![Simple-indifference-curves](https://upload.wikimedia.org/wikipedia/commons/f/f6/Simple-indifference-curves.svg)
In
economics
Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and intera ...
, 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
In psychology, economics and philosophy, preference is a technical term usually used in relation to choosing between alternatives. For example, someone prefers A over B if they would rather choose A than B. Preferences are central to decision theo ...
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
As a topic of economics, 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 happiness as part of the theory of utilitarianism by moral philosopher ...
(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
preference
In psychology, economics and philosophy, preference is a technical term usually used in relation to choosing between alternatives. For example, someone prefers A over B if they would rather choose A than B. Preferences are central to decision theo ...
s rather than something from which preferences come.
The main use of indifference curves is in the
representation
Representation may refer to:
Law and politics
*Representation (politics), political activities undertaken by elected representatives, as well as other theories
** Representative democracy, type of democracy in which elected officials represent a ...
of potentially observable
demand patterns for individual consumers over commodity bundles.
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 slope of an indifference curve is called the MRS (marginal rate of substitution), and it indicates how much of good y must be sacrificed to keep the utility constant if good x is increased by one unit. Given a utility function u(x,y), to calculate the MRS, we simply take the partial derivative of the function u with respect to good x and divide it by the partial derivative of the function u with respect to good y. If the marginal rate of substitution is diminishing along an indifference curve, that is the magnitude of the slope is decreasing or becoming less steep, then the preference is convex.
History
The theory of indifference curves was developed by
Francis Ysidro Edgeworth
Francis Ysidro Edgeworth (8 February 1845 – 13 February 1926) 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 ...
, who explained in his 1881 book the mathematics needed for their drawing; later on,
Vilfredo Pareto was the first author to actually draw these curves, in his 1906 book. The theory can be derived from
William Stanley Jevons
William Stanley Jevons (; 1 September 183513 August 1882) was an English economist and logician.
Irving Fisher described Jevons's book ''A General Mathematical Theory of Political Economy'' (1862) as the start of the mathematical method in ec ...
'
ordinal utility 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 ...
theory, which posits that individuals can always rank any consumption bundles by order of preference.
Map and properties
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.
Indifference curves are typically represented 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 if not offset by a decrease in the quantity consumed of the other good (Y). Equivalently,
satiation Satiation may refer to:
* Satiety, feeling "full" and satisfied after eating; the cessation of hunger
* Economic satiation, where increasing the amount of a good reduces the worth of each individual unit of it
* Predator satiation, an anti-predator ...
, such that more of either good (or both) is equally preferred to no increase, is excluded. (If utility ''U = f(x, y)'', ''U'', in the third dimension, does not have a
local maximum
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given ran ...
for any ''x'' and ''y'' values.) 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
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
, 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 since the point(s) of intersection would have equal utility).
#
Transitive with respect to points on distinct indifference curves. That is, if each point on ''I
2'' is (strictly) preferred to each point on ''I
1'', and each point on ''I
3'' is preferred to each point on ''I
2'', each point on ''I
3'' is preferred to each point on ''I
1''. 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.
# (Strictly)
convex. With (2),
convex preferences 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.
Assumptions of consumer preference theory
*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''
:*''B'' is preferred to ''A'', formally written as ''B''
p ''A''
:*''A'' is indifferent to ''B'', formally written as ''A''
I ''B''
:This axiom precludes the possibility that the consumer cannot decide,
It assumes that a consumer is able to make this comparison with respect to every conceivable bundle of goods.
*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''
*Preferences are transitive
:*If ''A''
p ''B'' and ''B''
p ''C'', then ''A''
p ''C''.
:*Also if ''A''
I ''B'' and ''B''
I ''C'', then ''A''
I ''C''.
: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''.
:"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''.
: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'.
:*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"
The negative slope of the indifference curves represents the willingness of the consumer to make a trade off.
Application
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.
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.
A line connecting all points of tangency between the indifference curve and the
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 ...
is called the
expansion path.
Examples of indifference curves
File:Simple-indifference-curves.svg, Figure 1: An example of an indifference map with three indifference curves represented
File:Indifference-curves-perfect-substitutes.svg, 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.
File:Indifference-curves-perfect-complements.svg, 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 ''I
2'', and would rather be on ''I
2'' than ''I
1'', but does not care where he/she is on a given indifference curve. The slope of an indifference curve (in absolute value), known by economists as the
marginal rate of substitution, shows the rate at which consumers are willing to give up one good in exchange for more of the other good. For ''most'' goods the marginal rate of substitution is not constant so their indifference curves are curved. The curves are convex to the origin, describing the negative
substitution effect. As price rises for a fixed money income, the consumer seeks the less expensive substitute at a lower indifference curve. The substitution effect is reinforced through the
income effect
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 pref ...
of lower real income (Beattie-LaFrance). An example of a utility function that generates indifference curves of this kind is the Cobb–Douglas function
. The negative slope of the indifference curve incorporates the willingness of the consumer to make trade offs.
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
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 ...
) or would change quantity demanded from one end of the
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 ...
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.
Preference relations and utility
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.
Preference relations
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
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of ele ...
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
In mathematics, a relation on a set is transitive if, for all elements , , in , whenever relates to and to , then also relates to . Each partial order as well as each equivalence relation needs to be transitive.
Definition
A homog ...
if whenever
and
then
.
For any element
, the corresponding indifference curve,
is made up of all elements of
which are indifferent to
. Formally,
.
Formal link to utility theory
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
As a topic of economics, 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 happiness as part of the theory of utilitarianism by moral philosopher ...
theory, the
utility function
As a topic of economics, 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 happiness as part of the theory of utilitarianism by moral philosopher ...
of an
agent
Agent may refer to:
Espionage, investigation, and law
*, spies or intelligence officers
* Law of agency, laws involving a person authorized to act on behalf of another
** Agent of record, a person with a contractual agreement with an insuranc ...
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
In mathematics, a relation on a set is transitive if, for all elements , , in , whenever relates to and to , then also relates to . Each partial order as well as each equivalence relation needs to be transitive.
Definition
A homog ...
. This means that for each bundle
there is a unique relation,
, representing the
utility
As a topic of economics, 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 happiness as part of the theory of utilitarianism by moral philosopher ...
(satisfaction) relation associated with
. The relation
is called the
utility function
As a topic of economics, 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 happiness as part of the theory of utilitarianism by moral philosopher ...
. The
range
Range may refer to:
Geography
* Range (geographic), a chain of hills or mountains; a somewhat linear, complex mountainous or hilly area (cordillera, sierra)
** Mountain range, a group of mountains bordered by lowlands
* Range, a term used to i ...
of the function is a set of
real numbers
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
. 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 curve
In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is:
: L_c(f) = \left\~,
When the number of independent variables is two, a level set is calle ...
s 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
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
of the indifference curve at point
. This ratio is called the
marginal rate of substitution between
and
.
Examples
Linear utility
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.
Cobb–Douglas utility
A class of utility functions known as Cobb-Douglas utility functions are very commonly used in economics for two reasons:
1. They represent ‘well-behaved’ preferences, such as more is better and preference for variety.
2. They are very flexible and can be adjusted to fit real-world data very easily.
If the utility function is of the form
the marginal utility of
is
and the marginal utility of
is
.Where
. The
slope
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
of the indifference curve, and therefore the negative of the
marginal rate of substitution, is then
:
CES utility
A general CES (
Constant Elasticity of Substitution
Constant elasticity of substitution (CES), in economics, is a property of some production functions and utility functions. Several economists have featured in the topic and have contributed in the final finding of the constant. They include Tom McK ...
) 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.
Biology
As used in
biology
Biology is the scientific study of life. It is a natural science with a broad scope but has several unifying themes that tie it together as a single, coherent field. For instance, all organisms are made up of cells that process hereditary i ...
, 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.
Criticisms
Indifference curves inherit the
criticisms directed at utility more generally.
Herbert Hovenkamp
Herbert Hovenkamp (born 1948) is an American legal scholar serving as James G. Dinan University Professor at the University of Pennsylvania Law School and the Wharton School of the University of Pennsylvania. Prior to that he held the Ben and Do ...
(1991)
has argued that the presence of an
endowment effect
In psychology and behavioral economics, the endowment effect (also known as divestiture aversion and related to the mere ownership effect in social psychology) is the finding that people are more likely to retain an object they own than acquire th ...
has significant implications for
law and
economics
Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and intera ...
, 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
) rendering the neoclassical tools of welfare analysis useless, concluding that courts should instead use
WTA as a measure of value. Fischel (1995)
however, raises the counterpoint that using WTA as a measure of value would deter the development of a nation's infrastructure and
economic growth
Economic growth can be defined as the increase or improvement in the inflation-adjusted market value of the goods and services produced by an economy in a financial year. Statisticians conventionally measure such growth as the percent rate of ...
.
Austrian economist
Murray Rothbard criticised the indifference curve as "never by definition exhibited in action, in actual exchanges, and is therefore unknowable and objectively meaningless."
See also
*
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 ...
*
Community indifference curve
A community indifference curve is an illustration of different combinations of commodity quantities that would bring a whole community the same level of utility. The model can be used to describe any community, such as a town or an entire nation. I ...
*
Consumer theory
*
Convex preferences
*
Endowment effect
In psychology and behavioral economics, the endowment effect (also known as divestiture aversion and related to the mere ownership effect in social psychology) is the finding that people are more likely to retain an object they own than acquire th ...
*
Indifference price In finance, indifference pricing is a method of pricing financial securities with regard to a utility function. The indifference price is also known as the reservation price or private valuation. In particular, the indifference price is the price ...
*
Level curve
In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is:
: L_c(f) = \left\~,
When the number of independent variables is two, a level set is calle ...
*
Microeconomics
Microeconomics is a branch of mainstream economics that studies the behavior of individuals and firms in making decisions regarding the allocation of scarce resources and the interactions among these individuals and firms. Microeconomics fo ...
*
Rationality
Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abil ...
*
Substitute good
In microeconomics, two goods are substitutes if the products could be used for the same purpose by the consumers. That is, a consumer perceives both goods as similar or comparable, so that having more of one good causes the consumer to desire less ...
*
Utility–possibility frontier
In welfare economics, a utility–possibility frontier (or utility possibilities curve), is a widely used concept analogous to the better-known production–possibility frontier. The graph shows the maximum amount of one person's utility given e ...
Notes
References
Further reading
*
External links
{{commons category, Indifference curves
Anatomy of Cobb–Douglas Type Utility Functions in 3D
Family economics
Economics curves
Utility
1881 in economics