# Supply and demand: where a price comes from

> Line up buyers by what they would pay and sellers by what they would accept. The price where the two counts match is the one that leaves nobody turned away.

Series: Economics (https://www.monocurl.com/monocurl-essays/economics/)
Canonical: https://www.monocurl.com/monocurl-essays/economics/supply-and-demand/
Keywords: supply and demand, equilibrium price, market clearing price, demand curve, willingness to pay, price ceiling shortage, shift in demand vs quantity demanded

[Figure 1, animation: Eight buyers (blue staircase) and eight sellers (orange staircase) of one used textbook. As the price moves, the number who want to buy and the number who want to sell change. They are equal, at five each, only for prices between $50 and $55.]

Tickets for a 2000-seat concert go on sale at \$80 and are gone within a minute. By the afternoon the same seats are listed on resale sites at \$320. People blame greed, or bots, or a ticketing website that buckled. There is a plainer diagnosis, and to reach it we need to know where a price comes from in the first place.

So leave the concert for a few minutes and take a market small enough to count on your fingers. A calculus course requires a particular textbook. Eight students need a used copy, and eight students who finished the course last term each have one to sell. The copies are in the same condition, so nobody cares which one they get. What will a copy sell for?

## Eight buyers and what each would pay

Ask each buyer privately for the most she would pay before giving up and making do with the library copy.

| Buyer | Ana | Ben | Cho | Dev | Eli | Fay | Gus | Hoa |
|---|---|---|---|---|---|---|---|---|
| Would pay up to | \$55 | \$95 | \$35 | \$75 | \$25 | \$85 | \$45 | \$65 |

**Definition.** A buyer's **willingness to pay** for one unit of a good is the highest price at which she would still choose to buy it. We assume she buys when the price is at or below this number and does not buy when it is above.

The number reflects her alternatives as much as her taste. Ben may be no keener on calculus than Eli; perhaps he has no time to sit in the library. All the model needs is the number itself.

**Pause and try.** If copies cost \$60, which of the eight would buy? How many would buy at \$30?

**Answer.** At \$60 the buyers are those willing to pay at least 60: Ben (95), Fay (85), Dev (75) and Hoa (65), so four copies. At \$30 everyone buys except Eli (25), so seven copies.

Answering that question took a search through the table. Sorting the buyers from highest willingness to pay to lowest turns the search into reading a graph.

[Figure 2, steps: Each bar is one buyer, as tall as the most that buyer would pay. Sorted from tallest to shortest, the tops of the bars form the demand curve: at any price, the number of bars reaching that height is the number of copies buyers want.]

  - step: The eight buyers in the order of the table: Ana, Ben, Cho, Dev, Eli, Fay, Gus, Hoa.
  - step: Sort them from the highest willingness to pay to the lowest.
  - step: Trace the tops of the bars. This staircase is the demand curve.
  - step: Read it sideways. At a price of 60 dollars, four bars are tall enough: four copies are wanted.
**Definition.** The **quantity demanded** at a price $p$, written $Q_d(p)$, is the number of units that buyers want to buy at that price. **Demand** is the whole relationship between price and quantity demanded: the function $p \mapsto Q_d(p)$, or its graph, the **demand curve**.

The two terms sound alike and mean different things, and a later section depends on the difference. "Quantity demanded" is a single number, such as 4 copies at \$60. "Demand" is the entire staircase.

One oddity of the graph is a convention you will meet everywhere in economics: price goes on the vertical axis even though it is the input. To use the curve, start from a price on the left edge, move across until you hit the staircase, and read the quantity below.

## Eight sellers and what each would accept

Each seller also has a private number: the lowest price she would take. Below it she would sooner keep the book. Call that her **cost** of selling. The eight sellers' costs are \$10, \$20, \$30, \$40, \$50, \$60, \$70 and \$80, and a seller sells when the price is at or above her cost.

[Figure 3, image: The eight sellers sorted from the lowest cost to the highest. The tops of the bars form the supply curve: at any price, the number of bars no taller than the price is the number of copies offered.]

The **quantity supplied** $Q_s(p)$ is the number of units sellers want to sell at price $p$, and **supply** is the whole relationship. This staircase climbs, because a higher price satisfies more sellers.

Before putting the two staircases together, here is what has been assumed. Every statement that follows is a statement about this model.

- **One homogeneous good.** The copies are interchangeable, so only the price matters.
- **Unit trades.** Each buyer wants one copy and each seller has one.
- **Price-taking.** There is one going price, and each person decides only whether to trade at it. Nobody thinks she can move it alone.
- **Everything else fixed.** Willingness to pay and costs stay put while we vary the price. Economists say *ceteris paribus*, "other things equal".
- **Voluntary trade.** Nobody can be made to buy or sell.

## The price at which the two counts agree

**Pause and try.** Someone posts a notice: "used calculus books, \$30 each". Count the buyers who want a copy at that price and the sellers willing to part with one. Who ends up disappointed? Then look for a price at which nobody would be.

**Answer.** At \$30, seven buyers want a copy (all but Eli) and three sellers are willing (costs 10, 20, 30). Only three trades can happen, so four buyers who wanted a book at \$30 go without. At \$50 the willing buyers are Ben, Fay, Dev, Hoa and Ana, and the willing sellers are those with costs 10 to 50: five and five. Any price from \$50 to \$55 gives the same two lists.

| Price | \$20 | \$30 | \$40 | \$50 | \$60 | \$70 | \$80 |
|---|---|---|---|---|---|---|---|
| Quantity demanded | 8 | 7 | 6 | 5 | 4 | 3 | 2 |
| Quantity supplied | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| Copies traded | 2 | 3 | 4 | 5 | 4 | 3 | 2 |

The last row is the smaller of the two above it. Since nobody can be forced to trade, the scarcer side decides how many trades happen. Trade is largest at \$50, the one column in which the counts agree.

[Figure 4, steps: The book market at three prices. The red bar is the gap between the number of copies wanted and the number offered. Only between $50 and $55 is there no gap.]

  - step: At 30 dollars, seven copies are wanted and three are offered. Four willing buyers go without.
  - step: At 72 dollars, seven copies are offered and three are wanted. Four willing sellers find no buyer.
  - step: At 52 dollars, five are wanted and five are offered. Everyone who wants to trade can.
A price of \$30 would not last, because four buyers are shut out, and at least one of them is willing to pay \$65 or more, because only three of the seven hopefuls (Ana, Gus and Cho) have a willingness to pay below that. Say it is Hoa. She would gladly pay \$45 to the seller whose cost is \$40, and that seller would gladly accept. Both are better off than under the posted price, where neither trades. So the \$30 price is broken by a buyer who offers more. At \$72 the pressure runs the other way. Four willing sellers find no buyer, and at least one of them has a cost of \$40 or less. She offers her copy for \$60, and Hoa, who would not pay \$72, takes it. At \$52 no such deal exists. The buyers left out value the book at \$45 or less, the sellers left out need \$60 or more, and no price pleases both.

**Definition.** A **market-clearing price**, or **equilibrium price**, is a price $p$ with $Q_d(p) = Q_s(p)$. The common quantity is the **equilibrium quantity**. At a price where $Q_d(p) > Q_s(p)$ there is a **shortage** of $Q_d - Q_s$ units, and where $Q_s(p) > Q_d(p)$ there is an **excess supply** of $Q_s - Q_d$ units.

What we noticed in the table holds for any lists of numbers.

**Theorem.** Suppose $n$ buyers each want one unit and have willingness to pay $v_1 > v_2 > \dots > v_n$, and $n$ sellers each have one unit and costs $c_1 < c_2 < \dots < c_n$, with no $v_i$ equal to any $c_j$. A buyer wants to trade at price $p$ when $v_i \ge p$, a seller when $c_j \le p$, and nobody can be forced to trade. Let $q^*$ be the number of indices $j$ with $v_j > c_j$.

- If $Q_d(p) = Q_s(p)$, the common value is $q^*$, everyone who wants to trade at $p$ can, and no buyer and seller who are left out could both gain by trading with each other at any price.
- If $Q_d(p) \ne Q_s(p)$, at least $|Q_d(p) - Q_s(p)|$ people who want to trade at $p$ cannot.
- If fewer than $q^*$ trades take place, some buyer and some seller are both left out even though the buyer's willingness to pay exceeds the seller's cost.

**Proof.** The differences $v_j - c_j$ decrease as $j$ grows, so $v_j > c_j$ exactly when $j \le q^*$.

*Part 1.* Let $k = Q_d(p) = Q_s(p)$. The willing buyers are buyers $1$ to $k$ and the willing sellers are sellers $1$ to $k$, so they can be paired off and all of them trade. For $j \le k$ we have $v_j \ge p \ge c_j$, and since no value equals a cost, $v_j > c_j$; hence $k \le q^*$. If $k < n$, buyer $k+1$ is unwilling and so is seller $k+1$, which means $v_{k+1} < p < c_{k+1}$ and therefore $k + 1 > q^*$. (If $k = n$ then $q^* \le n = k$ anyway.) So $k = q^*$. Finally, anyone left out has $v_i < p$ or $c_j > p$, so a left-out buyer and a left-out seller have $v_i < c_j$. A trade between them at a price $r$ helps both only if $c_j \le r \le v_i$, and no such $r$ exists.

*Part 2.* Each trade uses one willing buyer and one willing seller, so at most $\min(Q_d, Q_s)$ trades occur, and at least $|Q_d - Q_s|$ people on the longer side are left over.

*Part 3.* If fewer than $q^*$ trades occur, at least one of buyers $1, \dots, q^*$ does not trade; call her buyer $i$. Likewise some seller $j \le q^*$ does not trade. Then $v_i \ge v_{q^*} > c_{q^*} \ge c_j$, so any price between $c_j$ and $v_i$ would leave both better off.

Match the parts to the book market. There $q^* = 5$, because $v_5 = 55 > c_5 = 50$ while $v_6 = 45 < c_6 = 60$. Part 2 is the four disappointed buyers at \$30. Part 3 is Hoa and the \$40 seller: at any price below $c_5 = 50$ fewer than five sellers are willing, at any price above $v_5 = 55$ fewer than five buyers are, and in both cases a trade that would help two people goes unmade.

**Key idea.** At a market-clearing price, the quantity buyers want equals the quantity sellers offer, everyone who wants to trade at that price does, and no buyer and seller who are left out could both gain from a deal with each other. At any other price the scarcer side sets the amount traded and somebody willing is turned away.

Be clear about the status of these claims. The theorem compares prices; it is arithmetic inside the model and it is proved. The claim that real prices *move* to the clearing level is a claim about behavior, and the model contains no mechanism for it. The Hoa story makes it plausible. The evidence is experimental. In classroom markets that Vernon Smith reported in 1962, and in many repetitions since, students were given private values and costs like our table and allowed to call out bids and offers, and trading prices typically settled near the clearing price within a few rounds.

## From staircases to curves

Eight buyers left a \$5 range of clearing prices. That slack is an artifact of the small numbers.

[Figure 5, steps: The same market with 8, 80 and 800 traders on each side, their values and costs spread evenly over the same ranges. As the steps shrink, the staircases become indistinguishable from two straight lines that cross at $50.]

  - step: Eight on each side: steps of 10 dollars, and any price from 50 to 55 dollars clears the market.
  - step: Eighty on each side: steps of 1 dollar. The clearing prices run only from 50 to 50.50 dollars.
  - step: Eight hundred on each side: steps of 10 cents, and 500 copies trade at a price within a nickel of 50 dollars.
With 800 traders on each side the staircases are, to within a few cents, the straight lines

$$
Q_d = 1000 - 10P \quad (20 \le P \le 100), \qquad Q_s = 10P \quad (0 \le P \le 80),
$$

where $P$ is the price in dollars and quantities are copies per term. The coefficients are [slopes](@algebra/slope-and-lines): each extra dollar on the price loses 10 buyers and brings in 10 sellers. This is why economists draw demand and supply as smooth curves. A curve is a staircase with too many steps to see.

**Worked example.** **Find the clearing price and quantity in the campus market.**

We want the price at which the two quantities agree, so set $Q_d = Q_s$:

$$
1000 - 10P = 10P \quad\Longrightarrow\quad 1000 = 20P \quad\Longrightarrow\quad P = 50.
$$

Then $Q = 10 \times 50 = 500$ copies. This is a pair of linear equations in two unknowns, solved by substitution as in [systems of equations](@algebra/systems-of-equations), and the solution is the point where the two lines cross.

*Check.* On the demand side, $1000 - 10 \times 50 = 500$ as well. At \$40 there would be a shortage of $600 - 400 = 200$ copies, and at \$60 an excess supply of $600 - 400 = 200$.

## A change in demand is not a change in quantity demanded

Next term the course enrolls more students, and at every price 200 more copies are wanted than before. Nothing about the sellers has changed.

**Pause and try.** The new demand is $Q_d = 1200 - 10P$ and supply is still $Q_s = 10P$. Find the new clearing price and quantity. Did the quantity supplied change? Did supply?

**Answer.** $1200 - 10P = 10P$ gives $P = 60$ and $Q = 600$. The quantity supplied rose from 500 to 600 copies, because the higher price satisfies 100 more sellers. Supply, the relationship $Q_s = 10P$, is exactly what it was.

[Figure 6, animation: Demand rises by 200 copies at every price, so the demand line slides to the right. The supply line does not move. The crossing point climbs along the supply line from $50 and 500 copies to $60 and 600 copies.]

Here the distinction from earlier does its work.

**Definition.** A **change in quantity demanded** is a movement along a fixed demand curve, caused by a change in the good's own price. A **change in demand** is a shift of the whole curve, a different quantity at every price, caused by something the curve was holding fixed: the number of buyers, their incomes or tastes, the prices of related goods, what they expect. The same two phrases apply to supply.

The rise in enrollment was a change in demand. It caused a change in price, which caused a change in quantity supplied. Each curve answers "what if only the price changed?", so a price change can never shift either of them.

**Common mistake.** **"Demand went up, so the price rose; the higher price reduces demand; so the price falls back."** The second step uses "demand" for a movement along the curve. The higher price lowers the *quantity demanded* along the new curve, from 700 (what buyers would want at the old \$50) to 600, and that reduction is how the market gets to the new crossing point. Nothing pushes the curve back, so the price stays at \$60.

## A price ceiling

Suppose the student union, wanting books to be affordable, rules that no used copy may sell for more than \$30.

**Pause and try.** In the campus market, how many copies are wanted at \$30, and how many are offered? How many students who would have bought a book at the clearing price of \$50 now cannot get one?

**Answer.** $Q_d = 1000 - 300 = 700$ and $Q_s = 300$. There is a shortage of 400 copies. Only 300 copies trade. Of the 500 students who would have bought at \$50, at least 200 go without, and more than 200 if some of the 300 copies go to students outside that group. Trade has fallen by 200 copies even though 200 more students now want one.

[Figure 7, animation: A legal maximum price is lowered from $50 to $30. Sellers offer fewer copies and buyers want more, and the shortage (red) widens to 400 copies.]

**Definition.** A **price ceiling** is a legal maximum price. It is **binding** if it lies below the market-clearing price. A ceiling at or above the clearing price changes nothing in this model.

The 300 students who get a book pay \$20 less than before, and they are better off. But the price no longer decides who those 300 are. Something else has to: who saw the notice first, who knows a seller, who is willing to slip the seller an extra \$20. And since fewer than $q^* = 500$ trades take place, part 3 of the theorem applies. Some student who would pay more than \$50 has no book while an owner who would sell for \$35 keeps hers, and the rule forbids the trade that would help both.

Now return to the concert. The hall has 2000 seats whatever the price, so supply is a vertical line. Suppose the number of tickets fans want is $Q_d = 10000 - 25P$.

[Figure 8, image: The concert. Supply is 2000 seats at any price. At the $80 face price fans want 8000 tickets, a shortage of 6000. The price at which exactly 2000 are wanted is $320.]

**Worked example.** **The sell-out.** The clearing price solves $10000 - 25P = 2000$, so $25P = 8000$ and $P = 320$. At the face price of \$80, the quantity demanded is $10000 - 25 \times 80 = 8000$. The face price is a ceiling the promoter imposed on herself, 8000 fans chase 2000 seats, and the seats go to whoever clicks fastest. Among the 6000 left out are fans willing to pay far more than \$80, and among the lucky 2000 there may be some who value the seat at \$100. By the reasoning of part 3, trades between them benefit both, and the resale market is where those trades happen, at about \$320: four times the face price.

So nothing broke. A price set well below the clearing level produces a shortage as a matter of arithmetic, and a shortage produces a scramble and a second market. The demand numbers here are invented to fit the story, and the model is silent on *why* a promoter would choose \$80. Loyalty to fans, fear of looking greedy and the hope of selling merchandise have all been proposed, and deciding among them takes evidence.

## What the model leaves out

Each assumption fails somewhere. Used cars are not homogeneous, and a buyer who cannot judge quality behaves differently from Ben. A town with one employer or one pharmacy has a trader who sets the price and knows it. And "other things equal" is a discipline: every prediction has the form "if only this changes, then that follows". The model is also silent about whether a clearing price is fair. Eli, who could pay only \$25, has no book at \$52, and whether that is acceptable is a question the arithmetic cannot answer.

**Check yourself.** A frost in Brazil raises the price of coffee. Tea is a substitute for coffee. In the market for tea, what happens?
Answer: Demand for tea rises, so the price and quantity of tea both rise The price of a related good is one of the things the tea demand curve holds fixed. When it changes, the curve shifts right, and the crossing point moves up along the supply curve.

The hard part of this lesson was keeping a number apart from a curve. A price selects one point on each curve, a clearing price is where those points coincide, and events outside the market move a whole curve and drag the crossing point with it. [The next lesson](@economics/elasticity) asks how strongly quantity responds to price, which decides whether a seller who raises her price ends up with more revenue or less.

**Recap.** - Willingness to pay is the highest price at which a buyer still buys; a seller's cost is the lowest price she accepts. Sorting buyers and sellers by these numbers gives the demand and supply curves.
- Quantity demanded is a number at one price. Demand is the whole curve. The same goes for supply.
- A market-clearing price has $Q_d(p) = Q_s(p)$. At any other price the scarcer side sets the amount traded, someone willing is turned away, and if trade falls short of $q^*$ a trade that would help both sides goes unmade.
- With straight lines, find the equilibrium by setting $Q_d = Q_s$ and solving.
- A change in the good's own price moves along the curves. A change in anything else shifts a curve.
- A binding price ceiling creates a shortage and leaves something other than price to decide who is served.
- The model assumes a homogeneous good, price-taking traders and other things equal. That prices move to the clearing level is an empirical claim, supported by market experiments.

## Problems

The first three repeat the lesson's moves, the next three combine them, and the last two ask for an argument.

**Problem 1.** Five buyers would pay up to \$12, \$10, \$9, \$7 and \$4 for a concert poster. Five sellers would accept no less than \$3, \$5, \$6, \$8 and \$11. Find $q^*$, the quantity demanded and supplied at \$7.50, and all market-clearing prices.

**Solution.** Compare in order: $12 > 3$, $10 > 5$, $9 > 6$, but $7 < 8$. So $q^* = 3$. At \$7.50 the willing buyers are those at 12, 10 and 9, and the willing sellers are those at 3, 5 and 6, so $Q_d = Q_s = 3$ and \$7.50 clears the market. The counts stay at three and three for every price above 7 and below 8. At exactly \$7 the fourth buyer is also willing ($Q_d = 4$), and at exactly \$8 the fourth seller is ($Q_s = 4$), so the clearing prices are those strictly between \$7 and \$8.

**Problem 2.** Demand for a brand of notebook is $Q_d = 600 - 20P$ and supply is $Q_s = 40P - 120$, with $P$ in dollars and quantities in notebooks per week. Find the equilibrium.

**Solution.** Set $600 - 20P = 40P - 120$. Then $720 = 60P$, so $P = 12$, and $Q = 600 - 240 = 360$. Check with supply: $40 \times 12 - 120 = 360$.

**Problem 3.** In the notebook market of the previous problem, a ceiling of \$9 is imposed. Find the shortage and the number traded. What would a ceiling of \$15 do?

**Solution.** At \$9, $Q_d = 600 - 180 = 420$ and $Q_s = 360 - 120 = 240$. The shortage is 180 notebooks and 240 are traded, down from 360. A ceiling of \$15 is above the clearing price of \$12, so it is not binding and the market stays at \$12 and 360.

**Problem 4.** For each event, say whether it is a change in demand, in quantity demanded, in supply or in quantity supplied in the used-textbook market: (a) the professor makes the book optional; (b) the price falls from \$50 to \$45 and more students buy; (c) a campus bookstore starts paying \$10 more for returned copies, so owners are less eager to sell to students; (d) after (a), fewer owners bother to sell.

**Solution.** (a) A change in demand: fewer copies are wanted at every price, so the curve shifts left. (b) A change in quantity demanded: a movement along the demand curve caused by the book's own price. (c) A change in supply: every owner's cost of selling to a student rises, so fewer are offered at each price and the curve shifts left. (d) A change in quantity supplied: the fall in demand lowers the price, and sellers move down along an unchanged supply curve.

**Problem 5.** A **price floor** is a legal minimum price. In the campus market ($Q_d = 1000 - 10P$, $Q_s = 10P$), a floor of \$70 is imposed. Find the quantity traded and the excess supply. Who is left disappointed?

**Solution.** At \$70, $Q_d = 1000 - 700 = 300$ and $Q_s = 700$. Buyers are the scarcer side, so 300 copies trade and the excess supply is 400. The disappointed are 400 owners willing to sell at \$70 who find no buyer. Trade is 300, the same as under the \$30 ceiling: a price on either side of \$50 reduces trade, because one side or the other withdraws.

**Problem 6.** In the campus market, enrollment adds 200 buyers at every price and, in the same term, a graduating class adds 200 sellers at every price. Find the new equilibrium and explain why the direction of the price change could not have been predicted from the directions of the shifts alone.

**Solution.** The new curves are $Q_d = 1200 - 10P$ and $Q_s = 10P + 200$. Setting them equal, $1000 = 20P$, so $P = 50$ and $Q = 700$. The price is unchanged and the quantity is up by 200. More demand pushes the price up, more supply pushes it down, and here the two pushes are equal. If supply had risen by 100 instead, $1100 = 20P$ gives $P = 55$; if by 300, $P = 45$. Both shifts raise the quantity, so that direction is certain, but the price depends on their sizes.

**Problem 7.** A newspaper writes: "Rents rose this year because of higher demand for apartments. But as rents rise, demand will fall, and so rents will soon come back down." Find the error and say what the supply and demand model predicts.

**Solution.** The first sentence describes a shift of the demand curve to the right. The second uses "demand will fall" for the response of renters to a higher price, which is a fall in quantity demanded along the new curve. That movement is already part of the adjustment to the new crossing point; it does not shift the curve back. The model predicts that rents stay at the new, higher level until something else changes, for instance new building, which would be a shift in supply.

**Problem 8.** In the eight-student market, the price is capped at \$30 and the three sellers with costs 10, 20 and 30 sell. Define the gain from a trade as the buyer's willingness to pay minus the seller's cost. (a) Find the total gain if the three books go to Ben, Fay and Dev. (b) Find it if they go to Ana, Gus and Cho. (c) Find the total gain at a clearing price. (d) In case (b), describe a resale that benefits both parties.

**Solution.** The three sellers' costs total $10 + 20 + 30 = 60$. (a) $95 + 85 + 75 - 60 = 195$. (b) $55 + 45 + 35 - 60 = 75$. (c) At a clearing price the five highest-value buyers trade with the five lowest-cost sellers: $(95 + 85 + 75 + 65 + 55) - (10 + 20 + 30 + 40 + 50) = 375 - 150 = 225$. The price itself does not appear, because what the buyer pays the seller receives. (d) Cho values her copy at \$35 and Ben would pay \$95. If Ben buys it from her for \$60, she gains \$25 and he gains \$35 compared with not trading. That resale is the concert's secondary market in miniature, and it moves the total from the outcome in (b) toward the one in (a). Even (a) falls short of 225, because under the cap only three books change hands.