Substitutes, complements, lexicographic, homothetic and quasi-linear preferences

Two goods are perfect substitutes if she is always willing to swap them at a fixed rate. Two brands of the same salt are the textbook case. The utility function is linear:

The indifference curves are straight lines (JNU 2010, 2011, 2016; Hyderabad 2019), with a constant slope −a/b. With u = x₁ + x₂ the MRS is 1 at every bundle, and the goods are perfect substitutes (Hyderabad 2020). The same is true of 2x + 3y, which has a constant MRS of 2/3 (CUET 2024). A monotonic transformation does not change it: √(x + y) also describes perfect substitutes (JNU 2019).

Because the MRS never changes, there is no tangency to find. She spends all her income on the good that gives more utility per rupee. That is Lesson 6.

Two goods are perfect complements if she wants them in a fixed ratio and extra of one alone is useless: left and right shoes, or coffee and sugar in Mr Tokai’s recipe. If she needs a units of x for every b of y,

The indifference curves are L-shaped (JNU 2021). The corner of each L sits on the ray where the goods come in the right ratio. On a vertical arm there is too much y and nothing is gained by more of it; on a horizontal arm there is too much x. So the substitution effect of a price change is zero, and the goods are always used in fixed proportions. Straight-line curves belong to substitutes, not complements (UGC NET 2023).

Mr Tokai’s utility is min{coffee, sugar/2}. With 9 and 2, that is min{9, 1} = 1 cup. Seven of his nine teaspoons of coffee are wasted.

Put the goods in order of importance. A bundle with more of the first good always wins, whatever it has of the second. Only when the first goods tie does the second good decide. This is how words are ordered in a dictionary, hence lexicographic. The Delhi School sets it in several papers.

Two bundles are equally good only if they are the same bundle, so there are no indifference curves: each bundle is a point on its own. Worse, no utility function represents these preferences (JNU 2016), because they break continuity (UGC NET 2025). Take the sequence of bundles (2 + 1/n, 0). Each beats (2, 100), because it has more of the first good. But as n grows the sequence tends to (2, 0), and (2, 0) loses to (2, 100). A limit of preferred bundles is not preferred, which continuity forbids. A utility function would be a continuous way of scoring, so none can exist.

The constant elasticity of substitution function covers several of these as special cases:

The question about U = (x₁^a + x₂^a)^b (CUET 2023) is a CES with ρ = a: its MRS is (x₁/x₂)^a−1, a function of the ratio alone. That detail matters in a moment.

Pick a kind of taste. The blue curves are the ones she reaches at incomes of 8, 16 and 24 when x costs Re 1 and y costs Rs 2; the gold dotted line is the income expansion path, which joins her choices as income grows. The badges test the properties: convex, homothetic (is the path a straight line from the origin?) and whether a utility function exists. Tick the box to shade every bundle better than a chosen one; for the lexicographic case, that is everything with more x plus the ray above.

Preferences are homothetic if her ranking of two bundles does not change when both are scaled by the same factor. Equivalently, the MRS depends only on the ratio y/x, so every indifference curve is a magnified copy of the others. The test is that, at fixed prices, as income rises her chosen bundle moves along a straight line from the origin: the income expansion path is a ray, and she buys goods in the same proportions at every income.

Formally, u = G(f(x, y)) with G increasing and f homogeneous of degree one. Cobb–Douglas, CES, perfect substitutes and perfect complements are all homothetic. A homothetic preference gives these three things, and withholds a fourth (JNU 2021):

• the income consumption curve is a straight line from the origin;
• both goods are normal (demand rises in proportion to income);
• the MRS depends only on the ratio in which the goods are consumed;
• but the price consumption curve is not a straight line from the origin in general. Do not confuse the two curves.

Jai spends at equal prices and keeps the same ratio as his income grows (JNU 2018). He may have Cobb–Douglas tastes, perfect complements or perfect substitutes: all homothetic, so none of the three can be ruled out. √(x² + y²) is homogeneous of degree one, so it is homothetic, and its curves are concave to the origin (Hyderabad 2020).

Preferences are quasi-linear if utility is linear in one good: u = g(x) + y. Raising utility by one lifts the indifference curve by exactly one unit of y, so the curves are vertical translates of one another, vertically parallel (JNU 2018). They are not radial copies, so these preferences are not homothetic (GATE 2025). The MRS, g′(x), depends on x alone.

What it does to demand is the reason it is a favourite. Once she is buying some of both goods, an extra rupee goes entirely on y, and her demand for x does not change with income. Lesson 6 derives it. For now, check the picture in the lab: the expansion path for √x + y becomes a vertical line.