what does utility hypothesis mean

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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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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 objections 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 maximizing expected utility is impossible, 3.2 maximizing expected utility is irrational, 4.1 economics and public policy, 4.3 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 suppositional 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 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 (but still permits) acts to have probabilities, and it recommends two-boxing in the Newcomb problem.

Causal decision theory comes in many varieties, but I’ll consider a representative version proposed by Savage (1972), which 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 of outcomes, which maps \(o\) to 1 if \(o\) results from performing \(A\) in state s , maps \(o\) to 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).

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.

We can generalze Savage’s proposal by letting \(f_{A,s}\) be a probability function that maps outcomes to real numbers in the \([0, 1]\) interval. Lewis (1981), Skyrms (1980), and Sobel (1994) equate \(f_{A,s}\) with the objective chance that \(o\) would be the outcome if state \(s\) obtained and the agent chose action \(A\).

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 three 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, Ch. 6) 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 \(B\), or that 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.

Defenders of expected utility theory typically require that utility be measured by a linear scale , where 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 \(U\).

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 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.

The strong law of large numbers states that where each trial has an expected value of \(\mu\), with probability 1, for any arbitrarily small real number \(\epsilon \gt 0\),as the number of trials increases, the gambler’s average winnings per trial will fall within \(\epsilon\) of \(\mu\). 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.

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, winning $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. Are they repeatable? Must they be wholly within the agent’s control

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 Ramsey

The idea of a representation theorem for expected utility dates back to Ramsey (1926). (His sketch of a representation theorem is subsequently filled in by Bradley (2004) and Elliott (2017).) Ramsey assumes that preferences are defined over a domain of gambles, which yield one prize on the condition that a proposition \(P\) is true, and a different prize on the condition that \(P\) is false. (Examples of gambles: you receive a onesie if you’re having a baby and a bottle of scotch otherwise; you receive twenty dollars if Bojack wins the Kentucky Derby and lose a dollar otherwise.)

Ramsey calls a proposition ethically neutral when “two possible worlds differing only in regard to [its truth] are always of equal value”. For an ethically neutral proposition, probability 1/2 can be defined in terms of preference: such a proposition has probability 1/2 just in case you are indifferent as to which side of it you bet on. (So if Bojack wins the Kentucky Derby is an ethically neutral proposition, it has probability 1/2 just in case you are indifferent between winning twenty dollars if it’s true and losing a dollar otherwise, and winning twenty dollars if it’s false and losing a dollar otherwise.)

By positing an ethically neutral proposition with probability 1/2, together with a rich space of prizes, Ramsey defines numerical utilities for prizes. (The rough idea is that if you are indifferent between receiving a middling prize \(m\) for certain, and a gamble that yields a better prize \(b\) if the ethically neutral proposition is true and a worse prize \(w\) if it is falls, then the utility of \(m\) is halfway between the utilities of \(b\) and \(w\).) Using these numerical utilities, he then exploits the definition of expected utility to define probabilities for all other propositions.

The rough idea is to exploit the richness of the space of prizes, which ensures that for any gamble \(g\) that yields better prize \(b\) if \(E\) is true and worse prize \(w\) if \(E\) is false, the agent is indifferent between \(g\) and some middling prize \(m\). This means that \(EU(g) = EU(m)\). Using some algebra, plus the fact that \(EU(g) = P(E)U(b) + (1-P(E))U(w)\), Ramsey shows that

2.2.2 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.

2.2.3 Savage

Instead of taking probabilities for granted, as von Neumann and Morgenstern do, Savage (1972) defines them in terms of preferences over acts. Savage posits three separate domains. Probability attaches to events , which we can think of as disjunctions of states, while utility and intrinsic preference attach to outcomes . Expected utility and non-intrinsic preference attach to acts .

For Savage, acts, states, and outcomes must satisfy certain constraints. Acts must be wholly under the agent’s control (so publishing my paper in Mind is not an act, since it depends partly on the editor’s decision, which I do not control). Outcomes must have the same utility regardless of which state obtains (so "I win a fancy car" is not an outcome, since the utility of the fancy car will be greater in states where the person I most want to impress wishes I had a fancy car, and less in states where I lose my driver’s license). No state can rule out the performance of any act, and an act and a state together must determine an outcome with certainty. For each outcome \(o\), there is a constant act which yields \(o\) in every state. (Thus, if world peace is an outcome, there is an act that results in world peace, no matter what the state of the world.) Finally, he assumes for any two acts \(A\) and \(B\) and any 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 world peace and the end of the world are both outcomes, then there is a mixed act that results in world peace if a certain coin lands heads, and the end of the world 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; I paraphrase here.

Suppose \(A\) and \(B\) are constant acts such that \(A\) is preferred to \(B\). Then \(E\) is at least as likely as \(F\) 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), or else 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 \(E\) at least as likely as \(F\) just in case she would not rather bet on \(F\) than on \(E\)).

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.

2.2.4 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.

2.2.5 Summary

Together, these four representation theorems above can be summed up in the following table.

Notice that the order of construction differs between theorems: Ramsey constructs a representation of probability using utility, while von Neumann and Morgenstern begin with probabilities and construct a representation of utility. Thus, although the arrows represent a mathematical relationship of representation, they cannot represent a metaphysical relationship of grounding. The Reality Condition needs to be justified independently of any representation 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. Objections to Expected Utility Theory

Ought implies can, but is it humanly possible to maximize expected utility? March and Simon (1958) point out that in order to compute expected utilities, an agent needs a dauntingly complex understanding of the available acts, the possible outcomes, and the values of those outcomes, and that choosing the best act is much more demanding than choosing an act that is merely good enough. Similar points appear in Lindblom (1959), Feldman (2006), and Smith (2010).

McGee (1991) argues that maximizing expected utility is not mathematically possible even for an ideal computer with limitless memory. In order to maximize expected utility, we would have to accept any bet we were offered on the truths of arithmetic, and reject any bet we were offered on false sentences in the language of arithmetic. But arithmetic is undecidable, so no Turing machine can determine whether a given arithmetical sentence is true or false.

One response to these difficulties is the bounded rationality approach, which aims to replace expected utility theory with some more tractable rules. Another is to argue that the demands of expected utility theory are more tractable than they appear (Burch-Brown 2014; see also Greaves 2016), or that the relevant “ought implies can” principle is false (Srinivasan 2015).

A variety of authors have given examples in which expected utility theory seems to give the wrong prescriptions. Sections 3.2.1 and 3.2.2 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 3.2.3 discusses examples where expected utility theory permits preferences that seem irrational. These examples suggest that maximizing expected utility is not sufficient for rationality. Section 3.2.4 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.

3.2.1 Counterexamples Involving Transitivity and Completeness

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 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.

3.2.2 Counterexamples Involving Independence

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 prize for tickets 12–100. They can be converted into lotteries \(B\) and \(D\) by replacing this $100 million prize 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 need 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, Ch. 5) 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.

3.2.3 Counterexamples Involving Probability 0 Events

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.

3.2.4 Counterexamples Involving Unbounded Utility

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 $\(2^n\). 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.

Nover and Hájek (2004) argue that in addition to the St. Petersburg game, which has infinite expected utility, there are other infinitary games whose expected utilities are undefined, even though rationality mandates certain preferences among them.

One response to these problematic infinitary games is to argue that the decision problems themselves are ill-posed (Jeffrey (1983, 154); another is to adopt a modified version of expected utility theory that agrees with its verdicts in the ordinary case, but yields intuitively reasonable verdicts about the infinitary games (Thalos and Richardson 2013) (Fine 2008) (Colyvan 2006, 2008) (Easwaran 2008).

4. Applications

In the 1940s and 50s, expected utility theory gained currency in the US for its potential to provide a mechanism that would explain the behavior of macro-economic variables. As it became apparent that expected utility theory did not accurately predict the behaviors of real people, its proponents instead advanced the view that it might serve instead as a theory of how rational people should respond to uncertainty (see Herfeld 2017).

Expected utility theory has a variety of applications in public policy. In welfare economics, Harsanyi (1953) reasons from expected utility theory to the claim that the most socially just arrangement is the one that maximizes total welfare distributed across a society society. The theory of expected utility also has more direct applications. Howard (1980) introduces the concept of a micromort , or a one-in-a-million chance of death, and uses expected utility calculations to gauge which mortality risks are acceptable. In health policy, quality-adjusted life years, or QALYs, are measures of the expected utilities of different health interventions used to guide health policy (see Weinstein et al 2009). McAskill (2015) uses expected utility theory to address the central question of effective altruism : “How can I do the most good?” (Utilties in these applications are most naturally interpreted as measuring something like happiness or wellbeing, rather than subjective preference satisfaction for an individual agent.)

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.

Utilitarians, along with their descendants contemporary consequentialists, hold that the rightness or wrongness of an act is determined by the moral goodness or badness of its consequences. Some consequentialists, such as (Railton 1984), interpret this to mean that we ought to do whatever will in fact have the best consequences. But it is difficult—perhaps impossible—to know the long-term consequences of our acts (Lenman 2000, Howard-Snyder 2007). In light of this observation, 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.

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.

The expected-utility-maximizing version of consequentialism is not strictly speaking a theory of rational choice. It is a theory of moral choice, but whether rationality requires us to do what is morally best is up for debate.

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.

In epistemic decision theory , expected utilities are used to assess belief states as rational or irrational. 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. The entry on epistemic utility arguments for probabilism includes an overview of expected utility arguments for a variety of epistemic norms, including conditionalization and the Principal Principle.

Kaplan (1968), argues that expected utility considerations can be used to fix a standard of proof in legal trials. A jury deciding whether to acquit or convict faces the following decision problem:

Kaplan shows that \(EU(convict) > EU(acquit)\) whenever

Qualitatively, this means that the standard of proof increases as the disutility of convicting an innocent person \((U(\mathrm{true~conviction})-U(\mathrm{false~acquittal}))\) increases, or as the disutility of acquitting a guilty person \((U(\mathrm{true~acquittal})-U(\mathrm{false~conviction}))\) decreases.

Critics of this decision-theoretic approach, such as Laudan (2006), argue that it’s difficult or impossible to bridge the gap between the evidence admissible in court, and the real probability of the defendant’s guilt. The probability guilt depends on three factors: the distribution of apparent guilt among the genuinely guilty, the distribution of apparent guilt among the genuinely innocent, and the ratio of genuinely guilty to genuinely innocent defendants who go to trial (see Bell 1987). Obstacles to calculating any of these factors will block the inference from a judge or jury’s perception of apparent guilt to a true probability of guilt.

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How to cite this entry . Preview the PDF version of this entry at the Friends of the SEP Society . Look up topics and thinkers related to this entry at the Internet 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 | Ramsey, Frank: and intergenerational welfare economics | 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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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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7.1 The Concept of Utility

Learning objectives.

• Define what economists mean by utility.
• Distinguish between the concepts of total utility and marginal utility.
• State the law of diminishing marginal utility and illustrate it graphically.
• State, explain, and illustrate algebraically the utility-maximizing condition.

Why do you buy the goods and services you do? It must be because they provide you with satisfaction—you feel better off because you have purchased them. Economists call this satisfaction utility .

The concept of utility is an elusive one. A person who consumes a good such as peaches gains utility from eating the peaches. But we cannot measure this utility the same way we can measure a peach’s weight or calorie content. There is no scale we can use to determine the quantity of utility a peach generates.

Francis Edgeworth, one of the most important contributors to the theory of consumer behavior, imagined a device he called a hedonimeter (after hedonism, the pursuit of pleasure):

“[ L ]et there be granted to the science of pleasure what is granted to the science of energy; to imagine an ideally perfect instrument, a psychophysical machine, continually registering the height of pleasure experienced by an individual…. From moment to moment the hedonimeter varies; the delicate index now flickering with the flutter of passions, now steadied by intellectual activity, now sunk whole hours in the neighborhood of zero, or momentarily springing up towards infinity” (Edgeworth, F. Y., 1967).

Perhaps some day a hedonimeter will be invented. The utility it measures will not be a characteristic of particular goods, but rather of each consumer’s reactions to those goods. The utility of a peach exists not in the peach itself, but in the preferences of the individual consuming the peach. One consumer may wax ecstatic about a peach; another may say it tastes OK.

When we speak of maximizing utility, then, we are speaking of the maximization of something we cannot measure. We assume, however, that each consumer acts as if he or she can measure utility and arranges consumption so that the utility gained is as high as possible.

Total Utility

If we could measure utility, total utility would be the number of units of utility that a consumer gains from consuming a given quantity of a good, service, or activity during a particular time period. The higher a consumer’s total utility, the greater that consumer’s level of satisfaction.

Panel (a) of Figure 7.1 “Total Utility and Marginal Utility Curves” shows the total utility Henry Higgins obtains from attending movies. In drawing his total utility curve, we are imagining that he can measure his total utility. The total utility curve shows that when Mr. Higgins attends no movies during a month, his total utility from attending movies is zero. As he increases the number of movies he sees, his total utility rises. When he consumes 1 movie, he obtains 36 units of utility. When he consumes 4 movies, his total utility is 101. He achieves the maximum level of utility possible, 115, by seeing 6 movies per month. Seeing a seventh movie adds nothing to his total utility.

Figure 7.1 Total Utility and Marginal Utility Curves

Panel (a) shows Henry Higgins’s total utility curve for attending movies. It rises as the number of movies increases, reaching a maximum of 115 units of utility at 6 movies per month. Marginal utility is shown in Panel (b); it is the slope of the total utility curve. Because the slope of the total utility curve declines as the number of movies increases, the marginal utility curve is downward sloping.

Mr. Higgins’s total utility rises at a decreasing rate. The rate of increase is given by the slope of the total utility curve, which is reported in Panel (a) of Figure 7.1 “Total Utility and Marginal Utility Curves” as well. The slope of the curve between 0 movies and 1 movie is 36 because utility rises by this amount when Mr. Higgins sees his first movie in the month. It is 28 between 1 and 2 movies, 22 between 2 and 3, and so on. The slope between 6 and 7 movies is zero; the total utility curve between these two quantities is horizontal.

Marginal Utility

The amount by which total utility rises with consumption of an additional unit of a good, service, or activity, all other things unchanged, is marginal utility . The first movie Mr. Higgins sees increases his total utility by 36 units. Hence, the marginal utility of the first movie is 36. The second increases his total utility by 28 units; its marginal utility is 28. The seventh movie does not increase his total utility; its marginal utility is zero. Notice that in the table marginal utility is listed between the columns for total utility because, similar to other marginal concepts, marginal utility is the change in utility as we go from one quantity to the next. Mr. Higgins’s marginal utility curve is plotted in Panel (b) of Figure 7.1 “Total Utility and Marginal Utility Curves” The values for marginal utility are plotted midway between the numbers of movies attended. The marginal utility curve is downward sloping; it shows that Mr. Higgins’s marginal utility for movies declines as he consumes more of them.

Mr. Higgins’s marginal utility from movies is typical of all goods and services. Suppose that you are really thirsty and you decide to consume a soft drink. Consuming the drink increases your utility, probably by a lot. Suppose now you have another. That second drink probably increases your utility by less than the first. A third would increase your utility by still less. This tendency of marginal utility to decline beyond some level of consumption during a period is called the law of diminishing marginal utility . This law implies that all goods and services eventually will have downward-sloping marginal utility curves. It is the law that lies behind the negatively sloped marginal benefit curve for consumer choices that we examined in the chapter on markets, maximizers, and efficiency.

One way to think about this effect is to remember the last time you ate at an “all you can eat” cafeteria-style restaurant. Did you eat only one type of food? Did you consume food without limit? No, because of the law of diminishing marginal utility. As you consumed more of one kind of food, its marginal utility fell. You reached a point at which the marginal utility of another dish was greater, and you switched to that. Eventually, there was no food whose marginal utility was great enough to make it worth eating, and you stopped.

What if the law of diminishing marginal utility did not hold? That is, what would life be like in a world of constant or increasing marginal utility? In your mind go back to the cafeteria and imagine that you have rather unusual preferences: Your favorite food is creamed spinach. You start with that because its marginal utility is highest of all the choices before you in the cafeteria. As you eat more, however, its marginal utility does not fall; it remains higher than the marginal utility of any other option. Unless eating more creamed spinach somehow increases your marginal utility for some other food, you will eat only creamed spinach. And until you have reached the limit of your body’s capacity (or the restaurant manager’s patience), you will not stop. Failure of marginal utility to diminish would thus lead to extraordinary levels of consumption of a single good to the exclusion of all others. Since we do not observe that happening, it seems reasonable to assume that marginal utility falls beyond some level of consumption.

Maximizing Utility

Economists assume that consumers behave in a manner consistent with the maximization of utility. To see how consumers do that, we will put the marginal decision rule to work. First, however, we must reckon with the fact that the ability of consumers to purchase goods and services is limited by their budgets.

The Budget Constraint

The total utility curve in Figure 7.1 “Total Utility and Marginal Utility Curves” shows that Mr. Higgins achieves the maximum total utility possible from movies when he sees six of them each month. It is likely that his total utility curves for other goods and services will have much the same shape, reaching a maximum at some level of consumption. We assume that the goal of each consumer is to maximize total utility. Does that mean a person will consume each good at a level that yields the maximum utility possible?

The answer, in general, is no. Our consumption choices are constrained by the income available to us and by the prices we must pay. Suppose, for example, that Mr. Higgins can spend just $25 per month for entertainment and that the price of going to see a movie is $5. To achieve the maximum total utility from movies, Mr. Higgins would have to exceed his entertainment budget. Since we assume that he cannot do that, Mr. Higgins must arrange his consumption so that his total expenditures do not exceed his budget constraint : a restriction that total spending cannot exceed the budget available.

Suppose that in addition to movies, Mr. Higgins enjoys concerts, and the average price of a concert ticket is $10. He must select the number of movies he sees and concerts he attends so that his monthly spending on the two goods does not exceed his budget.

Individuals may, of course, choose to save or to borrow. When we allow this possibility, we consider the budget constraint not just for a single period of time but for several periods. For example, economists often examine budget constraints over a consumer’s lifetime. A consumer may in some years save for future consumption and in other years borrow on future income for present consumption. Whatever the time period, a consumer’s spending will be constrained by his or her budget.

To simplify our analysis, we shall assume that a consumer’s spending in any one period is based on the budget available in that period. In this analysis consumers neither save nor borrow. We could extend the analysis to cover several periods and generate the same basic results that we shall establish using a single period. We will also carry out our analysis by looking at the consumer’s choices about buying only two goods. Again, the analysis could be extended to cover more goods and the basic results would still hold.

Applying the Marginal Decision Rule

Because consumers can be expected to spend the budget they have, utility maximization is a matter of arranging that spending to achieve the highest total utility possible. If a consumer decides to spend more on one good, he or she must spend less on another in order to satisfy the budget constraint.

The marginal decision rule states that an activity should be expanded if its marginal benefit exceeds its marginal cost. The marginal benefit of this activity is the utility gained by spending an additional $1 on the good. The marginal cost is the utility lost by spending $1 less on another good.

How much utility is gained by spending another $1 on a good? It is the marginal utility of the good divided by its price. The utility gained by spending an additional dollar on good X, for example, is

[latex]\frac{MU_x}{P_x}[/latex]

This additional utility is the marginal benefit of spending another $1 on the good.

Suppose that the marginal utility of good X is 4 and that its price is $2. Then an extra $1 spent on X buys 2 additional units of utility ( MUX/PX=4/2=2 ). If the marginal utility of good X is 1 and its price is $2, then an extra $1 spent on X buys 0.5 additional units of utility ( MUX/PX=1/2=0.5 ).

The loss in utility from spending $1 less on another good or service is calculated the same way: as the marginal utility divided by the price. The marginal cost to the consumer of spending $1 less on a good is the loss of the additional utility that could have been gained from spending that $1 on the good.

Suppose a consumer derives more utility by spending an additional $1 on good X rather than on good Y:

Equation 7.1

[latex]\frac{MU_X}{P_X} > \frac{MU_Y}{P_Y}[/latex]

The marginal benefit of shifting $1 from good Y to the consumption of good X exceeds the marginal cost. In terms of utility, the gain from spending an additional $1 on good X exceeds the loss in utility from spending $1 less on good Y. The consumer can increase utility by shifting spending from Y to X.

As the consumer buys more of good X and less of good Y, however, the marginal utilities of the two goods will change. The law of diminishing marginal utility tells us that the marginal utility of good X will fall as the consumer consumes more of it; the marginal utility of good Y will rise as the consumer consumes less of it. The result is that the value of the left-hand side of Equation 7.1 will fall and the value of the right-hand side will rise as the consumer shifts spending from Y to X. When the two sides are equal, total utility will be maximized. In terms of the marginal decision rule, the consumer will have achieved a solution at which the marginal benefit of the activity (spending more on good X) is equal to the marginal cost:

Equation 7.2

[latex]\frac{MU_X}{P_X} = \frac{MU_Y}{P_Y}[/latex]

We can extend this result to all goods and services a consumer uses. Utility maximization requires that the ratio of marginal utility to price be equal for all of them, as suggested in Equation 7.3 :

Equation 7.3

[latex]\frac{MU_A}{P_A} = \frac{MU_B}{P_B} = \frac{MU_C}{P_C} = \ .\ .\ . = \frac{MU_n}{P_n}[/latex]

Equation 7.3 states the utility-maximizing condition : Utility is maximized when total outlays equal the budget available and when the ratios of marginal utilities to prices are equal for all goods and services.

Consider, for example, the shopper introduced in the opening of this chapter. In shifting from cookies to ice cream, the shopper must have felt that the marginal utility of spending an additional dollar on ice cream exceeded the marginal utility of spending an additional dollar on cookies. In terms of Equation 7.1 , if good X is ice cream and good Y is cookies, the shopper will have lowered the value of the left-hand side of the equation and moved toward the utility-maximizing condition, as expressed by Equation 7.1 .

The Problem of Divisibility

If we are to apply the marginal decision rule to utility maximization, goods must be divisible; that is, it must be possible to buy them in any amount. Otherwise we cannot meaningfully speak of spending $1 more or $1 less on them. Strictly speaking, however, few goods are completely divisible.

Even a small purchase, such as an ice cream bar, fails the strict test of being divisible; grocers generally frown on requests to purchase one-half of a $2 ice cream bar if the consumer wants to spend an additional dollar on ice cream. Can a consumer buy a little more movie admission, to say nothing of a little more car?

In the case of a car, we can think of the quantity as depending on characteristics of the car itself. A car with a compact disc player could be regarded as containing “more car” than one that has only a cassette player. Stretching the concept of quantity in this manner does not entirely solve the problem. It is still difficult to imagine that one could purchase “more car” by spending $1 more.

Remember, though, that we are dealing with a model. In the real world, consumers may not be able to satisfy Equation 7.3 precisely. The model predicts, however, that they will come as close to doing so as possible.

Key Takeaways

• The utility of a good or service is determined by how much satisfaction a particular consumer obtains from it. Utility is not a quality inherent in the good or service itself.
• Total utility is a conceptual measure of the number of units of utility a consumer gains from consuming a good, service, or activity. Marginal utility is the increase in total utility obtained by consuming one more unit of a good, service, or activity.
• As a consumer consumes more and more of a good or service, its marginal utility falls.
• Utility maximization requires seeking the greatest total utility from a given budget.
• Utility is maximized when total outlays equal the budget available and when the ratios of marginal utility to price are equal for all goods and services a consumer consumes; this is the utility-maximizing condition.

A college student, Ramón Juárez, often purchases candy bars or bags of potato chips between classes; he tries to limit his spending on these snacks to $8 per week. A bag of chips costs $0.75 and a candy bar costs $0.50 from the vending machines on campus. He has been purchasing an average of 6 bags of chips and 7 candy bars each week. Mr. Juárez is a careful maximizer of utility, and he estimates that the marginal utility of an additional bag of chips during a week is 6. In your answers use B to denote candy bars and C to denote potato chips.

• How much is he spending on snacks? How does this amount compare to his budget constraint?
• What is the marginal utility of an additional candy bar during the week?

Case in Point: Changing Lanes and Raising Utility

Chris Brown – Traffic – CC BY 2.0.

In preparation for sitting in the slow, crowded lanes for single-occupancy-vehicles, T. J. Zane used to stop at his favorite coffee kiosk to buy a $2 cup of coffee as he headed off to work on Interstate 15 in the San Diego area. Since 1996, an experiment in road pricing has caused him and others to change their ways—and to raise their total utility.

Before 1996, only car-poolers could use the specially marked high-occupancy-vehicles lanes. With those lanes nearly empty, traffic authorities decided to allow drivers of single-occupancy-vehicles to use those lanes, so long as they paid a price. Now, electronic signs tell drivers how much it will cost them to drive on the special lanes. The price is recalculated every 6 minutes depending on the traffic. On one morning during rush hour, it varied from $1.25 at 7:10 a.m., to $1.50 at 7:16 a.m., to $2.25 at 7:22 a.m., and to $2.50 at 7:28 a.m. The increasing tolls over those few minutes caused some drivers to opt out and the toll fell back to $1.75 and then increased to $2 a few minutes later. Drivers do not have to stop to pay the toll since radio transmitters read their FasTrak transponders and charge them accordingly.

When first instituted, these lanes were nicknamed the “Lexus lanes,” on the assumption that only wealthy drivers would use them. Indeed, while the more affluent do tend to use them heavily, surveys have discovered that they are actually used by drivers of all income levels.

Mr. Zane, a driver of a 1997 Volkswagen Jetta, is one commuter who chooses to use the new option. He explains his decision by asking, “Isn’t it worth a couple of dollars to spend an extra half-hour with your family?” He continues, “That’s what I used to spend on a cup of coffee at Starbucks. Now I’ve started bringing my own coffee and using the money for the toll.”

We can explain his decision using the model of utility-maximizing behavior; Mr. Zane’s out-of-pocket commuting budget constraint is about $2. His comment tells us that he realized that the marginal utility of spending an additional 30 minutes with his family divided by the $2 toll was higher than the marginal utility of the store-bought coffee divided by its $2 price. By reallocating his $2 commuting budget, the gain in utility of having more time at home exceeds the loss in utility from not sipping premium coffee on the way to work.

From this one change in behavior, we do not know whether or not he is actually maximizing his utility, but his decision and explanation are certainly consistent with that goal.

Source: John Tierney, “The Autonomist Manifesto (Or, How I learned to Stop Worrying and Love the Road),” New York Times Magazine , September 26, 2004, 57–65.

Answers to Try It! Problems

• He is spending $4.50 (= $0.75 × 6) on potato chips and $3.50 (= $0.50 × 7) on candy bars, for a total of $8. His budget constraint is $8.

In order for the ratios of marginal utility to price to be equal, the marginal utility of a candy bar must be 4. Let the marginal utility and price of candy bars be MU B and P B , respectively, and the marginal utility and price of a bag of potato chips be MU C and P C , respectively. Then we want

[latex]\frac{MU_C}{P_C} = \frac{MU_B}{P_B}[/latex]

We know that P C is $0.75 and P B equals $0.50. We are told that MU C is 6. Thus

[latex]\frac{6}{0.75} = \frac{MU_B}{0.50}[/latex]

Solving the equation for MU B , we find that it must equal 4.

Edgeworth, F. Y., Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences (New York: Augustus M. Kelley, 1967), p. 101. First Published 1881.

Principles of Economics Copyright © 2016 by University of Minnesota is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.

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Study Notes

4.1.2.1 Consumer Behaviour - Utility Theory (AQA)

Last updated 10 Sept 2023

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This brief study note covers utility theory for AQA Economics.

Study Notes on Utility Theory and Maximization

Utility Theory: Total and Marginal Utility

• Utility refers to the satisfaction or pleasure that individuals derive from consuming goods and services.
• Total Utility (TU) is the overall satisfaction obtained from consuming a certain quantity of a good or service.
• Marginal Utility (MU) is the additional satisfaction gained from consuming one more unit of a good or service.

Hypothesis of Diminishing Marginal Utility

• The Hypothesis of Diminishing Marginal Utility states that as a person consumes more units of a good or service, the additional satisfaction (marginal utility) from each additional unit decreases.
• This principle explains why individuals usually do not consume infinite quantities of a good, as the marginal utility diminishes.

Utility Maximization

• Utility maximization is the economic concept that rational individuals seek to allocate their resources (money or time) in a way that maximizes their total utility.
• To achieve utility maximization, consumers compare the marginal utility of the last unit consumed to its price (or opportunity cost).

Key Takeaways:

• Total utility increases as consumption of a good or service rises, but at a diminishing rate.
• Marginal utility decreases as more units are consumed due to the Hypothesis of Diminishing Marginal Utility.
• Utility maximization occurs when individuals allocate resources to goods or services in a way that maximizes their total utility, given budget constraints.

Multiple Choice Questions:

Question 1: What does Total Utility (TU) represent in utility theory?

• A) The additional satisfaction from consuming one more unit.
• B) The overall satisfaction from consuming a certain quantity of a good.
• C) The price of a good.
• D) The satisfaction from the first unit of consumption.

Question 2: According to the Hypothesis of Diminishing Marginal Utility, what happens to marginal utility as consumption of a good increases?

• A) Marginal utility increases.
• B) Marginal utility remains constant.
• C) Marginal utility decreases.
• D) Marginal utility becomes negative.

Question 3: Utility maximization occurs when:

• A) Consumers aim to minimize their total utility.
• B) Consumers allocate resources to goods to maximize their total utility.
• C) Consumers ignore their budget constraints.
• D) Consumers aim to maximize marginal utility.

Question 4: What concept explains why individuals usually do not consume infinite quantities of a good?

• A) The Law of Demand
• B) The Law of Supply
• C) The Hypothesis of Diminishing Marginal Utility
• D) The Law of Diminishing Returns

Question 5: If a consumer decides to buy more of a good until the marginal utility equals the price of the good, what economic principle are they following?

• C) Utility maximization

Question 6: Which of the following best defines Marginal Utility (MU)?

• A) The overall satisfaction from consuming a certain quantity of a good.
• B) The additional satisfaction from consuming one more unit of a good.
• Utility Maximisation
• Marginal utility
• Expected Utility Theory
• Diminishing Marginal Utility
• Homo Economicus

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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
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• 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
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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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What Is Marginal Utility?

Understanding marginal utility, marginal utility vs. total utility.

• Calculation
• Applications
• Marginal Utility FAQs

The Bottom Line

• Guide to Microeconomics

Marginal Utilities: Definition, Types, Examples, and History

Erika Rasure is globally-recognized as a leading consumer economics subject matter expert, researcher, and educator. She is a financial therapist and transformational coach, with a special interest in helping women learn how to invest.

• A Practical Guide to Microeconomics
• Economists' Assumptions in their Economic Models
• 5 Nobel Prize-Winning Economic Theories
• Understanding Positive vs. Normative Economics
• What Factors Influence Competition in Microeconomics?
• How Does Government Policy Impact Microeconomics?
• Understanding Microeconomics vs. Macroeconomics
• Differentiate Between Micro and Macro Economics
• Microeconomics vs. Macroeconomics Investments
• Introduction to Supply and Demand
• Is Demand or Supply More Important to the Economy?
• Law of Demand
• Demand Curve
• Law Of Supply
• Supply Curve
• Price Elasticity of Demand
• Understanding Elasticity vs. Inelasticity of Demand
• Factors Determining the Demand Elasticity of a Good
• What Factors Influence a Change in Demand Elasticity?
• What Is the Concept of Utility in Microeconomics?
• What Is the Utility Function and How Is it Calculated?
• Total Utility
• Marginal Utility CURRENT ARTICLE
• Law Of Diminishing Marginal
• What Does the Law of Diminishing Marginal Utility Explain?
• Economic Equilibrium
• Income Effect
• Indifference Curve
• Consumer Surplus
• Comparative Advantage
• Economies of Scale: What Are They and How Are They Used?
• Perfect Competition
• Invisible Hand
• Market Failure

Marginal utility is the added satisfaction that a consumer gets from having one more unit of a good or service. The concept of marginal utility is used by economists to determine how much of an item consumers are willing to purchase.

Positive marginal utility occurs when the consumption of an additional item increases the total utility. On the other hand, negative marginal utility occurs when the consumption of one more unit decreases the overall utility.

Key Takeaways

• Marginal utility is the added satisfaction a consumer gets from having one more unit of a good or service.
• The concept of marginal utility is used by economists to determine how much of an item consumers are willing to purchase.
• The law of diminishing marginal utility is often used to justify progressive taxes.
• Marginal utility can be positive, zero, or negative.

Investopedia / Dennis Madamba

Economists use the idea of marginal utility to gauge how satisfaction levels affect consumer decisions. Economists have also identified a concept known as the law of diminishing marginal utility . It describes how the first unit of consumption of a good or service carries more utility than later units.

Although marginal utility tends to decrease with consumption, it may or may not ever reach zero depending on the good consumed.

Marginal utility is useful in explaining how consumers make choices to get the most benefit from their limited budgets. In general, people will continue consuming more of a good as long as the marginal utility is greater than the marginal cost . In an efficient market, the price equals the marginal cost. That is why people keep buying more until the marginal utility of consumption falls to the price of the good.

Types of Marginal Utility

There are multiple kinds of marginal utility. Three of the most common ones are as follows:

Positive Marginal Utility

Positive marginal utility occurs when having more of an item brings additional happiness. Suppose you like eating a slice of cake, but a second slice would bring you some extra joy. Then, your marginal utility from consuming cake is positive.

Zero Marginal Utility

Zero marginal utility is what happens when consuming more of an item brings no extra measure of satisfaction. For example, you might feel fairly full after two slices of cake and wouldn't really feel any better after having a third slice. In this case, your marginal utility from eating cake is zero.

Negative Marginal Utility

Negative marginal utility is where you have too much of an item, so consuming more is actually harmful. For instance, the fourth slice of cake might even make you sick after eating three pieces of cake.

History of Marginal Utility

The concept of marginal utility was developed by economists who were attempting to explain the economic reality of price, which they believed was driven by a product's utility. In the 18th century, economist Adam Smith discussed what is known as " the paradox of water and diamonds ." This paradox states that water has far less value than diamonds, even though water is vital to human life.

This disparity intrigued economists and philosophers around the world. In the 1870s, three economists—William Stanley Jevons, Carl Menger, and Leon Walras —each independently came to the conclusion that marginal utility was the answer to the water and diamonds paradox. In his book, The Theory of Political Economy , Jevons explained that economic decisions are made based on "final" (marginal) utility rather than total utility .

Example of Marginal Utility

David has four gallons of milk, then decides to purchase a fifth gallon. Meanwhile, Kevin has six gallons of milk and likewise chooses to buy an additional gallon. David benefits from not having to go to the store again for a few days, so his marginal utility is still positive. On the other hand, Kevin may have purchased more milk than he can reasonably consume, meaning his marginal utility might be zero.

The chief takeaway from this scenario is that the marginal utility of a buyer who acquires more and more of a product steadily declines. Eventually, there is no additional consumer need for the product in many cases. At that point, the marginal utility of the next unit equals zero and consumption ends.

Marginal utility measures the change in satisfaction from consuming one additional unit. Total utility, instead, measures the total amount of satisfaction of you get from all the units you consume of a good or service. Marginal utility affects total utility. Positive marginal utility causes total utility to increase, while negative marginal utility decreases total utility.

For example, if you go to five sessions with a personal trainer, you might get the highest level of satisfaction from the novelty and excitement of the first session. With each additional session, the marginal utility decreases because you are less excited and doing more strenuous work. But the marginal utility of each is positive, so your total utility is still increasing.

How to Calculate Marginal Utility

You can calculate marginal utility by dividing the change in total utility (TU) by the change in number of units (Q):

Change in total utility is found by subtracting the previous total utility from the current total utility (TU2-TU1)). Change in number of units is found by subtracting the previous number of units from the current number of units (Q2-Q1).

Applications of Marginal Utility

Marginal utility is used to make a variety of economic decisions by governments, businesses, and consumers.

Consumers seek out products with higher marginal utility. Because their satisfaction stays high with each additional unit purchased, they are more likely to purchase more. They are also more likely to buy similar products from the same company, expecting them to have a similarly high level of marginal utility.

Higher marginal utility often leads to greater customer satisfaction because consumers feel they are getting their money's worth. This can lead to brand loyalty over time, as well as word-of-mouth recommendations.

Products that offer a higher level of satisfaction over time, and after the first time they are used, offer a higher level of marginal utility. This makes them more valuable to customers, so they can be priced higher for greater profits. This can also serve as a guide for businesses to create better products and increase customer satisfaction by focusing on products that offer higher marginal utility.

Marginal utility can also guide businesses when deciding which products to innovate or upgrade. A product or service that already has a high level of marginal utility becomes even more valuable when it is improved, allowing businesses to continue increasing the price over time or for newer models. For example, if a car manufacturer has an SUV that is already a top seller, they can create trim levels with additional features or upgrades. Because the original version is already popular, with a high marginal utility, customers are more likely to pay the increased price for an even more premium version.

Governments

The law of diminishing marginal utility is often used to justify progressive taxes . The idea is that higher taxes cause less loss of utility for someone with a higher income. In this case, everyone gets diminishing marginal utility from money . Suppose that the government must raise $10,000 from each person to pay for its expenses. If the average income is $60,000 before taxes, then the average person would make $50,000 after taxes and have a reasonable standard of living.

However, asking people making only $10,000 to give it all up to the government would be unfair and demand a far greater sacrifice. That is why poll taxes, which require everyone to pay an equal amount, tend to be unpopular.

Also, a flat tax without individual exemptions that required everyone to pay the same percentage would impact those with less income more because of marginal utility. Someone making $15,000 per year would be taxed into poverty by a 33% tax, while someone making $60,000 would still have about $40,000.

What Is the Formula for Marginal Utility?

The formula for marginal utility is change in total utility (ΔTU) divided by change in number of units (ΔQ): MU = ΔTU/ΔQ.

What Is the Law of Diminishing Marginal Utility?

The law of diminishing marginal utility is a law of economics that states that as your consumption increases, the satisfaction you derive from each individual unit decreases. This is why consumers are willing to pay the most for the first unit of something they buy, but after a point, they often will not buy additional units without a decrease in price.

What Is Marginal Cost?

Marginal cost is the change in production cost from producing or making one additional unit. You can find it by dividing the change in production costs by the change in quantity produced. If the price per unit is higher than the marginal cost, a business can make a profit. Tracking marginal costs allows businesses to achieve economies of scale.

Marginal utility is the amount of additional satisfaction that a consumer gets from having one more unit of a good or service. This amount can be positive, negative, or zero. When marginal utility equals zero or becomes negative, the consumer will stop buying because the value of what they are buying has stopped increasing.

As an economic concept, marginal utility can be used by businesses to understand customer behavior, set prices for goods and services, and decide which products to innovate or upgrade.

Marginal utility is also used in economics to justify progressive taxes. According to marginal utility, each additional dollar is more valuable to those with lower incomes because they have fewer dollars in total. For those with higher incomes, the marginal utility of each additional dollar of income is lower. This is an application known as the law of diminishing marginal utility.

Internet Archive. " Adam Smith Wealth of Nations ," Page 30.

William Stanley Jevons. "The Theory of Political Economy." Sentry Press, Reprinted 1965.

Corporate Finance Institute. " What Is the Law of Diminishing Marginal Utility? "

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Utility: Theories and Models

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Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 306))

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The aim of this study is to look at utility theory from a broad perspective. The main hypothesis in the theory of decision is that the person who is in the position of deciding is entitled to the “economic man.” Also, the individual acts rationally. Thus, utility is the ability to satisfy (eliminate) human needs of goods and services. Utility is basically a psychological concept and also is the basis of economics and finance. Three types of utility take place in the economics and finance literature: marginal utility, total utility, and average utility. In addition, two main approaches fall within utility comparison: cardinal utility theory and ordinal utility theory. Furthermore, expected utility theory forms the basis of traditional finance. Expected benefit theory assumes that people choose risky or uncertain opportunities by comparing the expected benefits from them. Allais and Ellsberg paradoxes criticize expected utility theory. Tversky and Kahneman (Econometrica, 47: 263–291, 1979) present that the expected utility axioms are violated for more reasonable lottery alternatives than in the Allais paradox and put a link between finance and psychology. The prospect theory of Tversky and Kahneman forms the basis of behavioral finance.

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Barberis, N., & Xiong, W. (2012). Realization utility. Journal of Financial Economics, 104 (2), 251–271.

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Akkaya, M. (2021). Utility: Theories and Models. In: Mercangöz, B.A. (eds) Applying Particle Swarm Optimization. International Series in Operations Research & Management Science, vol 306. Springer, Cham. https://doi.org/10.1007/978-3-030-70281-6_1

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Falsifiability

Karl popper's basic scientific principle, karl popper's basic scientific principle.

Falsifiability, according to the philosopher Karl Popper, defines the inherent testability of any scientific hypothesis.

This article is a part of the guide:

• Inductive Reasoning
• Deductive Reasoning
• Hypothetico-Deductive Method
• Scientific Reasoning
• Testability

Browse Full Outline

• 1 Scientific Reasoning
• 2.1 Falsifiability
• 2.2 Verification Error
• 2.3 Testability
• 2.4 Post Hoc Reasoning
• 3 Deductive Reasoning
• 4.1 Raven Paradox
• 5 Causal Reasoning
• 6 Abductive Reasoning
• 7 Defeasible Reasoning

Science and philosophy have always worked together to try to uncover truths about the universe we live in. Indeed, ancient philosophy can be understood as the originator of many of the separate fields of study we have today, including psychology, medicine, law, astronomy, art and even theology.

Scientists design experiments and try to obtain results verifying or disproving a hypothesis, but philosophers are interested in understanding what factors determine the validity of scientific endeavors in the first place.

Whilst most scientists work within established paradigms, philosophers question the paradigms themselves and try to explore our underlying assumptions and definitions behind the logic of how we seek knowledge. Thus there is a feedback relationship between science and philosophy - and sometimes plenty of tension!

One of the tenets behind the scientific method is that any scientific hypothesis and resultant experimental design must be inherently falsifiable. Although falsifiability is not universally accepted, it is still the foundation of the majority of scientific experiments. Most scientists accept and work with this tenet, but it has its roots in philosophy and the deeper questions of truth and our access to it.

What is Falsifiability?

Falsifiability is the assertion that for any hypothesis to have credence, it must be inherently disprovable before it can become accepted as a scientific hypothesis or theory.

For example, someone might claim "the earth is younger than many scientists state, and in fact was created to appear as though it was older through deceptive fossils etc.” This is a claim that is unfalsifiable because it is a theory that can never be shown to be false. If you were to present such a person with fossils, geological data or arguments about the nature of compounds in the ozone, they could refute the argument by saying that your evidence was fabricated to appeared that way, and isn’t valid.

Importantly, falsifiability doesn’t mean that there are currently arguments against a theory, only that it is possible to imagine some kind of argument which would invalidate it. Falsifiability says nothing about an argument's inherent validity or correctness. It is only the minimum trait required of a claim that allows it to be engaged with in a scientific manner – a dividing line between what is considered science and what isn’t. Another important point is that falsifiability is not any claim that has yet to be proven true. After all, a conjecture that hasn’t been proven yet is just a hypothesis.

The idea is that no theory is completely correct , but if it can be shown both to be falsifiable  and supported with evidence that shows it's true, it can be accepted as truth.

For example, Newton's Theory of Gravity was accepted as truth for centuries, because objects do not randomly float away from the earth. It appeared to fit the data obtained by experimentation and research , but was always subject to testing.

However, Einstein's theory makes falsifiable predictions that are different from predictions made by Newton's theory, for example concerning the precession of the orbit of Mercury, and gravitational lensing of light. In non-extreme situations Einstein's and Newton's theories make the same predictions, so they are both correct. But Einstein's theory holds true in a superset of the conditions in which Newton's theory holds, so according to the principle of Occam's Razor , Einstein's theory is preferred. On the other hand, Newtonian calculations are simpler, so Newton's theory is useful for almost any engineering project, including some space projects. But for GPS we need Einstein's theory. Scientists would not have arrived at either of these theories, or a compromise between both of them, without the use of testable, falsifiable experiments. 

Popper saw falsifiability as a black and white definition; that if a theory is falsifiable, it is scientific , and if not, then it is unscientific. Whilst some "pure" sciences do adhere to this strict criterion, many fall somewhere between the two extremes, with  pseudo-sciences  falling at the extreme end of being unfalsifiable. 

Pseudoscience

According to Popper, many branches of applied science, especially social science, are not truly scientific because they have no potential for falsification.

Anthropology and sociology, for example, often use case studies to observe people in their natural environment without actually testing any specific hypotheses or theories.

While such studies and ideas are not falsifiable, most would agree that they are scientific because they significantly advance human knowledge.

Popper had and still has his fair share of critics, and the question of how to demarcate legitimate scientific enquiry can get very convoluted. Some statements are logically falsifiable but not practically falsifiable – consider the famous example of “it will rain at this location in a million years' time.” You could absolutely conceive of a way to test this claim, but carrying it out is a different story.

Thus, falsifiability is not a simple black and white matter. The Raven Paradox shows the inherent danger of relying on falsifiability, because very few scientific experiments can measure all of the data, and necessarily rely upon generalization . Technologies change along with our aims and comprehension of the phenomena we study, and so the falsifiability criterion for good science is subject to shifting.

For many sciences, the idea of falsifiability is a useful tool for generating theories that are testable and realistic. Testability is a crucial starting point around which to design solid experiments that have a chance of telling us something useful about the phenomena in question. If a falsifiable theory is tested and the results are significant , then it can become accepted as a scientific truth.

The advantage of Popper's idea is that such truths can be falsified when more knowledge and resources are available. Even long accepted theories such as Gravity, Relativity and Evolution are increasingly challenged and adapted.

The major disadvantage of falsifiability is that it is very strict in its definitions and does not take into account the contributions of sciences that are observational and descriptive .

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Martyn Shuttleworth , Lyndsay T Wilson (Sep 21, 2008). Falsifiability. Retrieved Aug 20, 2024 from Explorable.com: https://explorable.com/falsifiability

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