1. Economic utility theory seeks to quantify personal happiness to inform policy decisions, but relies on assumptions about cardinalizing utility that are difficult to extend between individuals. 2. Policymakers can simplify decisions by assuming utility is cardinal and comparable between people, but this ignores distributional concerns. Practical tools like cost-benefit analysis, QALYs, and the U-Index make different assumptions that prioritize either efficiency or equity. 3. There is no perfect system, as Arrow's Impossibility Theorem shows, so policymakers must understand the tradeoffs inherent in different decision rules regarding efficiency, equity, and interpersonal comparisons.

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Economic Utility
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Benjamin Daniels
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2. Introduction
Economics is the science of choice and scarcity. There's not enough stuff to go around, so
how do people choose what they buy? Perhaps even more importantly, how do governments
decide how to give out things like public schooling, tax rebates, and food stamps? Any decision
of this type requires comparing the costs and benefits of a particular transfer, but depending on
the underlying assumptions, it may not actually be possible to compare these things. Some
conceptions of utility space can clearly order sets of individual preferences but cannot describe
the distance between them, making the potential gains from trade difficult or impossible to
measure. How can comparisons and decisions be made in this environment? What does this
failing imply for theories of public choice? For theories of economic efficiency?
Economic utility theory is the branch of economics concerned with answering questions
about happiness. By examining these questions through the lens of utility theory, it is possible to
see that ordinality and cardinality play an important role in the theoretical constructions of utility
space and interpersonal comparisons within it. In particular, economic utility theory seeks to
answer crucial questions about personal happiness, then quantify the results in order to provide
useful numerical outputs to policymakers and analysts. However, economists differ on the
fundamental assumptions that can be made in the realm of utility theory. In order to build a
useful model to aggregate happiness, it is important to first begin from the fundamental
conceptions of personal utility, then determine whether they can be successfully extended to
interpersonal and aggregate comparisons.
In particular, an “indifference curves” approach is most useful when attempting to make
sense of commodity and utility spaces. By applying basic assumptions about the order of these
spaces, it is possible to generate a set of equivalence classes that fully describe the space. Due to
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3. the shape of these classes, it becomes straightforward to compare different goods and perform
Lagrangian optimization tasks across them. However, the indifference curves approach is less
well suited to social comparisons, unless it is possible to devise a method for making
interpersonal comparisons. Even then, optimization is much less straightforward, because the set
of inputs to a social utility function is much larger and their interactions more complex than the
components of individual utility.
Policymakers can use various underlying mathematical assumptions about the
cardinalization of utility space for individuals and for society as a whole in order to dramatically
simplify the decision-making process. Indeed, without these simplifications, it is impossible to
construct a decision rule that accurately reflects the desires of the society as a whole, as Kenneth
Arrow proved in 1950. However, politicians and policymakers must be aware of the assumptions
they are making when utilizing different simplifying rules, particularly when the assumptions
have implications for overall efficiency and distributional equity. After establishing the
theoretical basis for the analysis and demonstrating the conflicts inherent in it, this essay will
review several such rules and analyze the strengths and weaknesses they exhibit with respect to
the overall theory of welfare maximization.
From the popular construction of the theory to the practical implementation, one gap is
overwhelmingly obvious – a lack of concern for or understanding of problems of distributional
equity. While the theoretical formalization of the public choice problem readily admits this
failing, the practical application generally assumes it away, and in doing so often implicitly
selects the most utilitarian limit case. If society wishes to design programs that do not conform to
this measure of societal well-being, a more thorough analysis of the topic will be needed if
decisions are to be fully and completely analyzed.
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4. A full analysis of every proposed decision rule is crucial because Arrow’s Impossibility
Theorem, stated as follows, shows that any social decision rule will be imperfect: Any decision
rule that respects transitivity, independence of irrelevant alternatives, and unanimity is a
dictatorship. Transitivity is defined the usual way. Unanimity is defined to mean that if every
individual prefers A to B, then option A will be preferred by the decision rule to B.
Independence of irrelevant alternatives (IIA) is defined to mean that the societal relative
ordering of A and B is affected only by the relative orderings of A and B by individuals.
Proof. Let option B be chosen arbitrarily. By unanimity, in any preference profile in
which every voter puts B at the very top or very bottom of his rankings, so too must society.
Suppose to the contrary that for such a profile and for distinct A, B, and C, the social preference
ranked A > B and B > C. By IIA, this would hold even if every individual moved C above A,
because that could be arranged without changing the relative order of A and B or C and B
(because B is at the very top or very bottom of every profile). By transitivity the social ranking
would then place A > C, but by unanimity it should also place C > A, leading to a contradiction.
In order to avoid the contradiction, there must be some voter n who is pivotal in the sense
that by changing his vote he can move B from the bottom of the social profile to the top (whether
this happens immediately or after some 1-n voters have changed their votes is irrelevant). In this
case, voter n is a dictator over any arbitrary pair of options (A, C) where A, C ≠ B, since the
ordering of A and C relative to B, and therefore to each other, is determined by the location of A,
B, and C in voter n’s individual preferences. But because A, B, and C are chosen arbitrarily, we
can see that there must be such a dictator for any pair of options. In fact, this dictator will be
voter n for every pair, since the relative societal rankings of any other option B are dependent on
the rankings of A and C, which voter n also controls. So the system is a dictatorship.▪
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5. Ordering Preferences and Making Decisions
To avoid dictatorship, we must accept flawed systems. But to fully analyze these systems,
we must be able to discuss their impact on individuals. The basic building blocks of individual
utility are commodities. When economists wonder how happy a person is, it is generally
accepted to define the level of happiness as some function Uj of the various commodities
possessed by a person j. For simplicity, we examine only that subclass of commodities known as
“goods” – commodities for which the utility
Figure 1. Commodity space for guns and butter,
function has a positive-valued partial and an indifference curve IB.
derivative. The simplest possible case, and
perhaps the most illustrative, is the two-
good economy. In this case, the quantity of
each good can be mapped clearly onto a
commodity space as shown in Figure 1.
It is natural to make several assumptions that lead to the well-ordering of the commodity
space. First, order is imposed; that is, for any two points A and B, exactly one of the following
must be true: A is preferred to B, B is preferred to A, or the individual is indifferent between the
two points. Second, preferences are transitive; if A > B and B > C, then A > C. Third, because
we have restricted our examination to “goods”, preferences are positive; moving in a rightward
or upward direction (ie, obtaining more of one good without sacrificing any of another) is always
preferable to the reference point. Lastly, preferences are convex, illustrated by the “indifference
curve” moving away from Point B in the diagram; in order to convince an individual to sacrifice
marginal units of one good, it requires an increasing amount of the other good as compensation.
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6. Together, these rules yield a set of equivalence classes in commodity space known as
indifference curves and denoted by Iij with index i for the individual and j for the utility level.
These curves are dense in the space and, as one moves upward and rightward, represent ever-
increasing levels of personal utility; j increases with utility. The conception is readily generalized
to many-dimensional commodity spaces and may even be extended to include commodities for
which Uj has a negative partial derivative with some adjustment to the underlying assumption of
positivity. In addition, because indifference curves themselves represent a new well-ordered
space, utility space, it becomes trivial for a mathematically-inclined individual to maximize his
or her utility: a simple Lagrangian process incorporating the prices of various goods and a budget
constraint allows the individual to select an allocation that achieves the highest level of utility.
Although individual utility space is well-ordered as a result of these assumptions, it is not
yet a cardinalized space. That is, while we can easily say that Iim > Iin whenever m > n, we have
not established a method for determining the value of the ratio Iim/Iin. This is made more difficult
by the fact that indifference curves are dense, and therefore has cardinality equal to that of the
continuum, c; the indices are not natural numbers. To establish a base point for these ratios, it is
common to select an arbitrary reference curve Ii1, then cardinalize the space such that Iim/Iin =
m/n. By this method the ratio of utilities of any two commodity vectors can be determined,
although there is still no conception for determining ratios between individuals, such as I1m/I2m.
Given these relatively simple assumptions and the system they imply, the next step is to
attempt to extend the approach to a social level. Since not all the decisions of society are made in
open commodity markets, the characteristics and limitations of these methods when applied to
social decisions, public goods, and the like will be crucial for policymakers to understand the
impacts and restrictions of various types of analysis.
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7. Aggregating Preferences and Making Social Decisions
The social analysis begins from much the same place as the individual analysis, for
obvious reasons: society as an entity is made up only of the individuals in it. As far as
commodities are concerned, a society can utilize exactly the same process as an individual for
mapping preferences and optimizing outputs. However, this is not the whole story for a
collective decision to be made:
The distribution of wealth is important for determining values and shaping production, and it can
even be maintained that a country with one and the same amount of general wealth may be rich or
poor according to the manner in which that wealth is distributed. (Schumpeter)
So although society, like the individuals who constitute it, benefits from a greater number of
goods, it is also important to take into account the impact of those goods on various individuals
by way of a distributional analysis.
Given this information, the utility space for a society gains a number of dimensions equal
to the number of people in the society. This construction represents the fact that, in addition to
making tradeoffs among different types of goods, society must also make tradeoffs about the
allocation of those goods. However, it is not sufficient to quantify these tradeoffs among
individuals in commodity terms, or even in dollar terms; from individual indifference-curve
analysis it is clear that different individuals have different valuations for various goods. Instead,
the tradeoffs among people must be represented in utility terms. The crucial cardinal expression
needed for this construction – the slope Iim/Ijm – is presently undefined, because there is no
accepted conception of individual utility that allows interpersonal comparison. There exist
various methods for addressing this problem in practice, but in order to fully appreciate them,
their implications must first be examined through theories of distributional equity.
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8. One key question facing decision-makers is the shape of the social indifference curve
with respect to distribution of goods among individuals. Theoretically, this parameter is the most
immediate method for assessing the importance of distributional equity to the overall societal
utility function. The analysis dates back to the utilitarian conception advanced by Jeremy
Bentham, who asserted that distributional concerns were entirely useless; in this conception, a
good is valued at exactly the same societal worth, no matter who possesses it. The diametrically
opposed theory was later advanced by John Rawls, who claimed that society should only be
judged by its worst-off citizens; additional goods were worthless if they were distributed to
anyone else. Graphically and algebraically, these arguments represent the upper and lower
bounds on the space of indifference curves most simply expressed by the formula
Rawls demonstrates the limit as α approaches -∞ and Bentham the case as α approaches 1.
Figure 2 illustrates the space of indifference curves generated by this conception over a simple 2-
person society. A Rawlsian society more concerned with equality will distribute goods more
evenly whereas, in the limiting Benthamite case, the additional interpersonal dimensions are
rendered irrelevant to the maximization by the flat shape of the indifference curves.
Figure 2. Rawlsian and Benthamite societal indifference curves.
R represents the social choice made with Rawlsian assumptions
and B the Benthamite choice.
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9. In Practice: CBA, QALYs, and the U-Index
When designing actual social policies, legislators and executives must handle these twin
hurdles of utility measurement and distributional analysis. They must decide how to measure
individual utility, how to measure it against the utility of others, and what weight to put on the
competing concepts of maximization and equity. Three practical and theoretical mechanisms
highlight the various choices that could be made by a policymaker: cost-benefit analysis, the
QALY system, and the U-Index. Each of these emphasizes a different aspect of the social
decision and, together, they demonstrate the fundamental fact that tradeoffs must be made which
cannot be quantitatively compared.
The time-honored practice of cost-benefit analysis (CBA) calls on the most utilitarian
assumptions to make public choices. It weights total costs directly against total benefits, entirely
without regard for distributional impacts. Most public accounting, whether directly or indirectly,
utilizes CBA in order to justify budgets and often includes explicit costs and benefits only; for
example, Congressional Budget Office scores are often manipulated by legislators by shifting
expenditures to states or individuals or by making advantageous assumptions about the value of
various services. However, CBA is not without merits; it does, in the broadest sense, capture the
economic ideal of maximization by providing a clear quantitative guideline for the value of a
public project or a benefit schedule and allowing for sound derivative-based optimization. When
costs and benefits are clear and distribution is of minor concern, CBA can be an appropriate
choice for decision-makers.
Another method of decision-making is demonstrated by the QALY system, a method of
apportioning health benefits utilized by the British National Health Service. To construct the
QALY system, researchers used an extensive series of interviews to determine the relative values
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10. of various health states as judged by normal people; the results were averaged and normalized
such that one year of normal life has a value of one Quality-Adjusted Life Year (QALY). The
key assumption underlying this valuation is that every life has the same value, which sidesteps
the problem of interpersonal comparison entirely. The QALY values are calculated by
multiplying expected life quality produced by expected longevity, so
Value = Quality ∙ Longevity
The marginal value of a procedure is given by its expected impact on the value of a life, and
thanks to the inherent Lagrangian process we can use it to assess the distributional impacts of the
QALY system. The total derivative is given by:
∆Value = (Quality ∙ ∆Longevity) + (∆Quality ∙ Longevity)
The inherent bias of this total derivative is to target longevity-increasing treatments to those with
a good quality of life, and quality-enhancing treatments to those expected to live the longest.
While this seems sensible from the abstract efficiency criterion, it is important to note that it
would also be possible to design a system that attempted to improve the quality of life for the
worst-off class first – a system that would certainly be preferable for the very sick.
The U-Index is a theoretical system proposed in order to avoid cardinalizing personal
utility space altogether. In a similar manner as the QALY system, the U-Index obtains ordinal
and relative measures of well-being from participants. It converts the responses into directly
comparable cardinal measurements by characterizing a respondent into binary groups of “happy”
and “unhappy” based on responses, and then reports a “U-Index” for an activity based on the
proportion of people who are unhappy during that activity. Like CBA and the QALY system, the
U-Index provides a clear, cardinal, quantitative target for policymakers to optimize, but like the
other systems it dodges the question of distributional impact by assuming utilitarian equality.
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11. Conclusion
The examples used to demonstrate practical applications of utility and social choice
theory make clear one crucial shortcoming of the theory: it is extremely difficult to explicitly
value interpersonal comparisons of utility. Generally, policymakers elect to assume them away
entirely, choosing to implement programs on a mostly utilitarian basis, and giving weight to
distributional concerns only indirectly – by allowing measured benefits to increase with need.
Indeed this is the largest gap in the underlying theory as well. Despite the obvious need for a
method of valuing interpersonal exchanges from the lens of society, theory and practice have not
solved the key question of cardinalizing interpersonal comparisons such that I1m/I2m has a well-
defined value, nor have they attempted to devise simplifying assumptions that handle the
problem in a more nuanced manner than simply assuming the value to be equal to m/m = 1.
However, the shortcoming may not be entirely the fault of the policymakers. Even
theories that are relatively liberal – the British healthcare system, for example – have defaulted to
the utilitarian conception implicitly, almost without realizing it. In addition, academic literature
beyond Rawls and responses to him offers no guidance on how to better include equity concerns
or how to “optimize” equity. It is a far easier task to maximize output than to ensure that it winds
up in the right hands, and even today much work on inequality lacks the formal rigor that was
developed for efficiency by the early Chicago school. Policymakers motivated to address
inequality will need a stronger set of theoretical and mathematical tools with which to develop
and assess policy options if they are to use anything other than the simplest criteria for
distributional equity.
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12. References
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Well-Being.” J. of Econ. Perspectives 20 (2006). 3-24
Kaldor, Nicholas. “Welfare Propositions of Economics and Interpersonal Comparisons of
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Social Choice."Cowles Foundation Discussion Papers 1322, Cowles Foundation for
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Economic Review 96, No. 4 (Sep 2006). 1114-1136.
Mercuro, Nicholas, and Steven G. Medema. Economics and the Law, Second Edition: From
Posner to Postmodernism and Beyond. 1997.
Schumpeter, Joseph. “On the Concept of Social Value.” Quarterly J. of Econ. 23 (1908-9).
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