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A Thought Experiment on Taxes

18 April 2011

A lot of people think economics an inpenetrable fog, but it needn’t be that way.  A lot of elementary economics is common sense.  And one of the best ways qb knows to get common-sense ideas on the table is to perform “thought experiments.”  In a thought experiment, we ask a series of questions about a hypothetical situation, and we build our understanding incrementally; eventually, the general contours of the phenomenon we seek to describe become evident.

So let’s do that with a simple taxing scheme, and ask the question:  how much revenue does a taxing entity receive at a given tax rate?  And what is the relationship between the tax rate on any given individual and the amount of tax revenue received from that individual?  This is a great set of questions for a thought experiment.

Let’s begin with the most trivial, easily understood idea:  how much revenue is generated at a tax rate of 0%?  It should be obvious that the answer here is ALWAYS zero.  If no tax is levied, there is no revenue.  This results in the following chart, thus far, with the blue marker at (0%,$0):

Next, and only slightly more challenging, is this question:  how much revenue is generated at a tax rate of 100%, in which the taxing entity confiscates all of the individual’s income?  Here, we have to invoke a key assumption, to wit, that the individual in question is both (a) free and (b) self-interested.  The individual must not be utterly enslaved to the taxing entity in order for us to establish common ground with our situation in the U. S.  And the individual must have his/her own self-interest at heart, although that self-interest may include the interests of his/her dependents, as would be implied by listening to the Pastoral epistles:  taking care of one’s own family is a moral requirement, whether the breadwinner be the wife or the husband.  Let us be generous, then, and posit that our individual is the breadwinning woman and that her husband and children are dependent on her income for food, shelter, and clothing.

How much revenue will she produce if all of her income is confiscated by the taxing entity?  Again, the answer is zero; no matter how hard she works, she does not get to take anything home, so there is no incentive for her to engage in productive work.  She might as well stay home, help take care of the dwelling, and scavenge for resources, an activity which (we may presume) is beyond the reach of the taxing entity.  In any case, she will not work for pay lest her time and effort be wasted and lest the taxing entity confiscate all she earns.  So we can add one more piont to the chart, thusly, with the new green marker at (100%,$0):

We now have established the two endpionts of the relationship we seek.

At this piont, we need to think a little bit.  Both of our first two scenarios established that the tax revenue generated is zero.  It’s fair to ask, then:  is there any tax rate at which the tax revenue generated by the individual for the taxing entity is NOT zero?  Of course.  If the tax rate is 1%, the individual gets to keep 99% of what she earns and so has an incentive to earn an income; and the tax entity receives 1% of what she earns.  That revenue figure is small, of course, but the important thing is that the tax revenue is greater than zero.  Likewise with 2%, and so forth.  In fact, we can assume that all of the tax rates between 0% and 100%, not including the endpionts, generate a value for the tax revenue that is greater than zero dollars.

We now have to make a further assumption, to wit, that the series of pionts we place on the chart represents a continuous and reasonably smooth curve.  That is, the tax revenue generated at a tax rate of X% is not terribly different from the revenue generated at a tax rate of [X+1]% (or, equivalently, [X-1]%).  Neighboring values of the tax revenue are not equal, but they’re not very far apart, either.

Let’s summarize what we know.  The curve that represents the relationship between tax rate (X, or horizontal axis) and the tax revenue generated (Y, or vertical axis) is a SMOOTH, SLOWLY VARYING function Y(X), and its Y values are everywhere >$0 except at the endpionts X=0% and X=100%, at which pionts Y=$0.

What is the shape of a curve like that, which we denote as Y(X)?

At this piont, we need to invoke Sir William of Ockham, who urges us to select the simplest of all plausible answers, to avoid adding unnecessarily to the “complicatedness” of our answers.  That means, in our case, that the general shape of the function Y(X) looks something like this:

Now it may appear that qb has stacked the deck here and has insisted that the maximum revenue occurs at X=50%.  But qb has no idea whether or not that’s true.  We simply haven’t generated enough data pionts to say for sure.  Still, granting our assumptions, the question concerned the GENERAL shape of the curve, not the PRECISE shape.  And the simplest smooth, slowly varying function Y(X) that passes through the pionts (0%,$0) and (100%,$0) and elsewhere takes on Y-values greater than zero must have the general shape of that last chart.

At this piont, we can draw two simple but powerful conclusions, which are unarguably true (again, granting our assumptions)…or, if you prefer, which are strictly required by the intermediate conclusions we’ve reached thus far:

1.  There is at least one value of the tax rate, X=X(loc-max), which generates a local maximum in the amount of revenue generated, Y(loc-max).  By “local maximum” we mean that if we move to the left or to the right along the curve from X(loc-max), the resulting value of Y will be less than Y(loc-max).  Try it out!  In the chart above, go to X=50%, put your finger on the yellow circle that corresponds to it, then move your finger along the curve either to the right or the left of that piont, and you will see that the value of Y decreases either way.

In the general case, there may be more than one of these local maxima, but if we stay with Ockham’s Razor (the “principle of parsimony”), there’s only one…which means we can call it the “global maximum.”  So we can treat the chart immediately above as if the curve were a piece of spring steel, and push on it from the right or left to move the X-location of the global maximum, but we’ll still have one maximum, and it will be somewhere between X=0% and X=100% (not including those endpionts, as we said).  Again, we don’t have enough information to say that X(max-loc)=50%; all we know is that there is a value X(max-loc), and it lies somewhere between 0% and 100%.

2.  The more powerful observation, which follows from conclusion #1, is this:  There is at least one non-trivial region of the curve in which lowering the tax rate (i. e., moving to the left along the curve) generates MORE revenue than was generated at the starting piont.  The easiest way to see this is to start at the piont X=100%, and then move to the left and rejoin the curve.  Whether the value of X(loc-max) is at 50% or 75% or even 90%, we must conclude that the revenue generated at X=99% is greater than the revenue generated at X=100%.

Lots of our fellow citizens – VOTERS – cannot see this.  They assume that decreasing tax rates ALWAYS reduces tax revenue, which means they think the world works like the red circles in the following chart:

The red chart, however, denies the validity of the assumption we made about the self-interested behavior of the breadwinning mom in our thought experiment.  People who ascribe to the red chart don’t actually behave that way; every single one of us, with only the rarest of exceptions, would continue to work productively if all of one’s earnings were confiscated at the muzzle of the IRS’ gun.

It is true that reducing tax revenue does occur in SOME regions of the chart.  But not all!  We have just established, by a simple thought experiment, that at some values of X, lowering the tax rate would actually increase revenue.  And that, dear friends, is like presenting liberals with kryptonite.  The only real question for us to answer is this one:  which side of X(loc-max) are we really on?

Fair question.  In a subsequent post, we’ll see how in the case of the Reagan tax cuts of the early ’80s and the Bush tax cuts of the early 2000s, revenue to the Treasury actually increased.  That suggests strongly that we have been overtaxed…and that if BHO and the House Progressive Caucus carry the day, revenue to the Treasury will decrease, not increase, and our deficit will spiral upward still more.

By the way, what we have just done with our thought experiment has a name:  the “Laffer Curve,” named after the famous economist Art Laffer.  And you thought economics was an inpenetrable fog!

qb

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