
All of these pursuits use those dollars to provide greater utility than they would have provided to the person who saved them. This is tautological, else the person would not have saved them in the first place.

The conclusion disproves the argument given above in support of progressive taxation. In other words, it is not a priori evident that dollars taxed from a rich person provide greater utility when given to the poor, because in the absence of the tax, those dollars would have been "given" to others anyway, in the form of loans, to satisfy the greater utility they attach to them,

Thus while we have disproven the common specious argument for progressive taxation, we have not yet answered the question and must dig deeper.

Based on the above, we can view taxation and private investment as two alternatives for increasing the utility of wealth. They are somewhat mutually exclusive alternatives, since people only have a limited amount of left over after fulfilling their immediate needs (e.g. disposable income), so what is taken in taxes cannot be privately invested, and vice versa.

Thus the question of whether progressive taxation works, hinges on which of the 2 alternatives provides greater utility for the dollars consumed. This question, too, relies on the concept of marginal utility.

Consider the marginal utility of income to the government in the form of taxes. The government is expected to fulfill certain services to the people: military, courts, police, firemen, roads, etc. Of course the first tax dollars received are used to provide the most important and urgent services.

This may seem contrived but it is not - it becomes obvious when one considers a simple example: Suppose we increased taxes on everyone by $1. Whatever new product or service that revenue provides, must necessarily be less urgent or important than any of the services already being provided, else this new service would already be provided instead of something less urgent already being done.

Because of this effect, we will consider a revenue neutral approach to the question. That is, compare a progressive tax to a flat tax, in which both taxes generate the same total revenue.

The key is to focus on disposable income and contrast the poor with the rich. Under a flat tax, a larger percentage of total revenue comes from the poor, compared to a progressive tax - right?

Not necessarily.
Progressive taxation (a high marginal rate) makes it cost effective for the rich to spend considerable effort and money finding ways to avoid taxes (whether lawful or not). Lowering the top tax rates makes these schemes less cost effective - with less at stake, these schemes are no longer worth the effort and money. Thus lowering the marginal (highest) tax rates by X% does not reduce revenue from that tax bracket by X%. It reduces it by less than X%, and sometimes it can actually increase total revenue. This is because the lower tax encourages greater economic activity. A smaller percentage of a larger economy may be a bigger absolute number. But even if it is smaller, it is not X% smaller.

So the question is, what would people have done with their money had it not been taxed? For simplicity, we assume that under either plan even the poor can satisfy basic necessities, since this is true of the majority of people. Poor people have less disposable income and save less, so they are likely to have spent the money. Rich people have more disposable income and save more, so they are likely to have saved the money. Thus under a flat tax, the negative impact of the tax falls more heavily on spending - and under a progressive tax, the negative impact of the tax falls more heavily on saving. In other words, a greater portion of the total revenue generated by a progressive tax, comes from money that would have been saved rather than spent, compared to a flat tax.

In other words, progressive taxation rewards spending and punishes saving, compared to a flat tax.

So far, we have the following:

We can talk about whether a tax is "neutral". This means that the economic impact of the tax is spread more or less evenly across the economy. Taxes that are not neutral effectively transfer wealth from some people to others. For example, consider inflation which punishes savers and creditors but rewards spenders and borrowers. Thus it enriches some people at the expense of others.

Ideally we want the spending / saving proportion of the total revenue generated by a tax, to match the overall ratio of spending to savings in society. The Machiavellian way to look at is we can use the tax system to control people's saving / spending habits.