Mixed Consumer-Choice Problems with Interior and Boundary Optima

Choosing the Right Method for the Preference

A consumer-choice problem is not solved by applying one equation to every utility function. A smooth Cobb-Douglas utility function typically leads to an interior optimum, while perfect substitutes can lead to a budget endpoint. Perfect complements instead lead to a kink, where the consumer buys goods in a fixed proportion. The first task is therefore to identify the preference structure; the second is to use a method suited to that structure.

The problems here use two goods, $x$ and $y$, with prices $p_x>0$ and $p_y>0$, and income $m>0$. The consumer chooses a nonnegative bundle $(x,y)$ satisfying the budget constraint $p_xx+p_yy\leq m$. Assume the consumer prefers more to less, so an optimum spends all income. Utility represents the consumer's preference ranking: bundles with higher utility are preferred.

Definition: An interior optimum has $x>0$ and $y>0$. A boundary optimum has at least one good consumed in quantity zero, as at a budget endpoint. A kink optimum occurs at a point where the utility function is not differentiable, such as the fixed-proportion point for perfect complements. A candidate optimum must be affordable and must give at least as much utility as every other affordable bundle.

The main methods to keep distinct are the marginal-rate-of-substitution condition for a differentiable interior solution, a utility-per-dollar comparison for perfect substitutes, and direct maximization of the feasible utility level for perfect complements. Each method comes with its own check: verify the budget and confirm that the candidate is globally best under the stated preferences.

Worked Example: A Cobb-Douglas Interior Optimum

Worked Example: Finding and Verifying an Interior Bundle

A consumer has utility $U(x,y)=x^{0.4}y^{0.6}$, income $m=\$100$, and prices $p_x=\$4$ and $p_y=\$5$. Find the utility-maximizing bundle and verify that it is an interior optimum.

Method and assumptions. Both goods have positive exponents, prices are positive, and income is positive. Utility increases in either good. Thus the budget binds at an optimum. The Cobb-Douglas utility is differentiable for positive quantities, so we can find an interior candidate using the tangency condition and then check it against the budget.

The marginal utilities are $MU_x=0.4x^{-0.6}y^{0.6}$ and $MU_y=0.6x^{0.4}y^{-0.4}$. Therefore the marginal rate of substitution is $MU_x/MU_y=(0.4/0.6)(y/x)=(2/3)(y/x)$. At an interior optimum it equals the price ratio:

\frac{MU_x}{MU_y}=\frac{p_x}{p_y} \quad\Longrightarrow\quad \frac{2}{3}\frac{y}{x}=\frac{4}{5} \quad\Longrightarrow\quad y=\frac{6}{5}x.

Substitute this relationship into the binding budget: $4x+5(6x/5)=100$, so $4x+6x=100$, $10x=100$, and $x=10$. Then $y=(6/5)(10)=12$.

Budget check. The bundle costs $4(10)+5(12)=40+60=\$100$. Both quantities are positive, so it is interior. The spending shares are $\$40$ on $x$ and $\$60$ on $y$, consistent with the exponents 0.4 and 0.6. Because this Cobb-Douglas utility is concave and the budget set is convex, the feasible tangency gives a global maximum; strict concavity along the budget line makes this bundle unique.

Common mistake avoided. Reversing the price ratio would produce the wrong tangency. The condition is $MU_x/MU_y=p_x/p_y$, not $p_y/p_x$. The ratio describes the consumer's marginal willingness to exchange $y$ for $x$, while $p_x/p_y$ is the market cost of that exchange.

Why the Cobb-Douglas Spending Shares Work

The example's spending shares follow from the first-order condition, rather than being a guess. For $U(x,y)=x^\alpha y^\beta$, where $\alpha>0$, $\beta>0$, and $\alpha+\beta=1$, marginal utilities are $MU_x=\alpha x^{\alpha-1}y^\beta$ and $MU_y=\beta x^\alpha y^{\beta-1}$. Their ratio is $(\alpha/\beta)(y/x)$. Equating it to $p_x/p_y$ gives $\beta p_y y=\alpha p_xx$: expenditure on $y$ is $\beta/\alpha$ times expenditure on $x$.

Combining that relation with the budget determines the shares. If $E_x=p_xx$ and $E_y=p_yy$, then $E_y=(\beta/\alpha)E_x$ and $E_x+E_y=m$. Solving gives $E_x=\alpha m$ and $E_y=\beta m$. Hence the quantities are:

x^*=\frac{\alpha m}{p_x}, \qquad y^*=\frac{\beta m}{p_y}.

This derivation relies on positive exponents, positive income and prices, and an interior solution. In this model those conditions yield positive quantities, and the spending shares exhaust the budget. Do not apply the formula to a utility function with different preference structure merely because it also contains $x$ and $y$.

Worked Example: Perfect Substitutes and a Budget Endpoint

Worked Example: Comparing Utility per Dollar

A consumer has utility $U(x,y)=3x+2y$, income $m=\$60$, and prices $p_x=\$6$ and $p_y=\$3$. Find the best bundle and establish whether it is an interior choice.

Method and assumptions. The utility function represents perfect substitutes: each good adds a constant amount of utility, and the consumer can trade one for the other at a constant rate. With positive prices and utility increasing in both goods, the budget binds. Compare utility gained per dollar for each good.

One dollar spent on $x$ buys $1/6$ unit and gives $3(1/6)=1/2$ utility units. One dollar spent on $y$ buys $1/3$ unit and gives $2(1/3)=2/3$ utility units. Since $2/3>1/2$, spending on $y$ produces more utility for every dollar. The best bundle spends the full budget on $y$: $x=0$ and $y=60/3=20$.

Verification. This bundle costs $6(0)+3(20)=\$60$ and gives utility $3(0)+2(20)=40$. By contrast, spending the full budget on $x$ would buy $60/6=10$ units and give utility $3(10)=30$. More generally, replacing any dollar spent on $x$ with a dollar spent on $y$ increases utility by $2/3-1/2=1/6$. Thus no affordable bundle with positive $x$ can improve on the all-$y$ choice.

Common mistake avoided. Comparing the coefficients 3 and 2 alone would suggest choosing $x$, because its coefficient is larger. The relevant comparison is utility per dollar, which also accounts for prices. Here the less expensive good $y$ gives more utility per dollar.

For perfect substitutes $U(x,y)=ax+by$, with $a,b>0$, compare $a/p_x$ with $b/p_y$. If $a/p_x>b/p_y$, all income goes to $x$; if $a/p_x<b/p_y$, all income goes to $y$. If the two ratios are equal, every point on the budget line gives the same utility, so there is not a unique best bundle. This comparison follows directly from writing utility as $a(x)+b(y)$: each dollar allocated to a good contributes its coefficient divided by its price.

Worked Example: Perfect Complements and a Kink

Worked Example: Maximizing Utility at Fixed Proportions

A consumer has utility $U(x,y)=\min\{x,2y\}$, income $m=\$48$, and prices $p_x=\$3$ and $p_y=\$4$. Find and verify the best bundle.

Method and assumptions. This is a perfect-complements preference: utility depends on the amount of the less available component, and extra units of one good alone do not raise utility once the other component is limiting. The consumer can purchase nonnegative quantities and prefers a higher value of $U$. The relevant bundles are at the kink, where the two arguments of the minimum are equal.

At the kink, $x=2y$. Substituting into the budget gives $3(2y)+4y=48$, so $10y=48$, $y=4.8$, and $x=9.6$. The utility is $\min\{9.6,2(4.8)\}=9.6$, and the bundle costs $3(9.6)+4(4.8)=28.8+19.2=\$48$.

Global verification. To attain any utility level $u$, the consumer must have $x\geq u$ and $2y\geq u$, or $y\geq u/2$. The minimum cost of reaching that level is therefore $3u+4(u/2)=5u$. The budget requires $5u\leq48$, so $u\leq9.6$. The bundle found above reaches $u=9.6$, proving that it is optimal.

Common mistake avoided. Do not use a smooth marginal-rate-of-substitution tangency at the kink: this utility is not differentiable where $x=2y$. Nor should you spend extra on just one good after reaching the proportion. For example, if $x>2y$, the utility is $2y$, so additional $x$ without additional $y$ does not increase utility.

Checks That Distinguish the Three Methods

The following sequence helps prevent a correct-looking calculation from being applied to the wrong problem. It is a way to organize the solution, not a substitute for checking the candidate bundle.

Identify the utility shape.
For a differentiable Cobb-Douglas function, test for an interior tangency. For perfect substitutes, compare utility per dollar. For perfect complements, locate the fixed-proportion kink.

Use the matching condition.
Apply the marginal-rate-of-substitution condition only to an interior differentiable candidate; compare coefficient-price ratios for linear substitutes; or impose the complement ratio for the kink.

Check feasibility and optimality.
Substitute the quantities into the budget and utility function. Then use concavity, the per-dollar comparison, or the affordable-utility bound to establish that no feasible alternative is better.

These distinctions are useful because a candidate from the wrong method can satisfy the budget and still fail to maximize utility. A tangency is not automatically relevant when a good is zero or utility has a kink; conversely, a linear utility function may have a unique endpoint optimum even though the budget line itself contains many feasible bundles. This tutorial is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Key takeaway: Match the optimization method to the preference structure. Verify an interior Cobb-Douglas candidate with the tangency condition and budget, a perfect-substitutes choice with utility per dollar, and a perfect-complements choice with the maximum affordable utility level at the kink.

Check Your Understanding

For each problem, identify the appropriate method, find the optimal bundle or set of bundles, and check the budget.

A consumer has $U(x,y)=x^{0.25}y^{0.75}$, income $m=\$80$, and prices $p_x=\$4$, $p_y=\$4$. Find the interior optimum and the spending on each good. (Answer: $x=5$, $y=15$; spending is $\$20$ on $x$ and $\$60$ on $y$.)
A consumer has $U(x,y)=5x+3y$, income $m=\$72$, and prices $p_x=\$9$, $p_y=\$3$. Compare utility per dollar and find the optimal bundle and utility. (Answer: all income goes to $y$; $x=0$, $y=24$, and $U=72$.)
A consumer has $U(x,y)=\min\{3x,y\}$, income $m=\$90$, and prices $p_x=\$6$, $p_y=\$2$. Find the kink bundle and verify the maximum utility using the minimum cost of attaining utility $u$. (Answer: $x=7.5$, $y=22.5$, and maximum utility is $22.5$; the minimum cost of utility $u$ is $4u$, so $u\leq22.5$.)
A consumer has $U(x,y)=2x+4y$, income $m=\$50$, and prices $p_x=\$5$, $p_y=\$10$. Determine whether the optimum is unique and describe all optimal bundles. (Answer: both goods give $0.4$ utility per dollar; every bundle satisfying $5x+10y=50$, with $x,y\geq0$, is optimal and gives utility $20$.)
A consumer has $U(x,y)=x^{0.6}y^{0.4}$, income $m=\$120$, and prices $p_x=\$6$, $p_y=\$8$. Use the Cobb-Douglas spending shares to find quantities, verify the budget, and state whether the optimum is interior. (Answer: spending is $\$72$ on $x$ and $\$48$ on $y$, so $x=12$, $y=6$; the budget binds and the optimum is interior.)

What You'll Learn