In microeconomic theory, an indifference curve is a graph showing different bundles of goods, each measured as to quantity, between which a consumer is indifferent. That is, at each point on the curve, the consumer has no preference for one bundle over another. In other words, they are all equally preferred. 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]}
Contents 
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.
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. An indifference curve describes a set of personal preferences and so can vary from person to person. An indifference curve is like a contour line on a topographical map. Each point on the map 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 nonsatiation requirement means that you will never reach the "top", or a "bliss point" which is the best possible.
Indifference curves are typically represented to be:
Let a, b, and c be bundles (vectors) of goods, such as (x, y) combinations above, with possibly different quantities of each respective good in the different bundles. The first assumption is necessary for a welldefined representation of stable preferences for the consumer as agent; the second assumption is convenient.
Rationality (called an ordering relationship in a more general mathematical context): Completeness + transitivity. For given preference rankings, the consumer can choose the best bundle(s) consistently among a, b, and c from lowest on up.
Continuity: This means that you can choose to consume any amount of the good. For example, I could drink 11 mL of soda, or 12 mL, or 132 mL. I am not confined to drinking 2 liters or nothing. See also continuous function in mathematics.
Of the remaining properties above, suppose, property (5) (convexity) is violated by a bulge of the indifference curves out from the origin for a particular consumer with a given budget constraint. Consumer theory then implies zero consumption for one of the two goods, say good Y, in equilibrium on the consumer's budget constraint. This would exemplify a corner solution. Further, decreases in the price of good Y over a certain range might leave quantity demanded unchanged at zero beyond which further price decreases switched all consumption and income away from X and to Y. The eccentricity of such an implication suggests why convexity is typically assumed.
Note this assumption means that a consumer is able to make this comparison with respect to every conceivable bundle of goods.^{[7]}
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 less the expensive substitute at a lower indifference curve. The substitution effect is reinforced through the income effect of lower real income (BeattieLaFrance). An example of a utility function that generates indifference curves of this kind is the CobbDouglas function . The negative slope of the indifference curve incorporates the willingness of the consumer to means to make trade offs.^{[14]}
If two goods are perfect substitutes then the indifference curves will have a constant slope since the consumer would be willing to trade 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 Lshaped. An example would be something like if you had a cookie recipe that called for 3 cups flour to 1 cup sugar. No matter how much extra flour you had, you still could not make more cookie dough without more sugar. Another example of perfect complements is a left shoe and a right shoe. 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. 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 pricebudgetline change that kept a consumer in equilibrium on the same indifference curve:
Figure 1 encore: 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. 
Choice theory formally represents consumers by a preference relation, and use this representation to derive indifference curves.
The idea of an indifference curve is a straightforward one: If a consumer was equally satisfied with 1 apple and 4 bananas, 2 apples and 2 bananas, or 5 apples and 1 banana, these combinations would all lie on the same indifference curve.
Let
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 .
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:
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:
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 : 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 α < 1. 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 CobbDouglas 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.
