Indifference curves for perfect substitutes and for complementary goods

The usual shape, and what decides it. An indifference curve joins the bundles of two goods that leave a consumer equally satisfied. Its shape depends on the marginal rate of substitution (MRS): the quantity of one good she is willing to give up for an extra unit of the other while staying on the same curve, measured as |ΔY/ΔX|. For most pairs of goods the MRS falls as she acquires more of the first good, because each extra unit of it adds less to her satisfaction while each unit of the second good she still holds becomes more valuable. A falling MRS makes the curve convex to the origin, the most common shape. The case below is different because the MRS behaves differently.

Perfect substitutes. Two goods are perfect substitutes for a consumer when either can be used in place of the other and each gives her exactly the same satisfaction. Only the total quantity matters to her, not how it is made up.

Why the line is straight.

• However much of one good she already holds, she is willing to exchange the two at the same fixed rate, so the MRS does not diminish; it stays constant along the curve.
• A curve whose slope never changes is a straight line. The indifference curve for perfect substitutes is therefore a downward-sloping straight line.
• It still slopes downwards, because with monotonic preferences more of one good must be balanced by less of the other if satisfaction is to stay the same.
• Higher straight lines, further from the origin, represent larger totals and so greater satisfaction.

Example. Our own example. Meera, a commuter in Mumbai, uses either a paper ticket or a QR ticket on her phone for the same metro ride and does not care which she has. She is equally well off with each of these bundles of ten rides:

The MRS is 1 throughout, so the points lie on a straight line with a slope of −1 that meets both axes at ten rides. If the fixed rate were different, say one large bottle of a soft drink for two small ones of the same total quantity, the line would be steeper or flatter but still straight.

Perfect complementary goods. Complementary goods are goods that are used in combination, that is, consumed together, such as a gas stove and the LPG cylinder that fuels it. A rise in the price of one of them tends to reduce the demand for the other, because the two are bought for use together.

Conclusion. The usual indifference curve is convex because the MRS diminishes. For perfect substitutes the MRS is constant, so the curve is a downward-sloping straight line; complementary goods are those the consumer uses together.