In microeconomic theory, an indifference curve is a graph showing different bundles of goods
between which a consumer is indifferent. That is, at each point on the curve, the consumer
has no preference for one bundle over another. One can equivalently refer to each point on
the indifference curve as rendering the same level of utility 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. The main use of indifference
curves is in the representation 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 indifference curves, illustrated graphically,
is referred to as an indifference map.
History The theory of indifference curves was developed
by Francis Ysidro Edgeworth, who explained in his book "Mathematical Psychics: an Essay
on the Application of Mathematics to the Moral Sciences,” 1881, the mathematics needed
for its drawing; later on, Vilfredo Pareto was the first author to actually draw these
curves, in his book "Manual of Political Economy," 1906; and others in the first part of the
20th century. The theory can be derived from William Stanley Jevons's ordinal utility theory,
which posits that individuals can always rank any consumption bundles by order of preference.
Map and properties of indifference curves
A graph of indifference curves for an individual consumer associated with different utility
levels 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 map. Each point on the curve represents the
same elevation. If you move "off" an indifference curve traveling in a northeast direction 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.
Negatively sloped. That is, as quantity consumed of one good increases, total satisfaction
would increase if not offset by a decrease in the quantity consumed of the other good.
Equivalently, satiation, such that more of either good is equally preferred to no increase,
is excluded., U, in the third dimension, does not have a local maximum 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; 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, no two curves can intersect.
Transitive with respect to points on distinct indifference curves. That is, if each point
on I2 is preferred to each point on I1, and each point on I3 is preferred to each point
on I2, each point on I3 is preferred to each point on I1. 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, 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 ⇒ A p B
B is preferred to A ⇒ B p A A is indifferent to B ⇒ A I BNote that this
axiom precludes the possibility that the consumer cannot decide, and that a consumer is able
to make this comparison with respect to every conceivable bundle of goods.
Preferences are reflexive 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 transitiveIf A p B and B p C then A p C.
Also 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 infinitesimally
close to B then A is preferred to C. A p B & 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
is strong monotonicity which 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.
There are also many sub-assumptions: Irreflexivity - for no x is xpx
Negative transitivity - if xnot-py then for any third commodity z, either xnot-pz or znot-py
or both.
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. 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 is called the expansion path.
Examples of indifference curves
In Figure 1, the consumer would rather be on I3 than I2, and would rather be on I2 than
I1, but does not care where he/she is on a given indifference curve. The slope of an
indifference curve, 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 less the expensive substitute
at a lower indifference curve. The substitution effect is reinforced through the income effect
of lower real income. 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 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. 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 define on the set .
The statement
is described as ' is weakly preferred to .' That is, is at least as good as .
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, .
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 theory, the utility function of an agent is a function that ranks all pairs
of consumption bundles by order of preference 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 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,
where is the partial derivative of with respect to its first argument, evaluated at .
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 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 .
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
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
CES utility A general CES form is
where and . 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, 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.
See also Budget constraint
Community indifference curve Consumer theory
Convex preferences Endowment effect
Indifference price Level curve
Microeconomics Rationality
Utility–possibility frontier Notes
References
Further reading Beattie, Bruce R.; LaFrance, Jeffrey T.. "The
Law of Demand versus Diminishing Marginal Utility". Appl. Econ. Perspect. Pol. 28: 263–271.
doi:10.1111/j.1467-9353.2006.00286.x. External links
Anatomy of Cobb-Douglas Type Utility Functions in 3D
Anatomy of CES Type Utility Functions in 3D