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Expected Utility Hypothesis

Definition of expected utility hypothesis.

The Expected Utility Hypothesis is a theory in economics that suggests individuals choose between alternatives to maximize their expected utility—a measure of satisfaction or happiness derived from the outcomes of their choices. This hypothesis operates under the assumption that people are rational actors who make decisions based on the potential risks and benefits, by calculating and comparing the expected utility of different actions. The concept is rooted in the broader field of utility theory, which seeks to explain how individuals prioritize choices to achieve the highest level of personal satisfaction.

Consider Alice, a financial advisor, deciding whether to invest in stock A or stock B for her portfolio. Stock A offers a 60% chance of earning $100 and a 40% chance of losing $50, whereas stock B offers a 50% chance of earning $80 and a 50% chance of losing $40. Using the Expected Utility Hypothesis, Alice would calculate the expected utility for each stock based on her personal risk tolerance. If Alice is risk-averse, she might find the expected utility of the more stable stock B higher than that of stock A, despite the potential for higher gains with stock A, and therefore, choose stock B for her investment.

Why Expected Utility Hypothesis Matters

The Expected Utility Hypothesis is fundamental in economics and finance because it provides a structured way to analyze how individuals make decisions under uncertainty. By understanding this hypothesis, economists and policymakers can better predict consumer behavior, design more effective financial products, and develop policies that align with how people assess risk and make choices. For investors and financial professionals, applying the expected utility hypothesis can aid in constructing portfolios that better meet their risk tolerance and financial objectives.

Frequently Asked Questions (FAQ)

How does the expected utility hypothesis account for risk preferences.

The hypothesis incorporates risk preferences by adjusting the utility values based on the individual’s risk tolerance. A risk-averse person, who prefers to avoid risk, assigns higher utility to outcomes with more certainty. Conversely, a risk-seeking individual, who prefers riskier alternatives for the chance of higher rewards, assigns higher utility to outcomes with greater risk. By considering these preferences, the expected utility hypothesis explains why different individuals might make different choices under the same circumstances.

Can the expected utility hypothesis apply to non-financial decisions?

Yes, the expected utility hypothesis is not limited to financial decisions but can apply to any situation involving uncertainty. For example, a person deciding whether to take an umbrella on a day with a 50% chance of rain might weigh the inconvenience of carrying an umbrella against the discomfort of getting wet. Their decision will reflect their personal preferences and the expected utility of each option, taking into account their aversion to risk (in this case, getting wet).

How do real-world behaviors challenge the expected utility hypothesis?

While the expected utility hypothesis provides a valuable framework for understanding decision-making under uncertainty, real-world behaviors sometimes deviate from its predictions. Psychological factors, biases, and heuristics can influence decisions in ways that are inconsistent with pure rationality. For instance, the prospect theory, developed by Daniel Kahneman and Amos Tversky, highlights how people value gains and losses differently, leading to decision-making patterns that the expected utility hypothesis would not predict. Understanding these deviations is crucial for developing a more accurate model of human behavior.

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Expected Utility Theory

This is a theory which estimates the likely utility of an action – when there is uncertainty about the outcome. It suggests the rational choice is to choose an action with the highest expected utility.

This theory notes that the utility of a money is not necessarily the same as the total value of money. This explains why people may take out insurance. The expected value from paying for insurance would be to lose out monetarily. But, the possibility of large-scale losses could lead to a serious decline in utility because of the diminishing marginal utility of wealth.

Expected value

Expected value is the probability-weighted average of a mathematical outcome.

For example, suppose:

• A lottery ticket costs $20.
• The probability of winning the $2000 prize is 0.5%
• The likely value from having a lottery ticket will be the outcome x probability of the event occurring.
• Therefore, expected value = 0.005 x 2000 = $10

The expected value of owning a lottery ticket is $10. With an infinite number of events, on average, this is the likely payout. Of course, we may be lucky or maybe unlucky if we play only once.

Since the ticket costs $20, it seems an illogical decision to buy – because the expected value of buying a ticket is $10 – a smaller figure than the cost of purchase $20.

Expected value of a university degree

Suppose we decide to study for three years to try and gain an economic degree. A good degree is likely to lead to a higher paying job but there is no guarantee. We may fail the degree or the jobs market may turn against a surplus of graduates.

Therefore, we may estimate we have a 0.7 chance of gaining an extra $250,000 earnings in our lifetime. In this case, the expected utility of an economics degree is $175,000.

From expected value to expected utility

In 1728, Gabriel Cramer wrote to Daniel Bernoulli:

“the mathematicians estimate money in proportion to its quantity, and men of good sense in proportion to the usage that they may make of it.”

In other words, an extra $1,000 does not always have the same impact on our marginal utility .

If you are poor and your income rises from $1,000 a year to $2,000 a year this will have a big improvement in utility and your quality of life.

However, if you are already rich and your income rises from $100,000 to $101,000 a year, the improvement in utility is small.

Therefore, if you are earning $100,000 a year, it makes sense to be risk-averse about the small possibility of losing all your wealth. By spending $1,000 a year on insurance, you lose $1,000 but protect against that limited possibility of losing everything.

The loss in utility from spending that extra $1,000 is small. But, protecting against the loss of everything enables protection against a devastating loss of livelihood.

Bernoulli in Exposition of a New Theory on the Measurement of Risk (1738) argued that expected value should be adjusted to expected utility – to take into account this risk aversion we often see.

Bernoulli noted most would pay a risk premium (losing out on expected value) in order to insure against events of low probability but very potential high loss.

Logic of insurance

• Suppose the chance of house being destroyed by lightning is 0.0001, but if it is destroyed you lose $300,000.
• The expected value of your house is therefore 0.9999 x 300,000 = $299,970.
• The expected loss of your house is just $30.
• An insurance company may be willing to insure against the loss of your 300,000 house for $100 a year.
• According to the expected value, you should not insure your house. The cost of insurance $100 is far greater than the expected loss $30 from the house being destroyed.
• However, the expected utility is different.
• If you are wealthy, paying $100 only has a small marginal decline in utility.
• However, if you were unlucky and lost your house the loss of everything would have a corresponding greater impact on utility.

Risk aversion and the diminishing marginal utility of wealth

However, an increase in wealth from £70 to £80 leads to a correspondingly small increase in utility (30 to 31).

This concave graph shows the diminishing marginal utility of money and a justification for why people may exhibit risk aversion for potentially large losses with small probabilities.

• Marginal utility theory
• Diminishing marginal utility of wealth/income

Expected Utility Hypothesis

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The expected utility hypothesis – that is, the hypothesis that individuals evaluate uncertain prospects according to their expected level of ‘satisfaction’ or ‘utility’ – is the predominant descriptive and normative model of choice under uncertainty in economics. It provides the analytical underpinnings for the economic theory of risk-bearing, including its applications to insurance and financial decisions, and has been formally axiomatized under conditions of both objective (probabilistic) and subjective (event-based) uncertainty. In spite of evidence that individuals may systematically depart from its predictions, and the development of alternative models, expected utility remains the leading model of economic choice under uncertainty.

This chapter was originally published in The New Palgrave Dictionary of Economics , 2nd edition, 2008. Edited by Steven N. Durlauf and Lawrence E. Blume

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Non-expected Utility Theory

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Machina, M.J. (2008). Expected Utility Hypothesis. In: The New Palgrave Dictionary of Economics. Palgrave Macmillan, London. https://doi.org/10.1057/978-1-349-95121-5_127-2

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Expected Utility

The utility of an action or event over a time period when the circumstances are unknown

What is Expected Utility?

Expected utility is a theory in economics that estimates the utility of an action when the outcome is uncertain. It advises choosing the action or event with the maximum expected utility. At any point in time, the expected utility will be the weighted average of all the probable utility levels that an entity is expected to reach under specific circumstances.

• Expected utility is the utility of an action or event over a time period when the circumstances are unknown.
• The expected utility will be the aggregate of the products of possible outcomes with the probability of occurrence of the events.
• While analyzing uncertain situations, entities may or may not choose the action with the highest value of expected utility, depending on their risk aversion.

Understanding Expected Utility

Expected utility is used as a tool for decision-making under circumstances where the outcomes of decisions are not known. The entity computes the probability of outcomes and compares them with expected utility. The expected utility value is calculated by aggregating the products of possible outcomes with the probability of occurrence of the events.

The expected utility theory considers it a logical choice to choose the event with the maximum expected utility. However, in case of risky outcomes, decision-makers may not choose the action with a higher expected utility. The decision to choose an action will also depend on the entity’s risk aversion and other entities’ utility. While some entities choose the option with the riskier highest expected utility, some highly risk-averse entities prefer the low-risk option even if it shows a lower expected value.

Expected utility theory also helps to explain the reason for people taking out insurance policies . It is a situation where the payback is not immediate; however, insurance policies cover individuals for several risks. Insurance policyholders receive tax benefits and a certain income at the expiry of a predetermined period. Hence, when one compares the expected utility to be received from paying insurance premiums with the expected utility of investing the amount on other products, insurance appears to be a better choice.

The concepts of marginal utility and expected utility are related. The expected utility of wealth or a reward reduces when the entity possesses sufficient wealth. Such entities may go for the safer alternative instead of the riskier ones.

The addition of $1,000 to the income may not impact the marginal utility of two different entities in the same way. For example, if the annual income of a low-earning family is increased from $1,250 to $2,250, it will improve their quality of life as well as the marginal utility. On the contrary, if the income of a high-earning family increases from $120,000 to $121,000 in a year, there is a very small utility improvement.

Some Applications of Expected Utility

1. public and economics policy.

The expected utility theory finds application in public policy, as it explains that the social arrangement that maximizes the total welfare across society is the most socially right arrangement. The concept of micromort, introduced by American professor Ronald Howard in the 1980s, uses the expected utility concept to measure the acceptability of various mortality risks.

The expected utility concept is also used to guide health policies. The expected utilities of various health interventions are used while framing health policies. The area of insurance sales also uses expected utility theory to calculate risks with the goal of financial gain in the long term while taking into consideration the possibility of going bust temporarily.

Utilitarians believe that the result of an act determines whether or not the right action is taken. However, it is extremely difficult to establish the long-term consequence of an act. Hence, some authors argue that instead of the act that results in the best consequences, the act with the highest expected moral value should be considered as the right act.

Others argue that even if we should do what will have the best outcome, the expected utility theory may help in making decisions when the consequences of the acts become uncertain. The consequentialism version of maximizing the expected utility is a moral choice.

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Normative Theories of Rational Choice: Expected Utility

We must often make decisions under conditions of uncertainty. Pursuing a degree in biology may lead to lucrative employment, or to unemployment and crushing debt. A doctor's appointment may result in the early detection and treatment of a disease, or it may be a waste of money. Expected utility theory is an account of how to choose rationally when you are not sure which outcome will result from your acts. Its basic slogan is: choose the act with the highest expected utility.

This article discusses expected utility theory as a normative theory—that is, a theory of how people should make decisions. In classical economics, expected utility theory is often used as a descriptive theory—that is, a theory of how people do make decisions—or as a predictive theory—that is, a theory that, while it may not accurately model the psychological mechanisms of decision-making, correctly predicts people's choices. Expected utility theory makes faulty predictions about people's decisions in many real-life choice situations (see Kahneman & Tversky 1982); however, this does not settle whether people should make decisions on the basis of expected utility considerations.

The expected utility of an act is a weighted average of the utilities of each of its possible outcomes, where the utility of an outcome measures the extent to which that outcome is preferred, or preferable, to the alternatives. The utility of each outcome is weighted according to the probability that the act will lead to that outcome. Section 1 fleshes out this basic definition of expected utility in more rigorous terms, and discusses its relationship to choice. Section 2 discusses two types of arguments for expected utility theory: representation theorems, and long-run statistical arguments. Section 3 considers counterexamples to expected utility theory; section 4 discusses its applications in philosophy of religion, economics, ethics, and epistemology.

1.1 Conditional Probabilities

1.2 outcome utilities, 2.1 long-run arguments, 2.2 representation theorems, 3.1 counterexamples involving transitivity and completeness, 3.2 counterexamples involving independence, 3.3 counterexamples involving probability 0 events, 3.4 counterexamples involving unbounded utility, 4.1 philosophy of religion, 4.2 economics, 4.4 epistemology, other internet resources, related entries, 1. defining expected utility.

The concept of expected utility is best illustrated by example. Suppose I am planning a long walk, and need to decide whether to bring my umbrella. I would rather not tote the umbrella on a sunny day, but I would rather face rain with the umbrella than without it. There are two acts available to me: taking my umbrella, and leaving it at home. Which of these acts should I choose?

This informal problem description can be recast, slightly more formally, in terms of three sorts of entities. First, there are outcomes —objects of non-instrumental preferences. In the example, we might distinguish three outcomes: either I end up dry and unencumbered; I end up dry and encumbered by an unwieldy umbrella; or I end up wet. Second, there are states —things outside the decision-maker's control which influence the outcome of the decision. In the example, there are two states: either it is raining, or it is not. Finally, there are acts —objects of the decision-maker's instrumental preferences, and in some sense, things that she can do. In the example, there are two acts: I may either bring the umbrella; or leave it at home. Expected utility theory provides a way of ranking the acts according to how choiceworthy they are: the higher the expected utility, the better it is to choose the act. (It is therefore best to choose the act with the highest expected utility—or one of them, in the event that several acts are tied.)

Following general convention, I will make the following assumptions about the relationships between acts, states, and outcomes.

• States, acts, and outcomes are propositions, i.e., sets of possibilities. There is a maximal set of possibilities, \(\Omega\), of which each state, act, or outcome is a subset.
• The set of acts, the set of states, and the set of outcomes are all partitions on \(\Omega\). In other words, acts and states are individuated so that every possibility in \(\Omega\) is one where exactly one state obtains, the agent performs exactly one act, and exactly one outcome ensues.
• Acts and states are logically independent, so that no state rules out the performance of any act.
• I will assume for the moment that, given a state of the world, each act has exactly one possible outcome. (Section 1.1 briefly discusses how one might weaken this assumption.)

So the example of the umbrella can be depicted in the following matrix, where each column corresponds to a state of the world; each row corresponds to an act; and each entry corresponds to the outcome that results when the act is performed in the state of the world.

Having set up the basic framework, I can now rigorously define expected utility. The expected utility of an act \(A\) (for instance, taking my umbrella) depends on two features of the problem:

• The value of each outcome, measured by a real number called a utility .
• The probability of each outcome conditional on \(A\).

Given these three pieces of information, \(A\)'s expected utility is defined as:

where \(O\) is is the set of outcomes, \(P_{A}(o)\) is the probability of outcome \(o\) conditional on \(A\), and \(U(o)\) is the utility of \(o\).

The next two subsections will unpack the conditional probability function \(P_A\) and the utility function \(U\).

The term \(P_{A}(o)\) represents the probability of \(o\) given \(A\)—roughly, how likely it is that outcome \(o\) will occur, on the supposition that the agent chooses act \(A\). (For the axioms of probability, see the entry on interpretations of probability .) To understand what this means, we must answer two questions. First, which interpretation of probability is appropriate? And second, what does it mean to assign a probability on the supposition that the agent chooses act \(A\)?

Expected utility theorists often interpret probability as measuring individual degree of belief, so that a proposition \(E\) is likely (for an agent) to the extent that that agent is confident of \(E\) (see, for instance, Ramsey 1926, Savage 1972, Jeffrey 1983). But nothing in the formalism of expected utility theory forces this interpretation on us. We could instead interpret probabilities as objective chances (as in von Neumann and Morgenstern 1944), or as the degrees of belief that are warranted by the evidence, if we thought these were a better guide to rational action. (See the entry on interpretations of probability for discussion of these and other options.)

What is it to have a probability on the supposition that the agent chooses \(A\)? Here, there are two basic types of answer, corresponding to evidential decision theory and causal decision theory.

According to evidential decision theory, endorsed by Jeffrey (1983), the relevant conditional probability \(P_{A}(o)\) is the conditional probability \(P(o \mid A)\), defined as the ratio of two unconditional probabilities: \(P(A \amp o) / P(A)\).

Against Jeffrey's definition of expected utility, Spohn (1977) and Levi (1991) object that a decision-maker should not assign probabilities to the very acts under deliberation: when freely deciding whether to perform an act \(A\), you shouldn't take into account your beliefs about whether you will perform \(A\). If Spohn and Levi are right, then Jeffrey's ratio is undefined (since its denominator is undefined).

Nozick (1969) raises another objection: Jeffrey's definition gives strange results in the following Newcomb Problem. A predictor hands you a closed box, containing either $0 or $1 million, and offers you an open box, containing an additional $1,000. You can either refuse the open box (“one-box”) or take the open box (“two-box”). But there's a catch: the predictor has predicted your choice beforehand, and all her predictions are 90% accurate. In other words, the probability that you one-box, given that she predicts you one-box, is 90%, and the probability that you two-box, given that she predicts you two-box, is 90%. Finally, the contents of the closed box depend on the prediction: if the predictor thought you would two-box, she put nothing in the closed box, while if she thought you would one-box, she put $1 million in the closed box. The matrix for your decision looks like this:

Two-boxing dominates one-boxing: in every state, two-boxing yields a better outcome. Yet on Jeffrey's definition of conditional probability, one-boxing has a higher expected utility than two-boxing. There is a high conditional probability of finding $1 million is in the closed box, given that you one-box, so one-boxing has a high expected utility. Likewise, there is a high conditional probability of finding nothing in the closed box, given that you two-box, so two-boxing has a low expected utility.

Causal decision theory is an alternative proposal that gets around these problems. It does not require acts to have probabilities (though it permits them to have probabilities), and it recommends two-boxing in the Newcomb problem.

Causal decision theorists define expected utility using variants on a proposal by Savage (1972). Savage calculates \(P_{A}(o)\) by summing the probabilities of states that, when combined with the act \(A\), lead to the outcome \(o\). Let \(f_{A,s}(o)\) be a function that takes on value 1 if \(o\) results from performing \(A\) in state s , and value 0 otherwise. Then

On Savage's proposal, two-boxing comes out with a higher expected utility than one-boxing. This result holds no matter which probabilities you assign to the states prior to your decision. Let \(x\) be the probability you assign to the state that the closed box contains $1 million. According to Savage, the expected utilities of one-boxing and two-boxing, respectively, are:

As long as the larger monetary amounts are assigned strictly larger utilities, the second sum (the utility of two-boxing) is guaranteed to be larger than the first (the utility of one-boxing).

Some versions of causal decision theory look superficially different from Savage's. For instance, Gibbard and Harper (1978/1981) and Stalnaker (1972/1981) propose that \(P_{A}(o)\) be understood as the probability of the counterfactual conditional “If \(A\) were performed, outcome \(o\) would ensue”. But Lewis (1981) shows how the Gibbard/Harper/Stalnaker proposal, as well as a variety of other proposals, can be recast in Savage's terms.

Savage assumes that each act and state are enough to uniquely determine an outcome. But there are cases where this assumption breaks down. Suppose you offer to sell me the following gamble: you will toss a coin; if the coin lands heads, I win $100; and if the coin lands tails, I lose $100. But I refuse the gamble, and the coin is never tossed. There is no outcome that would have resulted, had the coin been tossed—I might have won $100, and I might have lost $100.

A number of proposals relax Savage's assumption by letting \(f_{A,s}\) be a probability function. Lewis (1981), Skyrms (1980), and Sobel (1994) all suggest that \(f_{A,s}\) is given by taking the function that would measure the objective chances if s obtained, and conditionalizing that function on \(A\).

In some cases—most famously the Newcomb problem—the Jeffrey definition and the Savage definition of expected utility come apart. But whenever the following two conditions are satisfied, they agree.

• Acts are probabilistically independent of states. In formal terms, for all acts \(A\) and states \(s\), \[ P(s) = P(s \mid A) = \frac{P(s \amp A)}{P(A)}. \] (This is the condition that is violated in the Newcomb problem.)
• For all outcomes \(o\), acts \(A\), and states \(s\), \(f_{A,s}(o)\) is equal to the conditional probability of \(o\) given \(A\) and \(s\); in formal terms, \[ f_{A,s}(o) = P(o \mid A \amp s) = \frac{P(o \amp A \amp s)}{P(A \amp s)}.\] (The need for this condition arises when acts and states fail to uniquely determine an outcome; see Lewis 1981.)

The term \(U(o)\) represents the utility of the outcome \(o\)—roughly, how valuable \(o\) is. Formally, \(U\) is a function that assigns a real number to each of the outcomes. (The units associated with \(U\) are typically called utiles , so that if \(U(o) = 2\), we say that \(o\) is worth 2 utiles.) The greater the utility, the more valuable the outcome.

What kind of value is measured in utiles? Utiles are typically not taken to be units of currency, like dollars, pounds, or yen. Bernoulli (1738) argued that money and other goods have diminishing marginal utility: as an agent gets richer, every successive dollar (or gold watch, or apple) is less valuable to her than the last. He gives the following example: It makes rational sense for a rich man, but not for a pauper, to pay 9,000 ducats in exchange for a lottery ticket that yields a 50% chance at 20,000 ducats and a 50% chance at nothing. Since the lottery gives the two men the same chance at each monetary prize, the prizes must have different values depending on whether the player is poor or rich.

Classic utilitarians such as Bentham (1789), Mill (1861), and Sidgwick (1907) interpreted utility as a measure of pleasure or happiness. For these authors, to say \(A\) has greater utility than \(B\) (for an agent or a group of agents) is to say that \(A\) results in more pleasure or happiness than \(B\) (for that agent or group of agents).

One objection to this interpretation of utility is that there may not be a single good (or indeed any good) which rationality requires us to seek. But if we understand “utility” broadly enough to include all potentially desirable ends—pleasure, knowledge, friendship, health and so on—it's not clear that there is a unique correct way to make the tradeoffs between different goods so that each outcome receives a utility. There may be no good answer to the question of whether the life of an ascetic monk contains more or less good than the life of a happy libertine—but assigning utilities to these options forces us to compare them.

Contemporary decision theorists typically interpret utility as a measure of preference, so that to say that \(A\) has greater utility than \(B\) (for an agent) is simply to say that the agent prefers \(A\) to \(B\). It is crucial to this approach that preferences hold not just between outcomes (such as amounts of pleasure, or combinations of pleasure and knowledge), but also between uncertain prospects (such as a lottery that pays $1 million dollars if a particular coin lands heads, and results in an hour of painful electric shocks if the coin lands tails). Section 2 of this article addresses the formal relationship between preference and choice in detail.

Expected utility theory does not require that preferences be selfish or self-interested. Someone can prefer giving money to charity over spending the money on lavish dinners, or prefer sacrificing his own life over allowing his child to die. Sen (1977) suggests that each person's psychology is best represented using the rankings: one representing the person's narrow self-interest, a second representing the person's self-interest construed more broadly to account for feelings of sympathy (e.g., suffering when watching another person suffer), and a third representing the person's commitments, which may require her to act against her self-interest broadly construed.

Broome (1991) interprets utilities as measuring comparisons of objective betterness and worseness, rather than personal preferences: to say that \(A\) has a greater utility than \(B\) is to say that \(A\) is objectively better than \(B\). Broome suggests that \(A\) is better than \(B\) just in case a rational person would prefer \(A\) to \(B\). Just as there is nothing in the formalism of probability theory that requires us to use subjective rather than objective probabilities, so there is nothing in the formalism of expected utility theory that requires us to use subjective rather than objective values.

Those who interpret utilities in terms of personal preference face a special challenge: the so-called problem of interpersonal utility comparisons . When making decisions about how to distribute shared resources, we often want to know if our acts would make Alice better off than Bob—and if so, how much better off. But if utility is a measure of individual preference, there is no clear, meaningful way of making these comparisons. Alice's utilities are constituted by Alice's preferences, Bob's utilities are constituted by Bob's preferences, and there are no preferences spanning Alice and Bob. We can't assume that Alice's utility 10 is equivalent to Bob's utility 10, any more than we can assume that getting an A grade in differential equations is equivalent to getting an A grade in basket weaving.

Now is a good time to consider which features of the utility function carry meaningful information. Comparisons are informative: if \(U(o_1) \gt U(o_2)\) (for a person), then \(o_1\) is better than (or preferred to) \(o_2\). But it is not only comparisons that are informative—the utility function must carry other information, if expected utility theory is to give meaningful results.

To see why, consider the umbrella example again. This time, I've filled in a probability for each state, and a utility for each outcome.

The expected utility of taking the umbrella is

while the expected utility of leaving the umbrella is

Since \(EU(\take) \gt EU(\leave)\), expected utility theory tells me that taking the umbrella is better than leaving it.

But now, suppose we change the utilities of the outcomes: instead of using \(U\), we use \(U'\).

The new expected utility of taking the umbrella is

while the new expected utility of leaving the umbrella is

Since \(EU'(\take) \lt EU'(\leave)\), expected utility theory tells me that leaving the umbrella is better than taking it.

The utility functions \(U\) and \(U'\) rank the outcomes in exactly the same way: free, dry is best; encumbered, dry ranks in the middle; and wet is worst. Yet expected utility theory gives different advice in the two versions of the problem. So there must be some substantive difference between preferences appropriately described by \(U\), and preferences appropriately described by \(U'\). Otherwise, expected utility theory is fickle, and liable to change its advice when fed different descriptions of the same problem.

When do two utility functions represent the same basic state of affairs? Measurement theory answers the question by characterizing the allowable transformations of a utility function—ways of changing it that leave all of its meaningful features intact. If we characterize the allowable transformations of a utility function, we have thereby specified which of its features are meaningful.

Some analogies from Suppes (2002, 110–118) will be useful here. Consider Moh's scale of mineral hardness, which assigns mineral \(m_1\) a higher score than mineral \(m_2\) iff \(m_1\) can scratch, but cannot be scratched by, \(m_2\). Where \(H(m)\) is the hardness of \(m\) according to Moh's scale, \(H(talc)=1\) while \(H(topaz) = 8\). Thus, topaz can scratch talc, but talc cannot scratch topaz. Moh's scale is ordinal , meaning that only the order of the numbers is meaningful. Instead of measuring hardness in terms of \(H\), we could just as easily measure hardness in terms of any of the following functions:

Transforming \(H\) into \(H'\), \(H''\), or \(H'''\) would leave all of its meaningful features intact. The allowable transformations of Moh's scale are all and only those that preserve ordering; hence it is called an ordinal scale .

Next, consider measures of length. Let \(L\) be a function that assigns to each object its length in inches. \(L\) provides information about which objects are longer than others: \(L(a) \gt L(b)\) iff \(a\) is longer than \(b\). But \(L\) is not merely an ordinal scale, since it provides additional information. Ratios between lengths are meaningful: \(L(a) + L(b) = L(c)\) iff laying object \(a\) end-to-end with object \(b\) produces a new, composite object exactly as long as \(c\). The allowable transformations of length in inches are those that preserve ratios—all and only those that result from multiplying every length by a constant. So the transformation \(f\) that converts \(L\) to a measure of length in centimeters, is allowable: \(f(L(a)) = 2.54\cdot L(a)\). But the transformations that result from adding 7 to \(L(a)\), or raising 10 to the power of \(L(a)\), are not allowable. This type of scale is called a ratio scale .

Defenders of expected utility theory typically require that utility be measured by a linear scale . For linear scales, the allowable transformations are all and only the positive linear transformations, i.e., functions \(f\) of the form

for real numbers \(x \gt 0\) and \(y\).

Positive linear transformations of outcome utilities will never affect the verdicts of expected utility theory: if \(A\) has greater expected utility than \(B\) where utility is measured by function \(U\), then \(A\) will also have greater expected utility than \(B\) where utility is measured by any positive linear transformation of \(U\).

Jeffrey (1983) is an exception to the generalization that expected utility theorists measure utility with a linear scale. Jeffrey requires that probability and utility together be measured by a fractional linear scale . Jeffrey's allowable transformations operate on probability and utility together : instead of mapping probability functions to probability functions and utility functions to utility functions, they map probability-utility pairs to probability-utility pairs.

2. Arguments for Expected Utility Theory

Why choose acts that maximize expected utility? One possible answer is that expected utility theory is rational bedrock—that means-end rationality essentially involves maximizing expected utility. For those who find this answer unsatisfying, however, there are two further sources of justification. First, there are long-run arguments, which rely on evidence that expected-utility maximization is a profitable policy in the long term. Second, there are arguments based on representation theorems, which suggest that certain rational constraints on preference entail that all rational agents maximize expected utility.

One reason for maximizing expected utility is that it makes for good policy in the long run. Feller (1968) gives a version of this argument. He relies on two mathematical facts about probabilities: the strong and weak laws of large numbers . Both these facts concern sequences of independent, identically distributed trials—the sort of setup that results from repeatedly betting the same way on a sequence of roulette spins or craps games. Both the weak and strong laws of large numbers say, roughly, that over the long run, the average amount of utility gained per trial is overwhelmingly likely to be close to the expected value of an individual trial.

The weak law of large numbers states that where each trial has an expected value of \(\mu\), for any arbitrarily small real number \(\epsilon \gt 0\), as the number of trials increases, the probability that the gambler's average winnings per trial fall within \(\epsilon\) of \(\mu\) converges to 1. In other words, as the number of repetitions of a gamble approaches infinity, the average gain per trial will become arbitrarily close to the gamble's expected value with probability 1. So in the long run, the average value associated with a gamble is virtually certain to equal its expected value.

The strong law of large number states that where each trial has an expected value of \(\mu\), for any arbitrarily small real numbers \(\epsilon \gt 0\) and \(\delta \gt 0\), there is some finite number of trials \(n\), such that for all \(m\) greater than or equal to \(n\), with probability at least \(1-\delta\), the gambler's average gains for the first \(m\) trials will fall within \(\epsilon\) of \(\mu\). In other words, in a long run of similar gamble, the average gain per trial is highly likely to become arbitrarily close to the gamble's expected value within a finite amount of time. So in the finite long run, the average value associated with a gamble is overwhelmingly likely to be close to its expected value.

There are several objections to these long run arguments. First, many decisions cannot be repeated over indefinitely many similar trials. Decisions about which career to pursue, whom to marry, and where to live, for instance, are made at best a small finite number of times. Furthermore, where these decisions are made more than once, different trials involve different possible outcomes, with different probabilities. It is not clear why long-run considerations about repeated gambles should bear on these single-case choices.

Second, the argument relies on two independence assumptions, one or both of which may fail. One assumption holds that the probabilities of the different trials are independent. This is true of casino gambles, but not true of other choices where we wish to use decision theory—e.g., choices about medical treatment. My remaining sick after one course of antibiotics makes it more likely I will remain sick after the next course, since it increases the chance that antibiotic-resistant bacteria will spread through my body. The argument also requires that the utilities of different trials be independent, so that winning a prize on one trial makes the same contribution to the decision-maker's overall utility no matter what she wins on other trials. But this assumption is violated in many real-world cases. Due to the diminishing marginal utility of money $10 million on ten games of roulette is not worth ten times as much as winning $1 million on one game of roulette.

A third problem is that the strong and weak laws of large numbers are modally weak. Neither law entails that if a gamble were repeated indefinitely (under the appropriate assumptions), the average utility gain per trial would be close to the game's expected utility. They establish only that the average utility gain per trial would with high probability be close to the game's expected utility. But high probability—even probability 1—is not certainty. (Standard probability theory rejects Cournot's Principle , which says events with low or zero probability will not happen. But see Shafer (2005) for a defense of Cournot's Principle.) For any sequence of independent, identically distributed trials, it is possible for the average utility payoff per trial to diverge arbitrarily far from the expected utility of an individual trial.

A second type of argument for expected utility theory relies on so-called representation theorems. We follow Zynda's (2000) formulation of this argument—slightly modified to reflect the role of utilities as well as probabilities. The argument has three premises:

The Rationality Condition. The axioms of expected utility theory are the axioms of rational preference.

Representability. If a person's preferences obey the axioms of expected utility theory, then she can be represented as having degrees of belief that obey the laws of the probability calculus [and a utility function such that she prefers acts with higher expected utility].

The Reality Condition. If a person can be represented as having degrees of belief that obey the probability calculus [and a utility function such that she prefers acts with higher expected utility], then the person really has degrees of belief that obey the laws of the probability calculus [and really does prefer acts with higher expected utility].

These premises entail the following conclusion.

If a person [fails to prefer acts with higher expected utility], then that person violates at least one of the axioms of rational preference.

If the premises are true, the argument shows that there is something wrong with people whose preferences are at odds with expected utility theory—they violate the axioms of rational preference. Let us consider each of the premises in greater detail, beginning with the key premise, Representability.

A probability function and a utility function together represent a set of preferences just in case the following formula holds for all values of \(A\) and \(B\) in the domain of the preference relation

Mathematical proofs of Representability are called representation theorems . Section 2.1 surveys three of the most influential representation theorems, each of which relies on a different set of axioms.

No matter which set of axioms we use, the Rationality Condition is controversial. In some cases, preferences that seem rationally permissible—perhaps even rationally required—violate the axioms of expected utility theory. Section 3 discusses such cases in detail.

The Reality Condition is also controversial. Hampton (1994), Zynda (2000), and Meacham and Weisberg (2011) all point out that to be representable using a probability and utility function is not to have a probability and utility function. After all, an agent who can be represented as an expected utility maximizer with degrees of belief that obey the probability calculus, can also be represented as someone who fails to maximize expected utility with degrees of belief that violate the probability calculus. Why think the expected utility representation is the right one?

There are several options. Perhaps the defender of representation theorems can stipulate that what it is to have particular degrees of belief and utilities is just to have the corresponding preferences. The main challenge for defenders of this response is to explain why representations in terms of expected utility are explanatorily useful, and why they are better than alternative representations. Or perhaps probabilities and utilities are a good cleaned-up theoretical substitutes for our folk notions of belief and desire—precise scientific substitutes for our folk concepts. Meacham and Weisberg challenge this response, arguing that probabilities and utilities are poor stand-ins for our folk notions. A third possibility, suggested by Zynda, is that facts about degrees of belief are made true independently of the agent's preferences, and provide a principled way to restrict the range of acceptable representations. The challenge for defenders of this type of response is to specify what these additional facts are.

I now turn to consider three influential representation theorems. These representation theorems differ from each other in three of philosophically significant ways.

First, different representation theorems disagree about the objects of preference and utility. Of the three theories I discuss below, one claims that they are lotteries (which are individuated in explicitly probabilistic terms), one claims that they are acts (which are entirely up to the agent), and one claims that they are arbitrary propositions.

Second, representation theorems differ in their treatment of probability. They disagree about which entities have probabilities, and about whether the same objects can have both probabilities and utilities.

Third, while every representation theorem proves that for a suitable preference ordering, there exist a probability and utility function representing the preference ordering, they differ how unique this probability and utility function are. In other words, they differ as to which transformations of the probability and utility functions are allowable.

2.2.1 Von Neumann and Morgenstern

Von Neumann and Morgenstern (1944) claim that preferences are defined over a domain of lotteries . Some of these lotteries are constant , and yield a single prize with certainty. (Prizes might include a banana, a million dollars, a million dollars' worth of debt, death, or a new car.) Lotteries can also have other lotteries as prizes, so that one can have a lottery with a 40% chance of yielding a banana, and a 60% chance of yielding a 50-50 gamble between a million dollars and death.) The domain of lotteries is closed under a mixing operation, so that if \(L\) and \(L'\) are lotteries and \(x\) is a real number in the \([0, 1]\) interval, then there is a lottery \(x L + (1-x) L'\) that yields \(L\) with probability \(x\) and \(L'\) with probability \(1-x\). They show that every preference relation obeying certain axioms can be represented by the probabilities used to define the lotteries, together with a utility function which is unique up to positive linear transformation.

2.2.2 Savage

Instead of taking probabilities for granted, as von Neumann and Morgenstern do, Savage (1972) defines them in terms of preference.

He begins with two separate domains for probability and utility. First, there are acts , which the agent can choose to perform, and which are objects of non-intrinsic preference and expected utility. Second, there are events, which are not dependent on the agent's choices, and which are objects of probability. Savage thinks of events as sets of states, but since we are thinking of states as sets of possibilities (see section 1), we can think of events as disjunctions of states. If there are four states, in which it rains, snows, drizzles, and stays sunny, then there will be sixteen events, including an event in which it rains, an event in which it rains or drizzles, a trivial event in which it rains, snows, drizzles, or stays sunny, and an empty event in which it does none of the above.

Savage also posits a third domain of outcomes , which are objects of intrinsic preference and utility. Outcomes have the same utility regardless of which state obtains. To illustrate the state-independence of outcomes, consider an example in which I am planning an outing with my friends, and have enough money for either a bathing suit or a tennis racquet but not enough for both. The following are not outcomes: “I have a bathing suit” and “I have a tennis racquet”. The bathing suit is worth more to me if the outing is to the beach, and less to me if the outing is to the tennis court, while for the tennis racquet, this relationship is reversed. A better way of individuating outcomes would be as follows: “I bring a bathing suit to the beach”; “I bring a bathing suit to the tennis courts”; “I bring a tennis racquet to the beach”; and “I bring a tennis racquet to the tennis courts”.

Acts, states, and outcomes are interrelated. Savage assumes that no state rules out the performance of any act, and that an act and a state together determine an outcome. He also assumes that for each outcome \(o\), there is a constant act which yields \(o\) in every state. (Thus, if “I bring a bathing suit to the beach” is an outcome, there is an act that results in my bringing a bathing suit to the beach, no matter what the state of the world.) Finally, he assumes for any two acts \(A\) and \(B\) and event \(E\), there is a mixed act \(A_E \amp B_{\sim E}\) that yields the same outcome as \(A\) if \(E\) is true, and the same outcome as \(B\) otherwise. (Thus, if “I bring a bathing suit to the beach” and “I bring a tennis racquet to the tennis courts” are both outcomes, then there is a mixed act that results in my bringing a bathing suit to the beach in the event that my friends go to the tennis courts, and in my bringing a tennis racquet to the tennis courts otherwise.)

Savage postulates a preference relation over acts, and gives axioms governing that preference relation. He then defines subjective probabilities, or degrees of belief, in terms of preferences. The key move is to define an “at least as likely as” relation between events. (For convenience, I define “more likely than” and “equally likely” relations. Given that \(E\) is at least as likely as \(F\) if and only if either \(E\) is more likely than \(F\) or \(E\) and \(F\) are equally likely, my definition is equivalent to Savage's.)

Suppose \(A\) and \(B\) are constant acts such that \(A\) is preferred to \(B\). Then \(E\) is more likely than \(F\) just in case the agent either prefers \(A_E \amp B_{\sim E}\) (the act that yields \(A\) if \(E\) obtains, and \(B\) otherwise) to \(A_F \amp B_{\sim F}\) (the act that yields \(A\) if \(F\) obtains, and \(B\) otherwise); and \(E\) and \(F\) are equally likely just in case the agent is indifferent between \(A_E \amp B_{\sim E}\) and \(A_F \amp B_{\sim F}\).

The thought behind the definition is that the agent considers \(E\) more likely than \(F\) just in case she would rather bet on \(E\) than on \(F\), and considers \(E\) and \(F\) equally likely just in case she is indifferent between betting on \(E\) and betting on \(F\).

Savage then gives axioms constraining rational preference, and shows that any set of preferences satisfying those axioms yields an “at least as likely” relation that can be uniquely represented by a probability function. In other words, there is one and only one probability function \(P\) such that for all \(E\) and \(F\), \(P(E) \ge P(F)\) if and only if \(E\) is at least as likely as \(F\). Every preference relation obeying Savage's axioms is represented by this probability function \(P\), together with a utility function which is unique up to positive linear transformation.

Savage's representation theorem gives strong results: starting with a preference ordering alone, we can find a single probability function, and a narrow class of utility functions, which represent that preference ordering. The downside, however, is that Savage has to build implausibly strong assumptions about the domain of acts.

Luce and Suppes (1965) point out that Savage's constant acts are implausible. (Recall that constant acts yield the same outcome and the same amount of value in every state.) Take some very good outcome—total bliss for everyone. Is there really a constant act that has this outcome in every possible state, including states where the human race is wiped out by a meteor? Savage's reliance on a rich space of mixed acts is also problematic. Savage has had to assume that any two outcomes and any event, there is a mixed act that yields the first outcome if the event occurs, and the second outcome otherwise? Is there really an act that yields total bliss if everyone is killed by an antibiotic-resistant plague, and total misery otherwise? Luce and Krantz (1971) suggest ways of reformulating Savage's representation theorem that weaken these assumptions, but Joyce (1999) argues that even on the weakened assumptions, the domain of acts remains implausibly rich.

2.2.3 Bolker and Jeffrey

Bolker (1966) proves a general representation theorem about mathematical expectations, which Jeffrey (1983) uses as the basis for a philosophical account of expected utility theory. Bolker's theorem assumes a single domain of propositions, which are objects of preference, utility, and probability alike. Thus, the proposition that it will rain today has a utility, as well as a probability. Jeffrey interprets this utility as the proposition's news value —a measure of how happy or disappointed I would be to learn that the proposition was true. By convention, he sets the value of the necessary proposition at 0—the necessary proposition is no news at all! Likewise, the proposition that I take my umbrella to work, which is an act, has a probability as well as a utility. Jeffrey interprets this to mean that I have degrees of belief about what I will do.

Bolker gives axioms constraining preference, and shows that any preferences satisfying his axioms can be represented by a probability measure \(P\) and a utility measure \(U\). However, Bolker's axioms do not ensure that \(P\) is unique, or that \(U\) is unique up to positive linear transformation. Nor do they allow us to define comparative probability in terms of preference. Instead, where \(P\) and \(U\) jointly represent a preference ordering, Bolker shows that the pair \(\langle P, U \rangle\) is unique up to a fractional linear transformation.

In technical terms, where \(U\) is a utility function normalized so that \(U(\Omega) = 0\), \(inf\) is the greatest lower bound of the values assigned by \(U\), \(sup\) is the least upper bound of the values assigned by by \(U\), and \(\lambda\) is a parameter falling between \(-1/inf\) and \(-1/sup\), the fractional linear transformation \(\langle P_{\lambda}, U_{\lambda} \rangle\) of \(\langle P, U \rangle\) corresponding to \(\lambda\) is given by:

Notice that fractional linear transformations of a probability-utility pair can disagree with the original pair about which propositions are likelier than which others.

Joyce (1999) shows that with additional resources, Bolker's theorem can be modified to pin down a unique \(P\), and a \(U\) that is unique up to positive linear transformation. We need only supplement the preference ordering with a primitive “more likely than” relation, governed by its own set of axioms, and linked to belief by several additional axioms. Joyce modifies Bolker's result to show that given these additional axioms, the “more likely than” relation is represented by a unique \(P\), and the preference ordering is represented by \(P\) together with a utility function that is unique up to positive linear transformation.

2.2.4 Summary

Together, the three representation theorems above can be summed up in the following diagram, where an arrow from \(A\) to \(B\) means that, given an \(A\) satisfying the appropriate constraints, there is a \(B\) that represents \(A\). Solid lines indicate that there is a unique \(B\) that represents each \(A\); dashed lines that \(B\) is unique only up to some allowable transformation weaker than the identity.

Thus, we can see that each pair consisting of a preference ordering and probability function which satisfy the appropriate axioms is represented by an expected utility function, unique up to positive linear transformation. (This is von Neumann and Morgenstern's representation theorem.) Likewise, we can see that each probability-utility pair is represented by a unique expected utility function. Each expected utility function determines a unique utility function over outcomes, as well as a unique preference ordering. Each preference ordering that obeys the appropriate axioms is represented by a probability function and a utility function that together are unique up to fractional linear transformations. (This is the Bolker-Jeffrey theorem.)

Suitably structured ordinal probabilities (the relations picked out by “at least as likely as”, “more likely than”, and “equally likely”) stand in one-to-one correspondence with the cardinal probability functions. Finally, the grey line from preferences to ordinal probabilities indicates that every probability function satisfying Savage's axioms is represented by a unique cardinal probability—but this result does not hold for Jeffrey's axioms.

Notice that it is often possible to follow the arrows in circles—from preference to ordinal probability, from ordinal probability to cardinal probability, from cardinal probability and preference to expected utility, and from expected utility back to preference. Thus, although the arrows represent a mathematical relationship of representation, they do not represent a metaphysical relationship of grounding. This fact drives home the importance of independently justifying the Reality Condition—representation theorems cannot justify expected utility theory without additional assumptions.

3. Counterexamples to Expected Utility Theory

A variety of authors have given examples in which expected utility theory seems to get things wrong. Sections 3.1 and 3.2 discuss examples where rationality seems to permit preferences inconsistent with expected utility theory. These examples suggest that maximizing expected utility is not necessary for rationality. Section 3.3 discusses examples where expected utility theory permits preferences that seem irrational. These examples suggest that maximizing expected utility is not sufficient for rationality. Section 3.4 discusses an example where expected utility theory requires preferences that seem rationally forbidden—a challenge to both the necessity and the sufficiency of expected utility for rationality—as well as examples where expected utility fails to yield any usable verdict at all.

Expected utility theory implies that the structure of preferences mirrors the structure of the greater-than relation between real numbers. Thus, according to expected utility theory, preferences must be transitive : If \(A\) is preferred to \(B\) (so that \(U(A) \gt U(B)\)), and \(B\) is preferred to \(C\) (so that \(U(B) \gt U(C)\)), then \(A\) must be preferred to \(C\) (since it must be that \(U(A) \gt U(C)\)). Likewise, preferences must be complete : for any two options, either one must be preferred to the other, or the agent must be indifferent between them (since of their two utilities, either one must be greater or the two must be equal). But there are cases where rationality seems to permit (or perhaps even require) failures of transitivity and failures of completeness.

An example of preferences that are not transitive, but nonetheless seem rationally permissible, is Quinn's puzzle of the self-torturer (1990). The self-torturer is hooked up to a machine with a dial with settings labeled 0 to 1,000, where setting 0 does nothing, and each successive setting delivers a slightly more powerful electric shock. Setting 0 is painless, while setting 1,000 causes excruciating agony, but the difference between any two adjacent settings is so small as to be imperceptible. The dial is fitted with a ratchet, so that it can be turned up but never down. Suppose that at each setting, the self-torturer is offered $10,000 to move up to the next, so that for tolerating setting \(n\), he receives a payoff of \(n {\cdot} {$10,000}\). It is permissible for the self-torturer to prefer setting \(n+1\) to setting \(n\) for each \(n\) between 0 and 999 (since the difference in pain is imperceptible, while the difference in monetary payoffs is significant), but not to prefer setting 1,000 to setting 0 (since the pain of setting 1,000 may be so unbearable that no amount of money will make up for it.

It also seems rationally permissible to have incomplete preferences. For some pairs of actions, an agent may have no considered view about which she prefers. Consider Jane, an electrician who has never given much thought to becoming a professional singer or a professional astronaut. (Perhaps both of these options are infeasible, or perhaps she considers both of them much worse than her steady job as an electrician). It is false that Jane prefers becoming a singer to becoming an astronaut, and it is false that she prefers becoming an astronaut to becoming a singer. But it is also false that she is indifferent between becoming a singer and becoming an astronaut. She prefers becoming a singer and receiving a $100 bonus to becoming a singer, and if she were indifferent between becoming a singer and becoming an astronaut, she would be rationally compelled to prefer being a singer and receiving a $100 bonus to becoming an astronaut.

There is one key difference between the two examples considered above. Jane's preferences can be extended , by adding new preferences without removing any of the ones she has, in a way that lets us represent her as an expected utility maximizer. On the other hand, there is no way of extended the self-torturer's preferences so that he can be represented as an expected utility maximizer. Some of his preferences would have to be altered. One popular response to incomplete preferences is to claim that, while rational preferences need not satisfy the axioms of a given representation theorem (see section 2.2), it must be possible to extend them so that they satisfy the axioms. From this weaker requirement on preferences—that they be extendible to a preference ordering that satisfies the relevant axioms—one can prove the existence halves of the relevant representation theorems. However, one can no longer establish that the each preference ordering has a representation which is unique up to allowable transformations.

No such response is available in the case of the self-torturer, whose preferences cannot be extended to satisfy the axioms of expected utility theory. See the entry on preferences for a more extended discussion of the self-torturer case.

Allais (1953) and Ellsberg (1961) propose examples of preferences that cannot be represented by an expected utility function, but that nonetheless seem rational. Both examples involve violations of Savage's Independence axiom:

Independence . Suppose that \(A\) and \(A^*\) are two acts that produce the same outcomes in the event that \(E\) is false. Then, for any act \(B\), one must have \(A\) is preferred to \(A^*\) if and only if \(A_E \amp B_{\sim E}\) is preferred to \(A^*_E \amp B_{\sim E}\) The agent is indifferent between \(A\) and \(A^*\) if and only if she is indifferent between \(A_E \amp B_{\sim E}\) and \(A^*_E \amp B_{\sim E}\)

In other words, if two acts have the same consequences whenever \(E\) is false, then the agent's preferences between those two acts should depend only on their consequences when \(E\) is true. On Savage's definition of expected utility, expected utility theory entails Independence. And on Jeffrey's definition, expected utility theory entails Independence in the presence of the assumption that the states are probabilistically independent of the acts.

The first counterexample, the Allais Paradox, involves two separate decision problems in which a ticket with a number between 1 and 100 is drawn at random. In the first problem, the agent must choose between these two lotteries:

• Lottery \(A\)
• • $100 million with certainty
• Lottery \(B\)
• • $500 million if one of tickets 1–10 is drawn
• • $100 million if one of tickets 12–100 is drawn
• • Nothing if ticket 11 is drawn

In the second decision problem, the agent must choose between these two lotteries:

• Lottery \(C\)
• • $100 million if one of tickets 1–11 is drawn
• • Nothing otherwise
• Lottery \(D\)

It seems reasonable to prefer \(A\) (which offers a sure $100 million) to \(B\) (where the added 10% chance at $500 million is more than offset by the risk of getting nothing). It also seems reasonable to prefer \(D\) (an 10% chance at a $500 million prize) to \(C\) (a slightly larger 11% chance at a much smaller $100 million prize). But together, these preferences (call them the Allais preferences ) violate Independence. Lotteries \(A\) and \(C\) yield the same $100 million dollar prize for tickets 12–100. They can be converted into lotteries \(B\) and \(D\) by replacing this $100 million dollars with $0.

Because they violate Independence, the Allais preferences are incompatible with expected utility theory. This incompatibility does not require any assumptions about the relative utilities of the $0, the $100 million, and the $500 million. Where $500 million has utility \(x\), $100 million has utility \(y\), and $0 has utility \(z\), the expected utilities of the lotteries are as follows.

It is easy to see that the condition under which \(EU(A) \gt EU(B)\) is exactly the same as the condition under which \(EU(C) \gt EU(D)\): both inequalities obtain just in case \(0.11y \gt 0.10x + 0.01z\)

The Ellsberg Paradox also involves two decision problems that generate a violation of the sure-thing principle. In each of them, a ball is drawn from an urn containing 30 red balls, and 60 balls that are either white or yellow in unknown proportions. In the first decision problem, the agent must choose between the following lotteries:

• Lottery \(R\)
• • Win $100 if a red ball is drawn
• • Lose $100 otherwise
• Lottery \(W\)
• • Win $100 if a white ball is drawn

In the second decision problem, the agent must choose between the following lotteries:

• Lottery \(RY\)
• • Win $100 if a red or yellow ball is drawn
• Lottery \(WY\)
• • Win $100 if a white or yellow ball is drawn

It seems reasonable to prefer \(R\) to \(W\), but at the same time prefer \(WY\) to \(RY\). (Call this combination of preferences the Ellsberg preferences .) Like the Allais preferences, the Ellsberg preferences violate Independence. Lotteries \(W\) and \(R\) yield a $100 loss if a yellow ball is drawn; they can be converted to lotteries \(RY\) and \(WY\) simply by replacing this $100 loss with a sure $100 gain.

Because they violate independence, the Ellsberg preferences are incompatible with expected utility theory. Again, this incompatibility does not require any assumptions about the relative utilities of winning $100 and losing $100. Nor do we any assumptions about where between 0 and 1/3 the probability of drawing a yellow ball falls. Where winning $100 has utility \(w\) and losing $100 has utility \(l\),

It is easy to see that the condition in which \(EU(R) \gt EU(W)\) is exactly the same as the condition under which \(EU(RY) \gt EU(WY)\): both inequalities obtain just in case \(1/3\,w + P(W)l \gt 1/3\,l + P(W)w\).

There are three notable responses to the Allais and Ellsberg paradoxes. First, one might follow Savage (101 ff) and Raiffa (1968, 80–86), and defend expected utility theory on the grounds that the Allais and Ellsberg preferences are irrational.

Second, one might follow Buchak (2013) and claim that that the Allais and Ellsberg preferences are rationally permissible, so that expected utility theory fails as a normative theory of rationality. Buchak develops an a more permissive theory of rationality, with an extra parameter representing the decision-maker's attitude toward risk. This risk parameter interacts with the utilities of outcomes and their conditional probabilities on acts to determine the values of acts. One setting of the risk parameter yields expected utility theory as a special case, but other, “risk-averse” settings rationalise the Allais preferences.

Third, one might follow Loomes and Sugden (1986), Weirich (1986), and Pope (1995) and argue that the outcomes in the Allais and Ellsberg paradoxes can be re-described to accommodate the Allais and Ellsberg preferences. The alleged conflict between the Allais and Ellsberg preferences on the one hand, and expected utility theory on the other, was based on the assumption that a given sum of money has the same utility no matter how it is obtained. Some authors challenge this assumption. Loomes and Sugden suggest that in addition to monetary amounts, the outcomes of the gambles include feelings of disappointment (or elation) at getting less (or more) than expected. Pope distinguishes “post-outcome” feelings of elation or disappointment from “pre-outcome” feelings of excitement, fear, boredom, or safety, and points out that both may affect outcome utilities. Weirich suggests that the value of a monetary sum depends partly on the risks that went into obtaining it, irrespective of the gambler's feelings, so that (for instance) $100 million as the result of a sure bet is more than $100 million from a gamble that might have paid nothing.

Broome (1991) raises a worry about this re-description solution. Any preferences can be justified by re-describing the space of outcomes, thus rendering the axioms of expected utility theory devoid of content. Broome rebuts this objection by suggesting an additional constraint on preference: if \(A\) is preferred to \(B\), then \(A\) and \(B\) must differ in some way that justifies preferring one to the other. An expected utility theorist can then count the Allais and Ellsberg preferences as rational if, and only if, there is a non-monetary difference that justifies placing outcomes of equal monetary value at different spots in one's preference ordering.

Above, we've seen purported examples of rational preferences that violate expected utility theory. There are also purported examples of irrational preferences that satisfy expected utility theory.

On a typical understanding of expected utility theory, when two acts are tied for having the highest expected utility, agents are required to be indifferent between them. Skyrms (1980, p. 74) points out that this view lets us derive strange conclusions about events with probability 0. For instance, suppose you are about to throw a point-sized dart at a round dartboard. Classical probability theory countenances situations in which the dart has probability 0 of hitting any particular point. You offer me the following lousy deal: if the dart hits the board at its exact center, then you will charge me $100; otherwise, no money will change hands. My decision problem can be captured with the following matrix:

Expected utility theory says that it is permissible for me to accept the deal—accepting has expected utility of 0. (This is so on both the Jeffrey definition and the Savage definition, if we assume that how the dart lands is probabilistically independent of how you bet.) But common sense says it is not permissible for me to accept the deal. Refusing weakly dominates accepting: it yields a better outcome in some states, and a worse outcome in no state.

Skyrms suggests augmenting the laws of classical probability with an extra requirement that only impossibilities are assigned probability 0. Easwaran (2014) argues that we should instead reject the view that expected utility theory commands indifference between acts with equal expected utility. Instead, expected utility theory is not a complete theory of rationality: when two acts have the same expected utility, it does not tell us which to prefer. We can use non-expected-utility considerations like weak dominance as tiebreakers.

A utility function \(U\) is bounded above if there is a limit to how good things can be according to \(U\), or more formally, if there is some least natural number \(sup\) such that for every \(A\) in \(U\)'s domain, \(U(A) \le sup\). Likewise, \(U\) is bounded below if there is a limit to how bad things can be according to \(U\), or more formally, if there is some greatest natural number \(inf\) such that for every \(A\) in \(U\)'s domain, \(U(A) \ge inf\). Expected utility theory can run into trouble when utility functions are unbounded above, below, or both.

One problematic example is the St. Petersburg game, originally published by Bernoulli. Suppose that a coin is tossed until it lands tails for the first time. If it lands tails on the first toss, you win $2; if it lands tails on the second toss, you win $4; if it lands tails on the third toss, you win $8, and if it lands tails on the \(n\)th toss, you win $ \(2^{(n-1)}\). Assuming each dollar is worth one utile, the expected value of the St Petersburg game is

It turns out that this sum diverges; the St Petersburg game has infinite expected utility. Thus, according to expected utility theory, you should prefer the opportunity to play the St Petersburg game to any finite sum of money, no matter how large. Furthermore, since an infinite expected utility multiplied by any nonzero chance is still infinite, anything that has a positive probability of yielding the St Petersburg game has infinite expected utility. Thus, according to expected utility theory, you should prefer any chance at playing the St Petersburg game, however slim, to any finite sum of money, however large.

Another puzzling example is the Pasadena game, proposed by Nover and Hájek (2004). Just as in the St Petersburg game, a coin is flipped until it lands tails. But in the Pasadena game, there are potential losses as well as positive wins. If the coin lands tails on the first toss, you win $2; if it lands tails on the second toss, you lose $2; if it lands tails on the third toss, you win $8/3; if it lands tails on the fourth toss, you lose $4; and if it lands tails on toss \(n\), you win $(−1) n –1 (2 n / n ), where winning a negative amount is equivalent to losing money.

Assuming again that each dollar is worth one utile, expected value of the Pasadena game is

This sum converges to \(\ln(2)\); assuming that one dollar is worth one utile, this works out to about 69¢. But there is a catch. The sum only conditionally converges. If every term is replaced with its absolute value, the resulting series diverges:

Then, by the Riemann rearrangement theorem, for every real value, the series can be rearranged so as to sum to that value; and other rearrangements of the series diverge to \(\infty\), and to \(-\infty\). So the Pasadena game seems to have no well-defined expected utility.

Consequently, in a variety of situations where we want to rank options against each other, expected utility theory fails to give any result. Just as any gamble with a small chance of yielding the St Petersburg game as a prize has infinite expected utility, so any gamble with a small chance of yielding the Pasadena game as a prize has undefined expected utility. Furthermore, consider the a second game proposed by Nover and Hajek (2004): the Altadena game. The Altadena game is exactly like the Pasadena game, except that each of the payoffs is a dollar higher. Since the Altadena game dominates the Pasadena game, it should count as better, but expected utility theory fails to yield this result.

One response to these problematic infinitary games is to argue that the decision problems themselves are ill-posed. Jeffrey (1983, 154), for instance, writes that “anyone who offers to let the agent play the St Petersburg game is a liar, for he is pretending to have an indefinitely large bank”. Nover and Hájek (2004) rebut Jeffrey, arguing that even if all utility functions are bounded as a matter of contingent fact, unbounded utility functions are a conceptual possibility, and that they may be useful for developing idealized models of actual agents. In later work, Hájek and Nover (2008) argue at length that the Pasadena game is well posed.

Another response is to claim that the values of these infinitary games are not adequately captured by their expected utilities. This solution is compatible with accepting utility theory for a large class of ordinary gambles. Thalos and Richardson (2013), Fine (2008), Colyvan (2006, 2008), and Easwaran (2008) propose theories that coincide with expected utility theory in a wide range of gambles, but also let us give intuitively reasonable evaluations of the St. Petersburg, Pasadena, and Altadena games.

4. Applications

One of the earliest applications of expected utility theory is Pascal's wager , an argument for the conclusion belief in God is rationally obligatory. Pascal presents a number of distinct arguments (see the article on Pascal's wager), but one particularly notable version relies on expected utility considerations. We lack decisive evidence about whether God exists. However, the question of whether to believe in God can be understood as a decision problem, in which belief and disbelief are both acts. Pascal argues that belief in God is the better act: “there is here an infinity of an infinitely happy life to gain, a chance of gain against a finite number of chances of loss, and what you stake is finite.” This looks like an argument based on expected utility theory. The matrix for the decision problem looks like this (where outcomes are specified according to their utilities, and f 1 , f 2 , and f 3 are all finite):

Where the probability that God exists is \(g \gt 0\), the expected utility of believing in God infinite \((g \infty + (1-g)f_1)\), while the expected utility of not believing in God is finite \((g f_2 + (1-g)f_3)\).

Many of the objections to Pascal's wager concern whether the decision matrix above correctly models the choice faced by someone deciding whether to believe in God. Are there acts which are alternatives to belief and disbelief (such as tossing a coin to decide whether to believe)? Are there states that the matrix fails to represent (such as states in which there are many gods)? Is the probability that God exists really greater than 0? Of particular decision interest to expected utility theory is the objection that no outcome can have infinite utility—a version of the issue with unbounded utility that arose for the St. Petersburg game. Pascal's Wager requires a stronger assumption than the St. Petersburg game—not just that there is a lottery with infinite expected value, but that there is an outcome with infinite value. (In St. Petersburg, all of the outcome values were finite.)

In economics, expected utility theory is often invoked as an account of how people actually make decisions in an economic context. These uses of expected utility theory are descriptive, and don't bear directly on the normative question of whether expected utility theory provides a good account of rationality.

However, there are some economic uses of expected utility theory which are normative. One—the application for which expected utility theory was originally designed—is setting prices for games of chance. The strong and weak laws of large numbers entail that charging more for a game than its expected monetary value is overwhelmingly likely to yield a profit for the casino in the long term, while charging less is overwhelmingly likely to result in a monetary loss. There is a complication, however: casinos have only finitely large banks, and will go out of business if it loses too much money in the short term. Therefore, the analysis of reasonable pricing for casino games must take into account not only the casino's expected profits or losses in the long run, but also its chance of going broke.

Another area where expected utility theory finds applications is in insurance sales. Like casinos, insurance companies take on calculated risks with the aim of long-term financial gain, and must take into account the chance of going broke in the short run.

Consequentialists hold that the rightness or wrongness of an act is determined by the moral goodness or badness of its consequences. Some consequentialists, (Railton 1984) interpret this to mean that we ought to do whatever will in fact have the best consequences. But it is difficult to know with any degree of certainty what long-term consequences our acts will have. Lenman (2000) gives the example of an ancient Visigoth, Richard, who spares the life of an innocent citizen in a town he raids. But unknown to Richard, the citizen turns out to be a distant ancestor of Adolf Hitler. And as Lenman points out, “Hitler's crimes may not be the most significant consequence of Richard's action.” Perhaps, had Richard killed the innocent citizen, some even more murderous dictator would have been conceived. We have no way of determining whether the consequences of Richard's action are overall better or worse than the consequences of alternatives he might have chosen.

Howard-Snyder (1997) argues that in most cases, we cannot act so as to bring about the best consequences—we do not know how. (Similarly, I cannot beat Karpov at chess, since I do not know how—even though there is a sequence of moves I can perform that would constitute beating Karpov at chess.) Since “ought” implies “can”, it must be false that we ought to act so as to bring about the best consequences. Wiland (2005) argues that even if we can perform the acts with the best consequences, the view that we ought to do so yields strange consequences: even the morally best people act immorally most of the time.

Jackson (1991) argues that the right act is the one with the greatest expected moral value, not the one that will in fact yield the best consequences. Jackson defends this view using the example of a doctor choosing which of three drugs to prescribe for a skin illness. Drug A will relieve the condition without completely curing it, while of drugs B and C, one will cure the patient completely, and the other will kill him. The doctor has no way of distinguishing the killer drug from the cure. In this case, it seems morally obligatory for the doctor to take the safe option and prescribe drug A, even though she knows that one of the alternative drugs will produce a better result.

As Jackson notes, the expected moral value of an act depends on which probability function we work with. Jackson argues that, while every probability function is associated with an “ought”, the “ought” that matters most to action is the one associated with the decision-maker's degrees of belief at the time of action. Other authors claim priority for other “oughts”: Mason (2013) favors the probability function that is most reasonable for the agent to adopt in response to her evidence, given her epistemic limitations, while Oddie and Menzies (1992) favor the objective chance function as a measure of objective rightness. (They appeal to a more complicated probability function to define a notion of “subjective rightness” for decisionmakers who are ignorant of the objective chances.)

Still others (Smart 1973, Timmons 2002) argue that even if that we ought to do whatever will have the best consequences, expected utility theory can play the role of a decision procedure when we are uncertain what consequences our acts will have. Feldman (2006) objects that expected utility calculations are horribly impractical. In most real life decisions, the steps required to compute expected utilities are beyond our ken: listing the possible outcomes of our acts, assigning each outcome a utility and a conditional probability given each act, and performing the arithmetic necessary to expected utility calculations.

Expected utility theory can be used to address practical questions in epistemology. One such question is when to accept a hypothesis. In typical cases, the evidence is logically compatible with multiple hypotheses, including hypotheses to which it lends little inductive support. Furthermore, scientists do not typically accept only those hypotheses that are most probable given their data. When is a hypothesis likely enough to deserve acceptance?

Bayesians, such as Maher (1993), suggest that this decision be made on expected utility grounds. Whether to accept a hypothesis is a decision problem, with acceptance and rejection as acts. It can be captured by the following decision matrix:

On Savage's definition, the expected utility of accepting the hypothesis is determined by the probability of the hypothesis, together with the utilities of each of the four outcomes. (We can expect Jeffrey's definition to agree with Savage's on the plausible assumption that, given the evidence in our possession, the hypothesis is probabilistically independent of whether we accept or reject it.) Here, the utilities can be understood as purely epistemic values, since it is epistemically valuable to believe interesting truths, and to reject falsehoods.

Critics of the Bayesian approach, such as Mayo (1996), object that scientific hypotheses cannot sensibly be given probabilities. Mayo argues that in order to assign a useful probability to an event, we need statistical evidence about the frequencies of similar events. But scientific hypotheses are either true once and for all, or false once and for all—there is no population of worlds like ours from which we can meaningfully draw statistics. Nor can we use subjective probabilities for scientific purposes, since this would be unacceptably arbitrary. Therefore, the expected utilities of acceptance and rejection are undefined, and we ought to use the methods of traditional statistics, which rely on comparing the probabilities of our evidence conditional on each of the hypotheses.

Expected utility theory also provides guidance about when to gather evidence. Good (1967) argues on expected utility grounds that it is always rational to gather evidence before acting, provided that evidence is free of cost. The act with the highest expected utility after the extra evidence is in will always be always at least as good as the act with the highest expected utility beforehand.

Another application of expected utility theory, discussed in (Greaves 2013) is the evaluation of probabilities themselves—where these probabilities are understood as individual degrees of belief. If we think of belief formation as a mental act, facts about the contents of the agent's beliefs as events, and closeness to truth as a desirable feature of outcomes, then we can use expected utility theory to evaluate degrees of belief in terms of their expected closeness to truth. Greaves and Wallace (2006) this approach to justify updating by conditionalization; Leitgeb and Pettigrew (2010) use it to justify a new rule for updating on uncertain evidence that conflicts with the more orthodox “probability kinematics” due to Jeffrey (1983).

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• Zynda, L., 2000, “Representation Theorems and Realism about Degrees of Belief”, Philosophy of Science 67: 45–69.

How to cite this entry . Preview the PDF version of this entry at the Friends of the SEP Society . Look up this entry topic at the Indiana Philosophy Ontology Project (InPhO). Enhanced bibliography for this entry at PhilPapers , with links to its database.

• Decisions, Games, and Rational Choice , materials for a course taught in Spring 2008 by Robert Stalnaker, MIT OpenCourseWare.
• Microeconomic Theory III , materials for a course taught in Spring 2010 by Muhamet Yildiz, MIT OpenCourseWare.
• Choice Under Uncertainty , class lecture notes by Jonathan Levin.
• Expected Utility Theory , by Philippe Mongin, entry for The Handbook of Economic Methodology.
• The Origins of Expected Utility Theory , essay by Yvan Lengwiler.

decision theory | decision theory: causal | Pascal's wager | preferences | probability, interpretations of | -->rational choice, normative: rivals to expected utility --> | risk

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II.1 Utility Hypothesis

• Published: November 1987
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This is the first of six chapters in Part II about demand and utility cost, a typical area for what is understood as choice theory. It discusses utility hypothesis and the theory of value. Its five sections are: needs of measurement (of utility); common practice and (William) Fleetwood; parallels in theory (as applied to utility construction); revealed preference (as applied to demand functions); and the classical case (of the utility function).

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Probability

Ghost of Statistics

Hempel’s Joke

Jaynes: Exercises

Markov; Chebyshev

Chernoff Bound

Fast Binomials

Information Theory

Symbol Codes

Stream Codes

Kelly Criterion

Penney’s Game

Daniel Bernoulli’s Utility

Consider the following game. Keep flipping a coin until it shows heads. You receive 2 n dollars where n is the number of tails you saw. For example, if you flip tails 4 times and then heads, then you get $16.

How much would you pay to play?

This is the famous St. Petersburg paradox . It’s a paradox because the expected payout is:

\[ \frac{1}{2} \times 1 + \frac{1}{4} \times 2 + \frac{1}{8} \times 4 + …​ = \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + …​ \]

which increases without bound, suggesting you should be willing to sacrifice everything just to play this game!

It’s like Zeno’s dichotomy paradox , but cleverly counteracts repeated halving with repeated doubling in order to shove infinity in our face.

The futility of utility

Daniel Bernoulli analyzed this problem in Specimen Theoriae Novae de Mensura Sortis in 1738. An English translation Exposition of a New Theory of the Measurement of Risk appears in Econometrica , Volume 22, Issue 1 (January 1954).

This work is famous for the wrong reasons. A common narrative is that Bernoulli resolved the St Petersburg paradox by defining utility and then arguing one should maximize expected utility rather than expected wealth. This is only partly true.

Bernoulli did indeed introduce a good definition of utility along with good reasons for maximizing utility instead of wealth. However, Bernoulli failed to resolve the paradox. As Karl Menger observed in the 1930s, utility merely moves the embarrassing infinite expectation elsewhere, which is immediately apparent if we change the rewards from dollars to utiles , a unit that measures utility. [Did you think a layer of indirection would fix the problem? This isn’t software engineering !]

One might try to dodge Menger’s trick by mandating a finite supply of utiles for one reason or another, but then why not insist upon a finite supply of dollars in the first place? Moreover, the latter is true, while the former feels like a made-up restriction on a made-up thing.

The largest lottery jackpot ever won in the US was about $2 billion, that is, under 2 31 dollars. Suppose we cap the St. Petersburg lottery so that the 32nd flip always counts as heads. Then the expected payoff is $16.50, a long way from infinity.

Although it turns out we should not actually pay $16.50 to play (though the obscenely rich should pay close to this amount), the point is that the paradox vanishes when we inject the tiniest dose of reality.

Centuries after Bernoulli, von Neumann and Morgenstern would strengthen the case for maximizing the expected value of some kind of utility function , though not necessarily Bernoulli’s utility. Their utility, and that of economists in general, is a magic number that tells us how to live our lives. This may be theoretically sound, but I suspect has less relevance in practice. For example, it is well-known that plenty of exercise and eating right leads to longer, healthier lives. This surely increases economic utility, yet how many of us follow this advice?

We deliberately limit our scope. We only focus on maximizing wealth, with the aid of Bernoulli’s utility. How should wealth be tied to well-being? How should it be spent? Not our problem!

Log Money is Power

What should Bernoulli’s work be remembered for?

In section 10, he states that, on a small gain in wealth:

The gain in utility is proportional to the gain in wealth.

The gain in utility is inversely proportional to existing wealth.

That is, if \(y\) is utility and \(x\) is wealth:

\[ dy = k \frac{dx}{x} \]

for some constant \(k\). Integrating yields:

\[ y = k \log x + c \]

for some constant \(c\). Changing \(k\) changes the base of the logarithm, while changing \(c\) changes a wealth scaling factor before taking logarithms. Neither matter if the goal is to maximize \(y\), or to calculate means. We can dodge any psychological effects from the choice of constants by converting utility back to ducats or dollars.

For now, choose \(k = 1\) and \(c = 0\) so utility is just the natural logarithm of wealth:

\[ y = \log x \]

This is what Bernoulli’s paper should be remembered for.

In the St. Petersburg paradox, utility is a red herring. In real life, Bernoulli’s utility is a good measure of the power of one’s money. When I deposit money in a bank, the interest I earn is a percentage of the amount. On the other end of the spectrum, when a market genius manages money in a skyrocketing fund, they still advertise the return as a percentage of the amount invested. Rewards are geometric, not arithmetic ("the rich get richer"). As Napier noted, in such cases, calculation is easier with logarithms.

Why do we often think the power of money is arithmetic? Perhaps it’s because everyday decisions are on tiny scales where linear approximations suffice. Six cans of green beans costs twice as much as three cans of green beans. But would the cost increase linearly if we wanted a million cans or so?

Log wealth figures might never catch on, but perhaps we can still do better when quoting dollar amounts. For instance, published figures showing the mean wealth of various groups typically use the arithmetic mean. A common complaint is that such statistics look misleadingly high due to a few outliers.

In fact, the arithmetic mean is already questionable with just two data points. Suppose Smith has $1 and Jones has $10000. Then the arithmetic mean is roughly $5000. Yet it’s much easier to grow $5000 into $10000 than it is to grow $1 into $5000 ("the first million is the hardest").

It may be better to examine the arithmetic mean of their log wealth, which we can convert back to dollars via exponentiation. The result is also known as the geometric mean:

\[ \exp \frac{\log x + \log y}{2} = \sqrt{xy} \]

In this case the geometric mean is $100. Saving $1 and $100 in accounts that ultimately grow one-hundred-fold will result in exactly $100 and $10000. In other words, the geometric mean is a reasonable halfway marker between Smith and Jones, because it is equally challenging for Smith to reach $100 as it is to start from $100 and end up keeping up with Jones.

Bernoulli’s paper walks through a few examples to demonstrate how his definition of utility can guide decisions.

Pro tip: familiarity with Jensen’s inequality can improve one’s intuition with utility, as the log function is concave.

Double Trouble

If our life savings amounts to 100 ducats, should we place an even money bet of 50 ducats on a coin toss? The expected log wealth is:

\[ \frac{\log 50 + \log 150}{2} \]

This is less than \(\log 100\), thus even though the bet is fair, our expected utility decreases. Bernoulli advises the only winning move is not to play, calling it "Nature’s admonition to avoid the dice altogether".

We exponentiate to express this number in ducats:

\[ \exp \left( \frac{\log 50 + \log 150}{2} \right) = 86.60…​ \]

namely, the geometric mean of 50 and 150. We can think of the bet as being as unpleasant as losing about 13 of our 100 ducats.

On a fair coin toss, if the penalty is to lose half our wealth, then according to Bernoulli, the reward must be at least double our wealth:

\[ \frac{\log \frac{x}{2} + \log 2x}{2} = \log x \]

Reportedly, some believe "risk-neutral" means that one should always accept a double-or-nothing bet, and even risk destroying the world so long there is at least a 50% chance of making it twice as good . In contrast, Bernoulli suggests no finite improvement is worth wagering the whole world, and if there is a 50% chance of making the world twice as good, the largest penalty a risk-neutral party can tolerate is making the world twice as bad. Double-or-half; not double-or-nothing.

What if we can bet less than the whole world? That is, how much should we wager on a fair coin toss where on losing, half our stake is generously returned to us? If we start wtih 100 ducats, we should maximize:

\[ \frac{\log (100 - \frac{x}{2}) + \log (100 + x)}{2} \]

which happens when \(x = 50\). In general, when offered this bet, rather than greedily attempt to double our wealth in one go, it’s best to risk only 25% of our wealth, and hope for a 50% increase.

Such lopsided outcomes might seem too good to be true. However, economists have modeled stock prices with geometric random walks, which would imply the stock market offers deals of this nature. Maximizing log wealth then corresponds to maintaining a fixed ratio of cash to stocks, a strategy that some call Shannon’s Demon.

A Sinking Feeling

Caius wants to ship goods from overseas worth 10000 rubles. There is a 5% chance that disaster will befall the ship. Should he pay 800 rubles to insure the goods?

Let \(x\) be Caius' current wealth, excluding the faraway goods. Without insurance, his expected utility is:

\[ 0.95 \log (x + 10000) + 0.05 \log x \]

With insurance, his utility will be \( \log (x + 9200) \).

These are equal when \(x \approx 5043 \), thus Caius should only buy insurance if he has less than this amount.

Bernoulli proceeds with a similar calculation to show that, on the other side, the insurer must have at least 14243 rubles or so for the deal to make sense.

(He then repeats this exercise with insurance that costs 600 rubles, and observes it would be insane to charge under 500 rubles!)

How do you like your eggs?

Semipronus has 4000 eggs in a safe location and wishes to bring home another 8000 eggs, which must be transported in baskets. Each basket has a one-in-ten chance of catastrophic failure that breaks all the eggs within.

The expected number of eggs he will eventually have is:

\[ 4000 + 0.9 \times 8000 = 11200 \]

no matter how many baskets are used.

However, Bernoulli’s utility tells a different story. (And Bernoulli himself also told a different story: for fun, I changed ducats to eggs.) If Semipronus puts all his eggs in one basket, his expected utility is:

\[ 0.9 \log 12000 + 0.1 \log 4000 \]

Exponentiating yields 10751.5…​ eggs.

If divided evenly into two baskets:

In eggs, this is 11033.5…​

In other words, from a log wealth viewpoint, extra baskets are worthwhile, even though the expected number of surviving eggs is identical.

As the number of baskets increases, the AM-GM inequality implies the expected utility converted back to eggs approaches but never exceeds 11200.

Money makes more converts than reason (and converts make more money)

Two centuries passed before log wealth was rescued from obscurity. See William Poundstone, Fortune’s Formula .

1951 - Henry Latané realized Bernoulli’s utility could be applied to stock portfolios.

1954 - Leonard Savage calls the log curve a "prototype for Everyman’s utility function", having being convinced by Latané.

1956 - John Larry Kelly Jr., publishes A New Interpretation of Information Rate , arguing for maximizing log wealth.

1959 - Harry Markowitz publishes a well-known finance book with a chapter dedicated to maximizing log wealth, citing Latané.

This promising start was sabotaged by economists led by Paul Samuelson, who eloquently (by economists' standards) criticized Bernoulli’s thinking in his journal article Why We Should Not Make Mean Log of Wealth Big Though Years to Act Are Long .

Not everyone was fooled by the economists. Some of these sensible few amassed spectacular fortunes, vindicating Bernoulli’s ideas.

Equally spectacular was LTCM, a bond fund allegedly run using state-of-the-art economic theory, whose founders included state-of-the-art economists. It lived so fast and died so young that an emergency ballout ensued.

So the tide may be turning. One LTCM insider estimated he had bet over 80% of his family’s liquid wealth on the venture. He later co-wrote a book he wished he had read before his ill-fated gamble: Victor Haghani and James White, The Missing Billionaires .

Paul would roll in his grave if he knew this book says: use log wealth to size bets right. Do not put all eggs in one place.

However, Haghani and White miss at least one mark. While they acknowledge the importance of Bernoulli’s utility, they perpetuate the myth that it resolves the St. Petersburg paradox. Funnily enough, they dismiss the utile version of the paradox because it is:

a combination of using a generic utility function that doesn’t quite match a person’s true utility at extremes of wealth, with setups that tend to exploit those extremes, such as having an infinitesimal probability of making more than all the money in the world or losing everything.

Agreed. But why not say this sort of thing about the original paradox?

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What Is Utility?

Understanding utility, ordinal utility, cardinal utility.

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In economics, utility is a term used to determine the worth or value of a good or service . More specifically, utility is the total satisfaction or benefit derived from consuming a good or service. Economic theories based on rational choice usually assume that consumers will strive to maximize their utility.

The economic utility of a good or service is important to understand because it directly influences the demand, and therefore price, of that good or service. In practice, a consumer's utility is usually impossible to measure or quantify. However, some economists believe that they can indirectly estimate what is the utility of an economic good or service by employing various models.

Key Takeaways

• Utility, in economics, refers to the usefulness or enjoyment a consumer can get from a service or good.
• Although the concept of utility is abstract, it is a useful way to explain how and why consumers make their decisions.
• "Ordinal" utility refers to the concept of one good being more useful or desirable than another.
• "Cardinal" utility is the idea of measuring economic value through imaginary units, known as "utils."
• Marginal utility is the utility gained by consuming an additional unit of a service or good.

Investopedia / Mira Norian

The utility definition in economics is derived from the concept of usefulness. An economic good yields utility to the extent to which it's useful for satisfying a consumer’s want or need. Various schools of thought differ as to how to model economic utility and measure the usefulness of a good or service.

Utility in economics was first coined by the noted 18th-century Swiss mathematician Daniel Bernoulli. Since then, economic theory has progressed, leading to various types of economic utility.

Early economists of the Spanish Scholastic tradition of the 1300s and 1400s described the economic value of goods as deriving directly from this property of usefulness and based their theories on prices and monetary exchanges.

This conception of utility was not quantified, but a qualitative property of an economic good. Later economists, particularly those of the Austrian School , developed this idea into an ordinal theory of utility, or the idea that individuals could order or rank the usefulness of various discrete units of economic goods.

Austrian economist Carl Menger, in a discovery known as the marginal revolution , used this type of framework to help him resolve the diamond-water paradox that had vexed many previous economists. Because the first available units of any economic good will be put to the most highly valued uses, and subsequent units go to lower-valued uses, this ordinal theory of utility is useful for explaining the law of diminishing marginal utility and fundamental economic laws of supply and demand .      

To Bernoulli and other economists , utility is modeled as a quantifiable or cardinal property of the economic goods that a person consumes. To help with this quantitative measurement of satisfaction, economists assume a unit known as a “util” to represent the amount of psychological satisfaction a specific good or service generates for a subset of people in various situations. The concept of a measurable util makes it possible to treat economic theory and relationships using mathematical symbols and calculations.

However, it separates the theory of economic utility from actual observation and experience, since “utils” cannot actually be observed, measured, or compared between different economic goods or between individuals.

If, for example, an individual judges that a piece of pizza will yield 10 utils and that a bowl of pasta will yield 12 utils, that individual will know that eating the pasta will be more satisfying. For the producers of pizza and pasta, knowing that the average bowl of pasta will yield two additional utils will help them price pasta slightly higher than pizza.

Additionally, utils can decrease as the number of products or services consumed increases. The first slice of pizza may yield 10 utils, but as more pizza is consumed, the utils may decrease as people become full. This process will help consumers understand how to maximize their utility by allocating their money between multiple types of goods and services as well as help companies understand how to structure tiered pricing.

Economic utility can be estimated by observing a consumer's choice between similar products. However, measuring utility becomes challenging as more variables or differences are present between the choices.

If utility in economics is cardinal and measurable, the total utility (TU) is defined as the sum of the satisfaction that a person can receive from the consumption of all units of a specific product or service. Using the example above, if a person can only consume three slices of pizza and the first slice of pizza consumed yields ten utils, the second slice of pizza consumed yields eight utils, and the third slice yields two utils, the total utility of pizza would be twenty utils.

Marginal utility (MU) is defined as the additional (cardinal) utility gained from the consumption of one additional unit of a good or service or the additional (ordinal) use that a person has for an additional unit.

Using the same example, if the economic utility of the first slice of pizza is ten utils and the utility of the second slice is eight utils, the MU of eating the second slice is eight utils. If the utility of a third slice is two utils, the MU of eating that third slice is two utils.

In ordinal utility terms, a person might eat the first slice of pizza, share the second slice with their roommate, save the third slice for breakfast, and use the fourth slice as a doorstop.

How Do You Measure Economic Utility?

While there is no direct way to measure the utility of a certain good for an individual consumer, it is possible to estimate utility through indirect observation. For example, if a consumer is willing to spend $1 for a bottle of water but not $1.50, economists can safely state that a bottle of water has economic utility somewhere between $1 and $1.50. However, this becomes difficult in practice because of the number of variables that are present in a typical consumer's choices.

What Are the 4 Types of Economic Utility?

In behavioral economics , the four types of economic utility are form utility, time utility, place utility, and possession utility. These terms refer to the psychological importance attached to different forms of utility. For example, form utility is the result of the design of a product or service, and time utility refers to the ability of a company to provide services when the customers need them.

How Do You Invest in Utilities?

Utilities are companies that operate in the electric, water, oil, or gas sectors. These companies play a major role in industrial economies and have a total global market capitalization of nearly $6.4 trillion as of 2024. In addition to investing in individual companies, there are also many targeted funds that are invested in a basket of utilities-sector companies.

Utility can be used to measure the usefulness of goods and services to consumers. While there are limitations when more variables and differences appear in the market, various types of economic utility continue to be examined. Not only can it help companies with structuring their tiered pricing but it can also help consumers learn how to boost the utility of their purchases.

E. Kwan Choi. " Chapter 7. Utility ." Principles of Microeconomics ." Department of Economics, Iowa State University.

Moscati, Ivan. " How Economists Came to Accept Expected Utility Theory: The Case of Samuelson and Savage ." Journal of Economic Perspectives , vol. 30, no. 2, Spring 2016, pp. 220.

Marjorie Grice-Hutchinson. " Early Economic Thought in Spain, 1177-1740 ," Pages 86-95. Liberty Fund, Inc., 2015.

Mises Institute. " Why Austrians Stress Ordinal Utility ."

The Library of Economics and Liberty. " Carl Menger ."

U.S. Energy Information Administration. " Alternative Measures of Welfare in Macroeconomic Models ." Pages 1-2.

LibreTexts Social Sciences. " 7.1 The Concept of Utility ." Principles of Economics . UC Davis Library, California State University.

Kentley Insights. " Utilities – 2024 Global Market Size & Growth Report ."

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This is what utility theory is all about: the degree of satisfaction or usefulness derived from consuming a product or service.

What is utility?

Utility is the level of satisfaction a person derives from consuming a good or service. When the product or service is useful to the consumer’s needs or wants, they can achieve a certain level of utility from consuming it. In economics , there are two different types of utility: expected utility and subjective utility.

Expected utility

Expected utility is the utility that an economic agent is expected to reach in the future given several probable outcomes. Expected utility value is a probability concept used when several future outcomes are possible. I t is calculated by multiplying each possible utility outcome by the probability of its occurrence and then adding them up. Expected utility theory deals with decision-making under uncertainty.

Subjective utility

Subjective utility is utility based on an individual's perceived level of satisfaction from consuming a good or service. S ubjective utility is not based on market judgment. It is based on how attractive an individual perceives the benefit of using a good or service.

Utility theory

Students choose to study because they want to pass their exams. We eat something because we're hungry . We drive a car to reach a certain destination. We sleep to give our bodies some rest. Utility is involved in everything we do and w e get satisfaction from consuming or using goods or services. This is what utility theory is concerned with: explaining individuals’ choices and measuring the satisfaction level from consuming a good or service.

The level of satisfaction is measured in units called ‘utils.’

Total utility and marginal utility

There are two different types of utility:

Marginal utility (MU)

Marginal utility i s the satisfaction that a person receives from consuming an additional unit of the same good or service.

If John is drinking his first glass of water and gets 10 units of satisfaction, the marginal utility he derives from the first glass is 10 units. He then has a second glass of water and gets 8 units of satisfaction. The marginal utility he derives from the second glass is 8 units. With the third glass, he gets only 7 units of satisfaction. Thus, the marginal utility he derives from the third glass is 7 units.

Total utility (TU)

Total utility is the aggregate satisfaction a person receives from the consumption of all the units of the same good or service.

Total utility is derived from adding every marginal utility from each additional unit.

Continuing with our previous example, where John derived 10, 8, and 7 units of utility from the glasses of water, the total utility that John would derive is 10 + 8 + 7 = 25 units.

The equation for total utility (TU) is:

T U = M U - 1 + M U - 2 + M U - 3 + . . . M U - N

Where MU-N is the marginal utility from consuming the N-th unit of a good.

Table 1. Relationship between marginal utility and total utility - Vaia.

As the table shows, the marginal utility decreases with the addition of further units, whereas the total utility increases until a certain point. At that point, which is 5 glasses of water, the total utility reaches its maximum and starts declining.

Figure 1 shows the relationship between marginal utility and total utility:

We can conclude the following relationship between MU and TU:

• As the number of units increases, MU decreases, and TU increases.
• When TU reaches its maximum level, MU is 0. At that unit, the marginal utility is 0.
• MU starts to get negative and TU starts decreasing.

The law of diminishing marginal utility

Economists believe that the utility reduces as the consumption of the same product or service increases.

The Law of diminishing marginal utility states that the level of satisfaction for an individual diminishes as the use of the same product increases. Eventually, the consumer either looks for an alternative or stops consuming the product.

According to the law of diminishing marginal utility, the consumption of the first unit gives the consumer maximum utility. Then, the level of satisfaction starts reducing as the units increase. The consumer starts getting negative utility after a particular unit of consumption, which may vary from consumer to consumer.

Suppose Alan is very hungry and decides to eat a hamburger. The first burger satisfies his hunger. However, he is still hungry, so he buys another burger. This further satisfies his hunger. However, not as much as the first burger. He goes on to have a third burger to fill the little hunger he still has and gets fully satisfied. Any further burger will not satisfy Alan's hunger and might be a bit too much for him to eat. It may make him feel too full and may also result in him feeling sick. Thus, the fourth burger may not give any satisfaction to Alan and instead give him a negative utility.

Utility maximisation

Utility maximisation means that a consumer will try to get the highest level of satisfaction for consuming something they paid for. T he utility may be different for every individual and cannot be stated as a single total unit.

Imagine you are paying a tutor to help you with maths five days per week. However, the tutor isn't available at least two or three times per week as initially agreed. Would you be happy with that? Would you be satisfied with the tutor and willing to pay the same price? The answer is generally no. If someone is paying for five days a week tuition fee, they will expect to receive the tutoring hours they paid for. This is utility maximisation.

However, even though consumers wish to have the maximum utility from the consumption of a product or service, sometimes they may have to make other choices due to constraints. Let's explore them:

Limited income

Even though someone may fancy having the best of all products because it gives them the highest satisfaction, limited income may stop them from buying it.

Richard wishes to have a Ferrari car. However, his income just covers his basic needs of food, clothing, shelter, and a comparatively cheaper car. He has no budget for a Ferrari. In his case, limited income stops him from having the car that will satisfy him the most.

A given set of prices

Some individuals like some products or services more than others. However, they may opt for a substitute or a similar product due to the set of prices. Although they will get the maximum utility from consuming the high-priced goods, they may not be willing to pay the given set of prices. Thus, they will look for alternatives.

Many people like McVities digestive biscuits. However, some decide not to buy them because they have a high price. They may opt for cheaper available alternatives like a supermarket's own brand of digestive biscuits.

Budget constraints

Consumers’ choices are subject to their budget constraints. Budget refers to the total amount of money an individual is willing to spend, save, and borrow. Budget constraints can also be understood as limited income.

If Richard, the man who wants a Ferrari, has limited savings and is not willing to borrow money, his budget constraint will restrain him from buying the luxury car. His budget constraints don't allow him to make the choice that would maximise his utility.

Limited time

Another constraint consumers may face while making choices is the availability of time.

Suppose an individual is willing to get the goods currently on sale. However, they could not go to the store to buy the goods on time for the sale. They will not enjoy the maximum utility they would have gained if they purchased the good when the price was affordable for them.

The importance of margin when making choices

Economists say that most choices are made at the margin. The m argin is the current state at which an individual is making choices. Margin helps the consumers decide how much they will gain or lose with the extra unit of a good or service. Hence, the consumer’s buying decision is on the margin.

Margin is important for these reasons:

• It is the point at which an individual decides whether to consume more or less.
• The margin determines the benefit that a consumer may receive by consuming an additional unit of a good or service. Therefore, it helps in deciding whether to consume additional units.
• Understanding consumers’ total marginal utility also lets the supplier decide whether the consumer will buy further goods or services.

Utility Theory - Key takeaways

Utility is the level of satisfaction a person derives from consuming a good or service.

Utility theory explains individuals’ choices and measures their level of satisfaction from consuming a good or service. The level of satisfaction is measured in units called ‘utils.’

Marginal utility is the satisfaction that a person receives from consuming an additional unit of the same good or service.

Total utility is the aggregate satisfaction a person receives from consuming all the units of the same good or service.

As the number of units increases, marginal utility decreases, and total utility increases. When the total utility reaches its maximum level, the marginal utility is zero. After that point, marginal utility starts to get negative and total utility starts decreasing.

The Law of diminishing marginal utility states that an individual’s level of satisfaction diminishes as the use of the same product increases. Eventually, the consumer either looks for an alternative or stops consuming the product.

Utility maximisation is the highest level of satisfaction a consumer is able to derive from the decision they made in return for the cost they paid.

Individual choices when trying to maximise utility may be limited by income, prices, time, and budget constraints.

Flashcards in Utility Theory 30

Utility is the level of satisfaction a person derives from the consumption of goods or services.

What are the units of measuring utility called? 

Total utility decreases as the units of consumption increases.

Marginal utility and total utility have the same curve.

Marginal utility decreases as the units of consumption increase.

What is marginal utility?

Marginal utility is the satisfaction that a person receives from consuming an additional unit of the same good or service. 

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Frequently Asked Questions about Utility Theory

What is utility theory in decision making?

Utility theory explains individuals' choices and measures the level of satisfaction they obtain from consuming a good or service. 

What is standard utility theory?

The standard utility theory says that the level of satisfaction is measured in units called ‘utils.’

What are the four types of utility? 

Expected utility, subjective utility, marginal utility, and total utility.

Marginal utility is the satisfaction that a person receives from consuming an additional unit of the same good or service. 

What is expected utility? 

Expected utility is the utility that an economic agent is expected to reach in the future given several probable outcomes. Expected utility value is a probability concept that is used when several future outcomes are possible. 

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What is Utility Theory?

Home › Economics › Macroeconomics › What is Utility Theory?

Definition: Utility theory is an economic hypothesis that postulates the fact that consumers make purchase decisions based in the degree of utility or satisfaction they obtain from a given item. This means that the higher the utility level the higher the item will be prioritized in the consumer’s budget.

• What Does Utility Theory Mean?

This theory states that consumers rank products in their minds whenever they are facing a purchase decision. These ranking function drives their budget allocation, which means that resources are poured into the purchases that will bring the highest degree of satisfaction. It is assumed that individual budgets are limited and therefore there is a limited amount of goods or services that can be purchased, taking this into account, an individual will weigh which of the options currently available within the open market is the best suit to fulfill his current set of needs or desires.

In these cases, preferences also play a key role and these can be defined as a set of predispositions that each individual possesses towards certain brands or products by elements such as colors, shapes, tastes or smells. Finally, there are four essential types of utility and these are form utility, time utility, place utility and possession utility.

Harold is a 45 year old computer engineer that was recently hired by a company called Tech Mogul Co. which is a firm that provide security solutions for information systems, mostly to the banking industry. Harold is considered to be a very sophisticated person who enjoys luxurious accessories and gadgets. His salary is big enough to allow him to purchase such items and he is normally up to date with new technological devices. Recently, Harold was presented with the new version of the smartphone he currently owns.

This new device costs $1,100 and it was offered to a few VIP clients of the firm. Even though there are other amazing smartphones available in the market, Harold prefers this new version because he is loyal to the brand. The utility he gets from this phone comes in the form of possession, as owning this new device makes him feel important and appreciated.

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Decision Utility

The basic idea, theory, meet practice.

TDL is an applied research consultancy. In our work, we leverage the insights of diverse fields—from psychology and economics to machine learning and behavioral data science—to sculpt targeted solutions to nuanced problems.

When you’re making a decision – whether it’s a life-changing choice or what you want to eat for lunch – you’re usually guided by the question: What would be the best for me?

In other words, you’re considering what the most satisfying or useful decision would be. Every time we make a decision, we evaluate the potential outcomes and how useful they might be. This concept of “utility”—how beneficial an outcome will be for us—lies at the heart of the behavioural economic and psychological study.

Utility is a key term in economics that describes the benefit an agent receives from the consumption of goods or services. In traditional economics, people are generally expected to act rationally and make decisions based on maximizing an outcome’s utility. In theory, this process makes sense. In practice, utility is often difficult to quantify in real life.

In order to further refine the concept of utility, psychologists and economists have differentiated between two types of utility: our perceptions of utility before we experience it, or  decision utility , and the actual experienced utility of a choice, called  experienced utility .¹ Decision utility describes the usefulness that we perceive and use to make a decision, while experienced utility describes the lived consequences of the decision in reality. These different types of utility have driven new understandings of utility and its role in decision-making.

Utility is an important concept in economics, psychology, business, and our personal lives – it guides our every choice. If we can understand utility, we can understand why and how people come to their decisions, and even make predictions about how people will behave.

Maintaining one’s vigilance against biases is a chore—but the chance to avoid a costly mistake is sometimes worth the effort. – Daniel Kahneman

The beginnings

Utility as an economic principle goes back multiple centuries and was first described by 18th-century Swiss mathematician Daniel Bernoulli. Over time, however, economists and eventually psychologists developed more nuanced theories of utility, leading to the multiple conceptions of utility used today. Moreover, over the past century, a new branch in economics has emerged that developed a different understanding of our relationship to utility.

George Stigler, an American economist, and future Nobel laureate wrote a 1950 paper giving a historical survey of utility in economics.² His review of utility theory from 1776 to 1915 in this article served as a basis for many other researchers to build on. He begins with a theory developed by English philosopher Jeremy Bentham. In an influential 1789 paper, Bentham proposed measuring the amount of pleasure and pain in the context of developing a rationalist legal system. He gave four dimensions of these two feelings: intensity, duration, certainty, and propinquity. Bentham also realized that individual differences would change how a given person feels pleasure or pain in a specific situation. In this way, he described our process of evaluating utility as one of optimizing pleasure and minimizing pain. Although Bentham’s theory was justified by its convenience and approximating ability, it was not necessarily effective, since the philosopher did not provide a way to measure the pleasure and pain of a situation.

Utility theory did not become heavily discussed or studied in economics until the 1870s, when economists attempted to advance the idea of utility in different ways, such as studying the relationships between price and utility, and demand and utility. While various mathematical formulations and models were tested to estimate the utility of outcomes using variables like price, the quantity of product, and supply and demand, measurement remained a difficult goal of utility theory.

Expected utility theory

Then, in 1944, John Von Nuemann and Oskar Morgenstern developed the  expected utility hypothesis , based on Daniel Bernoulli’s first description of how we make decisions by estimating the probability and utility of an outcome. By multiplying the probability of an outcome by the expected benefit of that outcome, we get the expected utility of that choice. We can then use this to form our decision—we choose what will give us the best-expected utility. By using  Bayesian statistics  and probability, the theory suggested that we make precise calculations about the optimal outcome and decision even when the outcome is uncertain. While this theory became massively influential, it worked best in scenarios where the expected gains and probabilities are easily calculated. For instance, this framework can be applied to games like poker, but not easily to most life decisions, where we have trouble estimating the outcomes and the likelihood that we’ll get a particular outcome.³

Behavioral economics

In 1969, by the time the expected utility hypothesis was well known among economists, two economists undertook further research into applying the theory to real life situations. Intrigued by a psychologist’s observation that people followed this logical principle in their decisions by estimating basic probability,  Daniel Kahneman  and  Amos Tversky  performed studies to test how people actually behaved in comparison to the predictions made by decision analysts based on the expected utility hypothesis. They found that people often did not follow the statistical predictions decision analysts used, instead opting for a more intuitive approach.⁴ The concept of utility in the expected utility hypothesis was then either flawed or failed to take into account certain kinds of utility.

Kahneman and Tversky continued studying utility together, and into the later 20th century, economists distinguished between two different kinds of utility: decision utility and experienced utility.  Experienced utility  was connected back to the utility consisting of pleasure and pain that Bentham described, and characterized as a  hedonic quality , or relating to the pursuit of pleasure.  Decision utility , on the other hand, was conceived as the “weight of an outcome in a decision”⁵, or the value we optimize in a decision.

In modern economics, experienced utility was largely ignored because of arguments that it could not be observed or measured, and that choices reveal the utility of the outcomes because rational agents optimize their utility. In their paper, Kahneman, Peter Wakker, and Rakesh Sarin argued that experienced utility could, in fact, be measured and was distinct from decision utility. They suggested that normal human cognition could result in our perceived utility being different from our experienced satisfaction from an outcome. They proposed a utility framework consisting of 4 different types of utility: predicted utility, decision utility, experienced utility and remembered utility. Decision utility is the utility present at the time of the decision, meaning it drives our decision-making.⁵ As a result of these different utilities, we may not always act in a way that actually maximizes the expected utility of our decision —even if we think we are behaving logically at the time—although this is what traditional economics alleges. As a result,  behavioural economics developed as a separate branch of economics that accounts for psychological aspects of decision-making that may cause us to act irrationally, or away from the maximum utility.

Biological decision utility

Recently, the biological basis of decision utility has also been studied in neuroscience and connected to dopamine mechanisms in the brain. The importance of dopamine in motivation provides a biological basis for Bentham’s theory of “hedonic qualities” driving our decisions. Particular cues based on memory can also alter the utility of a particular action immediately after we encounter them, thanks to the release of dopamine.⁶ For example, when you get stressed, you might feel an overwhelming desire to smoke a cigarette. In other calm situations, however, we would feel no urge to do so. Our different reactions to these situations demonstrate how utility can change due to different levels of brain chemicals. The way our brains remember pleasurable experiences may drive us towards a desire, even if the experience fails to meet the remembered feeling.⁷

From a rather simplistic view in the late 18th century to a nuanced and humanistic perspective at the end of the 20th century, our understanding of utility has evolved dramatically. Today, our knowledge of utility as a complex, emotional, and changing quality can help us recognize short-sighted decisions and improve our choices.

Consequences

At the center of all decisions, utility is a core concept that we use every day, whether or not we’re conscious of it. Although it seems logical to assume we automatically resort to maximizing the utility of our actions, sometimes this does not appear to occur. As we experience the consequences of our decisions, big or small, it’s common to look back on ourselves and wonder “What were we thinking?”. It can seem like a different person was responsible for a poor decision, not us.

The distinction between decision utility and experienced utility can often be significant, so understanding the difference is critical to improving our decision-making. For instance, we tend to make faulty judgments of life decisions and their effect on overall satisfaction. One study about the perceived versus lived satisfaction of living in California demonstrated that we often fall prey to a “the grass is greener” mentality.⁸ The authors concluded that when thinking about differences in climate and culture, we overestimate the effect they will have on our satisfaction. In reality, these factors do not significantly impact our enjoyment of where we live—the study showed that we often believe that living in the sunny California weather will make us happier than it actually does.

The difference in decision utility and experienced utility can also be explained by something called a projection bias.⁹ We overestimate how much our future preferences will look like our current preferences. Understanding this tendency, we can recognize it when we make a poor prediction of what we need and account for the bias.

Additionally, the way decisions are framed can affect our perception of their utility. Kahneman and Tversky first demonstrated the influence of framing in a 1986 paper.¹⁰ We can know that the outcome will be the same—like if different discounts result in the same price reduction—yet,  we will still be more attracted to the higher discount percentage . The psychological aspect of buying something on sale, even if it is the same price as another product of equal quality that is not on sale, is another utility that traditional economical utility does not consider. As core parts of human decision-making, we have to take our cognitive biases into account if we are going to understand how we form ideas of utility and apply it to evaluate outcomes.

Controversies

A problem often brought up with traditional economics and expected utility theory is their assumption of rationality. The term “ Homo Economicus ” describes the agent implied by traditional economics. While real humans – Homo sapiens – are significantly affected by cognitive biases and emotions, Homo economicus is rational and economically-driven. Homo economicus may evaluate utility in a narrow way which disregards the social or emotional utility involved in a decision, for instance.

At the turn of this century,  Richard Thaler , an economist inspired by Kahneman and Tversky’s work, wrote a perspective piece on this issue, predicting that economics would pivot to incorporating human behaviour.¹¹ In the latter half of the 20th century, there had been  a shift towards accounting for irrational human behaviour , and Thaler was correct in predicting this future in the development of  behavioural economics.  That said, economists have been careful to specify that the movement away from traditional economics does not mean that we are not rational beings; rather, the existing conceptions of rational behaviour fail to describe the logic humans operate by.

We might expect in making decisions, we automatically maximize utility, or go with the choice that leads to the most useful outcome, considering that earlier economists working on utility theory in the late 19th century consistently arrived at this conclusion.³ In a 2006 paper, however, Kahneman and Thaler refuted this hypothesis.¹² They found that because we do not always know what we like, as demonstrated in the California example, we make errors in predicting the future utility of outcomes. As a result, we do not maximize the utility of our decisions because we make erroneous judgments about what will be useful to us. We make intuitive decisions without really thinking things through. When we go to the grocery store on an empty stomach, we often purchase much more food than we actually need – and more than what was on our grocery list – because of how we feel at that moment.

Intuitive decision-making

This error could also be explained by a process of substitution in intuitive thinking, where we wind up answering a different question than the one we intend to address. For instance, when we’re shopping and hungry, we may be making optimized utility decisions for ourselves in that moment, because we would like to eat the food we’re buying. Although we think we’re dealing with food decisions for the week ahead, we are really just addressing our immediate food desires.

In their paper, Kahneman and Thaler addressed four situations where “hedonic forecasting”, or our ability to know what we want in the future, resulted in errors in decision-making:

• Where the emotional or motivational state of the agent is very different at the time of the decision versus time of consumption
• Where the nature of the decision focuses attention on aspects of the outcome that will not be relevant when it is actually experienced
• When choices are made on the basis of flawed evaluations of past experiences
• When people forecast their future adjustment to new life circumstances.

The first instance, as in the example of shopping while hungry, has been proven to result in different outcomes. A similar case has been observed with the current weather influencing the clothes people buy—on an abnormally cold day, it is less likely that people will buy clothes for warm weather, even for future use by ordering on the phone. “Anchoring” in the present moment can result in us making a different decision that we’re not actually conscious of, but our attention becomes focused on our present needs, so we unknowingly make a decision fit for that problem.

In the second case, the way decisions are presented to us can influence how we evaluate the utility of those choices. This is where biases like  naive allocation  can result in us making a non-optimized decision in terms of utility. Given different assortments of options, we choose differently.

The third case, where we carry imperfect judgments of past experiences, has to do with the way we remember past pain and pleasure. The peak/end rule suggests that our retrospective evaluation of an incident will be composed of the average of our feelings at the most extreme point and at the end of the experience. In other words, we do not remember the beginning or less extreme aspects of an experience as well as the peak/end when thinking about past incidents. Based on studies on different hedonic experiences, such as measuring pain during medical procedures, people’s evaluations of painful experiences can indeed be altered by manipulating the extreme point of the experience or changing the ending. When a procedure ended more gradually and with a period of lesser pain, patients rated it as less painful than procedures that ended abruptly with pain, even though the only difference was the length of the procedure.

The final case describes what happens when people try to envision themselves living in California—we judge our future lives by metrics that won’t actually end up mattering to us. Kahneman also found that we adapt better to situations than we expect. In fact, we often think something will be worse than it actually is. Kahneman compared how paraplegics felt about their lives after they became paralyzed against asking non-paraplegics to estimate their feelings if they became paraplegic. Interestingly, he found that non-paraplegics greatly overestimated the negative effect of the disability and that paraplegics reported doing much better than people imagined.

Thus, utility can still describe the motivation for our decisions, but past models have failed to account for all that we consider useful.

Related TDL Content

Homo Economicus

This article goes into detail about the concept of Homo Economicus, or the completely logical species imagined by traditional economics. The difference between homo economicus and homo sapiens – or real human behaviour – is addressed, as well as the history of the term.

Behavioural Economics

This article on behavioural economics also highlights the differences between traditional and behavioural economics. As we saw with people’s failure to choose maximum utility, we need to account for our biases and flaws in rational decision-making in order to realistically understand how we make decisions, not just how we ought to make decisions. Behavioural economics, as opposed to traditional economics, takes our psychology and biases into account to study human decision-making.

Evolution of Decision Making: Current State

This article, the third part of a series, gives an overview of the current state of behavioural economics, including decision utility. The author also provides examples of how recent research can be incorporated in our professional and personal lives to improve our decision-making.

• Robson, A. & Samuelson, L. (2011). The evolution of decision and experienced utilities.  Theoretical Economics ,  6 (3), 311-339.  https://doi.org/10.3982/TE800 \
• Stigler, G. J. (1950). The Development of Utility Theory. I.  Journal of Political Economy ,  58 (4), 307–327.  http://www.jstor.org/stable/1828885
• von  Neumann, J ., and  Morgenstern, O . (1944).  Theory of Games and Economic Behavior . Princeton University Press.
• Kahneman, D., & Tversky, A. (1973). On the psychology of prediction.  Psychological Review, 80 (4), 237–251.  https://doi.org/10.1037/h0034747
• Kahneman, D., Wakker, P. P., & Sarin, R. (1997). Back to Bentham? Explorations of Experienced Utility.  The Quarterly Journal of Economics ,  112 (2), 375–405. http://www.jstor.org/stable/2951240
• Berridge, K. C., & Aldridge, J. W. (2008). Decision Utility, The Brain, and Pursuit of Hedonic Goals.  Social cognition ,  26 (5), 621–646.  https://doi.org/10.1521/soco.2008.26.5.621
• Berridge, K. C., & O’Doherty, J. P. (2014). From Experienced Utility to Decision Utility.
• Neuroeconomics ,  2 , 335-351,  https://doi.org/10.1016/B978-0-12-416008-8.00018-8 .
• Schkade, D. A. & Kahneman, D. (1998). Does Living in California Make People Happy? A Focusing Illusion in Judgments of Life Satisfaction. Psychological Science,  9 (5), 340-346.  http://www.jstor.org/stable/40063318
• Tversky, A., & Kahneman, D. (1986). Rational Choice and the Framing of Decisions. The Journal of Business,  59 (4), S251–S278.  http://www.jstor.org/stable/2352759
• Loewenstein, G., O’Donoghue, T., & Rabin, M. (2003). Projection Bias in Predicting Future Utility,  The Quarterly Journal of Economics , 118(4), 1209–1248.  https://doi.org/10.1162/003355303322552784
• Thaler, Richard, H. (2000). From Homo Economicus to Homo Sapiens.  Journal of Economic Perspectives , 14(1), 133-141.  https://www.aeaweb.org/articles?id=10.1257/jep.14.1.133
• Kahneman, D., & Thaler, R. H. (2006). Anomalies: Utility Maximization and Experienced Utility.  The Journal of Economic Perspectives ,  20 (1), 221–234.  http://www.jstor.org/stable/30033642

About the Author

Katharine Kocik

Katharine Kocik earned a Bachelor of Arts and Science from McGill University with major concentrations in molecular biology and English literature. She has worked as an English teacher and a marketing strategist specializing in digital channels. 

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Theory of Consumer’s Behaviour Utility Analysis

Read this article to learn about the Theory of Consumer’s Behaviour Utility Analysis, Meaning of Utility, Marshall’s Cardinal Utility Analysis, Limitations of the Law of Equimarginal Utility, Critical Evaluation of Marshall’s Cardinal Utility Analysis.

Theory of Consumer’s Behaviour Utility Analysis 

• Introduction to the Theory of Consumer’s Behaviour Utility Analysis
• Meaning of Utility
• Marshall’s Cardinal Utility Analysis
• Limitations of the Law of Equimarginal Utility
• Critical Evaluation of Marshall’s Cardinal Utility Analysis

1. Introduction to the Theory of Consumer’s Behaviour Utility Analysis:

The price of a product depends upon the demand for and the supply of it. In this part of the article we are concerned with the theory of demand, which explains the demand for a good and the factors determining it. Individual’s demand for a product depends upon the price of the product, income of the individual, and the prices of related goods.

It can be stated in the following functional form:

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D x = f (P x , I, P y , P z etc.)

Where D x stands for the demand for good X, P x for price of good X, I for individual’s income, P y , P z etc., for the prices of related goods. But among these determinants of demand, economists single out price of the good in question as the most important factor governing the demand for it. Indeed, the function of a theory of demand is to establish a relationship between quantity demanded of a good and its price and to provide an explanation for it.

From time to time, different theories have been advanced to explain consumer’s de­mand for a good and to derive a valid demand theorem. Cardinal utility analysis is the oldest theory of demand which provides an explanation of consumer’s demand for a product and derives the law of demand which establishes an inverse relationship between price and quantity demanded of a product.

Recently, cardinal utility approach to the theory of demand has been subjected to server criticisms and as a result some alternative theories, namely, Indifference Curves Analysis, Samuelson’s Revealed Preference Theory, have been propounded. Through cardinal utility approach to the theory of demand is very old, its final shape emerged at the hands of Marshall. Therefore, it is Marshallian utility analysis of demand which has been discussed in this article.

2. The Meaning of Utility:

People demand goods because they satisfy the wants of the people. The utility means want-satisfying power of a commodity. It is also defined as property of the commodity which satisfies the wants of the consumers. Utility is a subjective thing and resides in the mind of men. Being subjective it varies with different persons, that is, different persons derive different amounts of utility from a given good. People know utility of goods by their psycho­logical feeling.

The desire for a commodity by a person depends upon the utility he expects to obtain from it. The greater the utility he expects from a commodity, the greater his desire for that commodity. It should be noted that no question of ethics or morality is involved in the use of the word ‘utility’ in economics.

The commodity may not be useful in the ordinary sense of the term even then it may provide utility to some people. For instance, alcohol may actually harm a person but it possesses utility for a person whose want it satisfies. Thus, the desire for alcohol may be considered immoral by some people but no such meaning is conveyed in the economic sense of the term. Thus, in economics the concept of utility is ethically neutral.

Total Utility and Marginal Utility:

It is important to distinguish between total utility and marginal utility. Total utility of a commodity to a consumer is the sum of utilities which he obtains from consuming a certain number of units of the commodity per period. Consider Table 5.1 where a utility of a consumer from cups of tea per day is given. If the consumer consumes one cup of tea per day, he gets utility equal to 12 utils.

On consuming two units of the commodity per day his utility from the two units of the commodity rises to 22 utils and so on. When he takes 6 cups of tea per day, his total utility, that is, total utility of all the 6 units taken per day goes up to 41 utils. Generally, the greater the number of units of a commodity consumed by an individual, the greater the total utility he gets from the commodity. Thus, total utility is the function of the quantity of the commodity consumed.

It should however be noted that as the units of a commodity increases, total utility increases at a diminishing rate. When want of the consumer for a particular commodity is fully satisfied by consuming a certain quantity of the commodity, further increases in consumption of the commodity will cause a decline in total utility of the consumer. The number of units of commodity consumed at which a consumer is fully satisfied is called satiation quantity.

Beyond the satiation point, total utility decreases if more is consumed. It will be seen from Table 5.1 that total utility declines when the consumer consumes more than 6 units of the commodity. This happens because beyond satiety point more consumption of a good actually harms the consumer which causes a decline in utility or satisfaction from the commodity.

Marginal Utility:

Marginal utility of a commodity to a consumer is the extra utility which he gets when he consumes one more unit of the commodity. In other words, marginal utility is the addition made to the total utility when one more unit of a commodity is consumed by an individual. The concept of marginal utility can be easily understood from Table 5.1.

When the consumer takes two cups of tea instead of one cup his total utility increases from 12 to 22 utils. This means that the consumption of the second unit of the commodity has made addition to the total utility by 10 utils. Thus marginal utility is here equal to 10 utils.

Further when the number of cups of tea consumed per day from 2 to 3, the total utility increases from 22 to 30 utils. That is, the third unit of tea has made an addition of 8 utils to the total utility. Thus 8 is the marginal utility of the third of consump­tion of tea. Beyond 6 cups of tea consumption per day, total utility declines and therefore marginal utility becomes negative.

Marginal utility can be expressed as under:

We have joined the various rectangles by a smooth curve which is the curve of total utility which rises up to a point and then declines due to negative marginal utility. Moreover, the shaded areas of the rectangle representing marginal utility of the various cups of tea have also been shown separately in the figure 5.1.

We have joined the shaded rectangles by a smooth curve which is the curve of marginal utility. As will be seen, this marginal utility curve goes on declining throughout and even falls below the X-axis. Portion below the X-axis indicates the negative marginal utility.

This downward-sloping marginal utility curve has an important implication for consumer’s behaviour regarding demand for goods. We shall explain below how the demand curve is derived from marginal utility curve. The main reason why the demand curves for goods slope downward is the fact of diminishing marginal utility.

The significance of the diminishing marginal utility of a good for the theory of demand is that the quantity demanded of a good rises as the price falls and vice-versa. Thus, it is because of the diminishing marginal utility that the demand curve slopes downward.

If properly understood the law of diminishing marginal utility applies to all objects of desire including money. But it is worth mentioning that marginal utility of money is generally never zero or negative. Money represents general purchasing power over all other goods, that is, a man can satisfy all his material wants if he possesses enough money. Since man’s total wants are practically unlimited, marginal utility of money to him never falls to zero.

Applications and Uses of Diminishing Marginal Utility:

The marginal utility analysis has a good number of uses and applications in both economic theory and policy.

We explain below some of its important uses:

a. It Explains Value Paradox:

The law of diminishing marginal utility is of crucial significance in explaining determination of the prices of commodities. The discovery of the concept of marginal utility has helped to explain the paradox of value which troubled Adam Smith in The Wealth of Nations.

Adam Smith was greatly perplexed to know why water which is so very essential and useful to life has such a low price (indeed no price), while diamonds which are quite unnecessary, have such a high price. This value paradox is also known as water-diamond paradox. He could not resolve this water-diamond paradox. But modern economists can solve it with the aid of the concept of marginal utility.

According to the modern economists, the total utility of a commodity does not determine the price of a commodity and it is the marginal utility which is crucially important determinant of price. Now, the water is available in abundant quantities so that its relative marginal utility is very low or even zero. Therefore, its price is low or zero.

On the other hand, the diamonds are scarce and therefore their relative marginal utility is quite high and this is the reason why their prices are high.

Prof. Samuelson explains this paradox of value in the following words:

“The more there is of a commodity, the less the relative desirability of its last little unit becomes, even though its total usefulness grows as we get more of the commodity. So, it is obvious why a large amount of water has a low price. Or why air is actually a free good despite its vast usefulness. The many later units pull down the market value of all units.”

b. This Law Helps in Deriving Law of Demand:

Further, as shall be seen below, with the aid of the law of diminishing marginal utility, we are able to derive the law of demand and to show why the demand curve slopes downward. Besides, the Marshallian concept of consumer’s surplus is based upon the principle of diminishing marginal utility.

c. This Law Shows Redistribution of Income will Increase Social Welfare:

Another important use of marginal utility is in the field of fiscal policy. In the modern welfare state, the Governments redistribute income so as to increase the welfare of the people. This redistribution of income through imposing progressive income taxes on the rich sections of the society and spending the tax proceeds on social services for the poor people is based upon the diminishing marginal utility.

The concept of diminishing marginal utility demonstrates that transfer of income from the rich to the poor will increase the economic welfare of the community. Law of diminishing marginal utility also applies to the money; as the money income of a consumer increases, the marginal utility of money to him falls. How the redistribution of income will increase the welfare of the community, is illustrated in Fig. 5.2.

In the Fig. 5.2, money income is measured along X-axis and marginal utility of income is measured along Y-axis. MU is the marginal utility curve of money which is sloping downward. Suppose OL is the income of a poor person and OH is the income of a rich person. If rich person is subjected to the income tax and amount of money equal to HH’ is taken from him and the same amount of money LL’ (equal to HH’) is given to the poor man, it can be shown that the welfare of the community will increase.

As a result of this transfer of income, the income of the rich man falls by HH” and the income of the poor person rises by LL’ (HH’ = LL’). Now, it will be seen in Fig. 5.2 that the loss of satisfaction or utility of the rich man as a result of decline in his income by HH’ is equal to the area HDCH’. Further, it will be seen that the gain in satisfaction or utility by the increase of an equivalent amount of income LL’ of the poor man, is equal to LABL’.

It is thus obvious from the figure that the gain in utility of the poor person is greater than the loss of utility of the rich man. Therefore, the total utility or satisfaction of the two persons taken together will increase through transfer of some income from the rich to the poor.

Thus, on the basis Money Income of the diminishing marginal utility of money many economists and political scientists have advocated that Government must redistribute in­come in order to raise the economic welfare of the society. However, it may be pointed out that some economists challenge the validity of such redistribution of income to promote the social welfare.

They point out that the above analysis of marginal utility is based upon interpersonal comparison of utility which is quite inadmissible and unscientific. They argue that people differ greatly in their preferences and capacity to enjoy goods and, therefore, it is difficult to know the exact shapes of the marginal utility curves of the different persons. Therefore they assert that the losses and gains of utility of the poor and the rich cannot be measured and compared.

2. Law of Equimarginal Utility:

Principle of Equimarginal Utility: Consumer’s Equilibrium:

Principle of equimarginal utility occupies an important place in marginal utility analy­sis. It is through this principle that consumer’s equilibrium is explained. It is also called law of substitution because in it for reaching equilibrium position consumer substitutes one good for another. A consumer has a given income which he has to spend on various goods he wants.

Now, the question is how he would allocate his money income among various goods that is to say, what would be his equilibrium position in respect of the purchases of the various goods. It may be mentioned here that consumer is assumed to be ‘rational’, that is, he coldly and carefully calculates and substitutes goods for one another so as to maximise his utility or satisfaction.

Suppose there are only two goods X and Y on which a consumer has to spend a given income. The consumer’s behaviour will be governed by two factors Firstly, the marginal utilities of the goods and secondly, the prices of two goods. Suppose the prices of the goods are given for the consumer.

The law of equimarginal utility states that the consumer will distribute his money income between the goods in such a way that the utility derived from the last rupee spent on each good is equal. In other words, consumer is in equilibrium position when marginal utility of money expenditure on each good is the same. Now, the marginal utility of money expenditure on a good is equal to the marginal utility of a good divided by the price of the good.

In symbols:

Let the prices of goods X and Y be Rs. 2 and Rs. 3 respectively and the consumer has Rs. 24 to spend on the two goods. It is worth noting that in order to maximise his utility the consumer will not equate marginal utility of the goods because prices of the two goods are different. He will equate the marginal utility of the last rupee (i.e., marginal utility of money expenditure) spent on these two goods.

In other words, he will equate Mu x /P x with MU y /P y while spending his given money income on the two goods. Therefore, reconstructing the above Table 5.2 by dividing marginal utilities of X(MU x ) by Rs. 2 and marginal utilities of Y(MU y ) by Rs. 3 we get Table 5.3 which show marginal utility of money expenditure.

By looking at the Table 5.2 it will become clear that MU x /P x is equal to 5 utils when the consumer purchases 6 units of good X and MU y /P y is equal to 5 utils when he buys 4 units of good Y. Therefore, the consumer will be in equilibrium when he is buying 6 units of good X and 4 units of good Y and will be spending (Rs. 2 × 6 + Rs. 3 × 4) = Rs. 24 on them.

Thus, in the equilibrium position where he maximizes his utility:

Thus when the consumer is making purchases by spending his given income in such a way that MU x /P x = MU y /P y , he will not like to make any further changes in the basket of goods and will therefore be in equilibrium situation by maximizing his utility.

The above equimarginal condition for the equilibrium of the consumer can be stated in the following ways:

(i) A consumer is in equilibrium when he equalises the ratios of marginal utilities of goods and their prices with each other. In other words, he is in equilibrium when

Now, if the price of the good falls to OP 2 , the equality between the marginal utility and the price will be disturbed. Marginal utility MU X from good X at the quantity Oq 1 will be greater than the new price OP 2 . In order to equate the marginal utility with the lower price OP 2 , the consumer must buy more of the good. It is evident from Fig. 5.4 that when the consumer increases the quantity purchased to Oq 2 the marginal utility of the good falls to MU 2 and becomes equal to the new price OP 2 .

Hence, at price OP 2 , consumer demands Oq 2 amount of the commodity. Further, if the price falls to OP 3 , this is equal to the marginal utility MU 3 of the good at the larger quantity Oq 3 . Thus at price OP 3 , the consumer will demand Oq 3 quantity of the good X. It is in this way the downward-sloping marginal utility curve is transformed into the downward-sloping demand curve when we measure marginal utility of a good in terms of money. It is worth noting that negative segment of the marginal utility curve MU X will not constitute a part of the demand curve. This is because no rational consumer will buy any further units of the commodity which reduces his total utility and make marginal utilities negative.

It is thus clear that when the price of the good falls, the consumer buys more of the good so as to equate its marginal utility to the lower price. It follows therefore that the quantity demanded of a good varies inversely with price; the quantity rises when the price falls and vice-versa, other things remaining the same.

This is the famous Marshallian Law of Demand. It is quite evident that the law of demand is directly derived from the law of diminishing marginal utility. The downward-sloping marginal utility curve is transformed into the downward-sloping demand curve. It follows therefore that the force working behind the law of demand or the demand curve is the force of diminishing marginal utility.

Derivation of Law of Demand: Multi-Commodity Model:

We now proceed to derive the law of demand and the nature of the demand curve from the principle of equimarginal utility in case when a consumer spends his money income on more than one commodity. Consider the case of a consumer who has a certain given income to spend on a number of goods. According to the law of equimarginal utility, the consumer is in equilib­rium in regard to his purchases of various goods when marginal utilities of the goods are proportional to their prices.

Thus, the consumer is in equilibrium when he is buying the quantities of the two goods in such a way that satisfies the following proportionality rule: