Consumption 2: The Return (With New and Improved Definitions and Proofs!)

From: Jeff Bone (
Date: Thu Mar 29 2001 - 21:50:27 PST

(closeup Jeff)
(Jeff walks up to podium, knocks on mike...)
Jeff: Hello? Is this thing on?
Adam: (off camera) Hot mike, hot mike!
(closeup Adam)
(Adam twiddles knob on mix board)
Adam: (mutters) Jeff you goon...
(wide angle Jeff)
Jeff: Uh, good evening? Thanks for coming.
(pan audience. most are asleep. Lisa, Kent, a few others half-awake...)
(closeup Geege)
Geege rolls eyes.
(closeup Dave)
Dave arches eyebrow, yawns, glances at program.
(closeup Jeff)

Jeff: Our purpose this evening is to demonstrate that, in fact, straight
consumption taxes are not "regressive" in a logical and common-sensical


When we last tuned in, things were looking pretty grim for one of our heroes,
the straight consumption tax. It appeared definitionally doomed to being
considered "NOT EQUITABLE," and hence "regressive" and unacceptable. And while
the FairTax only made a cameo in the last episode, its future was questionable,

But don't touch that dial! Have we got a shocker for you, just as surprising as
tonight's Survivor! Go make yourself some popcorn, sit back, relax, and read
on. But first, a few words from and to (our sponsor? ;-) Lisa...


Lisa Dusseault wrote:

> I think the assumption of FairTax and perhaps others is that people who make
> more money spend more money. Can you make the system fail to be equitable
> for Cr>Cj?

A Damn Fine Question, right at the center of the controversy... the answer is
yes, in some cases, for the proof given. This question is critically important
to a better analysis, see below.

First, clarifications...

> The most intriguing parts of the ongoing tax discussion, to me, has been the
> opposition to taxes which allow the rich to get richer, or to increase the
> wealth gap.

This was hanging me up, too, Lisa, but I think you have it wrong --- as did I.
The opposition (mine, anyway) is not to taxes which allow the rich to get
richer, or which allow the wealth gap to increase. The opposition is to taxes
which artificially, simply by the construction of the system, allow the rich to
get richer at a faster rate than they would otherwise just by virtue of their
higher relative economic productivity. This is a "regressive" tax, one which
--- I'll admit --- unfairly advantages the higher earner. Hence, by the
definitions earlier, a fair or "EQUITABLE" tax system is one in which the
relative wealth accumulation of a high earner vs. a low earner is exactly equal
to the relative difference in economic productivity of the high earner vs. the
low earner. Slightly modifying my notation, for any given tax system the
following relation should hold:

   Wxy = Pxy

Just to be clear:

> If you define as "evil" any tax system which allows the rich to get richer,

I don't. I would define as "evil" --- or at least unfair --- any tax system
which by its particular construction allows anyone (rich or poor) to get richer
(accumulate wealth) faster than they otherwise would through the normal exercise
of capitalism. That's the heart of the fairness argument I've been fumbling
with since we began, kudos to Dave for helping me out there. It's what I was
trying to say in Bone's Lemma, which we've now made very precise I hope. This
is what the above equation captures.



We need to dig a bit deeper into the definition of EQUITABLE. Previously, we

     An EQUITABLE tax system is one where, given two parties x, y such that
     Ex > Ey,
     all other things being equal, the relative wealth accumulation of the
     parties is equivalent to the relative productivity of the two parties.
     following relation must hold: Wx / Wy = Pxy.

The problem here is this "all other things being equal" hocus. We took that to
mean that Cx = Cy was a given as well as Tx = Ty was a given. We also have a
pretty monolithic notion of earnings, E, and how they relate to wealth
accumulation. Let's dig in.

We can decompose earnings into two parts: consumption, C, and another quantity
we'll call "investment" or I. Consumption is defined as expenditure of earnings
on goods, services, and assets that will got generate returns, but rather will
either be consumed or depreciate over time. Investment is its complement:
investment is the expenditure or translation of earnings into assets and other
instruments which are expected to generate economic returns. So

   E = C + I
   C/E + I/E = 1

An equitable system should not give preferential treatment with respect to the
accumulation of wealth to any particular economic activity; consumption and
investment should be treated by the system as equally acceptable economic
activities. Ironically, this argues for a straight, flat consumption tax. Back
to Bone's Lemma:

     Any fair and equitable tax system should not inhibit someone from
     bettering their economic situation through increasing their economic
     productivity, i.e. income. The amount of increased individual economic
     productivity a person is able to achieve should directly and
     proportionally translate into corresponding betterment of their
     economic situation, regardless of the magnitudes involved.

Perhaps imperfectly stated, but you hopefully get the gist --- it's Wxy = Pxy.
Furthermore, this should hold true regardless of the absolute amounts allocated
to C (or I) by x and y, as long as they are equally proportional to Ex, Ey
respectively. My previous error was in dealing with absolute amounts rather
than ratios. If neither consumption nor investment is preferred by the system,
then the higher earner should be able to consume the same proportion of their
earnings as the lower earner without breaking Wxy = Pxy. So, we need a new
definition of EQUITABLE which makes Bone's Lemma true and reflects this new ---
and, I believe, quite fair --- thinking.


     Definition 1: (D1) An EQUITABLE-2 tax system is one where, given two
     parties x, y such that Ex > Ey, independent of the absolute
     allocations of earnings between consumption or investment but assuming
     that the individual ratios of these components over earnings are

        assuming Cx/Ex = Cy/Ey and Ix/Ex = Iy/Ey,
        then Wxy = Pxy must be true

THEOREM: A straight consumption tax is EQUITABLE-2

Here's the proof...


   W = E - C - TC (Eq. 1)
   Wxy = Wx / Wy (Eq. 2)
   Prj = Er / Ej (Eq. 3)

   Er = 10 (Ritchie earns $10) (G1)
   Ej = 2 (Joe earns $2) (G2)
   Cr = 5 (G3)
   Cj = 1 (Cr/Er must be given to be equal Cj/Ej, D1) (G4)
   Cr / Er = Cj / Ej (satisfying D1)
   T = .5 (50%, or whatever...) (G5)


   Wr = 10 - 5 - .5(5) = 2.5 (A1, from Eq. 1, G1, G3, G5)
   Wj = 2 - 1 - .5 = .5 (A2, from Eq. 1, G2, G4, G5)
   Wrj = Wr / Wj = 2.5 / .5 = 5 (A3, from Eq 2, A1, A2)

   Prj = 10 / 2 = 5 (A4, from Eq. 3, G1, G2)

   5 = 5, Wrj = Prj (A3, A4)

.: A straight consumption tax is EQUITABLE-2.


If you agree that the purpose of a tax system is not to "route around"
capitalism in any way, i.e. not to influence accumulation of wealth one way or
another, AND you believe that higher earners should not be hindered in enjoying
the "fruits of their labors" or penalized for being a higher earner, THEN you
should be convinced by the above reasoning and believe the following
conclusion. A straight consumption tax is an equitable (i.e., EQUITABLE-2)
system which does not artificially influence wealth accumulation in any way. A
straight consumption tax is not regressive under reasonable assumptions.


Ironically, turns out the FairTax --- isn't. The FairTax too dramatically
favors investment; while dressing itself up in a very "progressive" way, if you
work similar equations for FairTax you find artificial acceleration of wealth,
if i.e. Cx/Ex < Cy/Ey then Wxy > Pxy. The amount of acceleration that occurs is
also tied to the absolute magnitude of the numbers involved, rather than the
proportions. If anybody wants to see the math, drop me a line, I'll send it to
you OOB.


Once more, with gusto:

     A straight consumption tax is an equitable (i.e., EQUITABLE-2) system
     which does not artificially influence wealth accumulation in any way.
     A straight consumption tax is not regressive under reasonable

So, I retract my previous retraction, while simultaneously provisionally
"suspending" my support for FairTax pending further analysis. The straight
consumption tax, which turns out not to be "regressive" at all, takes the lead.
Other than that, I'm not sure yet... But I promise to report back!



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