The indifference curve: definition and properties

Definition. An indifference curve is the locus of all bundles of two goods among which a consumer is indifferent, because every bundle on it gives her the same level of satisfaction. The idea belongs to ordinal utility analysis: the consumer is not asked to put a number on her utility, only to rank bundles as better, worse or equally good. A whole family of such curves for one consumer, one curve for each level of satisfaction, is called her indifference map.

Example (our own). Arjun, a student in Jaipur, is equally content with each of the following weekly bundles of cups of masala chai and samosas. Plotted with chai on the horizontal axis and joined, they give one indifference curve.

What the curve assumes. Only two goods are considered; the consumer can compare any two bundles, either preferring one or being indifferent between them; and her preferences are monotonic, so that she prefers a bundle having more of at least one good and no less of the other. Equivalently, the marginal utility of each good is taken to be positive.

Properties.

(i) A higher curve stands for greater satisfaction. A curve lying further from the origin contains bundles with more of one good, or of both, and no less of either. As long as each good has positive marginal utility, such bundles are preferred, so a higher curve represents a higher level of satisfaction.

(ii) The curve slopes downwards. If Arjun gets one more cup of chai and gives up no samosas, monotonicity puts him on a higher curve. To stay equally well off, an increase in one good must be matched by a fall in the other, so the curve falls from left to right.

(iii) The curve is convex to the origin. The marginal rate of substitution (MRS) is the quantity of the second good the consumer is willing to give up for one more unit of the first while her satisfaction stays the same; it equals the absolute value of the curve's slope, |ΔY/ΔX|. As she gets more of the first good its marginal utility falls, and as she holds less of the second good its marginal utility rises, so she is willing to give up smaller and smaller amounts (4, then 2, then 1 samosa in the table). This diminishing MRS makes the curve bow in towards the origin, which is its usual shape.

(iv) Two curves cannot cut each other. Suppose two curves met at a bundle A, with B on the first curve and C on the second, and B holding the same quantity of one good as C but more of the other. A is as good as B and also as good as C, so B and C would be equally good; yet B has more of one good and no less of the other, so B is better. The two conclusions contradict each other, so the curves cannot intersect.

The special case of perfect substitutes. When two goods can be used in place of each other and give exactly the same satisfaction, the MRS does not diminish but stays constant, so the indifference curve is a downward-sloping straight line rather than a convex one.

Conclusion. An indifference curve joins equally satisfying bundles of two goods. Under monotonic preferences it slopes downwards, higher curves mean more satisfaction and no two curves intersect; diminishing MRS makes it convex to the origin, except for perfect substitutes, where it is a straight line.