I know that money, just like any other commodity, has diminishing
marginal utility.

My question involves getting some quantative data on this topic.  Has
anyone done any studies on exactly how much the marginal utility of a
dollar goes down as people get richer?

Ideally I'd like as specific numbers/equations as possible for current
US citizens.

Hello and thank you for your interesting question.

A good starting point is the historical analysis that underlies the
expected utility hypothesis:
Bernoulli and the St. Petersburg Paradox
http://cepa.newschool.edu/het/essays/uncert/bernoulhyp.htm
I cannot set forth its arguments here, because of the unavailability
of greek fonts in Google Answers.  Please take a look at the
discussion on that page.

Current thinking on the issue uses risk and aversion to risk as the
measure of wealth's marginal utility:
Rabin, Diminishing Marginal Utility of Wealth Cannot Explain Risk
Aversion
http://repositories.cdlib.org/iber/econ/E00-287/

Here's a lay summary of Rabin's arguments that you may find useful:
Averse to reality
http://www.carleton.ca/~kacheson/Economist_decision_making_uncertainty_9au1.htm

Personally I'm not too taken with Prof. Rabin's arguments, but clearly
I'm in the minority here, since his paper is very heavily cited, e.g.
http://tinyurl.com/7ds3
(122 references to Rabin's paper)

I prefer Prof. Reisman on the subject (although it is not
mathematically precise:
"George Reisman has said that Galbraith 'makes an enormous
equivocation between the importance of a concrete amount of wealth
as the total amount of wealth increases and the importance of
acquiring wealth as its total amount increases. For while the
importance of the former diminishes with the increase in the amount
of wealth, the importance of the latter does not. The very purpose
of acquiring wealth and the source of the importance of doing so
consist precisely in the reduction of the marginal utility of wealth
... For the ability to achieve an ever lower marginal utility of
wealth is identical with the ability to make an ever greater and
more complete provision for the maintainance and enhancement of
one's life and well-being.'"
George Reisman, The Revolt Against Affluence: Galbraith's 
Neo-Feudalism, p.5, quoted in
J. K. GALBRAITH'S THE AFFLUENT SOCIETY: A CRITIQUE
http://216.239.57.100/search?q=cache:Qkl1k7fzDjgC:www.capital.demon.co.uk/LA/economic/affsoc.txt+reisman+%22marginal+utility+of+wealth%22&hl=en&ie=UTF-8
[Google cache]

See also:
Diminishing Marg. Utility of Wealth
http://objectivism.cx/~atlantis/objectivism-l/msg00631.html
"Everybody agrees that the Law of Diminishing Marginal Utility applies
to
such things as pizza, water, and grain.  We will expend our first
gallon of
water (or unit of grain or whatever) on our most important need, our
second
on our second-most important need, and so on, the implication being
that
each additional unit of the good becomes less important to us.
There is somewhat less agreement as to whether the Law of Diminishing
Marginal Utility (DMU for short) applies to wealth per se."


A related concept is the marginal efficiency of wealth:
"[T]he marginal efficiency of wealth (or the diminishing utility of
concentrated money at the margin). This is still an extremely
under-discussed and under-researched aspect of standard economics.
Phil Hyde has expressed it like this - a million dollars in the hands
of one man works a lot less hard in terms of providing employment than
a thousand dollars in the hands of a thousand people...."
The Many Names and Faces of the Economic Puzzle
http://www.timesizing.com/3ecpuzz.htm

Here's an odd application of the concept:
RULES OF ENGAGEMENT 
http://www.schrag.info/tushnet/law/rules.html
"The declining marginal utility of wealth means that a mandatory rule
of return will more readily deter a woman from ending a bad engagement
than it will ease a man's decision to do so, because she will forfeit
relatively more than he will regain."


Search terms used:
wealth economics "marginal utility"
rubin "marginal utility" wealth
reisman "marginal utility" wealth

Thank you again for your interesting question.  If you need any more
material of this sort, please request clarification.  I would
appreciate it if you would hold off on rating my answer until I have
an opportunity to respond.

Sincerely,
Google Answers Researcher
Richard-ga

---

Request for Answer Clarification by harryh-ga on 13 Mar 2003 13:40 PST

Wow.  That looks like a lot of great information Richard.  And you
responded really quickly.  Give me a day to digest it all before I
rate your response (just in case I have a follow up).

Thx,
Harry

---

Clarification of Answer by richard-ga on 13 Mar 2003 14:13 PST

No problem - - thanks for the kind words.

-R

richard-ga's answer is comprehensive and detailed.  My own research
has picked up the following useful reference regarding the
"Econometric Models for Household Disposable Income"

http://www.stat.fi/isi99/proceedings/arkisto/varasto/uygu0945.pdf

Let's say you were to use the formulas therein or some empirical data
at various points on the utility curve.  You could easily demonstrate
diminishing utility from a mathematical model using, as richard-ga
suggests, the straightforward assumption that a hundred dollars to a
family at the poverty line is pocket change to a millionaire (let's
call this the utility approach).

If/when you do so, please remember to take into account fixed expenses
and the way they rise with an increasing standard of living.  For
example, when one is young and beginning to earn a wage while still
living at home, one may have more _disposable_ income than a person
with a mortgage and family, despite having a lower salary.

Another interesting spin is the cost of capital/ opportunity cost of
investment approach to assessment. The utility model above assumes
that all income earned is utilized.  Therefore, unused income is
'wasted'.  This is a fallacy from a financial point of view (let's
call this the return on equity approach). Money is unlike other
commodities, because it can earn a rate of return.  Therefore:

(1) An individual with net assets exceeding a million dollars will be
able to take advantage of investment opportunities yielding a higher
rate of return than his poorer contemporary.

(2) At the same time, the millionaire will have a substantially lower
borrowing cost than for a less fortunate individual. That is to say,
the cost of borrowing is lower for the rich.

So, the millionaire has a lower cost of borrowing and higher expected
rate of return on investment.  Therefore, through leverage, his equity
(net assets) can perform even better.  Using this approach, you can
show that a rich individual can obtain more benefit from a dollar than
a poor individual.

So which model should you use?

This all comes down to the practicalities of the situation.  If you
are trying to decide how to divide up a charitable grant, obviously,
you will find that the poorest members of society need money more than
the rich.

However, for corporate or high net-worth situations, where there is a
surplus of funds, you should be considering an investment-based model.

In the case of the average individual, where some regular level of
investment is made, but there are typical day-to-day financial
pressures, your model becomes extremely complicated.  I would
recommend you consider the investment model only to the extent which a
typical individual of that income level would do. That is to say,
apply a weighted utility model.

richard-ga also mentions a further empirical approach: the risk
aversion model.  As I understand it, the idea is that the more money
an individual has, the more tolerant to risk he/she should be.  This
method is a theoretical model that works well for certain
applications, such as the risk management of trading accounts, but
does falls down in the face of human nature.  People act irrationally
with respect to risk.  People under financial pressure may choose to
buy a lottery or Keno ticket where the odds are unfavorable in the
hope of a big payout, whereas a rich person may stick to hedge funds
or financial instruments that have a strongly positive expectation.

I hope my theoretical musings have added some value to Richard-ga’s
already carefully researched answer.

Regards,

asiatechnicals-ga