Market Demand by Horizontal Summation

Why Market Demand Is a Horizontal Sum

A market’s demand at a given price depends on how much each buyer is willing and able to purchase at that same price. The market quantity is therefore found by adding buyers’ quantities across individuals, not by adding prices. This operation is called horizontal summation: choose a price, read each individual quantity at that price, and add those quantities.

The calculation is especially important when consumers have demand functions with different intercepts or slopes. A consumer whose quantity has fallen to zero does not contribute to market demand at that price, so the aggregate formula can change when the price crosses an individual’s choke price. The method below makes those active price ranges explicit.

Definition: Suppose a market contains $n$ consumers, and $q_i(p)$ is consumer $i$’s quantity demanded at price $p$, with other demand conditions held fixed. Market demand is the sum of individual quantities at the common market price: $$ Q(p)=\sum_{i=1}^{n}q_i(p). $$ For a linear individual demand $q_i(p)=\max\{0,a_i-b_i p\}$, where $a_i>0$ and $b_i>0$, the choke price is $a_i/b_i$. At or above that price, the consumer demands zero.

This definition treats the consumers and their other demand conditions—such as income and preferences—as fixed while the market price varies. The quantity for each person is the result of that person’s demand at the common price. The aggregate is measured in the same units and over the same time period as the individual quantities.

How to Build the Aggregate Schedule

For linear demands, the key step is to determine which consumers are active at each price. Consumer $i$ is active when $a_i-b_i p>0$, or equivalently when $p<a_i/b_i$. Add the linear expressions for the active consumers only. The resulting market-demand function is generally piecewise linear, with possible changes in slope at individual choke prices.

Find each choke price.
For $q_i(p)=\max\{0,a_i-b_i p\}$, calculate $a_i/b_i$. These prices mark where individual demand reaches zero.

Order the price thresholds.
Use the choke prices to divide the nonnegative price range into intervals. The set of consumers with positive demand stays fixed within each interval.

Add quantities within each interval.
Sum the demand expressions of active consumers and set the other consumers’ quantities to zero.

Check endpoints and totals.
At a choke price the affected consumer demands zero. Check that adjacent formulas agree at each threshold and that the total equals the individual quantities.

Why the Addition Works

At price $p$, each consumer chooses a quantity $q_i(p)$. A market total counts all units purchased, regardless of which consumer buys them. If consumer 1 buys $q_1(p)$ units and consumer 2 buys $q_2(p)$ units, the total is $q_1(p)+q_2(p)$; extending the same accounting to $n$ consumers gives the summation formula.

For linear demands, this also explains how the slope changes. Let $A(p)$ be the set of consumers whose demand is positive at price $p$. On any price interval where this set does not change,

Q(p)=\sum_{i\in A(p)}(a_i-b_i p) =\left(\sum_{i\in A(p)}a_i\right)-p\left(\sum_{i\in A(p)}b_i\right).

Thus, within an interval, the market intercept is the sum of active consumers’ intercepts and the market slope is the negative sum of their slope coefficients. When a consumer reaches the choke price, that consumer drops out of the sum; the market intercept and slope may then change. This derivation relies on adding quantities at a shared price. Adding individual prices at a shared quantity is a different operation and does not produce market demand.

Worked Example: Add Two Demands Across Their Active Price Ranges

Two consumers have demands $q_1(p)=\max\{0,18-3p\}$ and $q_2(p)=\max\{0,12-p\}$, where price is in dollars per unit and quantities are units per week. Construct the market-demand function and find total demand at $p=\$4$.

Method. Find the choke prices, divide the price range at those thresholds, and horizontally sum the active individual quantities.

Choke prices. Consumer 1’s demand reaches zero when $18-3p=0$, so $p=6$. Consumer 2’s demand reaches zero when $12-p=0$, so $p=12$.

For $p\geq12$, neither consumer has positive demand, so $Q(p)=0$. For $6\leq p<12$, consumer 1 demands zero and consumer 2 is active: $Q(p)=12-p$. For $0\leq p<6$, both consumers are active: $Q(p)=(18-3p)+(12-p)=30-4p$.

Q(p)= \begin{cases} 30-4p, & 0\leq p<6,\\ 12-p, & 6\leq p<12,\\ 0, & p\geq12. \end{cases}

At $p=4$, consumer 1 demands $18-3(4)=18-12=6$ units. Consumer 2 demands $12-4=8$ units. The total is $6+8=14$ units per week. The market formula gives $30-4(4)=30-16=14$, the same result.

Check and common mistake avoided. At $p=6$, the first branch gives $30-24=6$, while the second gives $12-6=6$; the pieces join correctly. Summing both linear expressions at $p=8$ would be wrong because consumer 1 is inactive there: the correct market quantity is $12-8=4$, not $(18-24)+(12-8)$ or a sum that treats consumer 1 as buying a negative quantity.

Inverse Market Demand

An inverse demand function states the price associated with a market quantity. It is obtained by solving the appropriate market-demand branch for $p$, while keeping track of the quantity range belonging to that branch. A branch’s algebraic formula alone is not enough: using it outside its valid range can assign a price even though a different set of consumers should be active.

Worked Example: Derive Inverse Demand with One Consumer Leaving

Two consumers have weekly demands $q_A(p)=\max\{0,24-2p\}$ and $q_B(p)=\max\{0,9-p\}$. Find the inverse market-demand schedule for positive quantities and the price at which market demand is 15 units.

Method. Order the choke prices, derive market demand on each interval, and invert each branch together with its quantity range.

Consumer A’s choke price is $24/2=12$; consumer B’s is $9/1=9$. Between $p=9$ and $p=12$, only A is active, so $Q=24-2p$. At $p=12$, quantity is zero; at $p=9$, quantity is $24-18=6$. Thus this branch covers $0\leq Q\leq6$. Solving for price gives $p=12-Q/2$.

At $p=9$, consumer B begins to have positive demand as price falls. For $0\leq p\leq9$, both are active, so $Q=(24-2p)+(9-p)=33-3p$. This branch runs from $Q=6$ at $p=9$ to $Q=33$ at $p=0$. Solving for price gives $p=(33-Q)/3$.

p(Q)= \begin{cases} 12-\dfrac{Q}{2}, & 0\leq Q\leq6,\\ \dfrac{33-Q}{3}, & 6\leq Q\leq33. \end{cases}

For $Q=15$, the applicable range is $6\leq Q\leq33$. Therefore $p=(33-15)/3=18/3=\$6$. At that price, A demands $24-2(6)=12$ units and B demands $9-6=3$ units, giving $12+3=15$ units.

Check and common mistake avoided. Both inverse branches give $p=9$ at $Q=6$, so they meet at the threshold. Using $12-Q/2$ at $Q=15$ would give $p=4.5$, but that branch applies only through $Q=6$. Respecting the branch’s quantity range avoids that error.

A Market Schedule with Several Thresholds

With more consumers, the same process applies: sort all choke prices and recompute the active sum after each threshold. A consumer with a larger choke price may remain active after another consumer has already left the market. The aggregate slope becomes less negative as consumers leave, although the exact change depends on the slope coefficient of the consumer who exits.

Worked Example: Combine Three Consumers with Different Slopes

Three buyers have weekly demand functions $q_1(p)=\max\{0,20-2p\}$, $q_2(p)=\max\{0,15-3p\}$, and $q_3(p)=\max\{0,8-p\}$. Construct market demand and calculate total demand at prices $p=\$6$ and $p=\$4$.

Method. Sort the choke prices and sum only the consumers with positive demand in each interval. The choke prices are $20/2=10$ for buyer 1, $15/3=5$ for buyer 2, and $8/1=8$ for buyer 3. In descending order, the thresholds are 10, 8, and 5.

For $p\geq10$, all three demand zero. For $8\leq p<10$, only buyer 1 is active, giving $Q=20-2p$. For $5\leq p<8$, buyers 1 and 3 are active, giving $Q=(20-2p)+(8-p)=28-3p$. For $0\leq p<5$, all three are active, giving $Q=(20-2p)+(15-3p)+(8-p)=43-6p$.

Q(p)= \begin{cases} 43-6p, & 0\leq p<5,\\ 28-3p, & 5\leq p<8,\\ 20-2p, & 8\leq p<10,\\ 0, & p\geq10. \end{cases}

At $p=6$, buyer 1 demands $20-12=8$, buyer 2 demands zero because $15-18=-3$, and buyer 3 demands $8-6=2$. The total is $8+0+2=10$ units per week. The applicable market branch confirms $28-3(6)=10$.

At $p=4$, the individual quantities are $20-8=12$, $15-12=3$, and $8-4=4$. Their sum is $12+3+4=19$ units per week. The applicable branch gives $43-6(4)=43-24=19$.

Check and common mistake avoided. At the thresholds, adjacent expressions agree: at $p=8$, both relevant branches give 4; at $p=5$, both give 13. A common error is to retain buyer 2’s expression $15-3p$ when $p=6$, even though it is negative. Demand is truncated at zero, so that buyer contributes zero, not $-3$ units.

Common Traps and Why the Method Matters

Market demand is useful for describing how total purchases respond to a price change while individual demand conditions are held fixed. For example, a hypothetical insurer analyzing demand for an optional add-on might estimate separate quantities for a set of buyer segments and add those quantities at each offered price. The example is an application of the aggregation method, not a claim about actual insurer data. The same logic applies to any market where individual demand schedules are available.

Adding prices instead of quantities. Horizontal summation fixes the price and adds the quantities demanded at that price.
Ignoring the zero-quantity condition. For a demand written with a maximum and zero, an inactive consumer contributes zero, never a negative quantity.
Using the wrong active set. First identify the price interval, then add the consumers active on that interval.
Forgetting branch ranges in inverse demand. Solve each market branch for price and retain the quantity interval over which that branch applies.
Failing to check totals. At a chosen price, calculate the individual quantities and verify that their sum matches the aggregate formula.

Key takeaway: Market demand at price $p$ is the sum of individual quantities demanded at that same price. For linear demands truncated at zero, each consumer’s choke price helps determine the active set, and the market schedule is built piece by piece from those active consumers.

Check Your Understanding

For each problem, identify the active consumers before adding quantities. Treat prices and quantities as nonnegative.

Two consumers have demands $q_1(p)=\max\{0,14-2p\}$ and $q_2(p)=\max\{0,8-p\}$. Find market demand at $p=3$. (Answer: 13 units.)
Consumers A and B have demands $q_A(p)=\max\{0,30-3p\}$ and $q_B(p)=\max\{0,10-2p\}$. Find market demand at $p=4$, and state both choke prices. (Answer: 20 units; choke prices 10 and 5.)
Individual demands are $q_1(p)=\max\{0,16-2p\}$ and $q_2(p)=\max\{0,6-p\}$. Give both inverse market-demand branches and their quantity ranges. (Answer: $p=8-Q/2$ for $0\leq Q\leq4$; $p=(22-Q)/3$ for $4\leq Q\leq22$.)
Consumers A and B demand $q_A(p)=\max\{0,27-3p\}$ and $q_B(p)=\max\{0,18-2p\}$. Find the price at which market demand is 20 units and verify each consumer’s quantity. (Answer: $p=5$; A demands 12 units and B demands 8 units.)
Three consumers have demands $q_1(p)=\max\{0,25-5p\}$, $q_2(p)=\max\{0,12-2p\}$, and $q_3(p)=\max\{0,9-p\}$. Construct the piecewise market-demand schedule and find demand at $p=4$. (Answer: $Q=46-8p$ for $0\leq p<5$, $Q=21-3p$ for $5\leq p<6$, $Q=9-p$ for $6\leq p<9$, and $Q=0$ for $p\geq9$; at $p=4$, $Q=14$ units.)

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