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Utility - Wikipedia
equivalence means that maximizing expected utility is equivalent to maximizing the probability of success. In many contexts, this makes the concept of utility easier to justify and to apply. For example, a firm's utility might be the probability of meeting uncertain future customer expectations.[7][8][9][10]
Indirect utility An indirect utility function gives the optimal attainable value of a given utility function, which depends on the prices of the goods and the income or wealth level that the individual possesses.
Money One use of the indirect utility concept is the notion of the utility of money. The (indirect) utility function for money is a nonlinear function that is bounded and asymmetric about the origin. The utility function is concave in the positive region, reflecting the phenomenon of diminishing marginal utility. The boundedness reflects the fact that beyond a certain point money ceases being useful at all, as the size of any economy at any point in time is itself bounded. The asymmetry about the origin reflects the fact that gaining and losing money can have radically different implications both for individuals and businesses. The non-linearity of the utility function for money has profound implications in decision making processes: in situations where outcomes of choices influence utility through gains or losses of money, which are the norm in most business settings, the optimal choice for a given decision depends on the possible outcomes of all other decisions in the same time-period.[11]
Some simple utility functions
Written by Jüri Eintalu in 05. - 08. December 2019
As it follows, x denotes the amount of one's assets (for example, one's money minus one's debts). A common but a strong assumption shall be used:
Linear utility function In the present section, it is assumed that in a given decision problem, the possible outcomes are finite and known. Linear transformations x → ax + b do not change the preference ordering of monetary expectations. Changing the money scale or changing the origin of the money axis neither influences that preference ordering. It is comfortable to make such a transformation that x = 0 corresponds to the minimal possible outcome. Then, it is possible to choose such an utility function
that , and
at the maximal value of x.
If the maximal possible value is 20, then the corresponding linear utility function is shown on the Figure below. All preferences of monetary expectations remain the same, if to use linear utility graphs like the one shown on the Figure. That utility can be interpreted as a probability. The utility u(x) is the win probability in an utility-fair gamble, in which one can loose all one's money x and can achieve, as a result of win, the final assets equal to 20. For example, u(12) = 0.6 . Thus, if one's initial assets are x = 12 , then the following gamble would be monetary fair: with probability p = 0.6 one wins 20 ‒ 12 = 8 units and with probability 1 ‒ p = 0.4 one loses 12 units of money. https://en.wikipedia.org/wiki/Utility#Some_simple_utility_functions
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Linear utility function
However, the linear utility has some shortages. First, if there is no upper bound of the possible wealth, then the linear utility meets with mathematical obstacles. If the point A on the Figure moves to the right, then the angle Îą decreases. In the limit, it diminishes to zero. Second, the linear utility does not take into account the effect of risk aversion. Generally, the decision makers tend to avoid monetary fair bets. Losing 1 unit of money is considered as being more serious change than winning the same amount of money. Also, when earning additional money, every next unit of money earned is considered as being less important than the previous unit earned. Both of these obstacles can be overcame, if to assume that the utility function is concave - if to assume the diminishing marginal utility.
Logarithmic utility function In 1738, Daniel Bernoulli[12] (for the English translation see Bernoulli 1954[13]) introduced the logarithmic utility function (also known as log utility):
This function is concave and it yields to the risk aversion. More importantly, it also yields to the diminishing risk aversion. According to Bernoulli, it is important, that than richer one is, the lesser is one's reluctance against tossing a coin in a fair gamble with the same fixed bet. An infinitely rich man should behave as if one's utility function is linear. It is crucial to understand that the effect of risk aversion is the result of the concaveness of the utility function, while the effect of diminishing risk aversion is not. Unfortunately, Bernoulli's utility function is without lower and upper bounds, which creates shortages. It is possible to use the improved Bernoulli's function
Everything remains the same, except that the utility function has a lower bound now (see the Figure). Unfortunately, this improved utility still lacks the upper bound. The risk aversion indicator or the Arrow-Pratt measure[14] can be defined as follows:
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(In some sources, the right side of this formula has been multiplied with a positive constant.) If Îť > 0, a local risk aversion appears.
In
the
case
of
logarithmic utility u = ln(x + 1),
the
risk
aversion
is
decreasing in x and it is diminishing to zero:
Logarithmic utility function u = ln(x + 1)
See the Figure.
Exponential utility function Often, the following simple exponential
utility
is
presented:
where This
is a parameter. utility
function
is
bounded: Arrow-Pratt measures for logarithmic utilities u = ln(x) and u = ln(x + 1). and it can be interpreted as a probability. If x is the value of the assets of the decision maker, and u(x) above is one's utility function, then one regards one's assets x as equivalent to the gamble, in which with the probability u(x) one wins an infinite amount of money, while otherwise one loses everything. To prove this claim, let p be the win probability in such an utility-fair gamble. Then , therefore , therefore , which concludes the proof. In the case
which can be achieved by changing the money unit, that utility function has a particularly simple form (see the Figure) https://en.wikipedia.org/wiki/Utility#Some_simple_utility_functions
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Exponential utility function with a constant risk aversion Îą = ln(2)
in which case the disutility
is bisected after every additional unit of money obtained. Such an utility is also called as an utility with constant risk aversion. The indicator of risk aversion (the Arrow-Pratt measure) Îť is constant:
This result shows that while the concavity of the utility function produces the effect of risk aversion, it is not enough to secure that this risk aversion is decreasing in x and that it is diminishing to zero in the infinity of x. However, this is a serious obstacle when trying to model mutually motivated contracts, for example, mutually motivated insurance contracts.
Utility function u = x/(x + 1) Already in 1738, Daniel Bernoulli[12], who introduced the notion of utility function, regarded it as important to be able to explain the insurance contracts. Why a rich man is willing to provide the insurance and a poor man is not? The answer consists not in the existence of the risk aversion, but in the effect of decreasing risk aversion. Incidentally, Bernoulli's logarithmic utility u = ln(x) is not only concave, providing the risk aversion, but also with decreasing risk aversion, as was shown above. This risk aversion is diminishing in the infinity. However, Bernoulli's utility function is unbounded. Since then, it has often been asked about whether there was any simple utility function that was bounded and which had a decreasing risk aversion, moreover, a risk aversion diminishing to zero in the infinity (see, for example, Pratt 1964[14]). It has been asked about "everyman's utility function", but usually only the logarithmic or exponential utilities have been mentioned. However, there is a simple utility function , where
is a parameter.
This utility function is bounded: https://en.wikipedia.org/wiki/Utility#Some_simple_utility_functions
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Utility - Wikipedia
Utility function u = x/(x + 1)
and it can be interpreted as a probability. By changing the money unit, the condition b = 1 can be achieved and this utility function obtains a particularly simple form (see the Figure above):
The Arrow-Pratt measure (or the indicator of the risk aversion) is also very simple and it is diminishing in x (see the Figure below):[15]
Utility function u = x/(x + 1), its marginal utility, and its diminishing risk aversion
One group of authors Ikefuji, et al 2013[16] has derived a set of functions they call as "Pareto utilities". The function can be considered as a special case of Pareto utilities. There have been presented various utility functions earlier (for example, in Pratt 1964[14]), having as a special case the function
.
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The function
The utility function u = x/(x + 1) constructed as if a linear utility, if there is always 1 cent missing
has one amazing feature. Technically, at every point x, the value of that function u(x) can be calculated as the value of the linear utility in the case when there is still exactly one unit of money missing from the "ultimate happiness" corresponding to the value characterized as "There is always one cent
missing".[15]
(see the Figure). Therefore, that utility function can been However, as yet, there is no deeper explanation of this feature,
which might as well turn out to be a coincidence. Written by JĂźri Eintalu in 05. - 08. December 2019
Discussion and criticism Cambridge economist Joan Robinson famously criticized utility for being a circular concept: "Utility is the quality in commodities that makes individuals want to buy them, and the fact that individuals want to buy commodities shows that they have utility"[17]:48 Robinson also pointed out that because the theory assumes that preferences are fixed this means that utility is not a testable assumption. This is so because if we take changes in peoples' behavior in relation to a change in prices or a change in the underlying budget constraint we can never be sure to what extent the change in behavior was due to the change in price or budget constraint and how much was due to a change in preferences.[18] This criticism is similar to that of the philosopher Hans Albert who argued that the ceteris paribus conditions on which the marginalist theory of demand rested rendered the theory itself an empty tautology and completely closed to experimental testing.[19] In essence, demand and supply curve (theoretical line of quantity of a product which would have been offered or requested for given price) is purely ontological and could never been demonstrated empirically. Another criticism comes from the assertion that neither cardinal nor ordinal utility is empirically observable in the real world. In the case of cardinal utility it is impossible to measure the level of satisfaction "quantitatively" when someone consumes or purchases an apple. In case of ordinal utility, it is impossible to determine what choices were made when someone purchases, for example, an orange. Any act would involve preference over a vast set of choices (such as apple, orange juice, other vegetable, vitamin C tablets, exercise, not purchasing, etc.).[20][21]
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Other questions of what arguments ought to enter into a utility function are difficult to answer, yet seem necessary to understanding utility. Whether people gain utility from coherence of wants, beliefs or a sense of duty is key to understanding their behavior in the utility organon.[22] Likewise, choosing between alternatives is itself a process of determining what to consider as alternatives, a question of choice within uncertainty.[23] An evolutionary psychology perspective is that utility may be better viewed as due to preferences that maximized evolutionary fitness in the ancestral environment but not necessarily in the current one.[24]
See also Law of demand Marginal utility Utility maximization problem Decision-making software
References 1. Debreu, Gérard (1954), "Representation of a preference ordering by a numerical function", in Thrall, Robert M.; Coombs, Clyde H.; Raiffa, Howard (eds.), Decision processes, New York: Wiley, pp. 159–167, OCLC 639321 (htt ps://www.worldcat.org/oclc/639321). 2. Marshall, Alfred (1920). Principles of Economics. An introductory volume (8th ed.). London: Macmillan. 3. Von Neumann, J.; Morgenstern, O. (1953). Theory of Games and Economic Behavior (https://archive.org/details/t heoryofgameseco00vonn) (3rd ed.). Princeton University Press. 4. Kahneman, D.; Tversky, A. (1979). "Prospect Theory: An Analysis of Decision Under Risk" (http://www.immagic.c om/eLibrary/ARCHIVES/GENERAL/JOURNALS/E790301K.pdf) (PDF). Econometrica. 47 (2): 263–292. doi:10.2307/1914185 (https://doi.org/10.2307%2F1914185). 5. Grechuk, B.; Zabarankin, M. (2016). "Inverse Portfolio Problem with Coherent Risk Measures". European Journal of Operational Research. 249 (2): 740–750. doi:10.1016/j.ejor.2015.09.050 (https://doi.org/10.1016%2Fj.ejor.201 5.09.050). 6. Ingersoll, Jonathan E., Jr. (1987). Theory of Financial Decision Making (https://archive.org/details/theoryoffinancia 1987inge). Totowa: Rowman and Littlefield. p. 21 (https://archive.org/details/theoryoffinancia1987inge/page/21). ISBN 0-8476-7359-6. 7. Castagnoli, E.; LiCalzi, M. (1996). "Expected Utility Without Utility" (https://iris.unive.it/bitstream/10278/4143/1/LiC alzi-96-ExpectedUtilityWithoutUtility.pdf) (PDF). Theory and Decision. 41 (3): 281–301. doi:10.1007/BF00136129 (https://doi.org/10.1007%2FBF00136129). 8. Bordley, R.; LiCalzi, M. (2000). "Decision Analysis Using Targets Instead of Utility Functions". Decisions in Economics and Finance. 23 (1): 53–74. doi:10.1007/s102030050005 (https://doi.org/10.1007%2Fs10203005000 5). hdl:10278/3610 (https://hdl.handle.net/10278%2F3610). 9. Bordley, R.; Kirkwood, C. (2004). "Multiattribute preference analysis with Performance Targets". Operations Research. 52 (6): 823–835. doi:10.1287/opre.1030.0093 (https://doi.org/10.1287%2Fopre.1030.0093). 10. Bordley, R.; Pollock, S. (2009). "A Decision-Analytic Approach to Reliability-Based Design Optimization". Operations Research. 57 (5): 1262–1270. doi:10.1287/opre.1080.0661 (https://doi.org/10.1287%2Fopre.1080.066 1). 11. Berger, J. O. (1985). "Utility and Loss". Statistical Decision Theory and Bayesian Analysis (2nd ed.). Berlin: Springer-Verlag. ISBN 3-540-96098-8. 12. Bernoulli, Daniel (1738). "Specimen Theoriae Novae de Mensura Sortis". Commentarii Academiae Scientiarum Imperialis Petropolitanae. 5: 175–192. 13. Bernoulli, Daniel (1954). "Exposition of a New Theory on the Measurement of Risk" (https://www.jstor.org/stable/1 909829?origin=crossref&seq=1). Econometrica. 22: 23–36 – via JSTOR. 14. Pratt, John (1964). "Risk Aversion in the Small and in the Large" (https://www.jstor.org/stable/1913738?origin=cro ssref&seq=1). Econometrica. 32(1/2): 122–136 – via JSTOR. 15. Eintalu, J. (2019). Utility Function u = x/(x + 1). Berlin: Lambert Academic Publishing. ISBN 978-620-0-30723-1. https://en.wikipedia.org/wiki/Utility#Some_simple_utility_functions
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16. Ikefuji, Masako (2013). "Pareto Utility" (https://link.springer.com/article/10.1007%2Fs11238-012-9293-8). Theory and Decision. 75 (1): 43–57 – via Springer. 17. Robinson, Joan (1962). Economic Philosophy. Harmondsworth, Middle-sex, UK: Penguin Books. 18. Pilkington, Philip (17 February 2014). "Joan Robinson's Critique of Marginal Utility Theory" (https://fixingtheecono mists.wordpress.com/2014/02/17/joan-robinsons-critique-of-marginal-utility-theory/). Fixing the Economists. Archived (https://web.archive.org/web/20150713112846/https://fixingtheeconomists.wordpress.com/2014/02/17/jo an-robinsons-critique-of-marginal-utility-theory/) from the original on 13 July 2015. 19. Pilkington, Philip (27 February 2014). "utility Hans Albert Expands Robinson's Critique of Marginal Utility Theory to the Law of Demand" (https://fixingtheeconomists.wordpress.com/2014/02/27/hans-albert-expands-robinsons-cri tique-of-marginal-utility-theory-to-the-law-of-demand/). Fixing the Economists. Archived (https://web.archive.org/w eb/20150719164628/https://fixingtheeconomists.wordpress.com/2014/02/27/hans-albert-expands-robinsons-critiq ue-of-marginal-utility-theory-to-the-law-of-demand/) from the original on 19 July 2015. 20. "Revealed Preference Theory" (https://web.archive.org/web/20110716090409/http://www.societies.cam.ac.uk/cuji f/ABSTRACT/980606.htm). Archived from the original (http://www.societies.cam.ac.uk/cujif/ABSTRACT/980606.ht m) on 16 July 2011. Retrieved 11 December 2009. 21. "Archived copy" (https://web.archive.org/web/20081015203300/http://elsa.berkeley.edu/~botond/mistakeschicago. pdf) (PDF). Archived from the original (http://elsa.berkeley.edu/~botond/mistakeschicago.pdf) (PDF) on 15 October 2008. Retrieved 9 August 2008. 22. Klein, Daniel (May 2014). "Professor" (http://econjwatch.org/file_download/826/CompleteIssueMay2014.pdf) (PDF). Econ Journal Watch. 11 (2): 97–105. Archived (https://web.archive.org/web/20141005153234/http://econjw atch.org/file_download/826/CompleteIssueMay2014.pdf) (PDF) from the original on 5 October 2014. Retrieved 15 November 2014. 23. Burke, Kenneth (1932). Towards a Better Life. Berkeley, Calif: University of California Press. 24. Capra, C. Monica; Rubin, Paul H. (2011). "The Evolutionary Psychology of Economics" (https://doi.org/10.1093/ac prof:oso/9780199586073.003.0002). Applied Evolutionary Psychology. Oxford University Press. doi:10.1093/acprof:oso/9780199586073.003.0002 (https://doi.org/10.1093%2Facprof%3Aoso%2F978019958607 3.003.0002). ISBN 9780191731358.
Further reading Anand, Paul (1993). Foundations of Rational Choice Under Risk (https://archive.org/details/foundationsofrat00ana n). Oxford: Oxford University Press. ISBN 0-19-823303-5. Bernoulli, Daniel (1738). "Specimen Theoriae Novae de Mensura Sortis". Commentarii Academiae Scientiarum Imperialis Petropolitanae (5): 175–192. Bernoulli, Daniel (1954). "Exposition of a New Theory on the Measurement of Risk". Econometrica (22): 23–36. doi:10.2307/1909829 (https://doi.org/10.2307%2F1909829). JSTOR 1909829 (https://www.jstor.org/stable/190982 9). Fishburn, Peter C. (1970). Utility Theory for Decision Making. Huntington, NY: Robert E. Krieger. ISBN 0-88275736-9. Georgescu-Roegen, Nicholas (August 1936). "The Pure Theory of Consumer's Behavior". Quarterly Journal of Economics. 50 (4): 545–593. doi:10.2307/1891094 (https://doi.org/10.2307%2F1891094). JSTOR 1891094 (http s://www.jstor.org/stable/1891094). Gilboa, Itzhak (2009). Theory of Decision under Uncertainty. Cambridge: Cambridge University Press. ISBN 9780-521-74123-1. Ikefuji, Masako (2013). "Pareto Utility". Theory and Decision. 75 (1): 43–57. doi:10.1007/s11238-012-9293-8 (http s://doi.org/10.1007%2Fs11238-012-9293-8). Jensen, Niels Erik (1967). "An Introduction to Bernoullian Utility Theory: I. Utility Functions". The Swedish Journal of Economics. 69 (3): 163–183. doi:10.2307/3439089 (https://doi.org/10.2307%2F3439089). JSTOR 3439089 (htt ps://www.jstor.org/stable/3439089). Kreps, David M. (1988). Notes on the Theory of Choice. Boulder, CO: West-view Press. ISBN 0-8133-7553-3. Lindley, Dennis (1985). Making Decisions. London: Wiley. Nash, John F. (1950). "The Bargaining Problem". Econometrica. 18 (2): 155–162. doi:10.2307/1907266 (https://do i.org/10.2307%2F1907266). JSTOR 1907266 (https://www.jstor.org/stable/1907266). Neumann, John von & Morgenstern, Oskar (1944). Theory of Games and Economic Behavior (https://archive.org/ details/in.ernet.dli.2015.215284). Princeton, NJ: Princeton University Press. Nicholson, Walter (1978). Micro-economic Theory (Second ed.). Hinsdale: Dryden Press. pp. 53–87. ISBN 0-03020831-9. https://en.wikipedia.org/wiki/Utility#Some_simple_utility_functions
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