Equilibrium With Discrete Quantities

Why Discrete Quantities Need Careful Equilibrium Analysis

Many introductory supply-and-demand calculations treat quantities as if they could take any value. In some markets, however, units are indivisible: a buyer may purchase whole machines, policies, or delivery slots, and a seller may offer only whole units. Quantities then move in steps rather than smoothly. A schedule may show excess demand at one listed price and excess supply at another without showing equality at either price.

The key is to distinguish what the schedule actually specifies from what might be assumed between its entries. A table that gives quantities only at certain prices does not, by itself, tell us the quantities at unlisted prices. When a schedule is defined at every price, it may still jump over equality: integer quantities do not guarantee that an exact market-clearing quantity exists.

This analysis is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. It builds on the earlier material on market equilibrium by applying the same balance condition to discrete quantities.

Definition: At a given price $P$, excess demand is $Q_d(P)-Q_s(P)$ when this difference is positive, and excess supply is $Q_s(P)-Q_d(P)$ when that difference is positive. A price is market-clearing when $Q_d(P)=Q_s(P)$. For discrete quantities, this equality may hold at one price, at a range of prices, or at no price in the schedule’s allowed domain.

Read a Discrete Schedule as Written

Suppose a table lists aggregate quantity demanded and quantity supplied at several prices. At each listed price, subtract the smaller quantity from the larger one and name which side is larger. Do not treat the table entries as points on straight lines unless the schedule explicitly says to interpolate linearly. A jump in a quantity schedule is not evidence that every quantity between the two observed values is available.

If the schedule permits only the prices shown in a table, equilibrium can occur only at one of those prices. If it permits prices in a continuous range, additional information must specify how quantities behave at unlisted prices. Even with that information, discrete jumps can leave no price at which quantities are equal.

Worked Example: No Equality at the Listed Prices

A market table gives the following quantities of a hypothetical indivisible storage unit. The only permitted prices are the four listed prices. Determine the imbalance at each price and whether the table contains an equilibrium.

Price per unit	Quantity demanded	Quantity supplied
$8	7	2
$10	6	3
$12	4	5
$14	3	7

Method. At each allowed price, compare the two quantities directly. At $8, $7-2=5$, so there is excess demand of 5 units. At $10, $6-3=3$, so there is excess demand of 3 units. At $12, $5-4=1$, so there is excess supply of 1 unit. At $14, $7-3=4$, so there is excess supply of 4 units.

There is no listed price with equal quantities: the four demand-supply pairs are $(7,2)$, $(6,3)$, $(4,5)$, and $(3,7)$. Therefore, the table has no market-clearing price among its permitted prices. In particular, the imbalance at $10 and the opposite imbalance at $12 do not establish what happens at $11; $11 is not an allowed price in this problem.

Check. The differences $Q_d-Q_s$ at the listed prices are $5,3,-1,-4$. None is zero. The common mistake avoided here is interpolating between table entries even though the problem permits only the listed prices.

Bids, Asks, and the Price Conditions for Clearing

A useful way to represent discrete quantities is to list individual buyers’ maximum bids and individual sellers’ minimum asks. Assume one unit per buyer and seller, buyers purchase if the price is no greater than their bid, and sellers offer their units if the price is no less than their ask. These conventions include a buyer at a price equal to the bid and a seller at a price equal to the ask.

Definition: For bids $b_1,\ldots,b_n$ and asks $a_1,\ldots,a_m$, at price $P$, quantity demanded is the number of bids at least $P$, and quantity supplied is the number of asks at most $P$. Thus $Q_d(P)=\#\{j:b_j\ge P\}$ and $Q_s(P)=\#\{r:a_r\le P\}$. The symbol $\#$ means “number of elements in the set.”

This definition gives a direct way to determine whether an interval of prices clears the market. For example, sort the bids from highest to lowest and the asks from lowest to highest. To have exactly $k$ buyers willing to buy, the price must include the $k$-th highest bid but exclude the next bid. To have exactly $k$ sellers willing to sell, it must include the $k$-th lowest ask but exclude the next ask. The clearing prices are those satisfying both conditions at once.

\begin{aligned} Q_d(P)=k &\quad\Longleftrightarrow\quad b_k\ge P>b_{k+1},\\ Q_s(P)=k &\quad\Longleftrightarrow\quad a_k\le P<a_{k+1}. \end{aligned}

These conditions assume that the bids and asks are ordered as described and that the relevant adjacent values are distinct. Ties can cause a quantity to jump by more than one, so the number of buyers or sellers at a boundary must then be counted directly. The inequalities also show why endpoint inclusion matters: a buyer bidding exactly $P$ is counted, while a seller asking exactly $P$ is counted.

Why a Clearing Interval Is an Intersection

A price clears the market precisely when it produces the same quantity on both sides. If the common quantity is $k$, the price must belong to the set of prices giving demand $k$ and also to the set giving supply $k$. Therefore, the clearing-price set for that quantity is the intersection of those two sets. This is a set-based condition; it does not presume that every price in a table’s range has been observed or specified.

Choose a candidate quantity.
For each possible integer $k$, check whether the schedule can produce quantity $k$ on both sides.

Find the demand-price set.
Use the bids to identify prices for which exactly $k$ buyers are willing to buy.

Find the supply-price set.
Use the asks to identify prices for which exactly $k$ sellers are willing to sell.

Intersect and check endpoints.
A clearing interval exists for quantity $k$ only if the two sets overlap. Test each included or excluded endpoint using the bid and ask rules.

Worked Example: Finding a Market-Clearing Price Interval

Four buyers have bids of $50, $42, $34, and $26. Four sellers have asks of $15, $25, $32, and $45. Prices may be any amount in dollars. Find every price at which exactly three units are demanded and supplied.

Method. The bids are already in descending order. Exactly three buyers are willing to buy when the price is no greater than the third bid, $34, but is greater than the fourth bid, $26. Thus the demand condition is $26<P\le34$.

The asks are in ascending order. Exactly three sellers are willing to supply when the price is at least the third ask, $32, but less than the fourth ask, $45. Thus the supply condition is $32\le P<45$.

Both conditions hold on their intersection, $32\le P\le34$. Therefore, every price from $32 through $34, including both endpoints, clears the market at exactly three units.

Endpoint check. At $32, bids of $50, $42, and $34 are included, so demand is 3; asks of $15 and $25 are included, and the $32 ask is included as well, so supply is 3. At $34, the same three bids are included, and the $32 ask is included, so quantities are again 3 and 3. Just above $34, the $34 bid is excluded, leaving demand 2; just below $32, the $32 ask is excluded, leaving supply 2. The common mistake avoided is assuming an endpoint is included without checking the equality rule.

Opposite Imbalances Do Not Guarantee Equality

In a continuous, suitably specified schedule, a change from excess demand to excess supply may suggest that equality occurs in between. But a discrete schedule can jump from a positive difference to a negative difference without ever taking the value zero. And a table specified only at its listed prices gives no information about the imbalance at unlisted prices. The direction of two endpoint imbalances alone establishes the imbalance at those endpoints—and nothing more about the intervening schedule.

Worked Example: A Step Schedule With No Clearing Price

Three buyers have bids of $50, $30, and $30. Two sellers have asks of $10 and $20. Prices may be any nonnegative dollar amount, and the inclusion rules for bids and asks are as defined above. Determine whether any price clears the market.

Method. For $P\le30$, all three bids are at least the price, so $Q_d=3$. At any such price, at most two sellers are willing to supply, so $Q_s\le2$. There is excess demand, not equality.

For $30<P\le50$, only the $50 bid is at least the price, so $Q_d=1$. Both asks, $10 and $20, are at most the price, so $Q_s=2$. There is excess supply of 1 unit.

For $P>50$, no buyer has a bid at least the price, so $Q_d=0$. Both sellers still supply, so $Q_s=2$. This is excess supply of 2 units. These cases cover all nonnegative prices. There is no price at which $Q_d=Q_s$.

At exactly $30, the two $30 bids are included, so demand remains 3; just above $30, both are excluded at once and demand drops to 1. Supply stays at 2 across that jump. The difference $Q_d-Q_s$ therefore changes from $1$ to $-1$ without becoming zero. The common mistake avoided is concluding that opposite imbalances imply a clearing price between them.

Common Mistakes and a Practical Check

Interpolating without a rule. A table of discrete prices does not automatically define quantities between its entries. Use an interpolation rule only when the problem supplies one.
Assuming a crossing must include equality. A step can jump from excess demand to excess supply, skipping equal quantities entirely.
Using the wrong boundary convention. With the stated rules, a bid equal to the price counts toward demand, and an ask equal to the price counts toward supply.
Reporting a range without checking its endpoints. Write the inequalities first, intersect them, and then test every boundary price against the original schedule.
Confusing clearing quantity with total potential trades. A market-clearing price means scheduled demand equals scheduled supply; it does not, by itself, specify how units are allocated if buyers or sellers are tied or rationed.

A reliable solution records the allowed price domain, calculates both quantities using the stated inclusion rules, and reports either an exact clearing price, a clearing-price set, or no clearing price. If the only data are a few listed prices, state the result only for those prices; do not infer an unobserved path between them.

Key takeaway: For discrete schedules, equilibrium is an exact equality to verify, not a smooth crossing to assume. A market-clearing interval is the intersection of the prices that produce the same integer quantity demanded and supplied; if the sets do not overlap, no price clears the market.

Check Your Understanding

For each problem, apply the stated price domain and inclusion conventions rather than assuming quantities vary smoothly.

A table permits only prices $4, $6, and $8. At those prices, demand is 9, 7, and 5 units, while supply is 3, 5, and 7 units. Find the imbalance at each price and state whether a listed price clears the market. (Answer: At $4 there is excess demand of 6; at $6 there is excess demand of 2; at $8 there is excess supply of 2; no listed price clears the market.)
Four bids are $60, $48, $40, and $29. Four asks are $18, $33, $39, and $50. Prices may be any dollar amount. Find the prices at which exactly three units are demanded and supplied. (Answer: The demand condition is $29<P\le40$, the supply condition is $39\le P<50$, and their intersection is $[39,40]$, with three units.)
Three bids are $70, $45, and $45; three asks are $10, $25, and $60. Prices may be any nonnegative dollar amount. Determine whether a clearing price exists, paying particular attention to $45. (Answer: No clearing price exists. At $P\le45$, demand is 3 and supply is at most 2; for $45<P<60$, demand is 1 and supply is 2; at $P\ge60$, demand is at most 1 and supply is 3.)
Four bids are $52, $44, $36, and $28. Four asks are $16, $30, $34, and $48. Find the continuous clearing-price interval for exactly three units, then list the allowed prices if prices must be multiples of $0.50. (Answer: The interval is $[34,36]$; the allowed prices are $34.00, $34.50, $35.00, $35.50, and $36.00.)
A schedule is reported to have excess demand at one listed price and excess supply at the next listed price. Is that enough to assert a clearing price between them? State what additional information, if any, would permit that conclusion. (Answer: No. The endpoint imbalances alone say nothing about unlisted prices. The intervening schedule must be specified; even a specified discrete step schedule may jump across equality without attaining it.)

What You'll Learn