Consumer Composition and Market Demand

Why the Mix of Buyers Matters

A market can have exactly the same number of buyers today as it did yesterday and still have a different demand schedule. The reason may be that the buyers are not alike: some plan to purchase more of a product than others at a given price. If the share of buyers with each purchase plan changes, total market demand changes even though the number of people in the market does not.

For example, suppose a fixed population includes two groups with different demand schedules. If some buyers move from the group that plans smaller purchases to the group that plans larger purchases, the market quantity demanded at a given price may rise. The key comparison is between what those buyers would demand under each group’s purchase plan—not between the new and old population totals.

Definition: Consumer composition is the distribution of buyers across groups with different purchase plans. For groups $1,\ldots,k$, let $n_j$ be the number of buyers in group $j$, and let $q_j(P)$ be the quantity each buyer in that group plans to purchase at price $P$. Market demand is the sum of group quantities: $$ Q_d(P)=\sum_{j=1}^{k}n_jq_j(P). $$ A change in composition changes one or more group counts $n_j$. If the total number of buyers is fixed, then $\sum_{j=1}^{k}n_j$ remains constant.

This is horizontal aggregation: at a specified price, add the quantities demanded by all buyers. The group counts are weights on each group’s purchase plan. It is not generally correct to average group quantities without weighting by the number of buyers in each group.

To isolate a composition effect, hold the product and price fixed and assume that each group’s purchase plan is unchanged. Other influences on demand are also held constant. The question is then narrow: how does market quantity at that price differ because buyers are distributed differently across the groups?

How to Calculate a Composition Effect

For two groups, call them A and B, suppose $m$ buyers move from B to A. Before the change, the group counts are $n_A$ and $n_B$; afterward, they are $n_A+m$ and $n_B-m$. Total buyers are unchanged. At a fixed price, the market quantity change is:

\begin{aligned} \Delta Q_d(P) &=(n_A+m)q_A(P)+(n_B-m)q_B(P)\\ &\quad-\left[n_Aq_A(P)+n_Bq_B(P)\right]\\ &=m\left[q_A(P)-q_B(P)\right]. \end{aligned}

The equation shows why the composition calculation compares the two group plans. Each transferred buyer contributes $q_A(P)$ under the new mix but would have contributed $q_B(P)$ under the old mix. The net change per transferred buyer is their difference. Multiplying that difference by the number transferred gives the change in market demand.

Result: When $m$ buyers move from B to A, market quantity demanded at price $P$ rises if $q_A(P)>q_B(P)$, falls if $q_A(P)<q_B(P)$, and is unchanged if the two groups plan to purchase the same amount. This result compares quantities at the same price; it does not describe a movement along a demand curve caused by a price change.

The size and even the sign of the effect can depend on price. If the group demand schedules cross, one group may plan to buy more at one price and less at another. A change in composition can therefore raise market quantity at some prices and lower it at others. Do not decide the direction just from a label such as “high-demand group”; compare the groups’ quantities at the price in question.

Worked Examples

Worked Example: A Purchase Schedule With a Fixed Buyer Total

A hypothetical market has 80 buyers. At prices of $10, $8, and $6, each buyer in group A plans to purchase 1, 2, and 3 units, respectively. Each buyer in group B plans to purchase 0, 1, and 2 units, respectively. Initially, 50 buyers are in A and 30 are in B. Later, 20 buyers move from A to B, leaving 30 in A and 50 in B. Calculate market demand at each listed price before and after the change.

Method. At each price, multiply each group’s per-buyer quantity by its group count, then add the group totals. The buyer total is 80 both before and after.

At $10: Initially, market demand is $50(1)+30(0)=50+0=50$ units. After the change, it is $30(1)+50(0)=30+0=30$ units. The change is $30-50=-20$ units.

At $8: Initially, market demand is $50(2)+30(1)=100+30=130$ units. After the change, it is $30(2)+50(1)=60+50=110$ units. The change is $110-130=-20$ units.

At $6: Initially, market demand is $50(3)+30(2)=150+60=210$ units. After the change, it is $30(3)+50(2)=90+100=190$ units. The change is $190-210=-20$ units.

Check. At every listed price, 20 buyers move from a plan that involves one more unit per buyer to a plan involving one fewer unit. Thus the change is $20( q_B-q_A)=20(-1)=-20$ units at each price, matching the schedule calculations. The common mistake this method avoids is assuming that a fixed number of buyers means market demand cannot change.

Worked Example: The Effect Depends on Price

A market has 40 buyers: initially 20 in group H and 20 in group L. At prices $P$ from $0 through $6, each H buyer plans to purchase $q_H(P)=12-2P$ units, and each L buyer plans to purchase $q_L(P)=10-P$ units. Eight buyers move from L to H. Find the change in market demand at $P=$1$ and $P=$4$, and determine the price in this range at which the composition change has no effect.

Method. Use the net contribution per transferred buyer, $q_H(P)-q_L(P)$, and multiply by the eight buyers transferred. The stated price range keeps both quantities nonnegative.

At $P=$1$, one H buyer plans $12-2(1)=10$ units, while one L buyer plans $10-1=9$ units. Therefore: $$ \Delta Q_d(1)=8(10-9)=8\text{ units}. $$ The composition change raises market demand by 8 units at this price.

At $P=$4$, one H buyer plans $12-2(4)=4$ units, while one L buyer plans $10-4=6$ units. Therefore: $$ \Delta Q_d(4)=8(4-6)=-16\text{ units}. $$ The composition change lowers market demand by 16 units at this price.

For the price with no effect, set the two group quantities equal: $$ 12-2P=10-P \quad\Longrightarrow\quad 2=P. $$ At $P=$2$, each group plans $12-2(2)=8$ units, which also equals $10-2=8$. Transferring buyers between the groups therefore leaves market quantity unchanged.

Check. At $P=$1$, initial demand is $20(10)+20(9)=200+180=380$ units; new demand is $28(10)+12(9)=280+108=388$, an increase of 8. At $P=$4$, initial demand is $20(4)+20(6)=80+120=200$; new demand is $28(4)+12(6)=112+72=184$, a decrease of 16. The method avoids assuming the same direction of change at every price.

Worked Example: Deriving Both Market Demand Equations

A hypothetical market has 120 buyers divided between groups C and D. A buyer in C plans to purchase $q_C(P)=15-P$ units, and a buyer in D plans to purchase $q_D(P)=9-0.5P$ units, for prices from $0 through $15. Initially, there are 30 C buyers and 90 D buyers. Later, there are 70 C buyers and 50 D buyers. Derive each market demand equation, compare demand at $P=$8$ and $P=$14$, and find the price at which the two market quantities are equal.

Initial composition. Multiply each group’s plan by its initial count and add: $$ Q_{d,\mathrm{initial}}(P)=30(15-P)+90(9-0.5P) =450-30P+810-45P =1260-75P. $$

New composition. Use the new counts, which still total 120: $$ Q_{d,\mathrm{new}}(P)=70(15-P)+50(9-0.5P) =1050-70P+450-25P =1500-95P. $$

At $P=$8$, initial demand is $1260-75(8)=1260-600=660$ units. New demand is $1500-95(8)=1500-760=740$ units, an increase of $740-660=80$. Checking the groups directly, each C buyer demands 7 units and each D buyer demands 5: initially $30(7)+90(5)=210+450=660$; afterward $70(7)+50(5)=490+250=740$.

At $P=$14$, initial demand is $1260-75(14)=1260-1050=210$ units. New demand is $1500-95(14)=1500-1330=170$ units, a decrease of $170-210=-40$. Directly, C buyers demand 1 unit and D buyers demand 2: initially $30(1)+90(2)=210$; afterward $70(1)+50(2)=170$.

To find the price at which quantities are equal, set the equations equal: $$ 1260-75P=1500-95P \quad\Longrightarrow\quad 20P=240 \quad\Longrightarrow\quad P=12. $$ At $P=$12$, initial demand is $1260-900=360$ units and new demand is $1500-1140=360$ units. This is also consistent with each C buyer demanding $15-12=3$ units and each D buyer demanding $9-0.5(12)=3$ units. The group plans match at that price, so composition has no effect on market quantity there.

Check. Both compositions contain 120 buyers, and direct group calculations agree with the aggregate equations at both tested prices. The common mistake this method avoids is multiplying a group’s per-buyer quantity by the total population rather than by that group’s count.

Common Mistake: Counting Transferred Buyers Twice

Suppose 8 buyers move from L to H, and at a particular price an H buyer plans to purchase 4 units while an L buyer plans to purchase 2. A tempting but incorrect calculation is $8(4)=32$, as if the 8 buyers were newly added to the market. They were already buyers under the old composition. The correct net change is:

8(4-2)=8(2)=16\text{ units}.

The incorrect calculation counts the new H-plan quantity without subtracting what those same buyers would have purchased under the L plan. A quick check is to compare group counts before and after: if total buyers are unchanged, every increase in one group must be offset by a decrease in another.

Key takeaway: To measure the demand effect of a changed buyer mix, hold price fixed, calculate each group’s quantity per buyer, weight by the group counts, and compare the resulting totals. When buyers are transferred between groups, count only the difference between their old and new purchase plans.

Check Your Understanding

For each problem, calculate market demand or the composition effect at the stated price. Assume the stated purchase plans remain unchanged.

A market has 80 buyers: initially 30 are in A and 50 are in B. At a price of $5, each A buyer plans to purchase 3 units and each B buyer 1 unit. Ten buyers move from B to A. By how much does market demand change? (Answer: An increase of 20 units.)
A market has 90 buyers, initially 25 in M and 65 in N. At $P=$3$, each M buyer demands $9-P$ units and each N buyer demands $5-P$ units. Ten buyers move from N to M. Find the change in market demand. (Answer: An increase of 40 units.)
A market has 60 buyers, initially 20 in A and 40 in B. At $P=$6$, each A buyer plans to buy 4 units and each B buyer 1 unit. The new composition is 35 in A and 25 in B. Find initial demand, new demand, and the change. (Answer: Initial demand 120 units; new demand 165 units; increase of 45 units.)
There are 100 buyers, initially 50 in A and 50 in B; the new mix is 70 in A and 30 in B. At $P=$4$, per-buyer quantities are $q_A(P)=8-P$ and $q_B(P)=4-0.5P$. Calculate market demand under each mix and the change. (Answer: Initial demand 300 units; new demand 340 units; increase of 40 units.)
Twenty buyers move from A to B in a market with a fixed total population. Their group plans are $q_A(P)=14-2P$ and $q_B(P)=10-P$, for prices from $0 through $7. At what price is the composition effect zero, and does it raise or lower demand at $P=$2$ and $P=$6$? (Answer: Zero effect at $P=$4$; it lowers demand by 40 units at $P=$2$ and raises it by 40 units at $P=$6$.)

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