Four utility functions cover most of what you will meet, and each has an MRS (from The Marginal Rate of Substitution) that explains its shape. Learning these four shapes now saves trouble later, because the rule for the best bundle depends on which one you face.
| Family | Utility | MRS | Indifference curves | Example |
|---|---|---|---|---|
| Perfect substitutes | $ax + by$ | $a/b$ (constant) | Straight lines | Two brands of bottled water |
| Perfect complements | $\min\{ax, by\}$ | Undefined at the corner; 0 or "infinite" elsewhere | L shapes with corners on the ray $ax = by$ | Left and right shoes |
| Cobb-Douglas | $x^a y^b$ | $\dfrac{a}{b}\cdot\dfrac{y}{x}$ | Smooth, bowed toward the origin | Food and housing |
| Quasilinear | $v(x) + y$ | $v'(x)$ (no $y$) | Vertical shifts of one curve | A small item and "all other spending" |
Key idea
Substitutes trade at a fixed rate, so their curves are straight lines. Complements are used in fixed proportions, so their curves are L-shaped with a corner where $ax = by$. Cobb-Douglas curves are smooth and bowed with an MRS that falls as $x/y$ rises. Quasilinear curves have an MRS that depends only on $x$.
Why it works
Each shape comes straight from the MRS. A constant MRS means a constant slope: a line. For complements, extra $x$ beyond the corner adds nothing (the min is set by $y$), so the curve runs flat to the right of the corner and straight up above it. For Cobb-Douglas the MRS is proportional to $y/x$: steep where $y$ is plentiful, flat where $x$ is. For quasilinear utility, $MU_y = 1$ and $MU_x = v'(x)$, so the slope at a given $x$ is the same on every curve: each curve is the one below it shifted up.
Worked example: Substitutes
Femi treats a large pizza ($x$) as worth exactly two small ones ($y$): $U = 2x + y$. What is his MRS at $(1, 8)$ and at $(6, 0)$?
- $MU_x = 2$ and $MU_y = 1$, constants.
- $MRS = 2/1 = 2$ at every bundle, including both of these.
Answer: 2 at both.
Worked example: Complements
For $U = \min\{x, 2y\}$, find the corner of the indifference curve $U = 10$, and the utility of $(14, 5)$.
- At the corner both terms equal 10: $x = 10$ and $2y = 10$, so $(10, 5)$.
- For $(14, 5)$: $\min\{14, 10\} = 10$. The 4 extra units of $x$ add nothing; $(14, 5)$ lies on the flat arm of the same curve.
Answer: Corner $(10, 5)$; $U(14, 5) = 10$.
Common mistake
Expecting every indifference curve to be smooth and bowed, and so applying "MRS = price ratio" everywhere. With perfect substitutes the MRS never changes, so it usually never equals the price ratio, and the best choice is a corner (all of one good). With perfect complements the MRS doesn't exist at the corner where the best choice sits. Check: before setting MRS equal to anything, identify the family. Corners and Kinks shows what to do instead.