Indifference curve
Indifference Curve
What Does Indifference Curve Mean?
A diagram depicting equal levels of utility (satisfaction) for a consumer faced with various
combinations of goods.
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An example of an indifference map with three indifference curves represented
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 (satisfaction)
for 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]
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.
Contents
[hide]
• 1 History
• 2 Map and properties of indifference curves
• 3 Assumptions of consumer preference theory
o 3.1 Application
o 3.2 Examples of indifference curves
• 4 Preference relations and utility
o 4.1 Preference relations
o 4.2 Formal link to utility theory
o 4.3 Examples
4.3.1 Linear utility
4.3.2 Cobb-Douglas utility
4.3.3 CES utility
• 5 See also
• 6 Footnotes
• 7 Notes
• 8 References
• 9 External links
[edit] History
The theory of indifference curves was developed by Francis Ysidro Edgeworth, Vilfredo
Pareto and others in the first part of the 20th century. The theory can be derived from
ordinal utility theory, which posits that individuals can always rank any consumption
bundles by order of preference.
[edit] Map and properties of indifference curves
An example of how indifference curves are obtained as the level curves of a utility
function
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 is like a contour line 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 (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:
1. Defined only in the non-negative quadrant of commodity quantities (i.e. the
possibility of having negative quantities of any good is ignored).
2. 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, 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 for any x and y values.) The
negative slope of the indifference curve reflects the law of diminishing marginal
utility. That is as more of a good is consumed total utility increases at a
decreasing rate - additions to utility per unit consumption are successively
smaller. Thus as you move down the indifference curve you are trading
consumption of units of Y for additional units of X.
3. 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).
4. Transitive with respect to points on distinct indifference curves. That is, if each
point on I2 is (strictly) 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.
5. (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.
[edit] Assumptions of consumer preference theory
• Preferences are complete
o 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[3]
B is preferred to A ⇒ B p A[3]
A is indifferent to B ⇒ A I B[3]
Note that this axiom precludes the possibility that the
consumer cannot decide,[4] and that a consumer is able to
make this comparison with respect to every conceivable
bundle of goods.[3]
• Preferences are reflexive
o Means that if A and B are in all respect identical the consumer will
recognize this fact and be indifferent in comparing A and B
A = B ⇒ A I B[3]
• Preference are transitive[nb 1]
o If A p B and B p C then A p C.[3]
o Also A I B and B I C then A I C.[3]
This is a consistency assumption.
• Preferences are continuous
o If A is preferred to B and C is infinitesimally close to B then A is
preferred to C.
o A p B & C → B ⇒ A p B.
"Continuous" means infinitely divisible - just like there are an
infinity of numbers between 1 and 2 all bundles are infinitely
divisible. This assumption makes indifference curves continuous.
• Preferences exhibit strong monotonicity.
o if A has more of both x and y than B then A is preferred to B
this is 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
• Indifference curves exhibit diminishing marginal rates of substitution
o This assumption assures that indifference curves are smooth and convex to
the origin.
o 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.
o The marginal rate of substitution tells how much y a person is willing to
sacrifice to get one more unit of x.
o This is also called the substitution assumption. This 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 other good is that amount which will
leave the consumer indifferent between the new and old situations"[6] The
negative slope of the indifference curves represents the willingness of the
consumer to make a trade off. [6]
• There are also many sub-assumptions:
o Irreflexivity - for no x is xpx
o Negative transivity if xnot-py then for any third commodity z, either xnot-pz or
znot-py or both.
[edit] Application
To maximise utility, a household should consume at (Qx, Qy). Assuming it does, a full
demand schedule can be deduced as the price of one good fluctuates.
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 gives 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.[7] 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.[8]
[edit] Examples of indifference curves
Figure 2: Three indifference
Figure 3: Indifference
curves where Goods X and Y are
curves for perfect
Figure 1: An example of perfect substitutes. The gray line complements X and Y. The
an indifference map with perpendicular to all curves elbows of the curves are
three indifference curves indicates the curves are mutually collinear.
represented parallel.
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 (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 less the expensive substitute at a lower indifference curve. The
substitution effect is reinforced through the income effect 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 means to make trade
offs.[9]
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 includes - 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 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]
[edit] 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.
[edit] Preference relations
Let
= a set of mutually exclusive alternatives among which a consumer can choose
and = 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 .
Consider a particular element of the set , such as . Suppose one builds the list of all
other elements of which are indifferent, in the eyes of the consumer, to . Denote the
first element in this list by , the second by and so on... The set forms
an indifference curve since for all .
[edit] 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 (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 .
[edit] Examples
[edit] 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.
[edit] Cobb-Douglas utility
If the utility function is of the form the marginal utility of is
and the marginal utility of is
.Where α < 1. The slope of the indifference curve,
and therefore the negative of the marginal rate of substitution, is then
[edit] CES utility
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.
