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]}
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.
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]}
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^{[vague]} represented^{[clarification needed]} to be:
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.
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 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 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.^{[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 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:
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
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,
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 : the indifference curves are straight lines.
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 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.
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