marginal analysis

Marginalism is a theory of economics that attempts to explain the discrepancy in the value of goods and services by reference to their secondary, or marginal, utility. It states that the reason why the price of diamonds is higher than that of water, for example, owes to the greater additional satisfaction of the diamonds over the water. Thus, while the water has greater total utility, the diamond has greater marginal utility.

Although the central concept of marginalism is that of marginal utility, marginalists, following the lead of Alfred Marshall, drew upon the idea of marginal physical productivity in explanation of cost. The neoclassical tradition that emerged from British marginalism abandoned the concept of utility and gave marginal rates of substitution a more fundamental role in analysis. Marginalism is an integral part of mainstream economic theory.

Important marginal concepts

Marginality

For issues of marginality, constraints are conceptualized as a ''border'' or ''margin''. The location of the margin for any individual corresponds to his or her ''endowment'', broadly conceived to include opportunities. This endowment is determined by many things including physical laws (which constrain how forms of energy and matter may be transformed), accidents of nature (which determine the presence of natural resources), and the outcomes of past decisions made both by others and by the individual.

A value that holds true given particular constraints is a ''marginal'' value. A change that would be affected as or by a specific loosening or tightening of those constraints is a ''marginal'' change.

Neoclassical economics usually assumes that marginal changes are infinitesimals or limits. Although this assumption makes the analysis less robust, it increases tractability. One is therefore often told that "marginal" is synonymous with "very small", though in more general analysis this may not be operationally true and would not in any case be literally true. Frequently, economic analysis concerns the marginal values associated with a change of one unit of a resource, because decisions are often made in terms of units; marginalism seeks to explain unit prices in terms of such marginal values.

Marginal use

The marginal use of a good or service is the specific use to which an agent would put a given increase, or the specific use of the good or service that would be abandoned in response to a given decrease.von Wieser, Friedrich; ''Über den Ursprung und die Hauptgesetze des wirtschaftlichen Wertes'' /nowiki>''The Nature and Essence of Theoretical Economics''/nowiki> (1884), p. 128.

Marginalism assumes, for any given agent, economic rationality and an ordering of possible states-of-the-world, such that, for any given set of constraints, there is an attainable state which is best in the eyes of that agent. Descriptive marginalism asserts that choice amongst the specific means by which various anticipated specific states-of-the-world (outcomes) might be affected is governed only by the distinctions amongst those specific outcomes; prescriptive marginalism asserts that such choice ''ought'' to be so governed.

On such assumptions, each increase would be put to the specific, feasible, previously unrealized use of greatest priority, and each decrease would result in abandonment of the use of lowest priority amongst the uses to which the good or service had been put.

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 '' 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",von Wieser, Friedrich; ''Der natürliche Werth'' /nowiki>''Natural Value''/nowiki> (1889), Bk I Ch V "Marginal Utility"
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. 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.Mc Culloch, James Huston; "The Austrian Theory of the Marginal Use and of Ordinal Marginal Utility", ''Zeitschrift für Nationalökonomie'' 37 (1973) #3&4 (September). On the other hand, none of the early marginalists insisted that utility were ''not'' quantified, 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 $S_1$ to state $S_2$ is
:$\Delta U=U(S_2)-U(S_1)\,$
Moreover, if $S_1$ and $S_2$ are distinguishable by values of just one variable $g\,$ which is itself quantified, then it becomes possible to speak of the ratio of the marginal utility of the change in $g\,$ to the size of that change:
:$\left.\frac\_$
(where “ c.p.” indicates that the ''only'' independent variable to change is $g\,$).

Mainstream neoclassical economics will typically assume that
:$\lim_$
is well defined, and use "marginal utility" to refer to a partial derivative
:$\frac\approx\left.\frac\_$

Law of diminishing marginal utility

The law of diminishing marginal utility, also known as a 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, sometimes as something proven by introspection, or sometimes as a mere 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 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.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''.

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 arithmetic subtraction. It is the movement from use of higher to lower priority, and may be no more than a purely ordinal change. Theodore-Angwenyi, Nicholas; "Utility", ''International Encyclopedia of the Social Sciences'' (1968).

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
:$\frac<0$
Neoclassical economics usually supplements or supplants discussion of marginal utility with indifference curves, which were originally derived as the level curves of utility functions,Edgeworth, Francis Ysidro
''Mathematical Psychics''
(1881). or can be produced without presumption of quantification, but are often simply treated as axiomatic. In the absence of complementarity of goods or services, diminishing marginal utility implies convexity of indifference curves, although such convexity would also follow from quasiconcavity of the utility function.

Marginal rate of substitution

The ''rate of substitution'' is the ''least favorable'' rate at which an agent is willing to exchange units of one good or service for units of another. The ''marginal'' rate of substitution (MRS) is the rate of substitution at the margin; in other words, given some constraint.

When goods and services are discrete, the least favorable rate at which an agent would trade A for B will usually be different from that at which she would trade B for A:
:$MRS_ \neq \frac1$

When the goods and services are continuously divisible in the limiting case
:$MRS_ = \frac1$
and the marginal rate of substitution is the slope of the indifference curve (multiplied by $-1$).

If, for example, Lisa will not trade a goat for anything less than two sheep, then her
:$MRS_ = \frac\text$

If she will not trade a sheep for anything less than two goats, then her
:$MRS_ = \frac\text \neq \frac = \frac1 = \frac1$

However, if she would trade one gram of banana for one ounce of ice cream ''and vice versa'', then
:$MRS_ = \frac = \frac1 = \frac1$

When indifference curves (which are essentially graphs of instantaneous rates of substitution) and the convexity of those curves are not taken as given, the "law" of diminishing marginal utility is invoked to explain diminishing marginal rates of substitution – a willingness to accept fewer units of good or service $A$ in substitution for $B$ as one's holdings of $A$ grow relative to those of $B$. If an individual has a stock or flow of a good or service whose marginal utility is less than would be that of some other good or service for which he or she could trade, then it is in his or her interest to effect that trade. As one thing is traded-away and another is acquired, the respective marginal gains or losses from further trades are now changed. On the assumption that the marginal utility of one is diminishing, and the other is not increasing, all else being equal, an individual will demand an increasing ratio of that which is acquired to that which is sacrificed. One important way in which all else might not be equal is when the use of the one good or service complements that of the other. In such cases, exchange ratios might be constant. If any trader can better his or her own marginal position by offering an exchange more favorable to other traders with desired goods or services, then he or she will do so.

Marginal cost

At the highest level of generality, a marginal cost is a marginal opportunity cost. In most contexts, marginal cost refers to marginal ''pecuniary

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