Deriving Market Supply by Horizontal Summation

From Individual Supply to Market Supply

A market may include many producers, each with a different supply response. To determine the quantity supplied to the whole market at a particular price, find the quantity each producer is willing to supply at that same price, then add those quantities. This procedure is called horizontal summation because quantities are measured along the horizontal axis of a standard supply graph.

The method relies on a common market price and a consistent product and time period. For example, quantities of the same product supplied per week can be added to obtain total weekly market supply. Quantities measured in different units or for different periods must first be made comparable. As in the earlier material on supply and quantity supplied, a supply function describes quantities offered at different prices while other relevant conditions are held fixed.

Definition: If $n$ producers supply quantities $q_1(p),q_2(p),\ldots,q_n(p)$ at the common price $p$, market quantity supplied is $$ Q_S(p)=\sum_{i=1}^{n}q_i(p). $$ The market supply curve is the set of price-quantity combinations $(Q_S(p),p)$ obtained as price varies, with other determinants held fixed.

This is an aggregation rule, not a claim that producers have identical costs or identical supply functions. A producer that supplies zero at a particular price contributes zero to the sum. If a producer’s supply is defined to be zero below a minimum price, that zero-supply range must be included when constructing the market function.

Why Quantities, Not Prices, Are Added

A supply schedule or supply curve reports a quantity for each price. Imagine drawing a horizontal line across a graph at a chosen price. Each producer’s curve gives a quantity at the point where that line meets the producer’s curve. The market quantity is the sum of those horizontal distances. The market point is then plotted at that total quantity and the original price.

The reason follows directly from what is being counted. At a fixed price, producer 1 supplies $q_1(p)$ units and producer 2 supplies $q_2(p)$ units. Together they offer $q_1(p)+q_2(p)$ units. Their prices are not separate quantities of the product: under the common-price assumption, both producers face $p$. Thus the price coordinate stays the same while the quantity coordinate changes.

\text{At fixed }p:\qquad Q_S(p)=q_1(p)+q_2(p)+\cdots+q_n(p).

If supply functions are written as inverse functions, with price expressed as a function of quantity, the same rule still applies: first determine each producer’s quantity at a common price, then add quantities. Adding inverse-supply prices at a common quantity would answer a different question and does not give market supply.

Deriving a Market Function and Schedule

For a small number of producers, derive the market function by identifying price ranges in which different producers supply positive quantities. Within each range, add the applicable individual functions. A producer whose function is zero below its threshold contributes nothing in that range. At a threshold, check the value from the original function to ensure the piecewise expressions agree.

Choose a common price.
Every individual quantity in the sum must be evaluated at the same price and for the same product and time period.

Evaluate each producer’s supply.
Apply any minimum-price condition. A producer supplying zero at the selected price contributes zero.

Add quantities and record the market point.
The total is $Q_S(p)$; pair it with the unchanged price $p$ to make the market schedule or graph point.

Worked Example: Three Producers with Different Entry Prices

Three producers supply the same product per week. Their supply functions are $q_A(p)=2p-4$ for $p\geq2$, $q_B(p)=p-5$ for $p\geq5$, and $q_C(p)=0.5p-3$ for $p\geq6$. Each producer supplies zero below its stated threshold. Derive the market supply function, construct a short schedule, and find market supply at price $p=8$.

Method and assumptions. Use horizontal summation: evaluate all three weekly quantities at the same price, treating all quantities as units of the same product. The thresholds matter because a producer supplies zero before reaching its threshold.

For $p<2$, all three producers supply zero, so $Q_S(p)=0$. For $2\leq p<5$, only A supplies: $Q_S(p)=2p-4$. For $5\leq p<6$, A and B supply, giving $Q_S(p)=(2p-4)+(p-5)=3p-9$. For $p\geq6$, all three supply: $Q_S(p)=(2p-4)+(p-5)+(0.5p-3)=3.5p-12$.

Q_S(p)= \begin{cases} 0, & p<2,\\ 2p-4, & 2\leq p<5,\\ 3p-9, & 5\leq p<6,\\ 3.5p-12, & p\geq6. \end{cases}

To make the schedule, substitute each listed price into the individual functions and add. At $p=2$, the quantities are $0,0,0$, so the market total is 0. At $p=4$, they are $4,0,0$, totaling 4. At $p=5$, they are $6,0,0$, totaling 6. At $p=6$, they are $8,1,0$, totaling 9. At $p=8$, they are $12,3,1$, totaling 16.

Price $p$	A quantity	B quantity	C quantity	Market quantity $Q_S$
2	0	0	0	0
4	4	0	0	4
5	6	0	0	6
6	8	1	0	9
8	12	3	1	16

Graph interpretation and check. Put quantity on the horizontal axis and price on the vertical axis. The market schedule gives the points $(0,2)$, $(4,4)$, $(6,5)$, $(9,6)$, and $(16,8)$. At $p=8$, the piecewise formula gives $3.5(8)-12=28-12=16$, agreeing with the direct sum $12+3+1=16$. The common mistake avoided here is leaving out a producer whose threshold has been reached, or counting a producer before its threshold.

Using Supply Functions Written as Price Rules

Some problems state a producer’s inverse supply rule: price as a function of the quantity that producer supplies. This form does not change the aggregation principle. Solve each rule for the quantity at the given common price, and then add. The following example shows why the phrase “horizontal summation” is important even when the information is initially presented in inverse form.

Worked Example: Converting Inverse Supply to Market Supply

Producer A’s inverse supply rule is $p=10+2q_A$ for prices at or above 10; below 10, A supplies zero. Producer B’s rule is $p=4+3q_B$ for prices at or above 4; below 4, B supplies zero. Derive market supply by price range, find total supply at $p=16$, and determine the price at which the market supplies 12 units.

Method. Rearrange each rule to express quantity in terms of price. For A, $q_A=(p-10)/2$ when $p\geq10$. For B, $q_B=(p-4)/3$ when $p\geq4$. Below each threshold, use zero supply.

For $4\leq p<10$, only B supplies, so $Q_S(p)=(p-4)/3$. For $p\geq10$, both supply: $Q_S(p)=(p-10)/2+(p-4)/3$. With a common denominator, this becomes $(3p-30+2p-8)/6=(5p-38)/6$. For $p<4$, both supply zero.

Q_S(p)= \begin{cases} 0, & p<4,\\ \dfrac{p-4}{3}, & 4\leq p<10,\\ \dfrac{5p-38}{6}, & p\geq10. \end{cases}

At $p=16$, A supplies $(16-10)/2=3$ and B supplies $(16-4)/3=4$. Therefore $Q_S(16)=3+4=7$ units. To find the price for 12 units, use the range $p\geq10$, since the target exceeds the maximum market quantity available below 10, which is 2 units from B. Solve $(5p-38)/6=12$: multiplying by 6 gives $5p-38=72$, so $5p=110$ and $p=22$.

Check. At $p=22$, A supplies $(22-10)/2=6$ and B supplies $(22-4)/3=6$, for a total of $6+6=12$. The common error avoided is adding the two inverse-rule prices. At this allocation, both firms face price 22; the market price is not $22+22$.

Finding a Price from a Market Quantity

Once a market function has been derived, it can be used in either direction: insert a price to find quantity, or solve for the price associated with a specified market quantity. In the second direction, first identify the relevant price range. Then use that range’s formula and verify that the resulting price actually lies in the range. This check prevents a mathematically valid solution from being used outside the conditions under which its formula applies.

Worked Example: Solving for Price and Checking the Result

A hypothetical market has three producers whose weekly supplies, in units, are $q_A(p)=4p-20$ for $p\geq5$, $q_B(p)=2p-8$ for $p\geq4$, and $q_C(p)=p-10$ for $p\geq10$. Each supplies zero below its threshold. Find market supply at $p=9$, then find the price at which market supply is 40 units.

Method and assumptions. The units refer to the same product and week, and every producer faces the same price. At $p=9$, A supplies $4(9)-20=16$, B supplies $2(9)-8=10$, and C supplies zero because $9<10$. The total is $16+10+0=26$ units.

For a total of 40 units, the price must be at least 10: below that threshold, C supplies zero, and at $p=10$, A and B together supply $20+12=32$, which is less than 40. Thus all three functions apply. Their sum is $Q_S(p)=(4p-20)+(2p-8)+(p-10)=7p-38$. Set $7p-38=40$, so $7p=78$ and $p=78/7$, approximately $11.14$.

Check. Using the exact price $78/7$, A supplies $172/7$, B supplies $100/7$, and C supplies $8/7$. Their sum is $280/7=40$ units. The price $78/7$ is greater than 10, so the three-producer formula used in the calculation is valid. The price is rounded to two decimal places only in the reported approximation; the exact value verifies the total without rounding.

Common Mistakes and Why the Method Matters

The most reliable way to keep the graph and arithmetic straight is to say explicitly what is held fixed: choose a price, evaluate all individual quantities at that price, and add. The resulting market quantity is then paired with that price. This is the same procedure whether the information is supplied as equations, a table, or points on individual curves. It is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Adding prices instead of quantities. Producers face the same market price in the aggregation step. Add their quantities, not their price coordinates.
Using different prices for different producers. A market quantity at $p=10$ must be built from every producer’s quantity at $p=10$, not from whichever prices make the arithmetic convenient.
Ignoring a threshold. Apply the stated zero-supply range before using a producer’s positive-supply formula.
Adding unlike quantities. Confirm that every firm’s quantity refers to the same product and time period before summing.
Failing to check a solved price. A piecewise formula applies only in its stated price range; verify the answer against that condition.

Key takeaway: Market supply is obtained by horizontal summation: at every common price, add the quantities supplied by all producers. The price remains fixed; the total quantity changes.

Check Your Understanding

For each problem, apply the individual supply rules at a common price before adding quantities. Treat any producer as supplying zero below its stated threshold.

Producer A supplies $q_A=p-3$ for $p\geq3$, and producer B supplies $q_B=2p-10$ for $p\geq5$. Find market supply at $p=8$, and state the market point as $(Q_S,p)$. (Answer: $Q_S=11$ units; market point $(11,8)$.)
Producer A supplies $q_A=3p-6$ for $p\geq2$, and producer B supplies $q_B=p-4$ for $p\geq4$. Find the two quantities and the market total at $p=5$. (Answer: A supplies 9 units, B supplies 1 unit, and $Q_S=10$ units.)
Producer A supplies $q_A=(p-6)/2$ for $p\geq6$; producer B supplies $q_B=(p-3)/3$ for $p\geq3$. Write the market supply function by price range, then find market supply at $p=15$. (Answer: $Q_S=0$ for $p<3$, $Q_S=(p-3)/3$ for $3\leq p<6$, and $Q_S=(5p-24)/6$ for $p\geq6$; at $p=15$, $Q_S=8.5$ units.)
Producer A supplies $2p-8$ for $p\geq4$, producer B supplies $3p-15$ for $p\geq5$, and producer C supplies $p-9$ for $p\geq9$. Find the market quantity at $p=10$, then solve for the price at which market supply is 28 units. (Answer: $Q_S(10)=28$ units; the price is $p=10$.)
Producer A supplies $q_A=(p-8)/2$ for $p\geq8$, and producer B supplies $q_B=(p-2)/3$ for $p\geq2$. Find the market quantity at $p=14$, then find the price that gives market supply of 10 units. Verify that the price is in the range where both producers supply. (Answer: $Q_S(14)=10$ units; a market quantity of 10 is reached at $p=14$, which is at least 8.)

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