Editing Marginalism (section)

=== Marginal utility ===
{{main|Marginal utility}}
The marginal utility of a good or service is the utility of its [[marginal use]].  Under the assumption of economic rationality, it is the utility of its least urgent possible use ''from'' the best feasible combination of actions in which its use is included.

In 20th century [[mainstream economics]], the term "[[utility]]" has come to be formally defined as a ''[[Measure (mathematics)|quantification]]'' capturing preferences by assigning greater quantities to states, goods, services, or applications that are of higher priority.  But marginalism and the concept of marginal utility predate the establishment of this convention within economics.  The more general conception of utility is that of ''use'' or ''usefulness'', and this conception is at the heart of marginalism; the term "marginal utility" arose from translation of the German "Grenznutzen",<ref name="wieser_ein" /><ref name="wieser_zwei">von Wieser, Friedrich; ''Der natürliche Werth'' <nowiki>[</nowiki>''Natural Value''<nowiki>]</nowiki> (1889), Bk I Ch V "Marginal Utility" ([http://praxeology.net/FW-NV-I-5.htm HTML]).</ref> which literally means ''border use'', referring directly to the marginal use, and the more general formulations of marginal utility do not treat quantification as an ''essential'' feature.<ref name="mc_culloch">Mc Culloch, James Huston; "The Austrian Theory of the Marginal Use and of Ordinal Marginal Utility", ''[[Zeitschrift für Nationalökonomie]]'' 37 (1973) #3&amp;4 (September).</ref> On the other hand, none of the early marginalists insisted that utility were ''not'' quantified,<ref>[[George Stigler|Stigler, George Joseph]]; "The Development of Utility Theory" ''[[Journal of Political Economy]]'' (1950).</ref><ref>[[George Stigler|Stigler, George Joseph]]; "The Adoption of Marginal Utility Theory" ''History of Political Economy'' (1972).</ref> some indeed treated quantification as an essential feature, and those who did not still used an assumption of quantification for expository purposes.  In this context, it is not surprising to find many presentations that fail to recognize a more general approach.

==== Quantified marginal utility ====
Under the [[special case]] in which usefulness can be quantified, the change in utility of moving from state <math>S_1</math> to state <math>S_2</math> is
:<math>\Delta U=U(S_2)-U(S_1)\,</math>
Moreover, if <math>S_1</math> and <math>S_2</math> are distinguishable by values of just one variable <math>g\,</math> which is itself quantified, then it becomes possible to speak of the ratio of the marginal utility of the change in <math>g\,</math> to the size of that change:
:<math>\left.\frac{\Delta U}{\Delta g}\right|_{c.p.}</math>
(where "[[ceteris paribus|c.p.]]" indicates that the ''only'' [[Dependent and independent variables|independent variable]] to change is <math>g\,</math>).

Mainstream neoclassical economics will typically assume that
:<math>\lim_{\Delta g\to 0}{\left.\frac{\Delta U}{\Delta g}\right|_{c.p.}}</math>
is well defined, and use "marginal utility" to refer to a [[partial derivative]]
:<math>\frac{\partial U}{\partial g}\approx\left.\frac{\Delta U}{\Delta g}\right|_{c.p.}</math>

==== Law of diminishing marginal utility ====
The law of diminishing marginal utility, also known as a [[Hermann Heinrich Gossen|Gossen]]'s First Law, is that ''[[ceteris paribus]]'', as additional amounts of a good or service are added to available resources, their marginal utilities are decreasing. This law is sometimes treated as a [[tautology (logic)|tautology]], sometimes as something proven by introspection, or sometimes as a mere [[Instrumentalism|instrumental]] assumption, adopted only for its perceived predictive efficacy. It is not quite any of these things, although it may have aspects of each. The law does not hold under all circumstances, so it is neither a tautology nor otherwise proveable; but it has a basis in prior observation.

An individual will typically be able to [[Partially ordered set|partially order]] the potential uses of a good or service. If there is [[scarcity]], then a rational agent will satisfy wants of highest possible priority, so that no want is avoidably sacrificed to satisfy a want of ''lower'' priority.  In the absence of complementarity across the uses, this will imply that the priority of use of any additional amount will be lower than the priority of the established uses, as in this famous example:
:A pioneer farmer had five sacks of grain, with no way of selling them or buying more. He had five possible uses: as basic feed for himself, food to build strength, food for his chickens for dietary variation, an ingredient for making whisky and feed for his parrots to amuse him. Then the farmer lost one sack of grain. Instead of reducing every activity by a fifth, the farmer simply starved the parrots as they were of less utility than the other four uses; in other words they were on the margin. And it is on the margin, and not with a view to the big picture, that we make economic decisions.<ref name="bawerk_capital">Böhm-Bawerk, Eugen Ritter von; ''Kapital Und Kapitalizns. Zweite Abteilung: Positive Theorie des Kapitales'' (1889). Translated as ''Capital and Interest. II: Positive Theory of Capital'' with appendices rendered as ''Further Essays on Capital and Interest''.</ref>
[[Image:Marginal-utility.png|thumb|right|Diminishing marginal utility, given quantification]]
However, if there ''is'' a complementarity across uses, then an amount added can bring things past a desired tipping point, or an amount subtracted cause them to fall short.  In such cases, the marginal utility of a good or service might actually be ''increasing''.

Without the presumption that utility is quantified, the ''diminishing'' of utility should not be taken to be itself an [[Elementary arithmetic|arithmetic]] [[subtraction]]. It is the movement from use of higher to lower priority, and may be no more than a purely [[Ranking|ordinal]] change.<ref name="mc_culloch" /><ref name="georgescu-rogen">[[Nicholas Georgescu-Roegen|Theodore-Angwenyi, Nicholas]]; "Utility", ''International Encyclopedia of the Social Sciences'' (1968).</ref>

When quantification of utility is assumed, diminishing marginal utility corresponds to a utility function whose ''[[slope]]'' is continually or continuously decreasing. In the latter case, if the function is also smooth, then the law may be expressed as
:<math>\frac{\partial^2 U}{\partial g^2}<0</math>
Neoclassical economics usually supplements or supplants discussion of marginal utility with [[indifference curve]]s, which were originally derived as the [[level curve]]s of utility functions,<ref name="edgeworth">Edgeworth, Francis Ysidro; [http://socserv.mcmaster.ca/econ/ugcm/3ll3/edgeworth/mathpsychics.pdf ''Mathematical Psychics''] (1881).</ref> or can be produced without presumption of quantification,<ref name="mc_culloch" /> but are often simply treated as axiomatic. In the absence of complementarity of goods or services, diminishing marginal utility implies [[Convex function|convexity]] of indifference curves,<ref name="mc_culloch" /><ref name="edgeworth" /> although such convexity would also follow from [[quasiconcavity]] of the utility function.