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. The main use of indifference curves is in the representation of potentially observable demand patterns for individual consumers over commodity bundles.
Indifference curve analysis is a purely technological model which cannot be used to model consumer behaviour. Every point on any given indifference curve must be satisfied by the same budget (unless the consumer can be indifferent to different budgets). As a consequence, every budget line for a given budget and any two products is tangent to the same indifference curve and this means that every budget line is tangent to, at most, one indifference curve (and so every consumer makes the same choices).
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, one takes 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.
History The theory of indifference curves was developed by Francis Ysidro Edgeworth, who explained in his 1881 book the mathematics needed for their drawing; later on, Vilfredo Pareto was the first author to actually draw these curves, in his 1906 book. Later economists such as J.R. Hicks and R.G.D. Allen further refined and popularized indifference curve analysis in consumer theory in the 1930s. 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.
Map and properties
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 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 the consumption of one good increases, to maintain constant utility, a lesser quantity of the other good just be consumed. This is equivalent to assuming Local non-satiation (an increase in the consumption of either good increases, rather than decreases, total utility). The counterfactual to this assumption is assuming a bliss point. 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 assumption of the monotonicity of consumer's preferences, which generates monotonically increasing utility functions, and the assumption of non-satiation (marginal utility for all goods is always positive); 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 (2), no two curves can intersect (otherwise non-satiation would be violated since the point(s) of intersection would have equal utility). 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. 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. Convex preferences are assumed in concordance with the principle of declining marginal utility.
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, formally written as A p B B is preferred to A, formally written as B p A A is indifferent to B, formally written as A I B This axiom precludes the possibility that the consumer cannot decide, It assumes that a consumer is able to make this comparison with respect to every conceivable bundle of goods. Preferences are reflexive This 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 transitive If A p B and B p C, then A p C. Also if 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 sufficiently close to B then A is preferred to C. A p B and 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 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.
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 (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. 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. Stated precisely, a set of indifference curve for representative of different price ratios between two goods are used to generate the Price-consumption curve in good-good vector space, which is equivalent to the demand curve in good-price vector space. The line connecting all points of tangency between the indifference curve and the budget constraint as the budget constraint changes is called the expansion path, and correlates to shifts in demand. The line connecting all points of tangency between the indifference curve and budget constraint as the price of either good changes is the price-consumption curve, and correlates to movement along the demand curve.
Examples of indifference curves