Why Consumer Differences Matter
Market demand combines the choices of many consumers, but consumers need not have the same income, preferences, or willingness to buy at a given price. A higher price may cause one group to reduce its purchases while another group stops buying altogether. The resulting market demand can have different slopes over different price ranges, even when each group’s own demand schedule is simple.
The key is to aggregate quantities at the same price, while keeping track of which consumers are active at that price. This makes it possible to describe both the total market quantity and how the composition of demand changes as price changes.
Group differences can be represented by different demand schedules. For example, income and preferences may affect a consumer’s willingness to pay or desired quantity at each price. A participation threshold is a price above which a consumer or group chooses zero quantity. When a group reaches that threshold, its contribution to market demand disappears. The total market schedule can therefore change slope at that price.
This treatment assumes a common good and a common market price, that quantities purchased by separate consumers can be added, and that each group’s demand schedule describes the same price and fixed background conditions. The group counts and schedules are held fixed while tracing the demand curve. If those conditions change, the demand schedule being aggregated may change too.
Aggregating Groups With Different Thresholds
Consider two groups with different reservation prices, meaning that the groups stop buying at different prices. A convenient representation is a linear demand schedule truncated at zero: \(q_g(p)=\max\{a_g-b_gp,0\}\), where \(a_g>0\) and \(b_g>0\). The group’s choke price, the price at which its quantity reaches zero, is \(a_g/b_g\). Above that price, the group contributes nothing to market demand.
Worked Example: Finding the Market Schedule Across Two Thresholds
A market has three consumers in group A, each with demand \(q_A(p)=\max\{12-2p,0\}\), and two consumers in group B, each with demand \(q_B(p)=\max\{8-p,0\}\). Find aggregate demand at every nonnegative price and calculate it at \(p=7\).
Method. Find each group’s choke price first, then add only the quantities of groups that are still buying. Group A’s choke price is \(12/2=6\). Group B’s choke price is \(8/1=8\).
For \(0\leq p<6\), both groups buy. Group A demand is \(3(12-2p)=36-6p\), and group B demand is \(2(8-p)=16-2p\). Thus \(Q(p)=52-8p\).
For \(6\leq p<8\), group A’s quantity is zero and group B still buys, so \(Q(p)=2(8-p)=16-2p\). For \(p\geq8\), neither group buys, so \(Q(p)=0\). At the boundaries, these expressions agree: at \(p=6\), both applicable formulas give \(Q=4\); at \(p=8\), group B’s quantity is also zero.
At \(p=7\), group A buys zero. Each group B consumer demands \(8-7=1\), so group B buys \(2(1)=2\) units. Therefore, \(Q(7)=2\).
Check. The result is nonnegative, and market demand falls to zero at the highest group choke price, \(p=8\). The common mistake is to use \(52-8p\) at \(p=7\), obtaining \(-4\). That expression incorrectly includes group A’s negative linear quantity after its choke price. Truncate that group’s quantity at zero before aggregating.
The example shows why a market demand curve may be piecewise. Below \(p=6\), both groups respond to price, so the market slope is \(-8\). Between \(p=6\) and \(p=8\), only group B responds, so the slope is \(-2\). The curve is continuous at the thresholds in this example, but its slope changes. A change in slope is a kink; the market demand curve need not have a single constant slope.
Group Shares and the Shape of Demand
Group size affects how strongly a group influences total demand. If one group becomes more numerous while its per-consumer demand stays the same, its contribution \(n_g q_g(p)\) receives more weight. This changes market quantity and, when groups have different responses to price, can also change the market curve’s slope and elasticity.
Worked Example: How Group Composition Changes Market Demand
There are two consumer types. Each type A consumer demands \(q_A(p)=\max\{10-p,0\}\); each type B consumer demands \(q_B(p)=\max\{16-2p,0\}\). Compare market demand when the population consists of three A consumers and one B consumer with demand when it consists of one A consumer and three B consumers. Find the market quantity and slope at \(p=4\) in each case.
At \(p=4\), each A consumer demands \(10-4=6\), and each B consumer demands \(16-2(4)=8\). Both types are active because their choke prices are 10 and 8, respectively.
With three A consumers and one B consumer, market quantity is \(3(6)+1(8)=26\). In the price range where both types remain active, the market schedule is \(3(10-p)+(16-2p)=46-5p\), so its slope is \(-5\).
With one A consumer and three B consumers, market quantity is \(1(6)+3(8)=30\). The schedule while both types are active is \((10-p)+3(16-2p)=58-7p\), so its slope is \(-7\).
The population change raises quantity at \(p=4\) from 26 to 30 and makes the local slope more negative, from \(-5\) to \(-7\). The B group has a larger price response per consumer, so increasing its share makes aggregate quantity more responsive in absolute units at this price.
Check. Substituting \(p=4\) into the two aggregate schedules gives \(46-5(4)=26\) and \(58-7(4)=30\), matching the group-by-group totals. The mistake to avoid is assuming that changing population shares only scales the entire market curve by one common factor; here it changes the slope because the two types have different price responses.
Market Elasticity as a Weighted Average
A useful result connects the price elasticity of market demand with the elasticities of the active groups. For a group whose demand is positive and differentiable at price \(p\), define its price elasticity as \(\varepsilon_g=(p/q_g)(dq_g/dp)\). Under ordinary downward-sloping demand this elasticity is negative. The market elasticity is \(\varepsilon_Q=(p/Q)(dQ/dp)\), where \(Q>0\).
The result follows directly from aggregation. With fixed group counts, \(Q=\sum_g n_g q_g=\sum_g Q_g\). At a price where each active schedule is differentiable, differentiation gives \(dQ/dp=\sum_g n_g(dq_g/dp)\). Multiply by \(p/Q\), then express each term using its group elasticity:
The weights \(Q_g/Q\) are group shares of total quantity, not shares of the population. They are nonnegative and sum to one when the active groups together account for all market demand. Thus market elasticity lies between the smallest and largest active group elasticities, provided those elasticities are defined at that price. A group with many consumers need not dominate the elasticity if its quantity share is small.
Worked Example: Calculating Market Elasticity With Unequal Group Responses
At price \(p=5\), there are three A consumers, each demanding \(q_A(p)=\max\{12-2p,0\}\), and two B consumers, each demanding \(q_B(p)=\max\{8-p,0\}\). Calculate the market elasticity and verify it using the group-weighted formula.
At \(p=5\), each A consumer demands \(12-2(5)=2\), giving \(Q_A=3(2)=6\). Each B consumer demands \(8-5=3\), giving \(Q_B=2(3)=6\). Total quantity is \(Q=6+6=12\).
The aggregate slope is the sum of the groups’ total slopes. Group A’s slope is \(3(-2)=-6\), and group B’s slope is \(2(-1)=-2\), so \(dQ/dp=-8\). Directly, \(\varepsilon_Q=(5/12)(-8)=-40/12=-10/3\), approximately \(-3.333\).
To verify, the group elasticities are \(\varepsilon_A=(5/2)(-2)=-5\) and \(\varepsilon_B=(5/3)(-1)=-5/3\). Each group supplies half of total quantity, since \(Q_A/Q=Q_B/Q=6/12=1/2\). Therefore the weighted result is \((1/2)(-5)+(1/2)(-5/3)=-5/2-5/6=-20/6=-10/3\), as before.
Check. The direct and weighted calculations agree. The common mistake is to weight group elasticities by population shares alone. The correct weights here are each group’s share of total quantity; those shares happen to be equal in this example, but need not be equal in general.
Interpreting Heterogeneity Carefully
Different incomes and preferences matter because they can lead consumers to have different desired quantities at the same price, different sensitivities to price, or different prices at which they stop buying. A larger share of a high-demand group can raise market quantity; a larger share of a price-sensitive group can make the market curve locally steeper. If the groups have different choke prices, changing the price can alter which groups contribute at all.
These are statements about the shape of an aggregate schedule, not claims that every consumer behaves like an “average” consumer. A single representative demand curve generally cannot be inferred by averaging group parameters without checking what is being averaged and how the groups’ quantities add. In particular, averages can conceal a participation threshold: some consumers may buy zero while others still buy positive quantities.
- Aggregate at a common price. Calculate each group’s quantity at that price, then add the quantities.
- Respect zero-demand regions. Never allow a linear formula to imply negative purchases beyond its choke price.
- Separate population shares from quantity shares. Population weights describe how many consumers are in a group; elasticity weights depend on how much that group buys.
- Check thresholds and boundaries. A formula valid while a group is active does not necessarily apply above its participation threshold.
Check Your Understanding
For each problem, identify active groups before aggregating. Answers are provided at the end of each question.
- Two consumers each have demand \(q_A(p)=\max\{9-p,0\}\), and three consumers each have demand \(q_B(p)=\max\{6-0.5p,0\}\). At \(p=4\), find total demand and the market slope in the range containing that price. (Answer: \(Q=22\), slope \(-3.5\).)
- There are four consumers with \(q_A(p)=\max\{14-2p,0\}\) and two with \(q_B(p)=\max\{12-p,0\}\). Write the full aggregate demand schedule for \(p\geq0\), including its thresholds. (Answer: \(Q=80-10p\) for \(0\leq p<7\); \(Q=24-2p\) for \(7\leq p<12\); \(Q=0\) for \(p\geq12\).)
- At \(p=3\), there are two A consumers with \(q_A(p)=\max\{8-p,0\}\) and three B consumers with \(q_B(p)=\max\{14-2p,0\}\). Find market quantity, market slope, and market elasticity. (Answer: \(Q=34\), slope \(-8\), elasticity \(-12/17\approx-0.7059\).)
- At \(p=4\), compare two consumers with \(q_A(p)=\max\{12-p,0\}\) and one with \(q_B(p)=\max\{9-0.5p,0\}\) against a population of one A consumer and three B consumers. Find quantity and elasticity for each composition. (Answer: First composition \(Q=23\), elasticity \(-10/23\approx-0.4348\); second composition \(Q=29\), elasticity \(-10/29\approx-0.3448\).)
- One consumer has \(q_A(p)=\max\{15-3p,0\}\), and two have \(q_B(p)=\max\{10-p,0\}\). Write the piecewise aggregate schedule and identify its kink price. (Answer: \(Q=35-5p\) for \(0\leq p<5\); \(Q=20-2p\) for \(5\leq p<10\); \(Q=0\) for \(p\geq10\); kink at \(p=5\).)