Perfect Substitutes and Utility-Maximizing Endpoints

Why Linear Utility Produces Endpoint Choices

When two goods are perfect substitutes for a consumer, the consumer is willing to exchange one good for the other at a constant rate. With a fixed budget and constant prices, the key question is not simply which good has the larger marginal utility. It is which good provides more utility per dollar spent. If one good provides more, the consumer puts the entire budget toward that good. If both provide the same utility per dollar, every bundle that spends the full budget is equally good.

This is a useful contrast with consumer-choice problems where an interior bundle balances changing marginal utilities. For linear utility, the marginal utilities are constant, so the comparison does not change as the consumer reallocates spending. The method is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Definition: For two goods $x$ and $y$, a linear utility function has the form $u(x,y)=ax+by$, where $a>0$ and $b>0$ are constant marginal utilities. The goods are called perfect substitutes in this model because the consumer’s marginal rate of substitution is constant: $MU_x/MU_y=a/b$. Here, $MU_x=a$ and $MU_y=b$.

Assume the consumer can buy nonnegative quantities $x$ and $y$, faces positive prices $p_x$ and $p_y$, and has income $m>0$. The budget set consists of bundles satisfying $p_xx+p_yy\leq m$. Since both utility coefficients are positive, the consumer can increase utility by spending any income left over on either good. Thus, an optimum spends the full budget: $p_xx+p_yy=m$.

Key rule: Compare $a/p_x$, the utility from one dollar spent on $x$, with $b/p_y$, the utility from one dollar spent on $y$. If $a/p_x>b/p_y$, the unique optimal bundle is the all-$x$ endpoint. If $a/p_x<b/p_y$, the unique optimal bundle is the all-$y$ endpoint. If $a/p_x=b/p_y$, every bundle on the budget line is optimal.

Why the Utility-per-Dollar Rule Works

The comparison follows by expressing each good’s quantity in terms of the dollars spent on it. Let $z_x=p_xx$ and $z_y=p_yy$, so $z_x$ and $z_y$ are the amounts of income allocated to the two goods. Utility can then be written as

u(x,y)=\frac{a}{p_x}z_x+\frac{b}{p_y}z_y, \qquad z_x+z_y\leq m.

Each dollar allocated to $x$ produces $a/p_x$ units of utility, while each dollar allocated to $y$ produces $b/p_y$. The budget limits the total dollars that can be allocated. Therefore, if one rate is higher, shifting any positive amount of spending from the lower-rate good to the higher-rate good raises utility. If the rates are equal, shifting spending between the goods leaves utility unchanged.

The same result can be stated as an upper bound. Let $r=\max(a/p_x,b/p_y)$. Then

u(x,y) =\frac{a}{p_x}z_x+\frac{b}{p_y}z_y \leq r(z_x+z_y) \leq rm.

The bound is attained by spending the full budget on a good whose utility-per-dollar rate is $r$. If one rate is strictly higher, equality requires all spending to go to that good. If the rates are equal, equality holds for every allocation of the full budget. This proves both the endpoint result and the tie result.

Write the model.
Identify $a$, $b$, $p_x$, $p_y$, and $m$, and confirm that both utility coefficients and both prices are positive.

Calculate utility per dollar.
Compute $a/p_x$ and $b/p_y$, keeping each good’s coefficient paired with its own price.

Choose the allocation pattern.
A strict comparison selects one endpoint; equality means every full-budget bundle is optimal.

Calculate and check.
Use the budget equation to find endpoint quantities or verify the spending and utility of a tied bundle.

Worked Examples

Worked Example: Equal Utility per Dollar and a Tie

A consumer has utility $u(x,y)=6x+10y$, income $m=\$120$, and prices $p_x=\$3$ and $p_y=\$5$. Determine the optimal bundle or bundles, and verify whether a bundle containing both goods can also be optimal.

Method. Compare the utility each good provides per dollar, then use the budget equation to calculate the endpoint quantities and test a mixed bundle. The common mistake to avoid is treating equal rates as evidence that only one endpoint is optimal.

Step 1: Calculate the rates. For good $x$, $MU_x/p_x=6/3=2$ utility units per dollar. For good $y$, $MU_y/p_y=10/5=2$ utility units per dollar. The rates tie.

Step 2: Calculate the endpoints. Spending all $\$120$ on $x$ buys $120/3=40$ units, giving utility $6(40)+10(0)=240$. Spending all $\$120$ on $y$ buys $120/5=24$ units, giving utility $6(0)+10(24)=240$.

Step 3: Check a mixed bundle. Consider $x=20$. It costs $3(20)=\$60$, leaving $\$60$, which buys $y=60/5=12$. Its total spending is $3(20)+5(12)=60+60=\$120$, and utility is $6(20)+10(12)=120+120=240$.

Conclusion and check. Both goods provide 2 utility units per dollar, so every bundle satisfying $3x+5y=120$ is optimal. The two endpoints and the mixed bundle all give utility 240, as the tie rule predicts.

Worked Example: A Lower Marginal Utility Can Still Win

A consumer’s utility is $u(x,y)=20x+12y$. Income is $\$120$, good $x$ costs $\$10$ per unit, and good $y$ costs $\$4$ per unit. Find the utility-maximizing bundle and compare its utility with the other endpoint.

Method. Compare marginal utility per dollar rather than marginal utility alone, then calculate and check both endpoint bundles. The trap is selecting $x$ just because its marginal utility coefficient, 20, exceeds 12.

Step 1: Compare rates. Good $x$ provides $20/10=2$ utility units per dollar. Good $y$ provides $12/4=3$ utility units per dollar. Thus $y$ provides more utility per dollar.

Step 2: Buy the higher-rate good. Spending all $\$120$ on $y$ buys $120/4=30$ units. The resulting bundle is $(x,y)=(0,30)$, with utility $20(0)+12(30)=360$.

Step 3: Check the alternative endpoint. Spending all $\$120$ on $x$ buys $120/10=12$ units. Its utility is $20(12)+12(0)=240$. The $y$-only endpoint gives 360, exceeding 240 by $360-240=120$.

Conclusion and check. The optimum is $(0,30)$. It spends $4(30)=\$120$, and its utility of 360 is greater than the other endpoint’s 240. The higher marginal utility of $x$ does not make $x$ the better purchase because $x$ also costs more.

Worked Example: How a Price Change Switches the Endpoint

A consumer has utility $u(x,y)=12x+18y$, income $m=\$90$, and price $p_y=\$6$. First, find the optimum when $p_x=\$3$. Then find the tie price for $x$, and determine the optimum when $p_x=\$4.50$.

Method. Recalculate both utility-per-dollar rates at each price and use the budget to find the quantities. The common trap is assuming that the initially preferred endpoint must remain optimal after a price changes.

Step 1: Initial price $p_x=\$3$. Good $x$ provides $12/3=4$ utility units per dollar; good $y$ provides $18/6=3$. The consumer spends all $\$90$ on $x$, buying $90/3=30$ units. Utility is $12(30)=360$.

Step 2: Find the tie price. A tie requires $12/p_x=18/6=3$. Solving gives $12/p_x=3$, so $p_x=12/3=\$4$. At this price, the all-$x$ endpoint buys $90/4=22.5$ units and gives utility $12(22.5)=270$. The all-$y$ endpoint buys $90/6=15$ units and gives utility $18(15)=270$. Every full-budget combination ties at utility 270.

Step 3: New price $p_x=\$4.50$. Now $x$ provides $12/4.50=8/3$, or about 2.67, utility units per dollar, while $y$ still provides 3. Because $3>8/3$, the consumer spends all $\$90$ on $y$, buying $90/6=15$ units. Utility is $18(15)=270$.

Conclusion and check. The preferred endpoint changes from all $x$ at $\$3$ to all $y$ at $\$4.50$, with a tie at $\$4$. At the final price, the $x$-only endpoint buys $90/4.50=20$ units and gives utility $12(20)=240$, below the $y$-only utility of 270. Each stated bundle spends exactly $\$90$.

Reading the Result and Avoiding Common Traps

For this model, the marginal rate of substitution is constant at $MU_x/MU_y=a/b$. The budget line has a constant price ratio $p_x/p_y$. Equality of the two goods’ utility per dollar,

\frac{a}{p_x}=\frac{b}{p_y},

is equivalent to $a/b=p_x/p_y$. Thus, when the rates tie, the consumer’s constant willingness to exchange the goods matches the market exchange rate. Unlike a typical case where a tangency identifies one special bundle, linear utility and a tie make the consumer indifferent across the whole budget line. No single interior bundle is preferred to the endpoints.

A strict rate comparison, by contrast, rules out an optimal bundle that spends a positive amount on the lower-rate good. Moving that spending to the higher-rate good increases utility while leaving total expenditure unchanged. This is why the optimum is at an endpoint rather than a bundle with positive quantities of both goods.

Do not compare coefficients alone. The relevant comparison is $a/p_x$ versus $b/p_y$, not $a$ versus $b$.
Do not confuse a tie with a unique solution. Equal rates make every bundle on the budget line optimal, not just the two endpoints.
Keep each good’s price with its own coefficient. Calculate $a/p_x$ and $b/p_y$, not $a/p_y$ and $b/p_x$.
Check the budget. The endpoint quantities are $m/p_x$ for all $x$ and $m/p_y$ for all $y$. Multiplying by the corresponding price should give $m$.

The rule assumes both goods have positive constant marginal utilities, positive prices, and freely divisible nonnegative quantities. If a good has zero marginal utility, or if a good is free, the assumptions behind the endpoint and full-spending conclusions need to be reconsidered. The method here is for the stated two-good linear-utility model.

Key takeaway: With positive linear utility and positive prices, compare each good’s marginal utility per dollar. The higher-rate good receives the entire budget; equal rates make every full-budget bundle optimal.

Check Your Understanding

For each problem, state the maximizing choice and show how the utility-per-dollar comparison determines it.

A consumer has $u(x,y)=5x+8y$, income $\$64$, and prices $p_x=\$2$, $p_y=\$4$. Which endpoint maximizes utility, and what is the maximum utility? (Answer: All $x$: $(32,0)$, with utility 160.)
A consumer has $u(x,y)=7x+12y$, income $\$96$, and prices $p_x=\$3$, $p_y=\$8$. Calculate both utility-per-dollar rates, identify the optimum, and compare the endpoint utilities. (Answer: Rates are $7/3$ and $12/8=1.5$; choose all $x$, giving $(32,0)$ and utility 224 rather than 144.)
A consumer has $u(x,y)=9x+15y$, income $\$75$, and prices $p_x=\$3$, $p_y=\$5$. Is a bundle with both goods optimal? Give its utility for $x=10$. (Answer: Both rates are 3, so every full-budget bundle is optimal; $x=10$ leaves $\$45$ for $y=9$, giving utility 225.)
A consumer has $u(x,y)=10x+16y$, income $\$120$, and $p_y=\$8$. Find the tie price for $x$, then identify the preferred good if $p_x=\$4$ and if $p_x=\$6$. (Answer: The tie price is $\$5$; at $\$4$, choose all $x$, utility 300; at $\$6$, choose all $y$, utility 240.)
A consumer has $u(x,y)=3x+14y$, income $\$84$, and prices $p_x=\$1.50$, $p_y=\$7$. Determine whether the optimum is unique and verify the utility of the bundle $x=20$ that spends the full budget. (Answer: Both rates are 2, so the optimum is not unique; $x=20$ spends $\$30$, leaving $\$54$ for $y=54/7$, and utility is $3(20)+14(54/7)=168$.)

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