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Comment on Professor Timberlake’s
Squared Rule for the Equilibrium
Value for the Marginal

Utility of Money

William Barnett II

ics, Professor Timberlake states his “squared” rule for the equilibrium value

| n both volume 1! and in volume 22 of The Review of Austrian Econom-
of the marginal utility of money, to wit:

MU.
MU,, = “pr ’

where:

MU, is the marginal uality of money,
MU. is the marginal utility of the “composite good™, and

P_ is the price of the composite good.*

This rule is based on the neoclassical theory of the consumer. Now, even
if we ignore the insurmountable analytical problems concerning the concepts
of the marginal utility of a composite good, and accept the analysis on its
own terms (i.e., using the neoclassical analytical apparatuses of comparative
statistics/equilibrium analysis), Professor Timberlake’s rule is erroneous. That
is, even if the concepts of a composite good and its marginal utility are ac-
cepted, his formulation remains a faulty application of neoclassical comparative
statics/equilibrium analysis. Thus I shall deal with it on its own terms.

According to the neoclassical theory of the consumer, each consumer is
assumed to maximize a utility function subject to a budget constraint. (Pro-
fessor Timberlake simplifies matters somewhat by considering a “typical
individual” Thus:

I am grateful to two anonymous referees for helpful comments.


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152 o The Review of Austrian Economics, Volume 3

maximize: U[X] fori = 1,...n,

subject to: Y, = 2 P, * X,,
where:

U(X,) is the typical individual’s utility function,

X; is the quantity of the i-th good purchased by the typical individual,
P, is the price of the i-th good,

Y, is the wealth in monetary terms of the typical individual, and

n is the number of goods.

Assuming the utility function exhibits the necessary mathematical condi-

tions in terms of continuity, differentiability, and so on, the first-order condi-
tions for maximization are:

MU, =Lfori=1...n,and
P;
where:

MU, is the typical consumer’s marginal utility of the i-th good and

L is a Lagrange multiplier.
From the first-order conditions, the following can be derived:

MU, MU, _ _ MU,

or, in more useful form:

MU Spo fori= 1m andiz

)

That is, the (subjective) rate at which the typical individual is willing to
substitute between any two goods (given by the ratio of that individual’s
marginal utilities of the goods) must be equal to the (objective to him) rate
at which other people are offering to exchange the same two goods (given by
the ratio of the prices of the goods).


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Comment: Timberlake's Squared Rule « 153

In his analysis, Professor Timberlake assumes (implicity) that the typical
consumer's utility function and budget constraint may be represented as in-
cluding but two goods (money and a composite good):

U = U(C,M) and
Y=P -C+P,-M
where:

C is the quantity of the composite good purchased by the typical consumer,
and

M is quantity of money purchased by the typical consumer.

Thus he (implicitly) derives the maximizing conditions as:

MU, _
5 =L
MU
——m _— I
P.

Y=P-C+P,-M,
from which he (implicitly) derives his equilibrium condition:

MU, _ MU,
P. P,

C

This is a legitimate neoclassical formulation provided that P,, is the nominal
price of money, that is, the price of money in terms of itself, to wit: one (1). This
condition is essential if the budget constraint is to be consistent. Thus, where:

Y is in terms of dollars ($),

P. is in terms of dollars per unit of the composite good ($/c),

C is in units of the composite good (c), and

M is in dollars ($),

$=8%/cc+P,-%Sor$=9%+P_, "8.

It is immediately obvious that for the budget constraint to be consistent,
the price of money must be dimensionless, specifically P, = 1. Thus,
substituting one (1) for the price of money in our budget constraint yields

$=%5+1-%o0r$=9%+8,


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154 « The Review of Austrian Economics, Volume 3

in which case the dimensions of the budget constraint are: consistent.
Note that Professor Timberlake's use of the reciprocal of the price of the
composite good [1/P] as the price of money, that is

P.=1/P. = 1/($/c) = ¢/8$,
results in the budget constraint having the following dimensions:

Y=P C+ P, M,
$ =8%/cc+c/$-%o0r$=9% +c,

which obviously are inconsistent.®

Unfortunately, Professor Timberlake missed this crucial point. This failure
resulted in his substitution of the reciprocal of the price of the composite good
for the price of money in his equilibrium condition, yielding:

MU, MU, MU,

p. = 1p op - MUs
or, in more useful form,
MU. _
MU, Pe

In this latter form, Professor Timberlake’s error and its source are clear.
The rate at which money and the composite good may be exchanged is given
by the money price of the composite good, not by its square. It was the im-
proper substitution of “the” real price of money (the reciprocal of the price
of the composite good) for the nominal price of money in the budget constraint
that caused the squared rule.

We see then that even within the framework of neo-classical analysis, Pro-
fessor Timberlake’s rule is untenable. In addition, from an Austrian perspec-
tive, there is an even more fundamental problem with the neoclassicists’ at-
tempted mathemadzation, via the use of the infinitesimal calculus, of economic
analysis. In this case, it appears in the form of the unwarranted and implicit
assumptions that the composite good be infinitely divisible, that the law of
diminishing marginal utility not hold with respect to subdivisions (new units?)
of the last unit of the composite good, and that marginal utilities be infinitely
divisible, whatever that may mean.

Consider a more appropriate formulation of the neoclassical theory. All
notation is as previously given. However, the utility function and budget
constraint are expressed in terms of “real variables” —that is, “real income”
and “real (money) balances.” Real balances is, of course, the name given to the


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Comment: Timberlake's Squared Rule o 155

purchasing power of actual (“nominal”) money balances. Since there is only one
nonmoney good (the composite good), both real income and real balances must
be, and are, in terms of units of the composite good. That is, their dimensions are:

M/P. = $/($/c) = c and
y =c
where

M/P, = real balances and

y = real income.
Thus, the utility function and budget constraint are:

U = U(C,M/P,) and
y = M/P. + C.

Note that the dimensions of the budget constraint are consistent.

The first-order conditions for a maximum are:

MU, = L,
MU (M/P.) = L, and
y = M/P, + C.

Thus
MU. = MU (M/P,).

That is, the marginal utility of the last “unit of real balances” purchased by
the consumer must be equal to the marginal utility of the last unit of the com-
posite good purchased by the consumer.

But, the problem is that units of real balances are multiples (or possibly
fractions) of units of actual money balances. Thus, to use Professor Timberlake’s
example of a tripling of the money stock, assume the following:

1. The initial stock of money is $100 (M = $100),
2. The initial stock of the composite good is 500c (C = 500c¢), and
3. The initial price of the composite good is $1/c, (P. = $1/c¢).

Then

y = $100/($1/c) + 500c or y = 600c.


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156 « The Review of Austrian Economics, Volume 3

Let the money stock triple and, in (assumed) consequence thereof, the price
of the composite good do likewise.

Then:

1. Then the new stock of money is $300,
2. The stock of the composite good is unchanged, and
3. The new price of the composite good is $3/c.

Note that real income is unchanged:
y = $300/($3/c) + 500c or y = 600c.

Then, in the initial situation the marginal utility of the five-hundredth unit
of C is equal to the marginal utility of the last unit of real balances, which is
the amount of actual money necessary to buy a unit of the composite good.
In this situation, the unit of real balances is $1. That is, the marginal utility
of the five-hundredth unit of C is equal to the marginal utility of the one-
hundredth dollar.

In the subsequent situation, the marginal utility of the five-hundredth unit
of C is still equal to the marginal utility of the last unit of real balances. However,
in this case, the unit of real balances is $3. That is, the marginal utility of the five-
hundredth unit of C is equal to the marginal utility of the two-hundred-ninety-
eighth through the three-hundredth dollars, taken as a unit. Unless the composite
good is infinitely divisible, it is meaningless to speak of the marginal utility of
the last unit of actual money, i.e., the three-hundredth dollar, for it will only
fetch one-third of a unit of C, if such may be purchased. And, even if this is
possible, the marginal utility of the three-hundredth dollar would be equal to
the marginal utility of the “last third” of the five-hundredth unit of C, and not
to one-third of the marginal utility of the five-hundredth unit of C. That is:

Initial situation: MU (100th $) = MU (500th C);
Subsequent situation: MU (298th, 299th, and 300th $s, as a unit) = MU (500th C).
However, this does not mean that
MU (300th $) = 1/3 + MU (500th C) but, rather that
MU (300th $) = MU (last 1/3 of 500th C).

Thus, for Professor Timberlake’s analysis to be correct, these latter two
equations must be identical. That is:

MU (last 1/3 of 500th C) = 1/3 + MU (500th C).


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Comment: Timberlake's Squared Rule « 157

And this must hold true for any possible increase or decease in the money
stock and attendant price change. But, as mentioned already, this requires that
the composite good be infinitely divisible, that the law of diminishing marginal
utility not hold with respect to subdivisions (new units?) of the last unit of
the composite good, and that marginal utilities be infinitely divisible.

The foregoing analysis demonstrates some of the errors in the analysis
leading to the squared rule. These errors are, unfortunately, typical manifesta-
tions of what is perhaps the most serious problem in monetary/business
cycle/macroeconomic analysis. This problem is inappropriate aggregation,
which arises out of attempts to make economic analysis mathematically tract-
able and to make economic data grist for the mill of statistical (i.e., econo-
metric) analysis.

Notes

1. Richard H. Timberlake, Jr., “A Critique of Monetarist and Austrian Doctrines
on the Utility and Value of Money,” The Review of Austrian Economics, vol. 1 (1987),
pp. 93-94.

2. Richard H. Timberlake, Jr., “Reply to Comment by Murray N. Rothbard,”
The Review of Austrian Economics, vol. 2 (1987), pp. 189-97.

3. Of course, except in the imaginations of some people, including many famous
economists, “composite good(s)” do not exist. The attempt to create mathematically
tractable models of the economy has led many to accept such fiction(s) as reasonable
approximations to reality. Unfortunately, this has resulted in much confusion and
misunderstanding, and in fact to the foisting of much relatively low level mathematical
exercises and games as advanced theoretical economics.

4. The system of notation used in this comment is somewhat different from
Timberlake's, but the reader should have no trouble reconciling them.

5. The Review of Austrian Economics, vol. 1, p. 93.

6. If the editors of journals required economists to assign dimensions to the
variables in their mathematical and statistical equations, many errors could be avoided.