Perfect Complements and Fixed-Proportion Demand

Why Fixed Proportions Matter

Some goods provide value only when used together in a particular ratio. A left shoe without a right shoe, for example, does not make a wearable pair; likewise, a hypothetical device may require one sensor for every two units of a particular component. For such goods, buying more of one while holding the other fixed does not increase the number of useful combinations. This is the defining feature of perfect complements.

The consumer’s choice is therefore not found by dividing expenditure between goods according to a smooth trade-off. Instead, the consumer selects the preferred fixed-proportion bundle that the budget can afford. The relative prices determine how much that bundle costs, while income determines how many such bundles can be purchased.

Definition: Suppose $x$ and $y$ are nonnegative quantities, and $a>0$ and $b>0$ are the quantities of $x$ and $y$ required for one complete bundle. The perfect-complements utility function is $u(x,y)=\min\{x/a,y/b\}$. The consumer’s utility is the number of complete bundles available. A bundle in the fixed proportion satisfies $x/y=a/b$, or equivalently $x=at$ and $y=bt$ for some $t\geq 0$.

For instance, with $u(x,y)=\min\{x/3,y/2\}$, one complete bundle requires three units of $x$ and two units of $y$. A bundle with $x=6$ and $y=4$ contains two complete bundles and gives utility 2. A bundle with $x=6$ and $y=2$ gives utility 1: the extra $x$ cannot make another complete bundle because there is not enough $y$.

Deriving the Fixed-Proportion Demand

Assume the consumer has income $m>0$, faces strictly positive prices $p_x$ and $p_y$, and can purchase any nonnegative quantities of both goods. The budget constraint is $p_xx+p_yy\leq m$. The utility function is the perfect-complements function just defined. Under these assumptions, the consumer chooses the affordable bundle that yields the greatest number of complete bundles.

If the consumer chooses $t$ complete bundles, the quantities needed are $x=at$ and $y=bt$. Their total cost is $p_xat+p_ybt$, or $t(ap_x+bp_y)$. Because prices are positive, spending on quantities beyond the required ratio cannot increase utility. The optimal bundle is therefore the fixed-proportion bundle whose cost uses the available income.

t^*=\frac{m}{ap_x+bp_y}, \qquad x^*=at^*=\frac{am}{ap_x+bp_y}, \qquad y^*=bt^*=\frac{bm}{ap_x+bp_y}.

Here $t^*$ is the maximum number of complete bundles, while $x^*$ and $y^*$ are the associated quantities of the two goods. The denominator $ap_x+bp_y$ is the price of one complete bundle: it includes $a$ units of $x$ and $b$ units of $y$. This formula gives the optimal quantities for the stated utility representation and budget assumptions; it does not describe preferences where the goods can be substituted for one another.

Why the formula holds: To attain utility at least $t$, a consumer must have both $x\geq at$ and $y\geq bt$, by the definition of the minimum function. Those minimum required quantities cost at least $p_xat+p_ybt=t(ap_x+bp_y)$. The budget therefore requires $t(ap_x+bp_y)\leq m$, so $t\leq m/(ap_x+bp_y)$. Choosing $x=at^*$ and $y=bt^*$ attains this bound and costs exactly $m$, proving that it is optimal.

This reasoning also shows why the optimal bundle lies at the kink of the indifference curve: both arguments of the minimum are equal there, so neither good is left in excess of the fixed proportion. With strictly positive prices and income, the bundle at that kink is the unique affordable bundle that attains the maximum utility. A bundle with extra units of just one good would either cost more than the budget at the same utility or require giving up enough of the other good to reduce utility.

Read the proportions.
Write the utility as $u(x,y)=\min\{x/a,y/b\}$. One complete bundle contains $a$ units of $x$ and $b$ units of $y$.

Find the cost per bundle.
Calculate $ap_x+bp_y$, using the quantities required for one bundle and the prices of the corresponding goods.

Find the number of bundles.
Divide income by the cost per bundle: $t^*=m/(ap_x+bp_y)$.

Recover both quantities and check.
Calculate $x^*=at^*$ and $y^*=bt^*$, then verify both the required ratio and the budget expenditure.

Worked Applications

Worked Example: A Price Decrease for One Complement

A hypothetical buyer has preferences $u(x,y)=\min\{x/3,y/2\}$, income $m=\$120$, and initial prices $p_x=\$4$ and $p_y=\$3$. The price of $x$ falls to $\$2$, while income and the price of $y$ remain unchanged. Find the initial and new optimal quantities, and check the new expenditure.

Method. The utility function requires $a=3$ units of $x$ and $b=2$ units of $y$ per complete bundle. Calculate the cost per bundle at each set of prices, then divide income by that cost.

Initially, one complete bundle costs $3(\$4)+2(\$3)=\$12+\$6=\$18$. The number of bundles affordable is $t^*=\$120/\$18=20/3$. Thus $x^*=3(20/3)=20$ and $y^*=2(20/3)=40/3$, or approximately $13.3333$ units of $y$. The initial bundle costs $4(20)+3(40/3)=\$80+\$40=\$120$.

After the price decrease, one complete bundle costs $3(\$2)+2(\$3)=\$6+\$6=\$12$. The number of bundles affordable is $t^*=\$120/\$12=10$. The new quantities are $x^*=3(10)=30$ and $y^*=2(10)=20$.

Check. The new quantities satisfy $30/3=20/2=10$, so they are in the required proportion. Their expenditure is $2(30)+3(20)=\$60+\$60=\$120$, exactly the available income. Both quantities rise because the decrease in the price of $x$ lowers the cost of each complete bundle.

Common mistake avoided. Do not use the cost per bundle as if it were the quantity of either good. The quantity of $x$ is $a$ times the number of bundles, and the quantity of $y$ is $b$ times that number.

Worked Example: Income and Price Change Together

A hypothetical production kit requires two units of component $x$ and five units of component $y$, with utility $u(x,y)=\min\{x/2,y/5\}$. Initially, a buyer has income $\$220$, with $p_x=\$5$ and $p_y=\$2$. Later, income is $\$180$, $p_x$ is unchanged, and $p_y$ rises to $\$4$. Calculate both optimal bundles and compare the quantities.

Method. The fixed proportions are $a=2$ and $b=5$. At each set of income and prices, calculate the full cost of a two-unit-$x$, five-unit-$y$ bundle.

Initially, the bundle costs $2(\$5)+5(\$2)=\$10+\$10=\$20$. The buyer can afford $\$220/\$20=11$ complete bundles. Therefore $x^*=2(11)=22$ and $y^*=5(11)=55$. Expenditure is $5(22)+2(55)=\$110+\$110=\$220$.

Later, one bundle costs $2(\$5)+5(\$4)=\$10+\$20=\$30$. With income of $\$180$, the buyer can afford $\$180/\$30=6$ bundles. The new quantities are $x^*=2(6)=12$ and $y^*=5(6)=30$. Expenditure is $5(12)+4(30)=\$60+\$120=\$180$.

Check and comparison. Both bundles retain the ratio $x:y=2:5$, and each uses all available income. The quantities fall from $(22,55)$ to $(12,30)$. Since income and one price changed at the same time, this comparison does not isolate the effect of either change on its own.

Common mistake avoided. The price of $y$ applies to all five units of $y$ in each complete bundle. Omitting those units would understate the cost per bundle and overstate the number of bundles the buyer can afford.

Worked Example: Checking a Proposed Bundle

A hypothetical buyer has $u(x,y)=\min\{x/2,y/3\}$, income $\$180$, and prices $p_x=\$6$ and $p_y=\$4$. A proposed purchase is $(x,y)=(18,18)$. Determine whether it is optimal, and find the optimal bundle.

Method. First calculate the utility of the proposed bundle. Then calculate the bundle cost and the maximum attainable utility using the cost-per-complete-bundle method.

At $(18,18)$, the utility is $\min\{18/2,18/3\}=\min\{9,6\}=6$. The proposed bundle spends $6(18)+4(18)=\$108+\$72=\$180$, so it is affordable and uses all income. But its quantities are not in the required $2:3$ ratio: for six complete bundles, the buyer needs $x=12$ and $y=18$. The remaining six units of $x$ do not increase utility.

One complete bundle costs $2(\$6)+3(\$4)=\$12+\$12=\$24$. The buyer can afford $\$180/\$24=7.5$ bundles, giving $x^*=2(7.5)=15$ and $y^*=3(7.5)=22.5$. These quantities cost $6(15)+4(22.5)=\$90+\$90=\$180$, and yield utility $\min\{15/2,22.5/3\}=\min\{7.5,7.5\}=7.5$.

Conclusion. The proposed bundle is not optimal: it uses all the income but achieves utility 6, below the maximum utility of 7.5. The optimal bundle aligns both quantities with the fixed proportion.

Common mistake avoided. Spending the entire budget is necessary for the optimum under these assumptions, but it is not sufficient. The purchase must also provide both goods in the required proportion.

How Income and Prices Affect Demand

The demand formulas make the effects of income and prices explicit. For fixed proportions and fixed prices, multiplying income by a positive factor multiplies both optimal quantities by that same factor. For example, doubling $m$ doubles $x^*$ and $y^*$, because the consumer can afford twice as many complete bundles.

A change in either price changes the cost of every complete bundle. With $D=ap_x+bp_y$, the derivatives of the demand quantities with respect to income and prices are:

\frac{\partial x^*}{\partial m}=\frac{a}{D}>0, \qquad \frac{\partial y^*}{\partial m}=\frac{b}{D}>0,

\frac{\partial x^*}{\partial p_x}=-\frac{a^2m}{D^2}<0, \qquad \frac{\partial y^*}{\partial p_x}=-\frac{abm}{D^2}<0,

\frac{\partial x^*}{\partial p_y}=-\frac{abm}{D^2}<0, \qquad \frac{\partial y^*}{\partial p_y}=-\frac{b^2m}{D^2}<0.

Thus, under this model, an increase in either price reduces the quantities demanded of both goods: it makes each complete bundle more expensive. A decrease in either price has the reverse effect. These are changes in the uncompensated optimal quantities as income and prices enter the budget problem; they do not mean the goods have become different proportions. The ratio $x^*/y^*=a/b$ remains fixed.

This fixed-proportion model also explains why the usual smooth tangency condition is not the method for finding demand here. At the kink, the consumer’s utility changes at the boundary between two limiting quantities; there is no smooth indifference curve with a single marginal rate of substitution at that point. The direct bundle-cost argument is both simpler and sufficient.

Common Errors and a Reliable Check

A frequent error is to reverse the required ratio. For $u(x,y)=\min\{x/3,y/2\}$, equalizing the arguments gives $x/3=y/2$, hence $x:y=3:2$, not $2:3$. For example, if $m=\$120$, $p_x=\$4$, and $p_y=\$3$, the incorrect ratio $x:y=2:3$ would give a budget-exhausting bundle $(240/17,360/17)$. Its utility is $\min\{80/17,180/17\}=80/17$, approximately $4.7059$. The correct bundle is $(20,40/3)$, with utility $20/3$, approximately $6.6667$, and expenditure $\$120$. The wrong ratio spends the budget but leaves $y$ in excess of the required proportion.

Other checks catch many calculation errors:

Check the ratio: verify $x^*/a=y^*/b$. Both values should equal the calculated number of complete bundles.
Check expenditure: substitute the quantities into $p_xx^*+p_yy^*$. The result should equal $m$ when income and both prices are positive.
Check units: multiply the number of bundles by $a$ to get $x$, and by $b$ to get $y$. Do not report the number of bundles as either good’s quantity.
Check utility: evaluate the minimum of the two scaled quantities. At the optimal fixed-proportion bundle, both arguments of the minimum are equal.

Key takeaway: For $u(x,y)=\min\{x/a,y/b\}$, one complete bundle costs $ap_x+bp_y$. Divide income by that cost to find the number of bundles, then multiply by $a$ and $b$ to obtain the optimal quantities. This treatment is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Check Your Understanding

For each problem, identify the fixed proportions before calculating. Verify the budget and the resulting bundle ratio.

A buyer has $u(x,y)=\min\{x/4,y/2\}$, income $\$120$, $p_x=\$3$, and $p_y=\$6$. Find the optimal quantities and verify the expenditure. (Answer: $t^*=5$, $x^*=20$, $y^*=10$; expenditure $3(20)+6(10)=\$120$.)
A hypothetical kit has $u(x,y)=\min\{x/2,y/3\}$. With income $\$210$, $p_x=\$5$, and $p_y=\$4$, find the optimal quantities. Check both the proportion and budget. (Answer: $t^*=105/11$, $x^*=210/11$, $y^*=315/11$; expenditure is $\$210$.)
For $u(x,y)=\min\{x/3,y\}$, income is $\$160$, $p_x=\$2$, and $p_y=\$5$. Find the optimal bundle and maximum utility. (Answer: $t^*=160/11$, $x^*=480/11$, $y^*=160/11$; maximum utility is $160/11$.)
A consumer has $u(x,y)=\min\{x, y/2\}$, prices $p_x=\$4$ and $p_y=\$3$, and income rises from $\$150$ to $\$180$. Find both bundles and the change in each quantity. (Answer: The cost per bundle is $\$10$; quantities rise from $(15,30)$ to $(18,36)$, increases of 3 units of $x$ and 6 units of $y$.)
A buyer has $u(x,y)=\min\{x/2,y/3\}$, income $\$240$, and $p_x=\$3$. The price of $y$ falls from $\$4$ to $\$2$. Find the initial and new bundles, and verify both expenditures. (Answer: Initially $t^*=40/3$, so $(x^*,y^*)=(80/3,40)$, with expenditure $\$240$. After the price fall, $t^*=20$, so $(x^*,y^*)=(40,60)$, also with expenditure $\$240$.)

What You'll Learn