Non-Convex Preferences and Multiple Optimal Bundles

Why a Tangency May Not Be Enough

For many consumer-choice problems, finding a bundle where the marginal rate of substitution equals the price ratio is a useful step. But a tangency is a candidate, not automatically a global maximum. If preferences are non-convex, an interior tangency may deliver less utility than a feasible boundary bundle, and two widely separated bundles may tie for the highest utility.

The central method is direct comparison: state the feasible budget set, calculate utility at relevant candidates, and compare their values. The examples use hypothetical goods and preferences to isolate the logic. This topic is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Definition: Preferences are convex if, whenever two bundles are each at least as good as a third bundle, every weighted average of those two bundles is also at least as good as the third. Equivalently, every upper contour set—the set of bundles that provide at least a specified utility level—is convex. Preferences are non-convex when this condition fails: a mixture of two bundles may be worse than each of them.

Convex preferences commonly represent a taste for balanced combinations. Non-convex preferences can instead represent a preference for concentration or specialization. Non-convexity does not mean that a consumer cannot rank bundles or that utility maximization is impossible. It means that familiar local conditions alone may not identify the best affordable bundle.

A Utility Model with Non-Convex Preferences

Consider two divisible goods, with quantities $x\geq 0$ and $y\geq 0$, and utility

$$ U(x,y)=x^2+y^2. $$

This utility is increasing in either good on the nonnegative range: increasing a quantity while holding the other fixed cannot lower utility. Yet the preferences are non-convex. For example, the bundles $(6,0)$ and $(0,6)$ each provide utility $36$, while their midpoint $(3,3)$ provides utility $18$. Thus the upper contour set at utility $36$ contains the two first bundles but not their midpoint, so that set is not convex.

With positive prices $p_x,p_y$ and income $m$, the consumer chooses from the budget set $p_xx+p_yy\leq m$. Since this utility is increasing, an optimum uses the full budget. At a differentiable interior tangency where both marginal utilities are positive, the usual condition is $MU_x/MU_y=p_x/p_y$. This condition identifies a stationary candidate; it does not, by itself, establish that the candidate is a maximum.

Key distinction: A tangency equation concerns the local slope of an indifference curve and the budget line. Global optimality concerns utility over the entire affordable set. Under non-convex preferences, check feasible boundary bundles and any other relevant candidates rather than stopping at tangency.

Worked Examples: Compare Utility Across the Budget Set

Worked Example: An Interior Tangency Is Worse Than Both Endpoints

A consumer has income $m=\$10$, and each unit of either good costs $\$1$. Quantities $x,y$ are nonnegative and utility is $U(x,y)=x^2+y^2$. Find the global maximum and assess the interior tangency.

Method. Use the binding budget to describe all affordable bundles, then compare the stationary point and the endpoints. The common mistake to avoid is treating tangency as proof of a global maximum.

The budget is $x+y=10$, so $y=10-x$ for $0\leq x\leq 10$. At an interior tangency, $MU_x=2x$, $MU_y=2y$, and $p_x/p_y=1$. The tangency condition $MU_x/MU_y=1$ gives $x/y=1$. Together with $x+y=10$, this gives $x=5$, $y=5$. Its utility is $5^2+5^2=50$.

Now compare the boundary bundles. At $(10,0)$, utility is $10^2+0^2=100$. At $(0,10)$, utility is $0^2+10^2=100$. Both are feasible and each gives more utility than the tangency.

To verify that no other bundle does better, substitute the budget into utility:

$$ U(x,10-x)=x^2+(10-x)^2=2x^2-20x+100. $$

The derivative is $4x-20$, which is zero at $x=5$; the second derivative is $4>0$, so this stationary point is a minimum along the budget line. The quadratic is convex on the whole interval, and its maximum on this interval is attained at an endpoint. Thus the global maximum utility is $100$, attained at both $(10,0)$ and $(0,10)$. The calculation checks: each endpoint spends $\$10$, and each has utility $100>50$.

Worked Example: Two Separated Bundles Tie for the Maximum

A consumer has income $\$12$. Good $x$ costs $\$2$ per unit, good $y$ costs $\$1$ per unit, and utility is $U(x,y)=x^2+\frac14y^2$. Find all utility-maximizing bundles and evaluate the interior tangency.

Method. Parameterize the budget line, then compare its stationary point with both endpoints. Avoid assuming that a unique tangency means a unique optimum.

Because utility is increasing, the budget binds: $2x+y=12$, so $y=12-2x$ and $0\leq x\leq6$. Substitution gives

U(x,12-2x)=x^2+\frac14(12-2x)^2 =x^2+36-12x+x^2 =2x^2-12x+36.

The derivative is $4x-12$, so the stationary point is $x=3$, $y=6$. At this bundle, $MU_x=2x=6$ and $MU_y=\frac12y=3$, so $MU_x/MU_y=2=p_x/p_y$: it is indeed a tangency. Its utility is $3^2+\frac14(6^2)=9+9=18$. The second derivative of the budget-line utility is $4>0$, so this stationary point is a minimum along the budget.

The endpoint with all income spent on $y$ is $(0,12)$, with utility $0+\frac14(144)=36$. The endpoint with all income spent on $x$ is $(6,0)$, with utility $36+0=36$. The quadratic is convex, so no interior point exceeds the larger endpoint value. Both endpoints are global optima, each using the full $\$12$, and they are separated rather than nearby bundles. The interior tangency has lower utility, $18$.

Worked Example: Prices Can Favor One Endpoint

A consumer has income $m=\$15$, with $p_x=\$1$ and $p_y=\$3$. The goods are nonnegative and utility is $U(x,y)=x^2+y^2$. Find the global optimum and compare it with the interior tangency.

Method. Express utility as a function of one quantity along the budget and evaluate the stationary point and both endpoints. The common mistake is to find a tangency and omit the boundary comparison.

The binding budget is $x+3y=15$, hence $y=(15-x)/3$ and $0\leq x\leq15$. Utility along the budget is

U\left(x,\frac{15-x}{3}\right) =x^2+\frac{(15-x)^2}{9} =\frac{10}{9}x^2-\frac{10}{3}x+25.

The derivative is $\frac{20}{9}x-\frac{10}{3}$. Setting it to zero gives $x=\frac{10}{3}\cdot\frac{9}{20}=1.5$, and then $y=(15-1.5)/3=4.5$. At this bundle, $MU_x=2x=3$ and $MU_y=2y=9$. Therefore $MU_x/MU_y=1/3=p_x/p_y$, as required at an interior tangency. Its utility is $1.5^2+4.5^2=2.25+20.25=22.5$. The second derivative is $20/9>0$, so this stationary point is a minimum along the budget.

At $(15,0)$, utility is $15^2=225$. At $(0,5)$, utility is $5^2=25$; this endpoint is feasible because $3(5)=15$. The budget-line utility is convex, so its maximum over the interval is at an endpoint. Therefore $(15,0)$ is the unique global optimum, with utility $225$. The check confirms that the chosen bundle spends $\$15$ and exceeds both the tangency utility and the other endpoint's utility.

Why the Endpoint Comparison Works Here

The examples above use quadratic utility, which makes direct comparison particularly transparent. On a straight budget line, substituting one quantity as a linear function of the other produces a quadratic function. When that quadratic is strictly convex, an interior stationary point is a minimum, not a maximum, and the greatest value on the closed feasible interval is attained at an endpoint.

For the first example, let $x$ range from $0$ to $10$, and write the budget-line utility as $f(x)=2x^2-20x+100$. Its positive second derivative means the graph bends upward. More formally, any feasible $x$ is a weighted average of the endpoint quantities $0$ and $10$: $x=(1-t)0+t(10)$, where $t=x/10$ and $0\leq t\leq1$. Convexity gives $f(x)\leq(1-t)f(0)+tf(10)$. Since both endpoint values are $100$, this implies $f(x)\leq100$. This is why comparing the endpoints proves global optimality in that example, rather than merely suggesting it.

This endpoint argument depends on the function along the budget being convex. It should not be applied automatically to every non-convex preference problem. In a different model, the budget-line utility might bend in another way or have more than one stationary point. The reliable general procedure is to identify feasible candidates and compare utility, using additional shape information only when it has been established for the particular model.

Write the feasible set.
Record the budget inequality, nonnegativity restrictions, and whether the budget binds under the stated utility assumptions.

Find relevant candidates.
Check interior stationary or tangency points and feasible boundary bundles. A stationary point may be a minimum or another non-maximizing candidate.

Compare utility values.
Evaluate the original utility at feasible candidates and use the model's shape, if known, to rule out any unexamined bundles.

Common Exam Trap

A frequent error is to solve $MU_x/MU_y=p_x/p_y$ and immediately label the result “the consumer optimum.” In the first worked example, this calculation produces $(5,5)$, with utility $50$. Calling it the maximum is wrong because the feasible endpoints each give utility $100$.

The correction is to treat tangency as a candidate condition, not a global comparison. Check that the candidate is feasible, determine whether it is a maximum along the budget or a minimum, and compare it with boundary alternatives. A second trap is to report only one optimum when two separated bundles tie. In the second example, both $(6,0)$ and $(0,12)$ maximize utility; reporting only one omits a global optimum.

Key takeaway: Under non-convex preferences, an interior tangency can fail to maximize utility, and multiple separated bundles can be globally optimal. State the feasible set and compare utility values before concluding which bundle is best.

Check Your Understanding

For each problem, compare feasible candidates rather than assuming a tangency is a maximum.

A consumer has $x+y=8$, with $x,y\geq0$, and $U=x^2+y^2$. Find all global optima and the utility at the midpoint of the budget line. (Answer: $(8,0)$ and $(0,8)$ are both optima with utility $64$; the midpoint $(4,4)$ has utility $32$.)
A consumer has $2x+y=10$, $x,y\geq0$, and $U=x^2+\frac14y^2$. Find the global optima and the utility at the interior stationary point. (Answer: $(5,0)$ and $(0,10)$ are both optima with utility $25$; the stationary point $(2.5,5)$ has utility $12.5$.)
A consumer has $x+2y=12$, $x,y\geq0$, and $U=x^2+y^2$. Find the interior stationary bundle and determine the global optimum. (Answer: The stationary bundle is $(2.4,4.8)$, with utility $28.8$; the global optimum is $(12,0)$, with utility $144$.)
For $U=x^2+y^2$, compare bundles $(6,0)$, $(0,6)$, and their midpoint. What does this show about the upper contour set at utility $36$? (Answer: The first two bundles have utility $36$, while the midpoint $(3,3)$ has utility $18$; the upper contour set at $36$ is non-convex.)
A consumer has $x+y=10$, $x,y\geq0$, and $U=4x^2+y^2$. Find the interior stationary bundle and the global optimum. (Answer: The stationary bundle is $(2,8)$, with utility $80$; the global optimum is $(10,0)$, with utility $400$.)

What You'll Learn