Perfect Substitutes and Corner Demand

Why Perfect Substitutes Produce Corner Choices

When two goods are perfect substitutes, the consumer is willing to exchange one for the other at a constant rate. If one good delivers more utility for each dollar spent, buying some of the less efficient good would reduce the utility attainable from the same budget. The consumer therefore chooses a corner bundle: all spending goes to the good with greater utility per dollar. If the two goods deliver exactly equal utility per dollar, every affordable bundle on the budget line is equally good.

This result is an important contrast with an interior choice. A tangency between an indifference curve and the budget line need not occur for perfect substitutes: their indifference curves are straight lines, so the consumer typically chooses a budget endpoint. The key is to compare marginal utility per dollar, not just the marginal utilities or prices separately.

Definition: Let $x$ and $y$ be quantities of two divisible goods, with utility $U(x,y)=\alpha x+\beta y$, where $\alpha>0$ and $\beta>0$. The goods are perfect substitutes in this model: each additional unit of $x$ adds $\alpha$ units of utility, and each additional unit of $y$ adds $\beta$ units of utility, regardless of the bundle already consumed. Given positive prices $p_x,p_y$ and income $m$, the consumer chooses $x,y\geq 0$ to maximize utility subject to $p_xx+p_yy\leq m$.

The marginal utilities are $MU_x=\alpha$ and $MU_y=\beta$. Dividing each by its price gives utility per dollar: $MU_x/p_x=\alpha/p_x$ for good $x$ and $MU_y/p_y=\beta/p_y$ for good $y$. Assume goods can be bought in fractional quantities, the consumer values both goods positively, and there are no other constraints or benefits from unspent income. With positive prices and positive marginal utilities, spending all available income is optimal.

The Demand Rule and Price Threshold

Compare $\alpha/p_x$ with $\beta/p_y$. If the first is larger, each dollar spent on $x$ produces more utility, so the consumer buys only $x$. If the second is larger, the consumer buys only $y$. Equality means either good produces the same utility per dollar, and the consumer is indifferent among all affordable combinations.

Theorem: For $U(x,y)=\alpha x+\beta y$, with $\alpha,\beta,p_x,p_y>0$ and $m>0$:

If $\alpha/p_x>\beta/p_y$, the unique optimum is $x=m/p_x,\ y=0$.
If $\alpha/p_x<\beta/p_y$, the unique optimum is $x=0,\ y=m/p_y$.
If $\alpha/p_x=\beta/p_y$, every bundle satisfying $p_xx+p_yy=m$, $x,y\geq0$, is optimal.

The switching threshold can also be written $p_x/p_y=\alpha/\beta$. Below that relative price, $x$ is preferred; above it, $y$ is preferred.

The threshold form follows by multiplying the comparison $\alpha/p_x>\beta/p_y$ by the positive quantity $p_xp_y$. This gives $\alpha p_y>\beta p_x$, or equivalently $p_x/p_y<\alpha/\beta$. At equality, the relative price exactly matches the rate at which the utility function values the goods.

Why the Rule Holds

The budget constraint explains why the higher utility-per-dollar good dominates. Since utility is increasing in both goods, the budget binds at an optimum, so $p_xx+p_yy=m$. Solve this equation for $y$ and substitute into utility:

y=\frac{m-p_xx}{p_y}, \qquad U(x,y)=\alpha x+\beta\left(\frac{m-p_xx}{p_y}\right) =\frac{\beta m}{p_y} +x\left(\alpha-\frac{\beta p_x}{p_y}\right).

The feasible values of $x$ range from $0$ to $m/p_x$. The expression for utility is a constant plus $x$ multiplied by $\alpha-\beta p_x/p_y$. If that coefficient is positive, utility rises as $x$ increases, so the maximum is at $x=m/p_x$. If it is negative, utility falls as $x$ increases, so the maximum is at $x=0$. If it is zero, utility is constant for every feasible $x$, which means every point on the budget line is optimal. This establishes both the corner choices and the entire set of optima at the threshold.

Calculate each utility per dollar.
Compute $\alpha/p_x$ and $\beta/p_y$, keeping the good and its price paired correctly.

Compare the two values.
The larger value identifies the good that receives all spending; equality identifies the indifference threshold.

Apply the budget constraint.
For a strict preference, divide income by the chosen good’s price to obtain its quantity. At equality, report the full budget-line set, not just one bundle.

Worked Examples

Worked Example: Selecting the Better Value per Dollar

A consumer has utility $U(x,y)=5x+3y$, income $m=\$60$, and prices $p_x=\$4$ and $p_y=\$3$. Determine the optimal bundle and verify it against the all-$y$ alternative.

Method: compare marginal utility per dollar. One dollar spent on $x$ buys $1/4$ unit of $x$, adding $5(1/4)=1.25$ units of utility. Thus $MU_x/p_x=5/4=1.25$. One dollar spent on $y$ buys $1/3$ unit of $y$, adding $3(1/3)=1$ unit of utility, so $MU_y/p_y=3/3=1$. Since $1.25>1$, all spending goes to $x$.

The quantity of $x$ is $m/p_x=60/4=15$, and $y=0$. The consumer spends $4(15)+3(0)=\$60$ and obtains utility $U(15,0)=5(15)+3(0)=75$. If the consumer instead bought only $y$, the quantity would be $60/3=20$, giving utility $U(0,20)=3(20)=60$. The chosen bundle gives greater utility and exhausts the budget.

Common mistake avoided: Comparing $5$ with $3$ alone would happen to point toward $x$ here, but it is not the correct method. Prices matter: the valid comparison is $5/4$ against $3/3$.

Worked Example: Finding the Switching Price

A consumer has utility $U(x,y)=2x+5y$, income $m=\$120$, and the price of $y$ is $p_y=\$10$. Find the price of $x$ at which the consumer switches between corner choices. Then find the optimal bundle when $p_x=\$3$, when $p_x$ equals the threshold, and when $p_x=\$6$.

Method: solve the equality of utility per dollar. The switching condition is $2/p_x=5/10$. Since $5/10=0.5$, the condition becomes $2/p_x=0.5$, so $p_x=2/0.5=\$4$. Equivalently, $p_x/p_y=\alpha/\beta=2/5$, giving $p_x=(2/5)(10)=\$4$.

When $p_x=\$3$, utility per dollar is $2/3\approx0.6667$ for $x$ and $5/10=0.5$ for $y$. The consumer chooses only $x$: $x=120/3=40,\ y=0$. Spending is $3(40)=\$120$, and utility is $2(40)=80$.

At the threshold $p_x=\$4$, both goods provide $0.5$ utility per dollar. The optimal set is every bundle satisfying $4x+10y=120$, with $x,y\geq0$. The endpoints are $(30,0)$ and $(0,12)$. Both give utility $60$, and so does every bundle between them.

When $p_x=\$6$, utility per dollar for $x$ is $2/6=1/3\approx0.3333$, less than $0.5$ for $y$. The consumer buys only $y$: $y=120/10=12,\ x=0$. Spending is $\$120$, and utility is $5(12)=60$.

The three cases confirm the direction of the switch: below $\$4$, $x$ is preferred; at $\$4$, all budget-line bundles tie; above $\$4$, $y$ is preferred. A common error is to invert the threshold and conclude that $p_x/p_y=\beta/\alpha$. The correct equality is $p_x/p_y=\alpha/\beta=2/5$.

Worked Example: Describing All Optimal Bundles at Indifference

For utility $U(x,y)=7x+4y$, a consumer faces prices $p_x=\$7$, $p_y=\$4$, and income $m=\$56$. Determine the complete set of optimal bundles and check an interior bundle from that set.

Method: test the threshold, then use the budget equation. The utility per dollar of $x$ is $7/7=1$, and that of $y$ is $4/4=1$. Therefore the consumer is exactly at the threshold. Every affordable bundle that spends the full $\$56$ is optimal, so the complete set is $7x+4y=56$, $x,y\geq0$.

Solving for $y$ gives $y=14-(7/4)x$. Nonnegative quantities require $0\leq x\leq8$. The endpoints are $(8,0)$ and $(0,14)$. Choose the interior bundle $x=4,\ y=7$: its cost is $7(4)+4(7)=28+28=\$56$, and its utility is $7(4)+4(7)=28+28=56$. At the first endpoint, utility is $7(8)=56$; at the second, it is $4(14)=56$. Thus the interior bundle and both endpoints give the same utility, as the equality condition predicts.

Common mistake avoided: Reporting only $(8,0)$ or only $(0,14)$ would omit other optimal choices. At equal utility per dollar, demand is a set of bundles, not a unique quantity pair.

Interpreting Corner Demand and Avoiding Traps

A corner choice does not mean the consumer is unable to consume the other good. It means that, at the given prices and utility weights, using any of the budget on the less productive good would lower attainable utility. A change in either price can reverse the comparison, while a change in income alone changes the quantity purchased at a strict corner but does not change which good is preferred. At the exact price threshold, higher income expands the set of affordable bundles but does not remove indifference along the budget line.

The utility coefficients measure the utility contribution of one unit of each good, not the amount of money the consumer should spend on it. The price must be included to turn that contribution into utility per dollar. Also, the price threshold is a relative price condition. Multiplying both prices by the same positive factor leaves their ratio unchanged, so it does not change which good is preferred.

The rule relies on the linear utility representation and the stated assumptions. If marginal utilities vary with quantities, the utility-per-dollar comparison need not be constant across the budget line, and the corner-demand conclusion cannot be inferred from this rule alone. For perfect substitutes, however, comparing two constant marginal utilities per dollar is sufficient: it determines the corner optimum or, at equality, the entire optimal budget line.

Key takeaway: For linear utility $U(x,y)=\alpha x+\beta y$, compare $\alpha/p_x$ with $\beta/p_y$. The larger ratio gets the whole budget. Equality occurs at $p_x/p_y=\alpha/\beta$, where every bundle on the budget line is optimal.

Check Your Understanding

For each problem, compare utility per dollar before applying the budget constraint. At a tie, identify the entire set of optimal bundles.

A consumer has $U(x,y)=3x+2y$, income $\$48$, and prices $p_x=\$6$, $p_y=\$4$. Determine whether the optimum is unique and state the optimal set. (Answer: Both goods provide $0.5$ utility per dollar; every bundle $6x+4y=48$, $x,y\geq0$, is optimal.)
For $U(x,y)=4x+9y$, prices $p_x=\$8$, $p_y=\$18$, and income $\$72$, determine the optimal set and verify its utility. (Answer: Both goods provide $0.5$ utility per dollar; every bundle $8x+18y=72$ is optimal, and each gives utility $36$.)
A consumer has $U(x,y)=6x+2y$, income $\$90$, and prices $p_x=\$9$, $p_y=\$6$. Find the optimal quantities, expenditure, and utility. (Answer: $MU_x/p_x=6/9=2/3>2/6=1/3$; $x=10,\ y=0$, expenditure $\$90$, utility $60$.)
For $U(x,y)=3x+8y$, income is $\$96$, and $p_y=\$12$. Find the switching price for $x$, then determine demand if $p_x=\$6$. (Answer: The threshold is $p_x/p_y=3/8$, so $p_x=\$4.50$. At $\$6$, $8/12>3/6$, so $x=0,\ y=8$, with utility $64$.)
A consumer has $U(x,y)=5x+2y$, income $\$100$, and $p_y=\$8$. Find the threshold price for $x$, and give the optimal choice when $p_x=\$16$ and when $p_x=\$20$. (Answer: The threshold is $p_x/p_y=5/2$, so $p_x=\$20$. At $\$16$, $5/16>2/8$, so $x=100/16=6.25,\ y=0$. At $\$20$, all bundles $20x+8y=100$, $x,y\geq0$, are optimal, each giving utility $25$.)

What You'll Learn