From: cbbrowne@csi.uottawa.ca (Christopher Browne)
Subject: Brief Mathematical Analysis of Taxation/Welfare Systems
Summary: Current Tax/Pay welfare method compared to "Negative Income Tax"
Organization: Dept. of Computer Science, University of Ottawa
Date: Sun, 3 Jan 93 21:35:07 GMT

A recent set of posts in Can.politics have argued (sometimes
pointlessly and commonly VERY voluminously) about one of the suggested
proposals for Welfare System reform in Canada, namely a "Negative
Income Tax."

There has been some confusion as to just what a "Negative Income Tax"
is, and why it might be considered a "good thing," or a "bad thing" as
compared to the sorts of "welfare evaluations" done currently.

Here's my attempt a more detailed *mathematical* comparison between
the current welfare system and the "Negative Income Tax" proposal.

If you want to evaluate a proposed reform, it is necessary to look at
it from this sort of quantitiative perspective AS WELL AS any purely
qualitative perspective.

I.E., this analysis suggests some comparisons or "tests" that any
proposed system would have to pass in order to be worthy of
consideration.

The current set of taxation "functions":

Let I be defined as "Taxable Income".  Typically, it will closely
resemble a person's "actual" income from sources other than government
subsidies.  This treatment ignores noise like Social Security
payments, and any potentially "weird" deductions.

T(I) is the "Taxation function", mapping from I, the person's income,
to the set T, of taxes that the person winds up paying.

As things stand now, there are essentially several taxation functions;
this analysis will examine two of them.

#1:  Income Taxes - denoted T_{i}(I).

T_{i}(I) is, for those in the lowest bracket, approximately equal to:
  T_{i}(I) = Max(0, 0.27 * I - $1743)

#2: Welfare - denoted T_{w}(I).  I understand that there's usually
some sort of means testing by which different individuals would have
different needs.  Ignoring this fact, I'll just baldly say that
there's a minimum income of $10,000.  This makes it easier to compare
the two situations under consideration.

T_{i}(I) is roughly equal to:
  T_{i}(I) = - Max (10000 - I, 0)

Note that this "tax" is always at zero or lower.  It isn't really a
tax, but rather an "entitlement."  Doesn't matter.  It's still a
direct flow of money back and forth between individuals and the
government.

There are other indirect programs such as rent subsidies; for
simplicity, I'll ignore them.  They wouldn't change the nature of the
conclusions.  In fact, they'd just strengthen the conclusions.

Adding the two programs together, we get:

  T_{total} = Max(0, 0.27 * I - $1743) - Max (10000 - I, 0)

Simplifying this is a LITTLE complicated.  I'll skip the display of
the work, and jump to the final answers.

There's two cases:

1) Income < 10000
--> T_{total} = Max(0, 0.27 * I - 1743) - (10000 - I)

Within the range [0..6455], the derivative d(T_{total})/d(I) is 1,
which indicates that under the current system, for every dollar that a
welfare recipient earns up to $6455, a dollar is taken off of their
welfare payment.  To put it simply, their time is in fact completely
wasted.  There's no ECONOMIC point to working.

Within the range [6455..10000], the derivative d(T_{total})/d(I) is
1.27, which indicates the even WORSE fact that for every dollar
earned in that range, the person loses $1.27, because not only will
(s)he get the earnings chopped off of the welfare payments, they'll
actually get to pay some taxes on the earnings.  Double whammy.
Simply put, such an individual would be economically BETTER OFF by not
working.

This is the range in which there is currently a TERRIBLE PROBLEM.

2) Income > $10000
--> T_{total} = Max(0, 0.27 * I - 1743)

Until I gets up to the $28K range, the derivative is 0.27, indicating
that of every $1 that one earns, you get to keep $1 - $0.27, or $0.73.
However, in order to actually get more after tax dollars than you
would with an income of $0, you'd have to earn $13699.  Thus, under
this system, the minimum POSSIBLE wage that could attract people would
be an offer of $13699.  Below that, welfare is a better deal.

Under an NIT system, the taxation function would be simpler, looking
more like:

--> T_{total}(I) = 0.35 * I - 10000.

The derivative in this case is the constant 0.35, with no
discontinuous ranges as are found in the current system.

Here's plots of the two tax functions:

#1: The current system:
   *************************************************************************** 
   *                        *                       *                        * 
   *                        *                       *                        * 
   *                        *                       *                      *** 
   *                        *                       *            *********** * 
   *************************************************************************** 
   *                        *                 ***********                    * 
   *                        *       ***********     *                        * 
   *                        *********               *                        * 
   *************************************************************************** 
   *                   ***  *                       *                        * 
   *                 ***    *                       *                        * 
   *               ***      *                       *                        * 
   *             ***        *                       *                        * 
   *************************************************************************** 
   *       ****             *                       *                        * 
   *     ***                *                       *                        * 
   *  ***                   *                       *                        * 
   * **                     *                       *                        * 
   0 -x- 30000  -10000 -y- 10000********************************************** 

Note that there are THREE sections to the graph:  In the first, the
slope is 1, indicating that increases in income provides NO benefit.
In the second, the slope is 1.27, indicating that increases in income
is in fact counterproductive to the individual.  In the third, the
slope is 0.27, indicating that earnings do provide a marginal benefit.

Graph #2:  NIT system

   *************************************************************************** 
   *                        *                       *                        * 
   *                        *                       *                        * 
   *                        *                       *                        * 
   *                        *                       *                        * 
   *                        *                       *                        * 
   *************************************************************************** 
   *                        *                       *             ******     * 
   *                        *                       *       ******           * 
   *                        *                       * *******                * 
   *                        *                    ******                      * 
   *                        *              *******  *                        * 
   *************************************************************************** 
   *                        *   ******              *                        * 
   *                      ******                    *                        * 
   *                ******* *                       *                        * 
   *           ******       *                       *                        * 
   *     *******            *                       *                        * 
   * *****                  *                       *                        * 
   0 -x- 30000  -10000 -y- 5000*********************************************** 

Note the constant slope of 0.35, indicating that for people at ALL
levels, it is beneficial to increase earnings through hopefully
productive activity.

There is NOTHING intrinsically socialistic about the NIT taxation
function; it's politically value-neutral.  Differing groups might
prefer differing parameters, or perhaps to substitute some
"progressive" taxation function, modifying it to:

T_{nit}(I) = - B + F(I) I

where:
B is the "Basic entitlement," also known as the "Minimum income
level", and

F(I) is a tax rate function whose value varies in the range [0..1].
They'd typically start F with some moderate value, say 20-30%, and
then increase it.  It would probably not increase to significantly
past 50%, which is the highest marginal income tax rate currently in
effect.  If it rose much past 50%, the "rich" would seriously rebel,
and you'd see assets and jobs heading out of the country in quantity.

An undeniable problem with the current system is the FACT that the
derivative heads to effective tax rates GREATER than 100%, of about
127% in the Canadian case.  This is a CLEAR cause of people staying on
welfare.  I'm sure that things are similar in the US.

-- 
Christopher Browne                |     PGP 2.0 key available
cbbrowne@csi.uottawa.ca           |======================================
University of Ottawa              | Genius may have its limitations, but
Master of System Science Program  | stupidity is not thus handicapped.
Newsgroups: can.politics