WikiMili The Free Encyclopedia

In economics, an **indifference curve** connects points on a graph representing different quantities of two goods, points between which a consumer is *indifferent*. That is, any combinations of two products indicated by the curve will provide the consumer with equal levels of utility, and the consumer has no preference for one combination or bundle of goods over a different combination on the same curve. One can also refer to each point on the indifference curve as rendering the same level of utility (satisfaction) for the consumer. In other words, an indifference curve is the locus of various points showing different combinations of two goods providing equal utility to the consumer. Utility is then a device to represent preferences rather than something from which preferences come.^{ [1] } The main use of indifference curves is in the representation of potentially observable demand patterns for individual consumers over commodity bundles.^{ [2] }

**Economics** is the social science that studies the production, distribution, and consumption of goods and services.

In economics and other social sciences, **preference** is the order that a person gives to alternatives based on their relative utility, a process which results in an optimal "choice". Instead of the prices of goods, personal income, or availability of goods, the character of the preferences is determined purely by a person's tastes. However, persons are still expected to act in their best interest.

Within economics, the concept of **utility** is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or satisfaction within the theory of utilitarianism by moral philosophers such as Jeremy Bentham and John Stuart Mill. The term has been adapted and reapplied within neoclassical economics, which dominates modern economic theory, as a **utility function** that represents a consumer's preference ordering over a choice set. It is devoid of its original interpretation as a measurement of the pleasure or satisfaction obtained by the consumer from that choice.

- History
- Map and properties of indifference curves
- Assumptions of consumer preference theory
- Application
- Examples of indifference curves
- Preference relations and utility
- Preference relations
- Formal link to utility theory
- Examples
- Criticisms
- See also
- Notes
- References
- Further reading
- External links

There are infinitely many indifference curves: one passes through each combination. A collection of (selected) indifference curves, illustrated graphically, is referred to as an **indifference map**.

The theory of indifference curves was developed by Francis Ysidro Edgeworth, who explained in his 1881 book the mathematics needed for their drawing;^{ [3] } later on, Vilfredo Pareto was the first author to actually draw these curves, in his 1906 book.^{ [4] }^{ [5] } The theory can be derived from William Stanley Jevons' ordinal utility theory, which posits that individuals can always rank any consumption bundles by order of preference.^{ [6] }

**Francis Ysidro Edgeworth** was an Anglo-Irish philosopher and political economist who made significant contributions to the methods of statistics during the 1880s. From 1891 onward, he was appointed the founding editor of *The Economic Journal*.

**Vilfredo Federico Damaso Pareto** was an Italian engineer, sociologist, economist, political scientist, and philosopher. He made several important contributions to economics, particularly in the study of income distribution and in the analysis of individuals' choices. He was also responsible for popularising the use of the term "elite" in social analysis.

**William Stanley Jevons** FRS was an English economist and logician.

A graph of indifference curves for several utility levels of an individual consumer is called an **indifference map**. Points yielding different utility levels are each associated with distinct indifference curves and these indifference curves on the indifference map are like contour lines on a topographical graph. Each point on the curve represents the same elevation. If you move "off" an indifference curve traveling in a northeast direction (assuming positive marginal utility for the goods) you are essentially climbing a mound of utility. The higher you go the greater the level of utility. The non-satiation requirement means that you will never reach the "top," or a "bliss point," a consumption bundle that is preferred to all others.

In economics, the **bliss point** is a quantity of consumption where any further increase would make the consumer less satisfied. It is a quantity of consumption which maximizes utility in the absence of budget constraint. In other words, it refers to the amount of consumption that would be chosen by a person so rich that money imposed no constraint on his or her decisions.

Indifference curves are typically^{[ vague ]} represented^{[ clarification needed ]} to be:

- Defined only in the non-negative quadrant of commodity quantities (i.e. the possibility of having negative quantities of any good is ignored).
- Negatively sloped. That is, as quantity consumed of one good (X) increases, total satisfaction would increase
^{[ clarification needed ]}if not offset by a decrease in the quantity consumed of the other good (Y). Equivalently, satiation, such that more of either good (or both) is equally preferred to no increase, is excluded.^{[ clarification needed ]}(If utility*U = f(x, y)*,*U*, in the third dimension, does not have a local maximum for any*x*and*y*values.)^{[ clarification needed ]}The negative slope of the indifference curve reflects the assumption of the monotonicity of consumer's preferences, which generates monotonically increasing utility functions, and the assumption of non-satiation (marginal utility for all goods is always positive); an upward sloping indifference curve would imply that a consumer is indifferent between a bundle A and another bundle B because they lie on the same indifference curve, even in the case in which the quantity of both goods in bundle B is higher. Because of monotonicity of preferences and non-satiation, a bundle with more of both goods must be preferred to one with less of both, thus the first bundle must yield a higher utility, and lie on a different indifference curve at a higher utility level. The negative slope of the indifference curve implies that the marginal rate of substitution is always positive; - Complete, such that all points on an indifference curve are ranked equally preferred and ranked either more or less preferred than every other point not on the curve. So, with (2), no two curves can intersect (otherwise non-satiation would be violated).
- Transitive with respect to points on distinct indifference curves. That is, if each point on
*I*is (strictly) preferred to each point on_{2}*I*, and each point on_{1}*I*is preferred to each point on_{3}*I*, each point on_{2}*I*is preferred to each point on_{3}*I*. A negative slope and transitivity exclude indifference curves crossing, since straight lines from the origin on both sides of where they crossed would give opposite and intransitive preference rankings._{1} - (Strictly) convex. With (2), convex preferences
^{[ clarification needed ]}imply that the indifference curves cannot be concave to the origin, i.e. they will either be straight lines or bulge toward the origin of the indifference curve. If the latter is the case, then as a consumer decreases consumption of one good in successive units, successively larger doses of the other good are required to keep satisfaction unchanged.

- Preferences are
**complete**. The consumer has ranked all available alternative combinations of commodities in terms of the satisfaction they provide him.

- Assume that there are two consumption bundles
*A*and*B*each containing two commodities*x*and*y*. A consumer can unambiguously determine that one and only one of the following is the case:*A*is preferred to*B*, formally written as*A*^{p}*B*^{ [7] }*B*is preferred to*A*, formally written as*B*^{p}*A*^{ [7] }*A*is indifferent to*B*, formally written as*A*^{I}*B*^{ [7] }

- This axiom precludes the possibility that the consumer cannot decide,
^{ [8] }It assumes that a consumer is able to make this comparison with respect to every conceivable bundle of goods.^{ [7] }

- Preferences are
**reflexive**

- This means that if
*A*and*B*are identical in all respects the consumer will recognize this fact and be indifferent in comparing*A*and*B**A*=*B*⇒*A*^{I}*B*^{ [7] }

- Preferences are
**transitive**^{ [nb 1] }

- If
*A*^{p}*B*and*B*^{p}*C*, then*A*^{p}*C*.^{ [7] } - Also if
*A*^{I}*B*and*B*^{I}*C*, then*A*^{I}*C*.^{ [7] }

- If
- This is a consistency assumption.

- Preferences are
**continuous**

- If
*A*is preferred to*B*and*C*is sufficiently close to*B*then*A*is preferred to*C*. *A*^{p}*B*and*C*→*B*⇒*A*^{p}*C*.

- If
- "Continuous" means infinitely divisible - just like there are infinitely many numbers between 1 and 2 all bundles are infinitely divisible. This assumption makes indifference curves continuous.

- Preferences exhibit
**strong monotonicity**

- If
*A*has more of both*x*and*y*than*B*, then*A*is preferred to*B*.

- If
- This assumption is commonly called the "more is better" assumption.
- An alternative version of this assumption requires that if
*A*and*B*have the same quantity of one good, but*A*has more of the other, then*A*is preferred to*B*.

It also implies that the commodities are **good** rather than **bad**. Examples of **bad** commodities can be disease, pollution etc. because we always desire less of such things.

- Indifference curves exhibit
**diminishing marginal rates of substitution**

- The marginal rate of substitution tells how much 'y' a person is willing to sacrifice to get one more unit of 'x'.
^{[ clarification needed ]} - This assumption assures that indifference curves are smooth and convex to the origin.
- This assumption also set the stage for using techniques of constrained optimization because the shape of the curve assures that the first derivative is negative and the second is positive.
- Another name for this assumption is the
**substitution assumption**. It is the most critical assumption of consumer theory: Consumers are willing to give up or trade-off some of one good to get more of another. The fundamental assertion is that there is a maximum amount that "a consumer will give up, of one commodity, to get one unit of another good, in that amount which will leave the consumer indifferent between the new and old situations"^{ [9] }The negative slope of the indifference curves represents the willingness of the consumer to make a trade off.^{ [9] }

- The marginal rate of substitution tells how much 'y' a person is willing to sacrifice to get one more unit of 'x'.

Consumer theory uses indifference curves and budget constraints to generate consumer demand curves. For a single consumer, this is a relatively simple process. First, let one good be an example market e.g., carrots, and let the other be a composite of all other goods. Budget constraints give a straight line on the indifference map showing all the possible distributions between the two goods; the point of maximum utility is then the point at which an indifference curve is tangent to the budget line (illustrated). This follows from common sense: if the market values a good more than the household, the household will sell it; if the market values a good less than the household, the household will buy it. The process then continues until the market's and household's marginal rates of substitution are equal.^{ [10] } Now, if the price of carrots were to change, and the price of all other goods were to remain constant, the gradient of the budget line would also change, leading to a different point of tangency and a different quantity demanded. These price / quantity combinations can then be used to deduce a full demand curve.^{ [10] } A line connecting all points of tangency between the indifference curve and the budget constraint is called the expansion path.^{ [11] }

- Figure 1: An example of an indifference map with three indifference curves represented
- Figure 2: Three indifference curves where Goods
*X*and*Y*are perfect substitutes. The gray line perpendicular to all curves indicates the curves are mutually parallel. - Figure 3: Indifference curves for perfect complements
*X*and*Y*. The elbows of the curves are collinear.

In Figure 1, the consumer would rather be on *I _{3}* than

If two goods are perfect substitutes then the indifference curves will have a constant slope since the consumer would be willing to switch between at a fixed ratio. The marginal rate of substitution between perfect substitutes is likewise constant. An example of a utility function that is associated with indifference curves like these would be .

If two goods are perfect complements then the indifference curves will be L-shaped. Examples of perfect complements include left shoes compared to right shoes: the consumer is no better off having several right shoes if she has only one left shoe - additional right shoes have zero marginal utility without more left shoes, so bundles of goods differing only in the number of right shoes they include - however many - are equally preferred. The marginal rate of substitution is either zero or infinite. An example of the type of utility function that has an indifference map like that above is the Leontief function: .

The different shapes of the curves imply different responses to a change in price as shown from demand analysis in consumer theory. The results will only be stated here. A price-budget-line change that kept a consumer in equilibrium on the same indifference curve:

- in Fig. 1 would reduce quantity demanded of a good smoothly as price rose relatively for that good.
- in Fig. 2 would have either no effect on quantity demanded of either good (at one end of the budget constraint) or would change quantity demanded from one end of the budget constraint to the other.
- in Fig. 3 would have no effect on equilibrium quantities demanded, since the budget line would rotate around the corner of the indifference curve.
^{ [nb 2] }

Choice theory formally represents consumers by a **preference relation**, and use this representation to derive indifference curves showing combinations of equal preference to the consumer.

Let

- be a set of mutually exclusive alternatives among which a consumer can choose.
- and be generic elements of .

In the language of the example above, the set is made of combinations of apples and bananas. The symbol is one such combination, such as 1 apple and 4 bananas and is another combination such as 2 apples and 2 bananas.

A preference relation, denoted , is a binary relation define on the set .

The statement

is described as ' is weakly preferred to .' That is, is at least as good as (in preference satisfaction).

The statement

is described as ' is weakly preferred to , and is weakly preferred to .' That is, one is *indifferent* to the choice of or , meaning not that they are unwanted but that they are equally good in satisfying preferences.

The statement

is described as ' is weakly preferred to , but is not weakly preferred to .' One says that ' is strictly preferred to .'

The preference relation is **complete** if all pairs can be ranked. The relation is a transitive relation if whenever and then .

For any element , the corresponding indifference curve, is made up of all elements of which are indifferent to . Formally,

.

In the example above, an element of the set is made of two numbers: The number of apples, call it and the number of bananas, call it

In utility theory, the utility function of an agent is a function that ranks *all* pairs of consumption bundles by order of preference (*completeness*) such that any set of three or more bundles forms a transitive relation. This means that for each bundle there is a unique relation, , representing the utility (satisfaction) relation associated with . The relation is called the utility function. The range of the function is a set of real numbers. The actual values of the function have no importance. Only the ranking of those values has content for the theory. More precisely, if , then the bundle is described as at least as good as the bundle . If , the bundle is described as strictly preferred to the bundle .

Consider a particular bundle and take the total derivative of about this point:

or, without loss of generality,

**(Eq. 1)**

where is the partial derivative of with respect to its first argument, evaluated at . (Likewise for )

The indifference curve through must deliver at each bundle on the curve the same utility level as bundle . That is, when preferences are represented by a utility function, the indifference curves are the level curves of the utility function. Therefore, if one is to change the quantity of by , without moving off the indifference curve, one must also change the quantity of by an amount such that, in the end, there is no change in *U*:

- , or, substituting
*0*into*(Eq. 1)*above to solve for*dy/dx*: - .

Thus, the ratio of marginal utilities gives the absolute value of the slope of the indifference curve at point . This ratio is called the marginal rate of substitution between and .

If the utility function is of the form then the marginal utility of is and the marginal utility of is . The slope of the indifference curve is, therefore,

Observe that the slope does not depend on or : the indifference curves are straight lines.

If the utility function is of the form the marginal utility of is and the marginal utility of is .Where . The slope of the indifference curve, and therefore the negative of the marginal rate of substitution, is then

A general CES (Constant Elasticity of Substitution) form is

where and . (The Cobb–Douglas is a special case of the CES utility, with .) The marginal utilities are given by

and

Therefore, along an indifference curve,

These examples might be useful for modelling individual or aggregate demand.

As used in biology, the indifference curve is a model for how animals 'decide' whether to perform a particular behavior, based on changes in two variables which can increase in intensity, one along the x-axis and the other along the y-axis. For example, the x-axis may measure the quantity of food available while the y-axis measures the risk involved in obtaining it. The indifference curve is drawn to predict the animal's behavior at various levels of risk and food availability.

Indifference curves inherit the criticisms directed at utility more generally.

Herbert Hovenkamp (1991)^{ [13] } has argued that the presence of an endowment effect has significant implications for law and economics, particularly in regard to welfare economics. He argues that the presence of an endowment effect indicates that a person has no indifference curve (see however Hanemann, 1991^{ [14] }) rendering the neoclassical tools of welfare analysis useless, concluding that courts should instead use WTA as a measure of value. Fischel (1995)^{ [15] } however, raises the counterpoint that using WTA as a measure of value would deter the development of a nation's infrastructure and economic growth.

- ↑ The transitivity of weak preferences is sufficient for most indifference-curve analyses: If
*A*is weakly preferred to*B*, meaning that the consumer likes*A**at least as much*as*B*, and*B*is weakly preferred to*C*, then*A*is weakly preferred to*C*.^{ [8] } - ↑ Indifference curves can be used to derive the individual demand curve. However, the assumptions of consumer preference theory do not guarantee that the demand curve will have a negative slope.
^{ [12] }

**Acoustic theory** is a scientific field that relates to the description of sound waves. It derives from fluid dynamics. See acoustics for the engineering approach.

A **centripetal force** is a force that makes a body follow a curved path. Its direction is always orthogonal to the motion of the body and towards the fixed point of the instantaneous center of curvature of the path. Isaac Newton described it as "a force by which bodies are drawn or impelled, or in any way tend, towards a point as to a centre". In Newtonian mechanics, gravity provides the centripetal force responsible for astronomical orbits.

The theory of **consumer choice** is the branch of microeconomics that relates preferences to consumption expenditures and to consumer demand curves. It analyzes how consumers maximize the desirability of their consumption as measured by their preferences subject to limitations on their expenditures, by maximizing utility subject to a consumer budget constraint.

In economics, a **budget constraint** represents all the combinations of goods and services that a consumer may purchase given current prices within his or her given income. Consumer theory uses the concepts of a budget constraint and a preference map to analyze consumer choices. Both concepts have a ready graphical representation in the two-good case.

In differential geometry, the **Ricci curvature tensor**, named after Gregorio Ricci-Curbastro, represents the amount by which the volume of a narrow conical piece of a small geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space. As such, it provides one way of measuring the degree to which the geometry determined by a given Riemannian metric might differ from that of ordinary Euclidean n-space. The Ricci tensor is defined on any pseudo-Riemannian manifold, as a trace of the Riemann curvature tensor. Like the metric itself, the Ricci tensor is a symmetric bilinear form on the tangent space of the manifold.

In the calculus of variations, a field of mathematical analysis, the **functional derivative** relates a change in a functional to a change in a function on which the functional depends.

A **substitute good** is a good that can be used in place of another. In consumer theory, substitute goods or **substitutes** are goods that a consumer perceives as similar or comparable, so that having more of one good causes the consumer to desire less of the other good. Formally, good is a substitute for good if, when the price of rises, the demand for rises. Let be the price of good . Then, is a substitute for if

In economics, the **marginal rate of substitution** (**MRS**) is the rate at which a consumer can give up some amount of one good in exchange for another good while maintaining the same level of utility. At equilibrium consumption levels, marginal rates of substitution are identical. The marginal rate of substitution is one of the three factors from marginal productivity, the others being marginal rates of transformation and marginal productivity of a factor.

In economics and particularly in consumer choice theory, the **income-consumption curve** is a curve in a graph in which the quantities of two goods are plotted on the two axes; the curve is the locus of points showing the consumption bundles chosen at each of various levels of income.

In microeconomics, a consumer's **Marshallian demand function** specifies what the consumer would buy in each price and income or wealth situation, assuming it perfectly solves the utility maximization problem. Marshallian demand is sometimes called **Walrasian demand** or **uncompensated demand function** instead, because the original Marshallian analysis refused wealth effects.

In economics, an **ordinal utility** function is a function representing the preferences of an agent on an ordinal scale. Ordinal utility theory claims that it is only meaningful to ask which option is better than the other, but it is meaningless to ask *how much* better it is or how good it is. All of the theory of consumer decision-making under conditions of certainty can be, and typically is, expressed in terms of ordinal utility.

In probability theory and statistics, the **inverse gamma distribution** is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to the gamma distribution. Perhaps the chief use of the inverse gamma distribution is in Bayesian statistics, where the distribution arises as the marginal posterior distribution for the unknown variance of a normal distribution, if an uninformative prior is used, and as an analytically tractable conjugate prior, if an informative prior is required.(Hoff, 2009:74)

The **mathematics of general relativity** refers to various mathematical structures and techniques that are used in studying and formulating Albert Einstein's theory of general relativity. The main tools used in this geometrical theory of gravitation are tensor fields defined on a Lorentzian manifold representing spacetime. This article is a general description of the mathematics of general relativity.

**Constant elasticity of substitution** (**CES**), in economics, is a property of some production functions and utility functions.

In physics, **Maxwell's equations in curved spacetime** govern the dynamics of the electromagnetic field in curved spacetime or where one uses an arbitrary coordinate system. These equations can be viewed as a generalization of the vacuum Maxwell's equations which are normally formulated in the local coordinates of flat spacetime. But because general relativity dictates that the presence of electromagnetic fields induce curvature in spacetime, Maxwell's equations in flat spacetime should be viewed as a convenient approximation.

In economics, **convex preferences** are an individual's ordering of various outcomes, typically with regard to the amounts of various goods consumed, with the property that, roughly speaking, "averages are better than the extremes". The concept roughly corresponds to the concept of diminishing marginal utility without requiring utility functions.

There are various **mathematical descriptions of the electromagnetic field** that are used in the study of electromagnetism, one of the four fundamental interactions of nature. In this article, several approaches are discussed, although the equations are in terms of electric and magnetic fields, potentials, and charges with currents, generally speaking.

In economics and consumer theory, **quasilinear utility** functions are linear in one argument, generally the numeraire. Quasilinear preferences can be represented by the utility function where is strictly concave. A nice property of the quasilinear utility function is that, the Marshallian/Walrasian demand for does not depend on wealth and therefore is not subject to a wealth effect. The absence of a wealth effect simplifies analysis and makes quasilinear utility functions a common choice for modelling. Furthermore, when utility is quasilinear, compensating variation (CV), equivalent variation (EV), and consumer surplus are algebraically equivalent. In mechanism design, quasilinear utility ensures that agents can compensate each other with side payments.

In continuum mechanics, a **compatible** deformation **tensor field** in a body is that *unique* tensor field that is obtained when the body is subjected to a continuous, single-valued, displacement field. **Compatibility** is the study of the conditions under which such a displacement field can be guaranteed. Compatibility conditions are particular cases of integrability conditions and were first derived for linear elasticity by Barré de Saint-Venant in 1864 and proved rigorously by Beltrami in 1886.

The **Cut-insertion theorem**, also known as **Pellegrini's theorem**, is a linear network theorem that allows transformation of a generic network N into another network N' that makes analysis simpler and for which the main properties are more apparent.

- ↑ Geanakoplos, John (1987). "Arrow-Debreu model of general equilibrium".
*The New Palgrave: A Dictionary of Economics*.**1**. pp. 116–124 [p. 117]. - ↑ Böhm, Volker; Haller, Hans (1987). "Demand theory".
*The New Palgrave: A Dictionary of Economics*.**1**. pp. 785–792 [p. 785]. - ↑ Francis Ysidro Edgeworth (1881).
*Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences*. London: C. Kegan Paul and Co. - ↑ Vilfredo Pareto (1919).
*Manuale di Economia Politica — con una Introduzione alla Scienza Sociale*[*Manual of Political Economy*]. Piccola Biblioteca Scientifica.**13**. Milano: Societa Editrice Libraria. - ↑ "Indifference curves | Policonomics" . Retrieved 2018-12-08.
- ↑ "William Stanley Jevons - Policonomics".
*www.policonomics.com*. Retrieved 23 March 2018. - 1 2 3 4 5 6 7 Binger; Hoffman (1998).
*Microeconomics with Calculus*(2nd ed.). Reading: Addison-Wesley. pp. 109–117. ISBN 0-321-01225-9. - 1 2 Perloff, Jeffrey M. (2008).
*Microeconomics: Theory & Applications with Calculus*. Boston: Addison-Wesley. p. 62. ISBN 978-0-321-27794-7. - 1 2 3 Silberberg; Suen (2000).
*The Structure of Economics: A Mathematical Analysis*(3rd ed.). Boston: McGraw-Hill. ISBN 0-07-118136-9. - 1 2 Lipsey, Richard G. (1975).
*An Introduction to Positive Economics*(Fourth ed.). Weidenfeld & Nicolson. pp. 182–186. ISBN 0-297-76899-9. - ↑ Salvatore, Dominick (1989).
*Schaum's Outline of Theory and Problems of Managerial Economics*. McGraw-Hill. ISBN 0-07-054513-8. - ↑ Binger; Hoffman (1998).
*Microeconomics with Calculus*(2nd ed.). Reading: Addison-Wesley. pp. 141–143. ISBN 0-321-01225-9. - ↑ Hovenkamp, Herbert (1991). "Legal Policy and the Endowment Effect".
*The Journal of Legal Studies*.**20**(2): 225. doi:10.1086/467886. - ↑ Hanemann, W. Michael (1991). "Willingness To Pay and Willingness To Accept: How Much Can They Differ? Reply".
*American Economic Review*.**81**(3): 635–647. doi:10.1257/000282803321455449. JSTOR 2006525. - ↑ Fischel, William A. (1995). "The offer/ask disparity and just compensation for takings: A constitutional choice perspective".
*International Review of Law and Economics*.**15**(2): 187–203. doi:10.1016/0144-8188(94)00005-F.

- Beattie, Bruce R.; LaFrance, Jeffrey T. (2006). "The Law of Demand versus Diminishing Marginal Utility" (PDF).
*Appl. Econ. Perspect. Pol.***28**(2): 263–271. doi:10.1111/j.1467-9353.2006.00286.x.

Wikimedia Commons has media related to . Indifference curves |

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.