Convex Preferences and Preferred Averages

Why Averages Matter

A consumer choosing between two goods may face a familiar trade-off: one bundle contains more of the first good, while another contains more of the second. Convex preferences capture a preference for balance in a precise way. If two bundles are equally good, their average is at least as good as either one. This is the formal basis for describing a consumer as preferring variety or diversification.

The average is a bundle of quantities, not an average of prices or utility scores. To test the idea, first calculate the quantities in the average bundle; then compare its ranking with the rankings of the original bundles. The distinction between “at least as good” and “strictly better” is important: convexity does not always imply that a balanced bundle is strictly preferred.

The definitions below use the preference notation introduced in the earlier material on consumption bundles and preference rankings. This treatment of convex preferences is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Convex Preferences and Preferred Averages

A preference relation is convex when a weighted average of two bundles that are each at least as good as a given bundle is also at least as good as that given bundle. In the particular comparison emphasized here, the consumer is indifferent between the two original bundles. Convexity then says that any weighted average of them is weakly preferred to either one. The weights may be any values from 0 to 1; equal weights give the midpoint.

The general definition compares bundles with a reference bundle \(z\). When \(x\sim y\), each original bundle is a suitable reference: both \(x\) and \(y\) are at least as good as \(x\), and both are at least as good as \(y\). Applying convexity to each comparison gives the preferred-average property. Completeness and transitivity, covered in the earlier material on preference rankings, make these comparisons consistent.

A weighted average combines the same fraction of each bundle in each good. If \(\lambda=0.25\), the resulting bundle contains 25% of the quantities in \(x\) and 75% of the quantities in \(y\), added good by good. It does not mean that the consumer receives an average of the two bundles’ utility scores. A score can be used to represent rankings, but the bundle quantities must be averaged first.

Why a Concave Utility Representation Supports Convexity

A utility function is a numerical representation of preferences if a higher value corresponds to a preferred bundle. The size of a utility difference does not generally measure the size of a consumer’s satisfaction; the useful information is the ranking. Some utility representations are concave, which means that the utility of a weighted average is at least the weighted average of the utilities.

This inequality explains why concavity can establish the preferred-average property. If \(x\sim y\), their utility values are equal, say \(u(x)=u(y)=c\). Substitution into the inequality gives:

Thus the weighted average has utility at least \(c\), the utility of either original bundle. This is a sufficient way to verify convex preferences when the utility function is concave. Convex preferences are about rankings, however, and do not require every possible utility representation to be concave.

A direct algebraic check illustrates the same idea for a particular family of bundles. Under \(u(x_1,x_2)=\sqrt{x_1x_2}\), take \(A=(a,b)\) and \(B=(b,a)\), with \(a,b>0\). Both have utility \(\sqrt{ab}\). Their midpoint is \(((a+b)/2,(a+b)/2)\), with utility \((a+b)/2\). Since a square is nonnegative, \((a-b)^2\geq0\), which rearranges to \((a+b)^2\geq4ab\). Both sides are nonnegative, so taking square roots gives \((a+b)/2\geq\sqrt{ab}\). The midpoint is therefore weakly preferred; if \(a\ne b\), the square inequality is strict and so is the preference comparison in this example.

Convex Does Not Always Mean Strictly Better

Convex preferences require the average to be at least as good. A preference may satisfy this condition even when the average is exactly as good as the original bundles. Strict convexity is the stronger condition that makes the average strictly better for distinct indifferent bundles and interior weights. Therefore, an example in which the average ties the original bundles can still be consistent with convex preferences.

For instance, a linear utility representation such as \(u(x_1,x_2)=x_1+x_2\) assigns the same score to every bundle with the same total quantity. The weighted average of two such bundles also has that total: \(\lambda(x_1+x_2)+(1-\lambda)(y_1+y_2)\) equals the common total when the original totals are equal. The average therefore ties them. It is weakly preferred, as convexity requires, but not strictly preferred.

A Practical Test and Common Traps

For a numerical problem, keep the bundle calculation separate from the preference comparison. If a utility representation is provided, apply it to the original bundles and the calculated average. If only rankings are provided, use the definition directly: the question is whether the average is at least as good as each original bundle when those bundles are indifferent.

The most common traps are reversing the weights, averaging utilities instead of quantities, and treating weak preference as strict preference. A useful check is to test the endpoints: at \(\lambda=1\), the weighted average is \(x\); at \(\lambda=0\), it is \(y\). For an interior weight, each good’s resulting quantity must lie between its quantities in the two original bundles.

Convexity describes the shape of preference rankings across bundles, not a claim that every consumer always chooses a balanced bundle regardless of prices or available options. It also differs from monotonicity, the more-is-better condition covered in the earlier material: monotonicity compares bundles by whether they contain more goods, while convexity compares rankings of mixtures. The two properties answer different questions.