MARGINAL UTILITY: SHORT-CUT IN EQUILIBRIUM AND

MARGINAL UTILITY: SHORT-CUT IN EQUILIBRIUM AND
DISEQUILIBRIUM
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Auke R. Leen
Wageningen Agricultural University
Department of General Economics
P.O. Box 8130
6700 EW Wageningen
The Netherlands
Abstract, JEL classification code: D11
The paper tries to formulate and formalize the abbreviating role of marginal utility in consumer calculations. The
abbreviation problem comes to the fore, in fact it is the answer to it, if one tries to find the real-world consumer's
counterpart to the process of solving a Langrangian. The problem how to abbreviate economic calculations was one
of the core problems for the older Austrians. A problem overlooked by the English and Swiss co-inventors of
marginalism, because of the mathematical mould of their theories. Within a household production framework, I use
an intertemporal model of consumer choice. Three groups of consumers are considered, first, emigrants on arrival in
a foreign country and people who get married and start a family, second, the so-called trendy consumers, and third,
the well-established consumers of traditional theory. For the first group the abbreviating role of marginal utility as a
short-cut in calculations is modelled as a beneficial rational addiction. For the second group it is modelled as a
myopic or harmful rational addiction. For the third group it acts as a watch-dog. A special interest is taken in the
producer's problem: Who are most susceptible to the advertizing of new goods? The theory predicts that emigrants
and just-married couples save more and buy relatively more new goods, though at a decreasing rate, than more
established people do. Trendy consumers, on the other hand, in buying new goods save less than the other groups
do.
I. Introduction
Suppose the present-day economist's model of consumer choice is not only a formal apparatus that helps
thinking but also reflects the maximization process of the consumer in reality. Then, what wriggles
most? It is neither the beginning nor the end. We all intuitively accept the start: a budget constraint and a
set of preferences (including Gossen's first law of decreasing marginal utility), and the result: the
optimal allocation of goods (Gossen's second law of equi-marginality). Didn't economists in the early
phases of the history of economic analysis, think that Gossen's first law signified a real-world
psychological phenomenon? No, what comes to the fore as highly unrealistic is the maximization
process that lies in between both Gossen's laws. Whatever it is the consumer does, it certainly is not
maximizing a Lagrangian function.
From a theoretical point of view calculations have to be very labor and time consuming. To
compute the marginal utilities, time and again, the consumer has, first, to state the needs which market
goods and services can satisfy, and second, to compare the utilities of all the needs with all the units of
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the goods he has at his disposal. A problem that becomes even more complex if we use a household
production function. For then he has to state not only the needs to be fulfilled by the goods themselves
but also by their intermediate products.
The problem how to abbreviate economic calculation was one of the core problems for the older
Austrians. A problem overlooked by the English and Swiss co-inventors of marginalism, because of the
mathematical mould of their theories. In Austrian economics marginal utility is not only the solution to a
problem, but also gives rise to a new problem. Marginalism, the solution to the well-known value
paradox, poses the problem of finding an overall goods allocation, with its astonishing amount of
individual comparisons. The comparisons are necessary to find the values of the different goods and the
optimal goods allocation. In the classical objective value theory the calculation problem had no place in
the core of economics and could be neglected by the theoretical economists. With subjective use value,
however, this changed. Value was no longer given from the outside, but had to be calculated by man
itself. The problem further aggravated with the transition from a cardinal to an ordinal utility concept:
the calculation problem changed from a quantitative to an even more difficult to handle qualitative one.
In this paper, first, the logical development of abbreviation in real-world consumer calculations is
given: the idea of the older Austrians that economic theory has to be in congruence with reality.
Abbreviation is related to the role of marginal utility in equilibrium situations and disequilibrium
processes. Second, I give a first formalization of abbreviation in an intertemporal model of consumer
choice. Saving and consumption patterns are predicted for different groups of consumers.
II. Short-cuts in equilibrium
According to Eugen von Böhm-Bawerk (1886, pp. 74-77), one of the founding fathers of Austrianism,
experience shows that calculations do not give us much trouble. Why? First, we do have a lot of practice
in making the calculations. We are great virtuosi in our value calculations. Calculations, after all, that
from a cost-benefit point of view do not have to be very precise. We do not have to be more exact than
is worthwhile. Second, we have all sorts of short-cuts and rules of thumb, e.g., our memory, things we
have learned from other people, and our everyday routines. All these reliefs can be used so long as our
situation does not drastically change. This is practically the case for most of the people most of the time.
In his summarizing article on marginal utility, Paul Rosenstein-Rodan ([1927],1960) works out the
conditions that permit an abbreviation of the allocation calculations. Though observation shows that the
majority of the needs usually remain constant, and the same holds for the supply of productive resources
(Rosenstein-Rodan, 1960, p. 84), we can ask: How do we find the initial equilibrium situation that holds
on in the first place, and is of relevance if nothing changes? According to Rosenstein-Rodan, we cannot
create this situation uno actu, it is the result of a historical process of trial and error. If, however, we are
in an equilibrium situation, the attained marginal utilities fulfil their function as a short-cut without there
simply could be no rational economic conduct. The marginal utilities are the sine qua non for any real-
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world economic calculations. "The utilities of the new uses which are contemplated are compared with
past marginal utilities in these uses; if they are larger, the new uses are expedient, if smaller, the new
uses are not expedient. The marginal utility in different need classes (kinds of uses) thus fulfills a watchdog function with respect to consumption" (Rosenstein-Rodan, 1960, p. 85).
For Rosenstein-Rodan the essential function of marginal utility is to abbreviate economic
calculations. To apply the abbreviating procedure we need: (1) an initial equilibrium situation; and (2)
relative constancy of needs and ends. The complementarity of utilities seems to pose a problem. It
suggests the necessity of a complete revision of the whole economic plan, even with a relatively small
change in the data. One is forced to go through all calculations again each time a situation changes.
This, however, is not the case. For the layers of goods allocation under discussion are the marginal ones;
complementarity is the lowest and the utilities are the smallest. Small changes in the data have no great
repercussions elsewhere in the system.
III. Short-cuts in disequilibrium
Leo Schönfeld-Illy, an Austrian of the third generation, made an important extension to these thoughts.
In two of his books (1924,1948), he investigates the consumer's abbreviation problem. For SchönfeldIlly marginal utility as an instrument of abbreviation is not only used in equilibrium situations but also in
disequilibrium processes. Abbreviation is no longer associated with marginal changes in an equilibrium
allocation, but with abrupt, great changes in means and ends. A theory that describes small changes in
equilibrium only, and not the origination of the equilibrium itself, is highly unsatisfying. It would be just
as unsatisfying as a theory that could explain price changes only, but not their origination. His favorite
examples of budget allocation are students who receive their first income after graduation, people who
marry and start a family, and emigrants on arrival in a foreign and strange country (Schönfeld-Illy, 1948,
pp. 125-126 and p. 321).
Schönfeld-Illy develops his interpretation of marginal utility as an instrument of abbreviation from
an inconsistency in Friedrich von Wieser's thought. Wieser, the brother in law of Böhm-Bawerk, takes
the definition of marginal utility from the last stage of calculation. "The measure of marginal utility is
got from the least useful of the most important uses ... conditional on the highest use of the supply and
the most careful inquiry of the needs" (Schönfeld-Illy, 1924, p. 75). On the other hand, he draws the
description of the function of marginal utility from the situation from which calculation starts. "All uses
that stand below are forbidden ... all uses that stand above or are equal are permitted" (o.c., p. 75).
Schönfeld-Illy wonders whether the last stage of calculation contains all the things that have been used
during the actual calculations. Isn't a definition based on the last stage of calculation too narrow? The
correct definition, according to Schönfeld-Illy, of marginal utility consistent with its function is: For a
particular consumer the marginal utility of a stock of goods is the utility of the economic smallest part of
the stock. The economic smallest part lies at the considered margin of use of the stock. It is of relevance
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to the whole stock regarding the plans one has for using it. Marginal utility is an instrument in a process
of trial and error. The consumers are discovering what their optimal consumption pattern looks like.
(Schönfeld-Illy, 1948, p. 67 and p. 126).
In other words, marginal utility fulfills a special role. It acts no longer primarily as a watch-dog
which precludes all utility under it and sustains all utility above it. Its primary function is its economic
relevance: the relevance of the economic use at a margin for a whole. A consumer rarely deals with
small pieces, he calculates mostly with stocks of goods which contain many different intensities of
utility. The price a consumer is willing to pay for a good at the margin of a certain stock of goods, based
on its marginal utility, has an economic relevance for all other units of the stock of goods. "A fortiore:
for the utility of these other units is higher than the utility of the marginal unit and therefore the price
paid for the other units certainly must be justified" (Schönfeld-Illy, 1948, p. 60). The cumulative validity
of the subjective demand price makes it possible to shorten the calculation process: we do not have to
state and compare the other utilities anymore.
Consequently, Schönfeld-Illy and Rosenstein-Rodan give a different interpretation of Wieser's
statement that the entire surplus utility can be neglected. For Rosenstein-Rodan the surplus utilities are
the utilities above the equilibrium marginal utilities (Rosenstein-Rodan, 1960, p. 85). For Schönfeld-Illy
the surplus utilities are the utilities above the marginal utilities of certain quantities of goods in a
disequilibrium situation (Schönfeld-Illy, 1948, p. 65). There is also a change in character of the proof of
the abbreviation procedure. For Rosenstein-Rodan the possibility of abbreviation is a direct result of
certain empirical, psychological facts: the unchanging world around and within us. For Schönfeld-Illy,
on the contrary, the possibility of abbreviation is a direct result of Gossen's first law of decreasing
marginal utility.
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IV. A first model of consumer abbreviation
To capture the difference between the abbreviating role of marginal utility in equilibrium and
disequilibrium, I look at the following groups of consumers. The first group are consumers in
disequilibrium by choice or by necessity. Examples of the former group are the so-called trendy
consumers and examples of the latter are emigrants on arrival in a foreign country, and people who
marry and start a family. These consumers are discovering what their optimal allocation pattern looks
like. For them the primary role of marginal utility is its economic relevance. The second group are the
familiar consumers from the traditional theory of consumer choice. They represent the established
consumers, who are in equilibrium, and for whom marginal utility acts as a watch-dog.
In a simple intertemporal model of consumer choice a household maximizes the utility function
(1) U = U ( Z t , Z t+1 ,..., Z t+n ),
where Zt stands for the composite commodity from which utility is
directly obtained in period t. The commodities are produced in a
household production function
(2) Z t = Z ( X t , T t , S t ),
with a composite market good, Xt , the consumer's time input, Tt , and the
consumer's human capital specific to calculation skill, St . All other
inputs
are
ignored.
I
assume that
∂ Zt
∂
∂
>0 , Zt >0 , Zt >0 ,
∂ Xt
∂Tt
∂ St
and also that
2
∂ Zt
>0 .
∂T t ∂ St
To analyze the role of marginal utility as a short-cut, a simple investment function is
adopted:
(3) S t = S ( Z t -1 , Z t -2 ,..., E t ) .
St is produced through "learning by doing." It accumulates the
effects of earlier commodity allocation. Et measures the effect of
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education and other human capital on calculation, i.e. abbreviation skills. I assume that
2
∂ St
∂S
∂ St
> 0, for all v in (3), t > 0, and
>0 .
∂ Z t -v
∂ Et
∂ Z t -v ∂ E t
I model the abbreviating role of marginal utility as a household production function models rational
habit formation (cf. Stigler and Becker 1977; Becker and Murphy 1988; Chaloupka 1991). The by now
familiar results are used and re-interpreted as follows.
The ratios of the marginal utilities and the shadow prices determine the optimal allocation of
consumption over time
∂U
M U zt
π
∂
(4)
= Z t = zt .
∂U
M U zt+n
π zt+n
∂ Z t +n
The term π T zt 8is the full price of commodity Zt and consist of two parts: the money price of the
composite commodity,
w
,
MPT zt
and the value of future time inputs from the effect of the production of Zt on subsequent S ,
 ∂ Z t+i ∂ Z t+i  dS zt+i
1
.
- w ∑ 
/
.
,

∂ T zt+i  dZ t (1+ r )i
i=1  ∂ S zt+i
n -t
where w is the wage rate, r the interest rate, and n the
length of life.
To be simplified to
(5) π Z t =
w
MPT zt
- At .
V. Different saving and consumption patterns
What predictions can be made for the consumption and saving patterns of the different groups of
consumers? For well-established consumers, the ones we know from traditional economic theory, the
last term of equation (5), At , is zero. The price becomes the familiar marginal cost formula. This group
of consumers is in equilibrium by definition: they move from one equilibrium to another and make only
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marginal comparisons in their calculations. From an abbreviation point of view the primary function of
marginal utility is its watch-dog function, its economic relevance is secondary. To develop an
abbreviation skill is neither intended nor necessary: they have it already.
For consumers who are in disequilibrium by neccessity, e.g., our emigrants, At is positive. This
means the relative price of consumption now declines versus the price of consumption in the future.
Present consumption of commodities tends to rise over time. In other words, the marginal utility of time
allocated to figuring out the present composite commodity is increased by the growth of the stock of
consumption capital. If the demand curve is sufficiently elastic, the consumption of goods will rise too
and savings decline. The abbreviating role of marginal utility as a short cut in calculations results in
consumption as a beneficial addiction.
Those who are in disequilibrium by choice, our trendy-consumers, do not see their allocation
process as a positive investment. On the contrary, after the discovery of a new consumption pattern the
human capital specific to abbreviation loses its value. Calculation is shortsighted, myopic, or farsighted.
In the first case At is zero, just as with the well-established consumers. Only this time, one is myopic, or
shortsighted, because one does not want to be farsighted. While with the well-established consumers
one is shortsighted because one does not need to be farsighted. In the last case abbreviation is a harmful
habit. Abbreviation breeds a habit which one does not want to have. It costs time and energy to get rid of
it. At is negative. For the myopic trendy consumer
w
12is the money price of the composite
MPT zt
commodity; the relative price of consumption now and in the future does not change. For the trendy
consumer who sees the abbreviation skill as a negative habit the value of saving in future inputs is
negative. The shadow price rises, it is better to consume now than in the future. With an inelastic
demand curve, however, the consumption of goods doesn't fall in the future. An inelastic demand curve
seems a realistic assumption: trends can only be shown by conspicuous consumption. The abbreviating
role of marginal utility as a short cut in calculations results in consumption as a harmful rational
addiction.
Now, let me look at the producer's problem: Who are the potential buyers of new goods? Who are
the most susceptible to the advertising of new goods? On the one hand, advertizing lowers the search
costs for all groups of consumers, on the other hand, it makes human capital obsolete. The influence of
advertizing on human capital can be brought in the investment function by taking into account a
depreciation rate, δ, of human capital specific to abbreviation, St ,
(6) S t = S ( Z t -1 , Z t -2 ,..., E t , δ ) .
For the well-established consumer who has the biggest investment in human capital δ is strongly
negative. Abbreviation skill is of the greatest importance: without it he cannot exist. For consumers who
are in disequilibrium by necessity, though δ is still negative, the odds at stake are much lower. Our
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emigrants and young couples are still discovering, by trial and error, their optimal consumption pattern.
For trendy consumers, who are in disequilibrium by choice, the fact that human capital specific to
abbreviation becomes obsolete is either of no concern, if he is shortsighted, or, in the case of
abbreviation as a negative addictive good, a positive fact. δ is zero for the myopic trendy consumer and
positive for the farsighted trendy consumer.
Because the relative prices change for all groups, saving and consumption patterns will change.
Well-established consumers buy the least new goods, shortsighted trendy consumers buy more, and
farsighted trendy consumers with a positive depreciation rate buy the most new goods. As time goes by,
emigrants and just married couples buy less new goods.
To conclude: the way I handled abbreviation may strike some readers as rather old-fashioned at its
best or completely wrong at its worst. Is not one of the blessings of modern economic theory that it is
not expected to be a description of reality? Shouldn't one look for the implications for observable
behavior of utility-maximizing theory? (Stigler, 1965, p.153). I believe, however, that the paper proves
that there is still something to be gained from looking at the early phases of the development of utility
theory. It gives us just what we want: testable predictions.
Literature
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Chaloupka, Frank. "Rational Addictive Behavior and Cigarette Smoking," J.P.E. 99 (August 1991):
722-42.
Rosenstein-Rodan, Paul (1927). "Marginal Utility." Reprinted in International Economic Papers, no.
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