Perfect-Complements Utility and Kinked Indifference Curves

Why Some Goods Are Complements

Some goods are most useful when consumed together in a particular proportion. A left shoe without a right shoe, for example, does not provide the same service as a pair; additional left shoes do not make up for a shortage of right shoes. In the perfect-complements model, one good in excess cannot compensate for too little of the other. The resulting preferences have right-angle indifference curves, with a kink at the balanced proportion.

This tutorial represents those preferences with a minimum function, finds the utility level sets and their kinks, and explains why the usual smooth marginal-rate-of-substitution calculation does not give one unique value at a kink. As with other utility representations, utility numbers rank bundles; they do not measure a consumer’s happiness on an absolute scale.

The parameters \(a\) and \(b\) set the proportion in which the goods are useful together. If \(a=2\) and \(b=3\), for instance, the balanced bundles satisfy \(2x=3y\), or \(y=\frac{2}{3}x\). The model assumes this proportion is fixed: extra units of one good cannot replace missing units of the other. It is an idealized preference model, not a claim that every pair of goods has exactly this relationship.

Utility Levels and Their Kinks

To find the indifference curve for a utility level \(u\), solve \(U(x,y)=u\). Since the utility is the smaller of \(a x\) and \(b y\), both scaled quantities must be at least \(u\), and at least one must equal \(u\). Thus the level set consists of the two rays \(x=\frac{u}{a},\,y\geq\frac{u}{b}\) and \(y=\frac{u}{b},\,x\geq\frac{u}{a}\). They meet at the kink \((\frac{u}{a},\frac{u}{b})\).

This form follows directly from the definition. If \(\min\{a x,b y\}=u\), neither scaled quantity can be below \(u\), because then the minimum would be below \(u\); and they cannot both exceed \(u\), because then the minimum would exceed \(u\). Therefore one scaled quantity equals \(u\) while the other is at least \(u\). Conversely, either stated condition makes the minimum exactly \(u\).

With \(x\) on the horizontal axis and \(y\) on the vertical axis, the first ray is vertical and the second is horizontal. Moving along either ray adds only the good that is already in excess, so utility does not change. The kink is the smallest bundle on that particular indifference curve in which the goods are balanced.

The Kink and the Marginal Rate of Substitution

For a differentiable utility function, marginal utilities are partial derivatives: \(MU_x=\frac{\partial U}{\partial x}\) and \(MU_y=\frac{\partial U}{\partial y}\). Where \(MU_y\neq0\), the smooth-curve convention is \(MRS_{xy}=MU_x/MU_y\), and the indifference-curve slope is \(-MRS_{xy}\). Perfect-complements utility is not differentiable at its kink, so this derivative-based formula cannot provide a unique smooth MRS there.

Away from the kink, the function is locally determined by whichever scaled quantity is smaller. If \(a x<b y\), then \(U=a x\): increasing \(x\) raises utility locally, while increasing \(y\) alone does not. If \(b y<a x\), then \(U=b y\): increasing \(y\) raises utility locally, while increasing \(x\) alone does not. At the kink, which good is limiting changes as the bundle moves from one side to the other.

The level-set geometry makes the failure of a unique smooth MRS visible. On the vertical branch, the slope in the \((x,y)\) plane is vertical; on the horizontal branch, it is zero. At their meeting point there is a corner, not a single tangent line. Equivalently, the relevant formula for utility switches between \(a x\) and \(b y\), so the partial derivatives are not defined at the switching point. One must not assign the kink a single finite MRS by taking the coefficients \(a\) and \(b\) and forming their ratio.

Interpreting the Fixed Proportion

The equality \(a x=b y\) describes the balanced ratio. Solving for either good gives \(y=\frac{a}{b}x\) or \(x=\frac{b}{a}y\). This ratio identifies the kink locations across utility levels; it is not the slope of an indifference curve along either branch. For every positive \(u\), the kink scales proportionally with \(u\), while each level curve retains the same right-angle shape.

The minimum function also makes it straightforward to decide which good limits a particular bundle. Compare \(a x\) with \(b y\), not just \(x\) with \(y\): different coefficients mean the raw quantities alone may be misleading. If \(a x<b y\), the bundle has too little \(x\) relative to its balanced proportion; if \(b y<a x\), it has too little \(y\). If they are equal, the bundle is precisely at the kink.

Common Misreadings

• Using a sum instead of a minimum. \(a x+b y\) represents a different utility function. For perfect complements, the smaller scaled quantity determines utility.
• Confusing the kink ratio with a curve slope. The ratio \(y=\frac{a}{b}x\) connects kink locations across curves. Each individual indifference curve has a vertical and a horizontal branch.
• Assuming the utility coefficients give a constant MRS. The coefficients scale the goods in the minimum. They do not remove the corner or establish a unique smooth trade-off at it.
• Ignoring the excess-good part of a level set. A fixed utility level contains entire rays, not only its kink. Adding the already abundant good along a ray leaves utility unchanged.

The model is useful whenever the key economic feature is a fixed-proportion requirement: extra units of one input cannot make up for too few units of the other. It is also a clear reminder that a utility function can represent preferences even when it is not differentiable everywhere. The geometric kink carries important information, but smooth derivative rules must be applied only where the required derivatives exist.