Fact-checked by Grok 2 weeks ago

Cardinal utility

Cardinal utility is an economic concept positing that the satisfaction or welfare derived from the consumption of goods and services can be measured quantitatively using numerical units, known as "utils," thereby allowing for the assessment of absolute differences in utility levels between alternatives or even across individuals.^[1]^[2] This approach assumes utility functions are unique up to positive affine transformations, meaning scales and origins can shift without altering preference intensities, akin to measurable quantities in physics.^[2]^[3] In contrast to ordinal utility, which merely ranks preferences without quantifying their strength, cardinal utility facilitates derivations of marginal utility diminishing with increased consumption and supports interpersonal welfare comparisons essential for certain policy evaluations.^[4]^[5] Its theoretical foundations trace to the marginalist revolution of the late 19th century, with economists like William Stanley Jevons and Léon Walras treating utility as a directly measurable magnitude to explain demand curves and equilibrium prices.^[6] A pivotal advancement occurred in the 20th century through the von Neumann-Morgenstern expected utility theorem, which axiomatizes cardinal utility for decisions under risk, deriving numerical utilities from consistent lottery preferences via independence and continuity axioms.^[7]^[2] Despite these contributions, cardinal utility has faced significant scrutiny for its empirical unobservability and reliance on strong assumptions about psychological measurability, prompting Vilfredo Pareto and later ordinalists like John Hicks to advocate preference rankings sufficient for most demand analysis without cardinal assumptions.^[6]^[2] It remains integral, however, to fields like risk theory—where it quantifies attitudes toward uncertainty—and welfare economics, though interpersonal comparisons invite debates over ethical foundations and potential biases in aggregating diverse utilities.^[6]^[7]

Definition and Core Principles

Fundamental Concept and Assumptions

Cardinal utility theory posits that the satisfaction, or utility, obtained from consuming goods and services can be quantified in absolute numerical terms, known as "utils," enabling meaningful comparisons of utility levels and differences rather than mere rankings. This measurability implies that the utility function assigns cardinal significance to values, such that intervals between utility numbers represent comparable intensities of preference, supporting analyses like marginal utility calculations for consumer equilibrium where the marginal utility per unit price equates across goods.^[8]^[9] The theory assumes a rational consumer who maximizes total utility subject to a fixed budget, with utilities from different goods being independent and additive to form an overall utility function. Total utility is expressed as the sum of utilities from individual commodities, under the condition of diminishing marginal utility, where additional units of a good yield progressively smaller increments in satisfaction.^[10]^[9] Central assumptions include the cardinal measurability of utility, allowing expression in precise numerical scales; the constancy of the marginal utility of money, which facilitates comparisons by treating money's utility as invariant during consumption adjustments; and the completeness and transitivity of preferences, ensuring consistent rankings that underpin the utility representation. These assumptions enable derivations such as consumer equilibrium at points where marginal utilities divided by prices equal the marginal utility of income, but they impose strong restrictions, including the additivity of utilities across independent goods.^[9]^[10]^[2] Cardinal utility is distinguished from ordinal utility by its capacity to quantify the intensity of preferences in numerical terms, rather than merely ranking alternatives. Ordinal utility, as developed in the indifference curve approach, assumes only that individuals can order bundles of goods by preference without measuring the degree of satisfaction derived from them, rendering utility functions unique only up to monotonic transformations.^[11] In contrast, cardinal utility assigns specific values—often in hypothetical units called "utils"—to levels of satisfaction, enabling analysis of marginal rates of substitution and the computation of utility differences, such as whether the satisfaction from consuming an additional apple exceeds that from an additional orange by a factor of two.^[5]^[12] This measurability underpins classical marginal utility theory, where diminishing marginal utility is expressed as a declining numerical increment per unit consumed.^[4] A key implication of cardinality is the potential for interpersonal utility comparisons, which ordinal utility explicitly rejects as unverifiable without cardinal scaling. Under cardinal assumptions, economists can theoretically assess whether redistributing resources increases total welfare by comparing aggregated utils across individuals, though such comparisons remain contentious due to the arbitrary units involved and lack of empirical observability.^[13] Ordinal frameworks, by design, avoid this by focusing solely on Pareto efficiency, where improvements require unanimous preference without weighing intensities or cross-personal trade-offs.^[14] Cardinal utility also differs from von Neumann-Morgenstern (vNM) utility, a framework derived axiomatically for decisions under uncertainty. vNM utility functions, satisfying axioms of completeness, transitivity, continuity, and independence, represent preferences over lotteries and are unique up to positive affine transformations (scaling and shifting), yielding an interval-scale measure suitable for expected utility maximization.^[15] Traditional cardinal utility in deterministic consumer choice, however, presumes a ratio-scale measurability closer to absolute quantities, as in early marginalist works, without requiring lottery-based elicitation; yet both share cardinality's core feature of preserving differences under transformations, distinguishing them from ordinality's mere order preservation.^[16] This vNM variant revived cardinal approaches in the mid-20th century by grounding them in observable choice behavior over risky prospects, bypassing direct introspection of utils.^[17]

Historical Development

Origins in Classical Economics

Jeremy Bentham laid the philosophical foundations for cardinal utility in his 1789 treatise An Introduction to the Principles of Morals and Legislation, where he defined utility as that property of an action or object which produces pleasure or averts pain, quantifiable through the hedonic calculus. This calculus assessed pleasure and pain along seven dimensions—intensity, duration, certainty, propinquity, fecundity, purity, and extent—implying a numerical measurability of satisfaction that could be summed and compared.^[18]^[19] Bentham's framework extended to economic decision-making, positing that individuals and legislators could maximize aggregate utility by calculating these hedonic units, influencing early economic analyses of welfare and choice.^[20] John Stuart Mill, building on Bentham's utilitarianism, integrated cardinal-like utility into classical economic theory in works such as Principles of Political Economy (1848), where he analyzed consumer demand and production through the lens of utility derived from labor and abstinence. Mill's "greatest happiness principle" presupposed interpersonal comparisons of utility intensity, treating happiness as aggregable despite his qualitative distinctions between higher intellectual pleasures and lower sensual ones in Utilitarianism (1861).^[19] This approach allowed Mill to evaluate economic policies by their capacity to enhance total welfare, assuming utility's magnitude could inform judgments on distribution and taxation.^[21] Nassau William Senior further advanced cardinal utility within classical economics in his 1836 An Outline of the Science of Political Economy, positing that exchange value arises from utility (desirableness) moderated by scarcity and transfer cost, with utility treated as a measurable source of demand. Senior's abstinence theory of interest equated profit to foregone consumption utility, implying cardinal scaling to explain why agents forgo present goods for future ones. These formulations by Bentham, Mill, and Senior established utility as an empirically grounded, quantifiable driver of economic behavior, predating formal marginalism while prioritizing observable satisfaction over labor costs alone.^[22]

Marginal Revolution and Early Formalization

The Marginal Revolution of the 1870s fundamentally transformed economic thought by replacing the labor theory of value with the subjective theory of value based on marginal utility. William Stanley Jevons, Carl Menger, and Léon Walras independently developed frameworks where the value of goods derives from their utility in the margin—the additional satisfaction from consuming one more unit—rather than total utility or production costs. This shift implied an early conception of utility as amenable to quantitative analysis, with diminishing marginal utility curves central to explaining exchange ratios and prices.^[23]^[24] Jevons formalized these ideas in The Theory of Political Economy (1871), treating utility as a psychic magnitude measurable in relative terms through sensations or trade-offs, akin to physical intensities. He employed differential calculus to model utility as a function u(q) of quantity q, where the marginal utility \frac{du}{dq} diminishes continuously, enabling derivations of demand functions from utility maximization under budget constraints. Jevons argued that while absolute utility units elude direct observation, ratios and increments could be inferred from behavior, providing a basis for interpersonal comparisons via common experiential scales.^[25]^[26] Walras advanced the formalization in Éléments d'économie politique pure (first edition 1874), integrating marginal utility into a system of general equilibrium equations. He defined utility in terms of "rareté" (intensity of want relative to supply), quantified via a numeraire good, and assumed additive separability across goods, which presupposes cardinal measurability for solving simultaneous equations of exchange. This mathematical structure treated utility differences as numerically significant, facilitating the computation of equilibrium prices where marginal utilities equalized across budgets.^[27] Menger's Grundsätze der Volkswirtschaftslehre (1871) emphasized ordinal rankings of subjective goods but incorporated marginal analysis through the valuation of least-important uses, implying cardinality in the graded satisfaction of needs. Unlike Jevons and Walras's explicit mathematization, Menger's causal-genetic approach traced utility to individual purposefulness, yet his marginal pairs (Güterpaare) supported quantitative intuitions about utility gradients. These contributions collectively established cardinal utility's early scaffolding, though later historiography debates their strict adherence to modern cardinal standards, noting reliance on psychophysical analogies rather than axiomatic measurement.^[28]^[29]

20th-Century Debates and the Ordinal Shift

In the early 20th century, Vilfredo Pareto advanced an ordinal conception of utility, arguing that while individuals could rank bundles of goods via indifference curves, the intensity or numerical magnitude of satisfaction need not be quantified for economic analysis.^[30] Pareto's critique targeted the cardinal assumptions of earlier marginalists like Walras, positing that utility differences between indifference classes were directionally ordered but not metrically comparable, thus rendering interpersonal utility comparisons unscientific.^[31] The interwar period intensified these debates, culminating in Lionel Robbins' 1932 An Essay on the Nature and Significance of Economic Science, which rejected cardinal utility and interpersonal comparisons as unverifiable psychological propositions unsuitable for positive economics.^[32] Robbins redefined economics as the science of scarce means and alternative ends, emphasizing ordinal preferences to avoid normative welfare judgments, thereby shifting focus from utility measurement to revealed choice under constraints.^[33] This ordinal turn was formalized in J.R. Hicks and R.G.D. Allen's 1934 article "A Reconsideration of the Theory of Value," which derived individual demand functions solely from indifference curve analysis and marginal rates of substitution, dispensing with cardinal utility's additive assumptions.^[34] Their framework demonstrated that Slutsky-type demand equations hold under ordinal utility, provided income effects are accounted for, establishing ordinalism as sufficient for consumer theory without invoking unobservable utility quanta.^[35] The ordinal shift marginalized cardinal utility in mainstream microeconomics by the mid-1930s, prioritizing behavioral rankings over introspective measurement, though critics noted it complicated welfare analysis by precluding direct aggregation of utilities.^[36] Subsequent work, such as Paul Samuelson's 1938 revealed preference theory, further entrenched ordinalism by grounding utility in observable choices rather than hypothetical scales.^[37] However, cardinal utility experienced partial revival in decision theory under risk through von Neumann and Morgenstern's 1944 axiomatic expected utility theorem, which implied a cardinal representation unique up to affine transformations for preferences satisfying continuity, independence, and transitivity.^[38] This demonstrated that ordinal preferences over lotteries necessitate cardinal scaling for risk attitudes, bridging the debate but confining the ordinal dominance to certainty contexts.^[39]

Theoretical Framework

Construction of Cardinal Utility Functions

The construction of cardinal utility functions in economic theory relies on eliciting numerical values from observed preferences, particularly under conditions of risk or uncertainty, to capture not just rankings but also the strength of preferences. This process is formalized through the von Neumann-Morgenstern (vNM) representation theorem, which assumes a preference relation over lotteries (probability distributions over outcomes) that satisfies four key axioms: completeness (every pair of lotteries is comparable), transitivity (preferences are consistent across chains), continuity (preferences are continuous in probabilities, ensuring intermediate lotteries can be approximated), and independence (preferences between lotteries remain unchanged when mixed with a common third lottery in fixed proportions).^[15]^[40] Under these axioms, preferences can be represented by an expected utility functional, where the utility of a lottery L = (x_1, p_1; \dots; x_n, p_n) is EU(L) = \sum_{i=1}^n p_i u(x_i), and the function u over outcomes is cardinal—unique up to positive affine transformations u' = a + b u with b > 0.^[15] To derive u, normalize reference points: assign u(x_{\min}) = 0 to the worst outcome and u(x_{\max}) = 1 to the best. For any intermediate outcome x, identify the probability p \in [0,1] at which the decision-maker is indifferent between x for certain and the lottery yielding x_{\max} with probability p and x_{\min} with $1-p; set u(x) = p. This elicitation can be repeated across outcomes, often using binary choices or certainty equivalents to approximate the function empirically.^[40]^[15] The independence axiom ensures the derived u extends consistently to compound lotteries, as preferences over mixtures preserve the ranking. For instance, if indifference holds between a 50-50 lottery over outcomes x_1 (worst) and x_3 (best) and the certain outcome x_2, then $0.5 u(x_1) + 0.5 u(x_3) = u(x_2); with normalization u(x_1) = 0 and u(x_2) = 1, solving yields u(x_3) = 2, illustrating how risk attitudes scale utilities beyond the [0,1] interval while maintaining affine invariance.^[15] In deterministic choice without lotteries, such construction lacks uniqueness, as ordinal utility (unique up to monotonic transformations) suffices; cardinality requires the additional structure of probabilistic choices to measure trade-offs in intensity.^[41] Empirical elicitation methods, such as tradeoff gambles, refine this by iteratively adjusting probabilities to minimize inconsistencies, though they assume axiom compliance and may introduce measurement error from behavioral deviations.^[42]

Mathematical Properties and Derivations

Cardinal utility functions possess the property of being unique up to positive affine transformations, such that if U represents an agent's preferences, then \tilde{U} = aU + b (with a > 0 and b \in \mathbb{R}) represents the same preferences, preserving the meaningfulness of utility differences and their ratios.^[43]^[44] This contrasts with ordinal utility functions, which are unique only up to strictly increasing transformations, rendering differences non-invariant. The affine invariance implies that cardinal utility admits an interval scale of measurement, where the unit and origin are fixed but scalable and shiftable positively, enabling derivations of concepts like absolute risk aversion coefficients, defined as r_A(x) = -\frac{U''(x)}{U'(x)}, which remain unchanged under such transformations.^[43] A key derivation of cardinal utility arises in the von Neumann-Morgenstern expected utility framework under uncertainty, where axioms of completeness, transitivity, continuity, and independence yield a utility function linear in probabilities: for a lottery L = \sum p_i x_i, the expected utility is EU(L) = \sum p_i U(x_i), with U cardinal due to the linearity imposing scale invariance only under affine shifts.^[44] To illustrate, normalize U(x_1) = 0 and U(x_2) = 1 for outcomes x_1 \prec x_2. An indifference between a compound lottery L_1' offering x_1 or x_3 each with probability 0.5 and the sure outcome x_2 implies $0.5 U(x_1) + 0.5 U(x_3) = U(x_2), simplifying to $0.5 U(x_3) = 1 and thus U(x_3) = 2. This process extends recursively, fixing the cardinal scale via probabilistic trade-offs without arbitrary normalization beyond the affine class.^[44] More generally, cardinal utility can be derived axiomatically from preference relations incorporating weak trade-off conditions, ensuring a representation unique up to location and unit, as in models where indifference curves or choice data satisfy solvability for utility differences.^[45] For differentiable cases, the marginal utility MU(x) = \frac{dU}{dx} follows directly, with properties like diminishing marginal utility (MU'(x) < 0) implying concave U, derivable from second-order conditions in optimization problems such as \max U(x) subject to budget constraints, yielding demand functions where \frac{MU_x}{p_x} = \frac{MU_y}{p_y}. These derivations underpin applications like deriving individual demand curves from equating marginal utility per dollar across goods, assuming measurable utils.^[45]

Economic Applications

Consumer Choice and Marginal Analysis

In cardinal utility theory, consumer choice involves selecting quantities of goods to maximize total utility subject to a budget constraint defined by income and prevailing prices. The consumer derives measurable satisfaction, or utility, from bundles of goods, with total utility represented as a function U(x_1, x_2, \dots, x_n) where x_i denotes the quantity of good i.^[9] Equilibrium occurs when no reallocation of expenditure can increase total utility, achieved through marginal analysis of incremental consumption.^[46] Marginal utility (MU_i) quantifies the additional utility from consuming one more unit of good i, holding other quantities constant, and is derived as the partial derivative \partial U / \partial x_i.^[47] The law of diminishing marginal utility posits that MU_i decreases as x_i increases, reflecting satiation and leading to concave utility functions in relevant ranges.^[48] This property explains why consumers diversify purchases rather than concentrating on a single good, as equalizing marginal utilities per dollar spent—MU_i / P_i = \lambda for all i, where P_i is the price of good i and \lambda is the marginal utility of income—maximizes satisfaction.^[47]^[9] For two goods x and y, with budget I = P_x x + P_y y, the first-order conditions from utility maximization yield MU_x / P_x = MU_y / P_y, implying that the last dollar expended on each good yields identical utility increments.^[47] A price decrease for x, for instance, raises MU_x / P_x initially, prompting substitution toward x until equality restores, generating a downward-sloping demand curve.^[49] Income effects similarly alter \lambda, shifting consumption based on whether goods are normal or inferior, with cardinal measurability enabling quantification of these responses via utility differences.^[50] This framework, rooted in equi-marginal utility, supports derivations of individual demand functions from specific utility forms, such as Cobb-Douglas U = x^a y^b, where expenditure shares align with utility exponents.^[51] Empirical applications, though challenged by ordinal dominance post-1930s, persist in contexts requiring intensity comparisons, like welfare analysis of price changes.^[49]

Welfare Economics and Interpersonal Comparisons

Cardinal utility underpins interpersonal comparisons in welfare economics by positing that individuals' utilities can be quantified on a common scale, enabling aggregation or weighting to assess overall social welfare beyond Pareto improvements.^[52] This approach contrasts with ordinal utility, which limits analysis to preference orderings and renders distributive trade-offs incommensurable without additional ethical assumptions.^[14] Social welfare functions, as formalized by Bergson in 1938 and Samuelson in 1947, treat social welfare as a function of individual cardinal utilities, implicitly requiring such comparisons to rank allocations where no Pareto-dominant option exists.^[53] Lionel Robbins, in his 1932 Essay on the Nature and Significance of Economic Science and subsequent 1938 comments, critiqued interpersonal utility comparisons as unscientific and normative, arguing they introduce value judgments unverifiable by empirical observation or logical deduction, thereby confining welfare economics to positive analysis of efficiency.^[54] ^[55] This perspective influenced the ordinalist shift, emphasizing Paretian criteria and market outcomes over cardinal aggregation. However, proponents like Harsanyi in 1953 defended cardinal utility for welfare applications, proposing that utilities could be interpersonally comparable via an "equiprobability model" where an impartial observer equates risks across individuals, yielding utilitarian social welfare as the sum of expected utilities.^[52] ^[14] In practice, cardinal assumptions facilitate cost-benefit analysis and policy evaluation, such as weighing marginal utilities in progressive taxation where diminishing marginal utility implies transfers from high- to low-income individuals increase total welfare.^[56] Yet, these comparisons remain ethically laden, as no objective metric exists to calibrate scales across diverse preferences or circumstances, leading some modern economists to rely on revealed preference data or hypothetical compensations rather than direct utility summation.^[14] Empirical challenges persist, with behavioral evidence suggesting utility functions may exhibit inconsistencies that undermine strict cardinality, though expected utility theory provides a foundational framework for risk-inclusive welfare assessments.^[52]

Decision Theory Under Uncertainty

In decision theory under uncertainty, agents select among lotteries—probability distributions over outcomes—requiring a framework to aggregate preferences probabilistically. Cardinal utility enables this through expected utility maximization, where the value of a lottery equals the sum of probabilities times utilities of outcomes: EU(L) = \sum p_i u(x_i).^[15] Ordinal utility, preserving only rankings of sure outcomes, fails here, as monotone transformations alter probabilistic mixtures inconsistently with preferences.^[57] The von Neumann–Morgenstern theorem, presented in 1944, proves that preferences over lotteries satisfying completeness (all pairs comparable), transitivity, continuity (intermediate lotteries preferred to extremes), and independence (preferences invariant to common mixtures) admit a cardinal utility representation unique up to affine transformations u' = a + b u with b > 0. This cardinal scale captures intensity of preferences, essential for mixed strategies in games and risk assessments, unlike ordinal scales inadequate for quantitative probability weighting.^[58] Cardinal utility thus facilitates deriving risk attitudes: a concave u implies risk aversion, where u(E) > EU(L) for non-degenerate lotteries, reflecting diminishing marginal utility.^[15] For instance, equating expected utilities of lotteries reveals utility differences, such as scaling u(x_3) = 2 from $0.5 u(x_3) = 1 under equal probabilities.^[57] These properties underpin economic models of insurance, investment, and portfolio choice under uncertainty.

Intertemporal Choice and Discounting

In intertemporal choice, agents face trade-offs between consumption or outcomes at different points in time, such as saving for future needs versus immediate spending. Cardinal utility enables rigorous modeling of these decisions by quantifying satisfaction levels, allowing future utilities to be discounted and aggregated into a lifetime objective function. This approach assumes utility is measurable on an interval scale, permitting operations like summation and scaling that ordinal utility cannot support without additional cardinal assumptions.^[2]^[31] Paul Samuelson formalized this in his 1937 discounted utility model, positing that rational agents maximize U = \sum_{t=0}^{\infty} \beta^t u(c_t), where u(c_t) represents cardinal utility from consumption c_t in period t, and \beta = 1/(1 + \rho) (with \rho > 0) captures impatience via exponential discounting. This yields time-consistent behavior, where marginal rates of substitution between periods equal the discount factor adjusted for interest rates, facilitating predictions of consumption smoothing and savings rates. The model's reliance on cardinal measurability stems from the need to compare utility intensities across time, a feature implicit in Samuelson's axiomatization of utility as uniquely determined up to positive affine transformations.^[59]^[60]^[61] Preceding Samuelson, Frank Ramsey's 1928 optimal growth model employed cardinal utility to maximize \int_0^\infty e^{-\rho t} u(c_t) \, dt subject to capital accumulation constraints, deriving the "Ramsey rule" where the interest rate equals the marginal product of capital plus adjustments for time preference and growth. This framework influenced modern macroeconomics, including the Ramsey-Cass-Koopmans model, by enabling quantitative welfare comparisons over infinite horizons—impossible under strict ordinality, which restricts analysis to relative rankings within finite periods, as in Irving Fisher's two-period indifference maps. Ordinal approaches falter in multi-period settings because they lack a basis for weighting temporal utilities, often requiring implicit cardinal elements for discounting.^[62]^[63] Empirical implementations calibrate discount rates from observed behaviors, such as bond yields or savings data; for instance, U.S. household surveys from 1983–2010 imply annual discount rates around 4–6% under exponential assumptions, though deviations like hyperbolic discounting challenge strict cardinal additivity. Despite such anomalies, cardinal discounting remains foundational for policy tools, including social cost of carbon estimates via the Ramsey formula r = \rho + \eta g, where \eta is the elasticity of marginal utility and g is growth, as applied in U.S. government guidelines updated in 2023.^[61]^[64]

Empirical Foundations and Modern Evidence

Neuroscientific and Experimental Support

Single-neuron recordings in the orbitofrontal cortex (OFC) of rhesus monkeys demonstrate neural encoding of subjective reward value on a cardinal scale. Neurons respond to the utility of liquid rewards, such as water or flavored solutions, with firing rates proportional to perceived value rather than objective quantity or flavor intensity; for example, activity equates 1 drop of water to approximately 4 drops of low-concentration Kool-Aid, predicting subsequent choices between offers independently of spatial location or motor demands.^[65] Human functional magnetic resonance imaging (fMRI) studies reveal that the dorsal striatum encodes marginal utility in intertemporal decisions, where activity correlates with diminishing marginal returns on reward magnitude (e.g., £1 to £100) and integrates it with hyperbolic temporal discounting (delays of 1 week to 1 year). Model-based analyses show this neural signal reflects cardinal discounted utility values, outperforming non-cardinal alternatives in explaining both behavior and brain activity (Akaike weight = 0.99).^[66] Experimental economics employs elicitation techniques grounded in von Neumann-Morgenstern expected utility theory to derive cardinal utility functions unique up to positive affine transformations. The tradeoff method, for instance, requires subjects to equate lotteries by adjusting outcomes or probabilities (e.g., balancing gains in duration or amount), yielding measurable utility scales; applications include assessing utility over life expectancy in samples of economics Ph.D. students, where elicited functions exhibit risk aversion consistent with cardinal properties.^[67] Behavioral experiments further link cardinal revealed preferences to introspective measures, constructing a choice-based utility index for money that aligns closely with direct, choiceless valuations of the same outcomes, supporting the internal consistency of cardinal representations across elicitation modes.^[36] Emerging neural approaches extend this to interpersonal comparisons, using fMRI signals from the anterior cingulate cortex and adjacent ventromedial prefrontal cortex to quantify subjective utility differences across individuals during reward tasks, providing a potential physiological basis for cardinal and comparable utility metrics.^[68]

Happiness Metrics and Survey-Based Measures

Survey-based measures of happiness, often termed subjective well-being (SWB), typically involve respondents rating their life satisfaction or happiness on numerical scales, such as the 0-10 Cantril Ladder used in the Gallup World Poll since 2005, where 0 represents the worst possible life and 10 the best.^[69] These metrics aggregate data from large-scale surveys like the Gallup World Poll (covering over 160 countries in more than 140 languages), the World Values Survey (since 1981, spanning about 100 countries), and the European Social Survey, enabling cross-national and temporal comparisons of reported well-being levels.^[69] Such scales provide cardinal data by quantifying intensity, as respondents consistently map verbal descriptors to numbers in a linear fashion, supporting arithmetic operations like averaging for interpersonal utility assessments.^[70] Empirical evidence suggests these measures capture underlying utility-like constructs, correlating with physiological indicators such as brain activity in pleasure centers and observable behaviors like smiling frequency.^[69] For instance, regression analyses of life satisfaction predictors (e.g., income, health) yield consistent coefficients across nations, implying shared scale interpretations and enabling cardinal treatment for welfare evaluations.^[70] Studies treating SWB scores as cardinal, rather than merely ordinal, better align with economic behaviors, such as income-happiness elasticities that facilitate utility function estimation.^[69] The World Happiness Report, drawing on Gallup data, ranks countries cardinally—Finland scored 7.84 in 2022—allowing aggregation for policy analysis, like assessing GDP's marginal contribution to well-being (approximately 0.3-0.5 points per doubling in some contexts).^[69] However, linking these metrics directly to cardinal utility faces challenges, as self-reports may reflect momentary elation or baseline mood rather than comprehensive preference satisfaction, with hedonic adaptation causing scores to revert post-events (e.g., within months after lottery wins or disasters).^[71] Cultural and response-style differences, such as vignette equivalency issues, can introduce non-linearities, though evidence from objective analogs like self-reported height (correlation 0.8 with actual) supports practical comparability.^[70] While not resolving all philosophical objections to interpersonal utility comparisons, these measures offer verifiable proxies when calibrated against objective correlates, outperforming ordinal approaches in predictive power for long-run welfare changes, as seen in post-crisis drops like Greece's life satisfaction falling from 67% in 2007 to 32% in 2012.^[69]^[71]

Recent Theoretical Reevaluations

In decision theory, recent theoretical work has sought to ground cardinal utility in observable choice behavior, moving beyond ordinal rankings to represent preference intensities empirically. Baccelli and Mongin (2016) propose a "choice-based cardinal utility" framework, distinguishing it from ordinalism by emphasizing utility functions that capture differences in preferences derivable from choice data under certainty, uncertainty, and stochastic settings. This approach, inspired by Patrick Suppes' earlier contributions, addresses classic objections to cardinalism—such as the lack of interpersonal comparability—by linking utility scales directly to empirical choices rather than introspective measures, thereby reviving cardinal representations as compatible with behaviorist foundations.^[43] Building on this, advancements in cardinal utility have incorporated psychological and needs-based structures to explain deviations from standard demand theory. In a 2024 analysis, Miller introduces the concept of "separate needs" into cardinal utility theory, positing that utility functions for commodities serving the same need exhibit weak (multiplicative) separability, while needs differ via strong (additive) separability. This yields "leaning-S-shaped" utility curves reflecting stages from deprivation to satiation, derived from a two-variable additive normal distribution function, which predicts phenomena like inferior and Giffen goods based on fulfillment levels and defines absolute poverty lines through specific indifference curve geometries. Such formulations challenge ordinal utility's inability to quantify these intensities, offering a refined cardinal model for consumer equilibrium that aligns marginal utilities per price across needs.^[72] These reevaluations reflect a broader theoretical resurgence, as noted by Moscati (2019), who traces a revival of cardinal utility interest since the early 2000s, driven by behavioral economics' emphasis on measurable satisfaction over mere rankings. While ordinalism remains dominant in core microeconomic models, these developments argue for cardinalism's superiority in contexts requiring intensity assessments, such as welfare comparisons or risk aversion derivations, without relying on untestable psychological assumptions. Critics, however, contend that deriving unique cardinal scales from choices still presupposes auxiliary independence conditions, limiting general applicability.^[73]

Criticisms and Debates

Methodological and Philosophical Objections

Critics of cardinal utility contend that its foundational assumption of quantitative measurability lacks empirical verifiability, as utility represents a subjective psychological state without observable units akin to physical magnitudes like length or weight.^[74] Unlike market prices or quantities, which can be directly observed and aggregated, attempts to assign numerical values to satisfaction—such as through psychophysical experiments inspired by Gustav Fechner's work in the late 19th century—failed to establish reliable, interpersonally consistent scales, leading early 20th-century economists to question its scientific status.^[75] A central methodological objection, articulated by Lionel Robbins in his 1932 Essay on the Nature and Significance of Economic Science, holds that cardinal utility invites unverifiable interpersonal comparisons, transforming economics from a positive science of means and ends into a normative exercise laden with ethical presuppositions.^[54] Robbins argued that statements equating one individual's utility gain to another's loss—essential for welfare theorems or cost-benefit analysis—cannot be tested empirically and instead reflect arbitrary value judgments about human equality or desert, rendering them unscientific.^[13] This critique prompted a shift toward ordinal utility in mainstream economics by the 1940s, as interpersonal aggregation proved indispensable yet unsubstantiated for policy applications like progressive taxation or public goods provision.^[76] Philosophically, cardinal utility presupposes a hedonistic or preference-based ontology where diverse ends of human action—ranging from aesthetic enjoyment to moral fulfillment—can be reduced to a common, additive metric, an assumption Austrian economists like Carl Menger and Ludwig von Mises deemed incompatible with methodological individualism.^[77] Menger, in Principles of Economics (1871), emphasized value's radical subjectivity and incommensurability across individuals, arguing that cardinal scales impose artificial homogeneity on heterogeneous valuations derived from ordinal rankings of ends.^[78] Later Austrians, including Murray Rothbard, reinforced this by rejecting cardinalism's reliance on introspective quantification, which they viewed as psychologistic pseudoscience prone to the same measurement paradoxes as historical attempts to quantify "psychic income." Such objections highlight cardinal utility's vulnerability to infinite rescaling, where utility functions remain empirically indistinguishable under affine transformations, undermining claims of unique, meaningful numerical representation.^[79]

Empirical and Practical Limitations

Empirical assessments of cardinal utility face fundamental challenges due to its inherent subjectivity and lack of direct observability. Utility, conceptualized as a quantifiable measure of satisfaction, cannot be empirically verified through objective metrics, as it manifests as an internal psychological state rather than a tangible quantity like price or quantity consumed. Experimental attempts to elicit cardinal utility values, such as through willingness-to-pay surveys or hypothetical choice scenarios, often yield inconsistent results influenced by framing effects, hypothetical bias, and individual variability in introspection, rendering such measures unreliable for aggregation or prediction.^[1]^[74] Behavioral economics provides stark empirical counterevidence against cardinal utility assumptions embedded in models like expected utility theory. The Allais paradox, first documented in 1953, demonstrates that individuals' risk preferences violate the independence axiom of von Neumann-Morgenstern utility, where subjects prefer certain gains over risky prospects in ways inconsistent with any fixed cardinal utility function, highlighting how certainty effects and probability weighting distort cardinal rankings. Similarly, prospect theory experiments since the 1970s reveal loss aversion and reference dependence, further undermining the empirical validity of globally consistent cardinal scales. These anomalies persist across diverse populations and contexts, with meta-analyses confirming violation rates exceeding 50% in standard tests.^[80]^[81] Practically, implementing cardinal utility in applied economics proves infeasible owing to the absence of a universal unit of measurement and the confounding role of diminishing marginal utility of money. Cardinal analysis assumes a stable metric for interpersonal comparisons, yet empirical proxies like income equivalents fail under varying wealth levels, as the marginal utility of a dollar decreases nonlinearly—evidenced by Engel curves showing expenditure patterns that deviate from constant marginal utility postulates. In policy contexts, such as cost-benefit analysis, reliance on cardinal welfare weights invites arbitrary scaling, exacerbating errors in redistributive decisions without ordinal safeguards. Neuroeconomic efforts to map utility via brain imaging, such as fMRI activations in reward centers, offer correlational insights but lack the precision for cardinal quantification, with signal noise and ethical constraints limiting scalability.^[82]^[10]

Ideological Critiques of Adoption Patterns

Some egalitarian economists and philosophers contend that the historical shift toward ordinal utility in the early 20th century, exemplified by Vilfredo Pareto's emphasis on ophelimity rankings and the Hicks-Allen framework of 1934, was partly ideologically motivated to constrain welfare economics within Pareto optimality, thereby obstructing arguments for redistribution that rely on interpersonal utility comparisons.^[83] This approach renders it impossible to deem a policy improving aggregate utility at the cost of some uncompensated losses as superior, even if cardinal measures indicate greater total satisfaction; critics argue this preserves unequal distributions under the guise of scientific neutrality, aligning with market-preserving ideologies that prioritize efficiency over equity.^[84] Amartya Sen, in his analysis of Paretian liberalism, highlighted how ordinal restrictions fail to address distributional inequities, implying that their dominance limits social choice to scenarios favoring the status quo unless all affected parties consent.^[85] Conversely, libertarian and methodological individualist thinkers critique the limited adoption of cardinal utility—confined largely to expected utility models like von Neumann-Morgenstern—as ideologically selective, avoiding its fuller implications for utilitarian policy that could rationalize coercive interventions under the banner of maximizing total welfare. John C. Harsanyi argued in 1955 that cardinal utility, grounded in ethical individualism, necessitates interpersonal comparisons for equitable social decisions, and its sidelining in standard theory reflects a bias against utilitarianism's potential to override individual rights for collective gains, such as through progressive taxation justified by diminishing marginal utility.^[52] This pattern, they claim, stems from a post-Robbinsian emphasis on positive economics that ideologically insulates market outcomes from normative scrutiny, despite cardinal approaches enabling rigorous risk and welfare analysis elsewhere.^[85] These critiques often intersect with broader debates on utility measurability; while ordinalism's proponents attribute its prevalence to empirical challenges—like the non-uniqueness of cardinal scales under affine transformations—the ideological lens posits that academic preferences reflect systemic biases toward non-interventionist conclusions, as evidenced by the ordinal core of neoclassical textbooks despite cardinal applications in fields like cost-benefit analysis.^[43] Empirical surveys of economists show divided views on interpersonal comparisons, with ordinal dominance persisting not purely on scientific grounds but amid ideological tensions over policy scope.^[13]

Comparative Analysis

Versus Ordinal Utility: Theoretical Differences

Cardinal utility theory assumes that the utility derived from consumption can be measured in absolute quantitative terms, typically using hypothetical units known as "utils," enabling assessments of the intensity or magnitude of satisfaction differences. This approach treats utility as a cardinal magnitude, comparable to physical quantities like length or weight, where differences and ratios between utility levels hold meaning independent of scaling.^[4] In opposition, ordinal utility theory rejects such measurability, asserting that utility functions need only preserve the ranking of preferences, with any strictly increasing transformation of the utility function yielding equivalent behavioral predictions.^[86] Thus, ordinal representations are unique only up to monotonic transformations, rendering utility differences and ratios undefined or arbitrary.^[86] A core theoretical divergence lies in the treatment of interpersonal utility comparisons. Cardinal utility permits aggregating or comparing utility levels across individuals, underpinning utilitarian welfare functions that maximize total or average utility, as in Benthamite calculations where societal well-being is the sum of individual utils.^[84] Ordinal utility, by contrast, precludes such comparisons, confining normative economics to Pareto efficiency—changes that improve at least one person's welfare without harming others—since no objective basis exists for weighing one person's gain against another's loss.^[84] This restriction arises from ordinalism's foundational axiom that preferences are private and non-comparable, avoiding the interpersonal aggregation challenges that cardinalism embraces but struggles to empirically ground.^[85] Under uncertainty, cardinal utility gains renewed theoretical justification through the von Neumann-Morgenstern axioms, which derive a cardinal utility scale from observed choices over lotteries, where expected utility calculations require preserving both orderings and affine properties to rationalize risk attitudes like diminishing marginal utility of wealth.^[43] Ordinal utility suffices for deterministic choice but fails to accommodate probabilistic decisions without additional structure, as it cannot distinguish between risk-averse, risk-neutral, or risk-loving behaviors beyond mere rankings.^[43] Consequently, cardinal approaches enable derivations of phenomena like the equity premium puzzle or insurance demand, whereas ordinalism demands supplementary assumptions, such as ad hoc risk functionals, to extend to stochastic environments.^[84] In consumer theory, cardinal utility implies testable predictions via marginal utility diminishing at quantifiable rates, supporting derivations of demand curves from equating marginal utility per dollar across goods, as in early Marshallian analysis.^[2] Ordinal utility, however, derives demand solely from indifference curve slopes and budget constraints, yielding equivalent behavioral outcomes without invoking intensity, thus simplifying analysis by eschewing unverifiable utility metrics.^[86] This parsimony fueled the ordinalist revolution around 1930–1940, led by Pareto and Hicks, who demonstrated that ordinal rankings alone suffice for positive economics, relegating cardinalism to normative or risky contexts where ordinalism proves insufficient.^[2]

Implications for Policy and Analysis

Cardinal utility theory underpins much of welfare economics by permitting interpersonal comparisons of well-being, thereby justifying policies aimed at redistribution on the grounds of diminishing marginal utility of income. For instance, if utility diminishes at higher income levels, transferring resources from wealthy to poorer individuals can increase aggregate social welfare, providing a rationale for progressive taxation systems observed in many economies since the early 20th century.^[87] This contrasts with ordinal approaches, which limit analysis to Pareto improvements and cannot quantify net gains from such transfers without additional ethical assumptions.^[84] In policy analysis, cardinal utility facilitates quantitative assessments in public economics, including evaluations of inequality and poverty alleviation programs. It allows for the construction of social welfare functions that aggregate individual utilities, such as utilitarian sums or weighted averages, to rank policy outcomes beyond mere efficiency.^[52] Empirical applications include using survey-based utility estimates to simulate the welfare effects of subsidies or minimum wage hikes, where ordinal rankings alone would fail to capture intensity of preferences or distributional impacts.^[88] Cost-benefit analysis in government projects often implicitly relies on cardinal assumptions by monetizing benefits and costs, treating money as a proxy for utility under constant or diminishing marginal returns. This approach supports interventions like infrastructure investments or health programs when projected utility gains exceed losses, as seen in U.S. federal guidelines for regulatory impact assessments since the 1980s.^[4] However, such analyses require verifiable scaling of utilities across individuals, which cardinal theory enables but demands robust empirical grounding to avoid overreliance on untested interpersonal equivalences.^[84]

Cases Where Cardinal Adds Unique Insights

In decision theory under uncertainty, cardinal utility offers unique insights by enabling the representation of preferences over lotteries through expected utility maximization, as formalized by the von Neumann-Morgenstern theorem. This theorem demonstrates that consistent choices among risky prospects imply a utility function unique up to positive affine transformations, preserving the curvature that distinguishes risk attitudes: concavity indicates risk aversion, linearity risk neutrality, and convexity risk loving. Ordinal utility, invariant only under monotonic transformations, cannot maintain this curvature, rendering it inadequate for analyzing how individuals weigh probabilities against outcomes or for deriving comparative measures like absolute or relative risk aversion.^[89]^[90] In welfare economics, cardinal utility facilitates interpersonal comparisons essential for aggregating individual well-being into social welfare functions, evaluating inequality, and assessing redistributive policies. Ordinal utility supports Pareto comparisons but cannot quantify the intensity of preferences across agents or justify transfers, as it provides no scale for determining whether gains to one outweigh losses to another. John C. Harsanyi contended that such comparisons require cardinal measures to incorporate utilitarian ethics, where total or average utility guides policy, enabling analyses of progressive taxation or poverty alleviation that ordinal approaches deem incommensurable.^[91]^[52] These applications highlight cardinal utility's role in contexts demanding quantitative trade-offs, such as insurance demand under risk or optimal income redistribution, where ordinal rankings alone yield indeterminate outcomes. Empirical implementations, like estimating risk premiums from choice data, further rely on cardinal scaling to infer underlying utility differences.^[92]

References

[1]
How Is Economic Utility Measured? - Investopedia
"Cardinal" utility measures economic value through imaginary units known as "utils." "Marginal" utility is what is gained by consuming an additional unit of a ...What Is Utility? · Utility Theory · Measuring Utility
[2]
[PDF] How cardinal utility entered economic analysis, 1909-1944
Jul 9, 2013 · As originally utility unique up to positive linear transformations was not called 'cardinal', the paper also investigates the terminological.
[3]
[PDF] Monotonic transformations: Cardinal Versus Ordinal Utility
A utility function could be said to measure the level of satisfaction associated to each commodity bundle. Nowadays, no economist really believes that a ...
[4]
Cardinal and Ordinal Utility - Economics Help
Cardinal Utility is the idea that economic welfare can be directly observable and be given a value. For example, people may be able to express the utility ...
[5]
Difference Between Cardinal And Ordinal Utility - BYJU'S
The cardinal utility states that the level of satisfaction a consumer acquires after consuming any goods and services can be measurable and expressed in ...
[6]
Cardinal Utility in Welfare Economics and in the Theory of Risk-taking
Cardinal utility is used in welfare economics for equal income distribution and in risk-taking theory, though the concepts may differ. It is also used in value ...
[7]
How Economists Came to Accept Expected Utility Theory
This feature addresses the history of economic terms and ideas. The hope is to deepen the workaday dialogue of economists, while perhaps also casting new light.
[8]
Cardinal Utility Analysis and its Assumptions - eNotes World
The cardinal utility approach believes that utility can be measured and compared to each other in terms of mathematical numbers like 1, 2, 3,..., n.
[9]
The Cardinal Utility Theory (Explained With Diagram)
Assumptions: 1. Rationality: The consumer is rational. He aims at the maximization of his utility subject to the constraint imposed by his given income.Missing: fundamental | Show results with:fundamental
[10]
Cardinal Utility Theory: Concept, Assumptions, Equilibrium ...
Jun 21, 2023 · First, the assumption of the cardinal utility approach that utility is measurable in utils and in monetary terms is very dissatisfying. The ...Missing: fundamental | Show results with:fundamental
[11]
[PDF] 2. Distinguish ordinal utility from cardinal utility. - GCWK
The indifference curve approach is called as ordinal utility approach. Meaning of utility: Utility' is the power of a commodity or service to satisfy a human ...
[12]
Difference Between Cardinal and Ordinal Utility - Testbook
Cardinal utility measures utility quantitatively, while ordinal utility ranks utility in order. Cardinal utility requires numerical assignment to utility, ...
[13]
Interpersonal Comparisons of Utility - Econlib
Feb 1, 2022 · One can attempt to characterize a valuation function (aka cardinal utility ... Economic Methods. Interpersonal Comparisons of Utility · Pierre ...
[14]
[PDF] INTERPERSONAL COMPARISONS OF UTILITY - Stanford University
A. Camacho (1986), “Individual Cardinal Utility, Interpersonal Comparisons, and Social Choice,” in Recent Developments in the Foundations of Utility and.
[15]
[PDF] 1 Basic Concepts
5 Cardinal utility. If in (7) we replace the ... If you choose either A, D or B, C, then your behavior is inconsistent with expected utility theory.
[16]
Von Neumann-Morgenstern Utilities and Cardinal Preferences
We show that with a finite number of choices there exist no continuous anonymous aggregation rules that respect unanimity, for such preferences or utilities.
[17]
Interpersonal comparison of utility - Economics Stack Exchange
Feb 12, 2021 · Von Neumann-Morgenstern cardinal utility requires that axioms of completeness, transitivity, continuity and independence are satisfied (see von ...
[18]
Jeremy Bentham - Stanford Encyclopedia of Philosophy
Mar 17, 2015 · In 1776, he first announced himself to the world as a proponent of utility as the guiding principle of conduct and law in A Fragment on ...
[19]
The Hedonistic Calculus - Philosophy Home Page
The main problem for the calculus is calculating the interpersonal utility comparison using cardinal utility measurement rather than ordinal measurement.
[20]
[PDF] Utility theory from Jeremy Bentham to Daniel Kahneman
They showed that just as an indifference map can be drawn from consistent choices between outcomes, so a cardinal utility function can be drawn from consistent ...
[21]
Political Economy (1850 ed.) - Online Library of Liberty
Nassau William Senior (author). This work is a summary statement of the nature of economic thought by one of the leading theorists of the English classical ...<|separator|>
[22]
[PDF] Nassau Senior Value and the Forces of Demand and Supply
Nassau Senior was an important economist in the development of classical economics in. Britain during the early 1800s. He was the son of a vicar and was ...
[23]
3.1 — The Marginalist Revolution - History of Economic Thought
Menger advocated subjectivism and a causal-genetic theory. Walras condoned cardinal utility, Menger & Jevons likely ordinal utility. Marginalist Revolution ...
[24]
[PDF] Rethinking the "Marginal Revolution" in the History of Economic ...
Beginning in the 1870s, economists began to formally accept marginal utility theory, on account of the work of three economists. Jevons in England, Menger in.
[25]
The Theory of Political Economy | Online Library of Liberty
Its final degree of utility is measured by that of any of the other commodities which he consumes. ... JEVONS (W. S.) The Theory of Political Economy. 8vo ...
[26]
Jevons on Utility - UT liberal arts
The intensity of utility of the third increment is measured either by pq, or p'q', and its utility is the product of the units in pp' multiplied by those in pq.
[27]
Jevons, Menger, and Walras on the Measurability of Utility, 1870–1910
Dec 20, 2018 · Chapter 2 discusses how William Stanley Jevons, Carl Menger, and Léon Walras addressed the issue of the measurability of utility.
[28]
[PDF] Were Jevons, Menger and Walras really cardinalists? On the notion ...
Abstract. The paper argues that the canonical dichotomy between cardinal utility and ordinal utility is inadequate to tell an accurate history of utility ...
[29]
Were Jevons, Menger, and Walras Really Cardinalists? On the ...
Sep 1, 2013 · This article argues that the canonical dichotomy between cardinal utility and ordinal utility is inadequate to tell an accurate history of ...
[30]
chapter 5 Ordinal Utility: Pareto and the Austrians, 1900–1915
Dec 20, 2018 · Chapter 5 deals with the ordinal revolution in utility analysis inaugurated by Vilfredo Pareto around 1900.Missing: critique | Show results with:critique
[31]
Cardinal Utility: How It Entered Economic Analysis from Pareto to ...
In 1924, Arthur Bowley, an English economist and statistician based at the London School of Economics, published The Mathematical Groundwork of Economics. His ...
[32]
[PDF] Lionel Robbins's essay on the nature and significance of ... - STICERD
In 1932 Robbins set out to inquire about that which defines the subject matter of economic analysis: what economists can and cannot say and the method by ...
[33]
lessons from Lionel Robbins's "Essay" - jstor
the ordinal utility theory that the Essay championed, was that of the ability of individuals to arrange their ends in order of preference (Robbins, 1935, pp.
[34]
A Reconsideration of the Theory of Value. Part I - jstor
A Reconsideration of the Theory of. Value. By J. R. HICKS and R. G. D. ALLEN. Part I. By J. R. HICKS. THE pure theory of exchange value, after a period of ...Missing: ordinalism | Show results with:ordinalism
[35]
[PDF] A Reconsideration of the Theory of Value. Part II. A Mathematical ...
A Reconsideration of the Theory of Value. Part II. A Mathematical Theory of Individual. Demand Functions. Author(s): J. R. Hicks and R. G. D. Allen.Missing: ordinalism | Show results with:ordinalism
[36]
Reconciling introspective utility with revealed preference
Since the beginning of the 20th century, after what has since become known as the ordinal revolution, utility has been taken as an ordinal concept, based solely ...
[37]
[PDF] Review of Ivan Moscati's Measuring Utility
The book reconstructs the history of utility measurement from the 1870s to 1985, arguing early marginalists used ratio-scale utility, not cardinal.<|control11|><|separator|>
[38]
Expected utility – John Quiggin
Dec 27, 2002 · Almost immediately, however, the concept of cardinal utility theory was revived by von Neumann and Morgenstern in their analysis of behavior ...
[39]
Cardinal Utility - jstor
This paper is concerned with the recent revival of the proposition that utility is ... (in my opinion, more neatly than the von Neumann-Morgenstern utility.
[40]
[PDF] Von Neumann-Morgenstern Utility Theory
Von Neumann-Morgenstern utility theory introduces one operation for the combining two lotteries into a third lottery: convex combination. The convex combination ...
[41]
Build your own von Neumann–Morgenstern utility function
May 13, 2018 · The von Neumann--Morgenstern utility theorem lets us turn an ordinal utility function into a cardinal utility function.
[42]
[PDF] Eliciting von Neumann-Morgenstern Utilities When Probabilities are ...
T his paper proposes a new method, the (gamble-)tradeoff method, for eliciting utilities in decision under risk or uncertainty. The elicitation of utilities ...
[43]
[PDF] Choice-Based Cardinal Utility - PhilSci-Archive
Jun 8, 2016 · We reexamine some of the classic problems connected with the use of cardinal utility functions in decision theory, and discuss Patrick Sup-.
[44]
[PDF] Giacomo Bonanno
An affine transformation is a function f : R → R of the form f(x) = ax+b ... referred to as a cardinal utility function. Theorem 3.1.1 guarantees the ...
[45]
Derivation of a Cardinal Utility Through a Weak Trade-off ...
Sep 6, 2013 · An axiomatic model is presented in which a utility function over consequences, unique up to location and unit, is derived.
[46]
Consumer's Equilibrium under Cardinal Utility Analysis - eNotes World
Consumer's equilibrium is the position in which the consumer reaches the highest level of satisfaction given his or her money income and the prices of goods.
[47]
[PDF] TWO
The assumptions on which the theory of cardinal utility is built are very restrictive. Equivalent conclusions can be deduced from much weaker assumptions.
[48]
Cardinal Utility Analysis and Law of Diminishing Returns
Mar 1, 2022 · Cardinal utility analysis attempts to explain the logic behind consumer behaviour by attaching value to the utility derived from the consumption of a commodity.
[49]
3.1 Utility Maximization and Consumer Choice - Fiveable
Consumers balance marginal utilities across different goods to maximize total utility; Marginal analysis helps determine optimal quantity of each good to ...<|separator|>
[50]
Utility Analysis : Total Utility and Marginal Utility - GeeksforGeeks
Jan 30, 2024 · The Cardinal Utility Approach employs the concept of "Utility" to determine the level of satisfaction. What is Utility? The characteristic of a ...
[51]
[PDF] Lecture 3 Axioms of Consumer Preference and the Theory of Choice
The problem with cardinal utility functions comes from the difficulty in finding the appropriate measurement index (metric).
[52]
[PDF] Cardinal Welfare, Individualistic Ethics, and Interpersonal ...
This paper examines the ethical meaning of a postulate, shows it expresses individualistic value judgment, and explores the logical basis for interpersonal ...<|separator|>
[53]
[PDF] Social-Welfare Functions - UNB Scholar - University of New Brunswick
Re-birth (in part), yes; but a return to the pre-Pareto doctrine of an additive cardinal-utility ... whether interpersonal comparisons should be made, but ijv.
[54]
[PDF] Lionel Robbins: Ogre or Prophet? - STICERD
the grounds that interpersonal comparisons of utility are economically meaningless. (Robbins [6]). On the second issue, I think he was a prophet, albeit of ...<|separator|>
[55]
[PDF] 1 Introduction and Outline 1.1 Interpersonal Comparisons
But no matter how the cardinal utility functions are constructed, we have seen how different individuals' ratios of utility differences and utility curvature.
[56]
[PDF] Welfare Economics and Interpersonal Utility Comparisons
Important parts of economics (including parts of welfare economics) do not rely on interpersonal comparisons of cardinal utility, such as 'positively'.
[57]
Decision Theory - Stanford Encyclopedia of Philosophy
Dec 16, 2015 · By the same token, in order to construct or conceptualise a cardinal utility function, one typically appeals to preferences over lotteries.
[58]
[PDF] Von Neumann - Morgenstern Utility Theory
Numbers which are chosen according to an interval scale reflecting the underlying preferences are called cardinal utilities. Cardinal utilities are necessitated ...
[59]
Note on Measurement of Utility | The Review of Economic Studies
Paul A. Samuelson; A Note on Measurement of Utility, The Review of Economic Studies, Volume 4, Issue 2, 1 February 1937 ... D15 - Intertemporal Household ...
[60]
[PDF] Decision over Time as a By-Product of a Measure of Utility - HAL-SHS
Nov 28, 2021 · This contribution aims to highlight a neglected aspect of Samuelson's famous 1937 paper. “A Note on Measurement of Utility”.
[61]
[PDF] Measuring Time Preferences - Keith Marzilli Ericson
Samuelson's discounted utility model has many useful properties first and foremost, it is tractable and parsimonious but it is not designed to explain ...
[62]
Ramsey and Intergenerational Welfare Economics
Jun 1, 2019 · His approach was to apply the Classical-Utilitarian calculus to identify the best match from among attainable and desirable utility streams over ...Production Possibilities in... · The Ramsey Rule and Its...Missing: cardinal | Show results with:cardinal
[63]
[PDF] The Fall and Rise
Section 3 examines the ordinal utility revolution and its con- sequences for intertemporal choice. Rae, Senior, and Jevons: Three Early Perspectives. John Rae, ...<|control11|><|separator|>
[64]
Going Empirical: The Econometric and Experimental Approaches to ...
Samuelson's discounted-utility model of 1937 (see chapter 6, section 6.7.1) explained intertemporal choices using cardinal utility. However, in the late ...
[65]
https://www.nature.com/articles/nature04676
[66]
Encoding of Marginal Utility across Time in the Human Brain - PMC
Marginal utility theory prescribes the relationship between the objective property of the magnitude of rewards and their subjective value.Missing: neuroscientific | Show results with:neuroscientific<|separator|>
[67]
Eliciting von Neumann-Morgenstern Utilities When Probabilities Are ...
This paper proposes a new method, the (gamble-)tradeoff method, for eliciting utilities in decision under risk or uncertainty. The elicitation of utilities, ...
[68]
Interpersonal Comparison of Utility by Measuring Neural Activity
**Summary of Method and Findings:**
[69]
Happiness and Life Satisfaction - Our World in Data
Our focus here will be on survey-based measures of self-reported happiness and life satisfaction. Here is a preview of what the data reveals. Surveys asking ...
[70]
A Happy Possibility About Happiness (And Other) Scales
There are doubts about whether subjective scales—such as happiness surveys—are cardinally comparable; it remains unclear whether these doubts are justified.
[71]
https://www.nber.org/papers/w31707
[72]
[PDF] The Concept of Separate needs in Cardinal Utility Theory
Jul 15, 2024 · The introduction of the concept of separate needs into cardinal utility theory requires two propositions. The first specifies that the shape ...
[73]
Measuring utility: from the marginal revolution to behavioral economics
Aug 7, 2019 · ... revival of interest in the past two decades. To the extent that ... In this context, the distinction between a cardinal utility ...
[74]
[PDF] The Problem with the Concept of Utility and its Measurement
Because of problems with measuring utility, contemporary neoclassical economics has largely retreated from using cardinal utility functions as the basic objects ...
[75]
How Cardinal Utility Entered Economic Analysis, 1909-1944 - SSRN
Jul 23, 2013 · The paper illustrates the methodological and analytical issues that characterized, as well as the personal and institutional aspects that ...Missing: objections | Show results with:objections
[76]
Value Judgements, Positivism and Utility Comparisons in Economics
... Robbins, utility comparisons were subsequently rejected as a value judgement. Robbins' methodological stance is still prevalent among mainstream economists.
[77]
[PDF] Austrian Debates on Utility Measurement from Menger to Hayek
This chapter examines how some of the main exponents of the Austrian school of economics addressed the issues related to the measurability of utility.<|control11|><|separator|>
[78]
[PDF] The Austrian Theory of the Marginal Use and of Ordinal Marginal Utility
When this is done, the Austrian wants-structure will imply that the resulting cardinal utility index on commodity space will be mathematically concave, and ...
[79]
(PDF) Cardinal Utility, Measurability and the Evolution of the ...
Mar 30, 2017 · PDF | This paper examines the evolution of the concept of utility among neoclassical authors from 1871 to 1930. After showing that the ...
[80]
Criticisms of Cardinal Utility Analysis-Microeconomics - eNotes World
The cardinal utility analysis has ignored the phenomenon related to inferior goods. The goods having negative income effects are known as inferior goods and ...Missing: 2010-2025 | Show results with:2010-2025
[81]
[Solved] State the limitations of the cardinal utility theory - Studocu
Inability to Explain Paradoxes: The cardinal utility theory struggles to explain certain paradoxes, such as the Allais paradox and the Ellsberg paradox, which ...
[82]
Reevaluating Cardinal Utility Analysis: A Modern Perspective
Dec 9, 2023 · Cardinal utility analysis proposes that consumers can assign specific numerical values to the satisfaction derived from goods and services.
[83]
Interpersonal Comparison of Utilities - SpringerLink
The exclusion of interpersonal comparisons of utility from formal decision theory stems partly from methodological, partly from ideological considerations.Missing: implications | Show results with:implications
[84]
Why do we need cardinal utility (instead of ordinal ranking)?
Sep 21, 2024 · Without using cardinal utility almost whole field of poverty, inequality, welfare and public economics becomes impossible. With ordinal utility ...Equality of ordinal vs. cardinal utility under transformationsWhat is the concept of ordinal utility? - Economics Stack ExchangeMore results from economics.stackexchange.com
[85]
Murphy on Interpersonal Utility Comparisons - Econlib
May 22, 2015 · With cardinal utility, this is very easy (violate in the area that ... interpersonal comparisons. Von Neumann showed that you could ...
[86]
[PDF] Lecture 3 - Axioms of Consumer Preference and the Theory of Choice
MRS of x for y is the marginal utility of x divided by the marginal utility ... In consumer theory, we choose to use ordinal not cardinal utility functions.
[87]
Cardinal Utility in Welfare Economics and in the Theory of Risk-Taking
Cardinal utility has been kept only in welfare economics to support the demand for a more equal income distribution.
[88]
The estimation of welfare levels of a cardinal utility function
A review. The Individual Welfare Function (IWF), introduced by Van Praag (1968), is a cardinal utility function. It can be measured by means of survey ...
[89]
[PDF] Chapter 2 Decision Theory - MIT OpenCourseWare
2.4 Attitudes Towards Risk. Here, we will relate the attitudes of an individual towards risk to the properties of his von-Neumann-Morgenstern utility function.
[90]
[PDF] Risk Aversion as a Foundation of Expected Utility*
Utility function U that has ex- pected utility representation exhibits risk aversion if and only if the von Neumann-. Morgenstern utility function v is concave.
[91]
Cardinal Utility in Welfare Economics and in the Theory of Risk-taking
Cardinal Utility in Welfare Economics and in the Theory of Risk-taking | Journal of Political Economy: Vol 61, No 5.
[92]
[PDF] CARDINALIST APPROACH AS A MEASURE OF SOCIAL WELFARE
With income subject to diminishing marginal utility, transfers of income from the rich to the poor will increase social welfare by satisfying the more intense ...