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 1 History 2 Map and properties of indifference curves 3 Assumptions of consumer preference theory 3.1 Application 3.2 Examples of indifference curves 4 Preference relations and utility 4.1 Preference relations 4.2 Formal link to utility theory 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 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. 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: 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 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. 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). 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. (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 A p B3 B is preferred to A B p A3 A is indifferent to B A I B3 Note that this axiom precludes the possibility that the consumer cannot decide4 and that a consumer is able to make this comparison with respect to every conceivable bundle of goods.3 Preferences are reflexive 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 B3 Preference are transitivenb 1 If A p B and B p C then A p C.3 Also A I B and B I C then A I C.3 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 an infinity of 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 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 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. The marginal rate of substitution tells how much 'y' a person is willing to sacrifice to get one more unit of 'x'. 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 another good in that amount which will leave the consumer indifferent between the new and old situations"5 The negative slope of the indifference curves represents the willingness of the consumer to make a trade off.5 There are also many sub-assumptions: Irreflexivity - for no x is xpx Negative transivity if xnot-py then for any third commodity z either xnot-pz or znot-py or both. 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.6 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.6 Examples of indifference curves

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 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 make trade offs.7 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 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 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 . 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 . 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 < 1. The slope 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) 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. See also Budget constraint Level curve Consumer theory Convex preferences Community indifference curve Endowment effect Microeconomics Rationality Footnotes The transitivity of weak preferences is sufficient for most IC analysis: If A is weakly preferred to B meaning that the consumer likes A at least as much as B and B is weakly preferred to C then A is weakly preferred to C.4 Indifference curves can be used to derive the individual demand curve. However the assumptions of consumer preference theory do not guarantee that the demand curve will have a negative slope.8 Notes Geanakoplis (1987) p. 117. Bhm and Haller (1987) p. 785. a b c d e f g Binger and Hoffman (1998). pp. 109-17. a b Perloff (2008). p. 62. a b Silberberg and Suen (2000). a b Lipsey (1975). pp. 182-186. Id. Binger and Hoffman (1998). p. 141-43. References Bruce R. Beattie and Jeffrey T. LaFrance "The Law of Demand versus Diminishing Marginal Utility" (2006). Review of Agricultural Economics. 28 (2) pp. 263271. Volker Bhm and Hans Haller (1987). "demand theory" The New Palgrave: A Dictionary of Economics v. 1 pp. 78592. John Geanakoplos (1987). "Arrow-Debreu model of general equilibrium" The New Palgrave: A Dictionary of Economics v. 1 pp. 11624. Binger and Hoffman (1998) Microeconomics with Calculus 2nd ed. Addison-Wesley. Perloff (2008). Microeconomics theory & Applications with Calculus. Addison-Wesley. Silberberg and Suen (2000). The Structure of Economics A Mathematical Analysis 3rd ed. McGraw-Hill. Lipsey Richard G. (1975). An introduction to positive economics (fourth ed.). Weidenfeld & Nicolson. pp. 2147. ISBN 0297768999.  External links Anatomy of Cobb-Douglas Type Utility Functions in 3D Anatomy of CES Type Utility Functions in 3D