Buyer Numbers and Market Demand

Introduction

Market demand depends not only on how much each buyer plans to purchase at a given price, but also on how many buyers are in the market. If a new group of customers becomes eligible to buy a product, the market total can rise even when every existing buyer’s plans remain unchanged. If buyers leave, their planned purchases no longer contribute to the market total.

The key is to compare total planned quantities at the same price. This tutorial develops that comparison and explains when entry or exit shifts market demand. The treatment is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Definition: A buyer’s individual demand schedule gives the quantity that buyer plans to purchase at each possible price, holding the buyer’s other relevant conditions fixed. Market demand at a given price is the sum of the quantities all buyers plan to purchase at that price. Adding a buyer adds that buyer’s planned quantity at each price; removing a buyer subtracts that quantity.

Adding and Removing Buyers

Let $q_i(p)$ denote buyer $i$’s planned quantity at price $p$, and let $N$ be the set of buyers in the market. Market demand is the horizontal sum of individual demands:

Q_D(p;N)=\sum_{i\in N}q_i(p).

The notation emphasizes that total demand depends both on price and on which buyers are included. If a new buyer, called $e$, enters, the new market quantity demanded at price $p$ is

Q_D^{\text{new}}(p)=Q_D^{\text{old}}(p)+q_e(p).

Since planned quantities are nonnegative, entry cannot reduce market quantity demanded at a fixed price under this model. It increases the quantity strictly at prices where the entrant plans to buy a positive amount. At a price where the entrant plans to buy zero, market quantity at that price is unchanged. Exit reverses the comparison: the departing buyer’s planned quantity is removed from the total.

These comparisons require a clear ceteris paribus condition. The continuing buyers’ individual demand schedules must remain the same, and the product, relevant time period, and price being compared must be the same. The entrant or departing buyer may have a different demand schedule from others; the change in market demand is precisely the addition or subtraction of that buyer’s planned quantities. This analysis isolates a change in buyer numbers rather than a change in the preferences, income, or other demand conditions of continuing buyers.

A change in buyer numbers changes the market demand schedule, not the price-induced movement along a continuing buyer’s individual schedule. For a fixed price, compare the old and new market totals. To compare market demand curves, make that calculation at each price. A rightward shift is strict at prices where the added buyers collectively plan to buy a positive amount; at other prices, the curves can coincide.

Worked Example: One New Buyer Joins the Market

Two buyers have individual demand schedules $q_A(p)=\max(0,18-2p)$ and $q_B(p)=\max(0,12-p)$, where quantity is measured in units and price is in dollars per unit. A third buyer enters with $q_C(p)=\max(0,10-p)$. Assume all three schedules refer to the same product and period, and the first two buyers’ schedules do not change. Find market quantity demanded before and after entry at prices of $2, $6, and $11.

Method. Calculate each buyer’s planned quantity at each price, then add horizontally. The maximum with zero means a buyer does not plan to buy a negative quantity.

At $p=2$. Buyer A plans $18-2(2)=14$ units, and buyer B plans $12-2=10$ units. Before entry, market quantity is $14+10=24$. Buyer C plans $10-2=8$ units, so after entry the market total is $24+8=32$ units.

At $p=6$. Buyer A plans $18-2(6)=6$ units, and buyer B plans $12-6=6$ units. The old market total is $6+6=12$. Buyer C plans $10-6=4$ units, making the new total $12+4=16$ units.

At $p=11$. Buyer A’s expression is $18-2(11)=-4$, so A plans zero units. Buyer B plans $12-11=1$ unit. The old total is $0+1=1$. Buyer C’s expression is $10-11=-1$, so C plans zero units; the new total remains 1 unit.

Check and common mistake. At each price the new total equals the old total plus C’s nonnegative quantity: $24+8=32$, $12+4=16$, and $1+0=1$. A common error is to extend a linear expression into negative quantities and subtract them from the market total. The correction is to apply the zero floor to each individual schedule before adding. Entry raises market quantity at $2 and $6, but does not change it at $11.

Why the Addition Rule Holds

The market total is defined as the sum of buyers’ planned quantities at a common price. If the original buyers are in set $N$ and the entrant is $e$, then the new set is $N\cup\{e\}$. Since $e$ was not already in $N$, separating its term from the sum gives:

\begin{aligned} Q_D^{\text{new}}(p) &=\sum_{i\in N\cup\{e\}}q_i(p)\\ &=\sum_{i\in N}q_i(p)+q_e(p)\\ &=Q_D^{\text{old}}(p)+q_e(p). \end{aligned}

Because $q_e(p)\geq 0$, it follows that $Q_D^{\text{new}}(p)\geq Q_D^{\text{old}}(p)$. The inequality is strict exactly when $q_e(p)>0$. For several entrants, the same reasoning adds each entrant’s quantity; the increase is strict at a price if at least one entrant plans to buy a positive quantity. For exit, removing buyer $j$ gives $Q_D^{\text{new}}(p)=Q_D^{\text{old}}(p)-q_j(p)$, so the total is weakly lower and is strictly lower when $q_j(p)>0$.

Worked Example: A Buyer Exits from an Identical-Buyer Market

Four buyers each have the unchanged schedule $q_i(p)=\max(0,20-2p)$, where price is dollars per unit and quantity is units per week. One buyer exits. Find the old and new market demand expressions for prices from $0 through $10, compare quantities at $p=5$ and $p=10$, and state which individual curves changed.

Method. First find one buyer’s quantity in the stated price range, then multiply by the number of buyers. For $0\leq p\leq10$, $20-2p\geq0$, so each buyer plans $20-2p$ units.

Before exit there are four buyers, giving

Q_D^{\text{old}}(p)=4(20-2p)=80-8p,\qquad 0\leq p\leq10.

After one buyer exits, three buyers remain, giving

Q_D^{\text{new}}(p)=3(20-2p)=60-6p,\qquad 0\leq p\leq10.

At $p=5$, each buyer plans $20-2(5)=10$ units. The old total is $4(10)=40$ units; the new total is $3(10)=30$ units. At $p=10$, each buyer plans $20-2(10)=0$ units, so both market totals are zero.

Check and common mistake. At $p=5$, the difference is $40-30=10$ units, exactly the departing buyer’s planned quantity. At $p=10$, there is no difference because that buyer planned to buy zero. It would be incorrect to say that the remaining buyers each reduce purchases after the exit: their individual curves are unchanged. Only the number of curves included in the market sum has changed.

Entry by Buyers with Different Demand Schedules

Buyers need not be identical. The addition rule applies to any entrant’s schedule, not just to a buyer who resembles those already in the market. Consequently, the market demand curve can change in shape as well as in its quantities. The exact change at each price is determined by the entrants’ own schedules.

Worked Example: Two Entrants with Different Plans

Two existing buyers have weekly demand schedules $q_E(p)=\max(0,15-3p)$ and $q_F(p)=\max(0,9-p)$. Two buyers enter with schedules $q_G(p)=\max(0,8-2p)$ and $q_H(p)=\max(0,12-2p)$. Price is in dollars per unit. Assume the existing buyers’ schedules remain fixed. Calculate market quantity before and after entry at prices of $2, $4, and $5.

Method. Evaluate each schedule separately at each price and then sum. This avoids assuming that all buyers purchase the same quantity.

At $p=2$. The existing buyers plan $q_E=15-3(2)=9$ and $q_F=9-2=7$ units, for an old total of $9+7=16$. The entrants plan $q_G=8-2(2)=4$ and $q_H=12-2(2)=8$ units. The increase is $4+8=12$, so the new total is $16+12=28$ units.

At $p=4$. The existing buyers plan $q_E=15-3(4)=3$ and $q_F=9-4=5$, for an old total of 8. Buyer G’s expression is $8-2(4)=0$, while buyer H plans $12-2(4)=4$. The new total is $8+0+4=12$ units.

At $p=5$. Buyer E’s expression is $15-3(5)=0$, and buyer F plans $9-5=4$, for an old total of 4. Buyer G’s expression is $8-2(5)=-2$, so G plans zero; buyer H plans $12-2(5)=2$. The new total is $4+0+2=6$ units.

Check and common mistake. The increases are 12, 4, and 2 units, respectively, matching $q_G+q_H$ at each price. A common error is to add the two entrants’ quantities at one price and use that same increase at every price. Their planned quantities vary with price, so the market total must be recomputed price by price.

Interpreting the Market-Demand Change

A useful way to organize an entry or exit calculation is to keep the price fixed, identify which buyers are included before and after, and then add or remove only the affected individual quantities. When the comparison is about an entire schedule, repeat the calculation for each relevant price. The following table summarizes the distinction:

Change	Effect at a fixed price	Condition for a strict change
One buyer enters	Add the entrant’s planned quantity	The entrant plans to buy a positive amount
Several buyers enter	Add their planned quantities	At least one entrant plans to buy a positive amount
One buyer exits	Subtract the departing buyer’s planned quantity	The departing buyer would have bought a positive amount

The table describes the effect on market quantity demanded at a given price. It does not say what happens to the market price. A price change would require information about supply as well as demand. Nor does buyer entry by itself imply that every buyer purchases more: the model holds the continuing buyers’ individual schedules fixed, and their quantities at a given price do not change.

For example, if an entrant’s schedule assigns zero quantity to prices above a particular threshold, entry creates no difference in market quantity at those prices. At lower prices, where the entrant plans positive purchases, the new market demand is larger. Thus, a careful description is that entry shifts market demand weakly right, with a strict increase at prices where the entrants’ combined planned quantity is positive. Exit shifts it weakly left, with a strict decrease where the departing buyers would have purchased.

Common Mistakes

Changing the remaining buyers’ schedules without a reason. Under the isolated buyer-number comparison, their individual demand curves are fixed. Add or remove the affected buyers’ quantities; do not alter the others.
Assuming every entry increases quantity at every price. An entrant may plan to buy zero at some prices. At those prices, entry leaves the market total unchanged.
Adding at different prices. Horizontal addition compares all buyers at the same price. Quantities from different prices cannot be summed to obtain market demand at one price.
Allowing negative individual quantities. If a schedule is defined with a zero floor, apply it before summing. A negative algebraic value is not a planned purchase.
Inferring a price change from demand alone. This analysis establishes how the demand schedule changes; determining a resulting market price also requires the supply side.

Key takeaway: Market demand is the sum of individual planned quantities at each common price. Entry adds the entrants’ quantities, and exit removes the departing buyers’ quantities. The individual schedules of continuing buyers remain unchanged when buyer numbers alone are changing.

Check Your Understanding

For each problem, compare quantities at the specified price or prices. Apply a zero floor to any individual schedule that would otherwise be negative.

An incumbent buyer has $q_1(p)=\max(0,14-2p)$. A new buyer enters with $q_2(p)=\max(0,6-p)$. At $p=3$, calculate market quantity before and after entry and the increase. (Answer: before 8 units, after 11 units, increase 3 units.)
Five identical buyers each plan $q_i(p)=\max(0,10-p)$ units at price $p$. One buyer exits. At $p=4$, find the market quantities before and after exit. (Answer: before 30 units, after 24 units.)
Two existing buyers have schedules $q_1(p)=\max(0,16-2p)$ and $q_2(p)=\max(0,8-p)$. An entrant has $q_3(p)=\max(0,12-3p)$. At $p=2$, find the old total, the entrant’s quantity, and the new total. (Answer: old total 18 units, entrant 6 units, new total 24 units.)
An incumbent market has quantity demanded of 20 units at $p=0$, 14 units at $p=2$, and 5 units at $p=5$. A new buyer’s schedule is $q_e(p)=\max(0,10-2p)$. Find the new market quantities at all three prices and identify where entry makes no change. (Answer: 30, 20, and 5 units, respectively; entry makes no change at $p=5$.)
Three buyers each plan 4 units at a price of $7. A fourth buyer enters but plans zero units at $7. State the change in market quantity at that price and explain whether the market demand schedule can nevertheless differ at other prices. (Answer: quantity at $7 is unchanged at 12 units; yes, the schedule can differ at other prices if the entrant plans a positive quantity there.)

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