Finding Equilibrium With Piecewise Schedules

Why Piecewise Schedules Need an Extra Check

A supply or demand schedule may follow one formula at low prices and another at higher prices. For example, a supplier might have limited capacity in one range and access to additional capacity in another. The market equilibrium is still the price and quantity at which planned quantity supplied equals planned quantity demanded. The extra step is to confirm that each algebraic solution was calculated using formulas that actually apply at that price.

A price that solves an equation is not automatically an equilibrium. If the equation uses a low-price formula but its solution falls in the high-price range, the calculation has used the wrong schedule there. The candidate must be checked against the interval attached to each formula.

Definition: A piecewise supply schedule or piecewise demand schedule gives different quantity formulas for different price ranges. A market equilibrium is a price $p^*$ and quantity $Q^*$ such that $Q_s(p^*)=Q_d(p^*)=Q^*$, using the supply and demand formulas applicable at $p^*$. Here $Q_s$ is quantity supplied and $Q_d$ is quantity demanded.

Unless a problem states otherwise, interpret price and quantity as nonnegative. Price ranges may include or exclude their endpoints: for instance, $p\leq 10$ and $p>10$ assign the breakpoint price of 10 to the first formula only. Some schedules are continuous at a breakpoint, while others may jump. Either way, use the stated endpoint convention rather than assuming both formulas apply at the boundary.

A Reliable Interval-by-Interval Method

The method is to split the price axis at every breakpoint in either schedule. Within each resulting interval, the applicable supply and demand formulas are fixed, so one can set them equal and solve. Then test whether the solution belongs to that interval. If it does, calculate the common quantity; if it does not, discard it as a candidate for that interval.

List the breakpoints.
Include every price at which either supply or demand changes formula.

Form the price intervals.
Respect whether each range includes its endpoint. On each interval, identify the active supply and demand formulas.

Set the active formulas equal.
Solve $Q_s(p)=Q_d(p)$ for that interval, without yet assuming the solution is valid.

Check the candidate.
Keep it only if its price lies in the interval used to derive it. Check breakpoint prices using the formula assigned at each breakpoint.

Verify the quantity.
Substitute the accepted price into both applicable schedules. They must give the same quantity, and the result should satisfy any stated quantity restrictions.

The interval check follows directly from the definition of a piecewise schedule. On a given interval, equality of the active formulas is necessary for equilibrium. It is also sufficient if the resulting price belongs to that interval and the common quantity is admissible. Thus the full test is not merely “solve an equation”; it is “solve the equation for a region, then check membership in that region.”

Q_s(p)=Q_d(p) \quad\text{and}\quad p\in I,

where $I$ is the interval on which the formulas used in the equality are valid. A solution outside $I$ does not establish equilibrium, because at that price at least one of the formulas in the equation is not the applicable schedule.

Worked Examples

Worked Example: A Demand Schedule With Two Price Ranges

A market has supply $Q_s=10+2p$. Demand is $Q_d=90-5p$ for $0\leq p\leq 8$, and $Q_d=58-p$ for $p>8$. Find the equilibrium. Quantities are measured in units per week and price is in dollars per unit.

Method. The demand breakpoint is $p=8$, so solve once using each demand range and check each resulting price against its range.

Low-price range, $0\leq p\leq 8$. Set the supply formula equal to the low-price demand formula:

10+2p=90-5p \quad\Longrightarrow\quad 7p=80 \quad\Longrightarrow\quad p=\frac{80}{7}\approx 11.43.

This candidate is not in the range $0\leq p\leq 8$, so it must be rejected for this branch.

High-price range, $p>8$. Use the other demand formula:

10+2p=58-p \quad\Longrightarrow\quad 3p=48 \quad\Longrightarrow\quad p=16.

Since $16>8$, this candidate is valid. Substitution gives $Q_s=10+2(16)=42$ and $Q_d=58-16=42$. The equilibrium is $p^*=\$16$ per unit and $Q^*=42$ units per week.

Check. Both schedules give 42 at the accepted price. The rejected candidate is the trap to avoid: it solves an equation, but not the schedule that applies at its price.

Worked Example: Both Supply and Demand Change Formula

Supply is $Q_s=12+2p$ for $0\leq p\leq 14$, and $Q_s=26+p$ for $p>14$. Demand is $Q_d=120-4p$ for $0\leq p\leq 10$, and $Q_d=80-2p$ for $p>10$. Find the equilibrium price and quantity.

Method. The breakpoints are 10 and 14. Together they divide the nonnegative prices into $0\leq p\leq 10$, $10<p\leq 14$, and $p>14$. Select the applicable formula from each schedule in each interval.

First interval, $0\leq p\leq 10$. The active formulas are $Q_s=12+2p$ and $Q_d=120-4p$:

12+2p=120-4p \quad\Longrightarrow\quad 6p=108 \quad\Longrightarrow\quad p=18.

Because $18$ is not in $0\leq p\leq 10$, reject this candidate for the first interval.

Second interval, $10<p\leq 14$. Demand has changed formula, but supply has not:

12+2p=80-2p \quad\Longrightarrow\quad 4p=68 \quad\Longrightarrow\quad p=17.

The candidate $17$ is not in $10<p\leq 14$, so reject it for this interval as well.

Third interval, $p>14$. Both schedules use their second formulas:

26+p=80-2p \quad\Longrightarrow\quad 3p=54 \quad\Longrightarrow\quad p=18.

Now $18>14$, so this candidate is valid. The common quantity is $Q_s=26+18=44$, and $Q_d=80-2(18)=80-36=44$. The equilibrium is $p^*=\$18$ and $Q^*=44$ units.

Check. This example shows why it is not enough to find a price that solves one of the equations. The price 18 was invalid for the first two intervals but valid when both second-branch formulas applied.

Worked Example: Checking a Breakpoint With a Jump in Demand

Supply is $Q_s=8+3p$ for all $p\geq 0$. Demand is $Q_d=50-2p$ for $0\leq p<6$, and $Q_d=38-p$ for $p\geq 6$. Find the equilibrium, paying particular attention to the breakpoint.

Method. The price range is split at 6. Because the first demand formula applies only when $p<6$, the price $p=6$ belongs to the second formula.

Below the breakpoint, $0\leq p<6$. Set the active schedules equal:

8+3p=50-2p \quad\Longrightarrow\quad 5p=42 \quad\Longrightarrow\quad p=8.4.

Since $8.4$ is not less than 6, reject it for this range.

At and above the breakpoint, $p\geq 6$. Use $Q_d=38-p$:

8+3p=38-p \quad\Longrightarrow\quad 4p=30 \quad\Longrightarrow\quad p=7.5.

The candidate satisfies $7.5\geq 6$. At this price, $Q_s=8+3(7.5)=8+22.5=30.5$, and $Q_d=38-7.5=30.5$. The equilibrium is $p^*=\$7.50$ and $Q^*=30.5$ units.

Breakpoint check. At $p=6$, the applicable demand formula gives $Q_d=38-6=32$, while supply gives $Q_s=8+3(6)=26$. These are unequal, so the breakpoint itself is not an equilibrium. The demand schedule jumps at the boundary: the first formula approaches 38 as price approaches 6 from below, while the second formula gives 32 at 6. Do not substitute the breakpoint into a branch that excludes it.

Common Mistakes and Useful Checks

The most common error is to solve only one branch and report its answer without checking the corresponding price range. In the first example, using the low-price formulas gives $p=80/7$, but that price exceeds 8. Reporting it as equilibrium would mean evaluating demand with a formula that is not valid at that price.

Common mistake: Solve $10+2p=90-5p$, obtain $p=80/7$, and report it as the equilibrium.

Correction: The equation used the demand formula valid only for $p\leq 8$, while $80/7\approx 11.43>8$. Reject that candidate, solve with the demand formula for $p>8$, and verify the valid result $p=16,\ Q=42$.

A related mistake is to assume that the schedule is continuous at a breakpoint. Continuity may hold, as it does in some examples, but it is not required by the definition of a piecewise schedule. Always use the stated boundary conditions and check the actual quantities at the breakpoint when it matters.

An efficient final verification has three parts: the price lies in the interval used, the applicable supply and demand formulas give equal quantities, and the units and restrictions match the question. If no interval produces a valid candidate, the stated schedules have no equilibrium at a price in the specified domain. If multiple intervals produce valid candidates, each must be checked against the original piecewise schedules before reporting the result.

This interval-checking approach is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. It is useful whenever schedules contain thresholds, capacity ranges, or other changes in formula: algebra finds candidates, while the schedule’s price ranges determine which candidates are meaningful.

Check Your Understanding

For each problem, solve on the relevant price intervals and check that the reported price uses the formulas from which it was obtained.

Supply is $Q_s=6+2p$. Demand is $Q_d=70-4p$ for $0\leq p\leq 10$ and $Q_d=40-p$ for $p>10$. Find equilibrium price and quantity. (Answer: $p^*=34/3$, $Q^*=86/3$.)
Demand is $Q_d=96-3p$. Supply is $Q_s=12+2p$ for $0\leq p\leq 12$ and $Q_s=24+p$ for $p>12$. Find equilibrium price and quantity. (Answer: $p^*=18,\ Q^*=42$.)
Demand is $Q_d=150-5p$ for $0\leq p\leq 10$ and $Q_d=110-p$ for $p>10$. Supply is $Q_s=10+3p$ for $0\leq p\leq 15$ and $Q_s=40+p$ for $p>15$. Find equilibrium price and quantity, checking all price regions. (Answer: $p^*=35,\ Q^*=75$.)
Supply is $Q_s=10+2p$. Demand is $Q_d=80-2p$ for $0\leq p<10$ and $Q_d=55-p$ for $p\geq 10$. Determine the equilibrium and state whether the breakpoint itself is an equilibrium. (Answer: $p^*=15,\ Q^*=40$; at $p=10$, supply is 30 and demand is 45, so the breakpoint is not an equilibrium.)
Demand is $Q_d=90-3p$ for $0\leq p\leq 12$ and $Q_d=66-p$ for $p>12$. Supply is $Q_s=6+2p$ for $0\leq p\leq 16$ and $Q_s=22+p$ for $p>16$. Find the valid equilibrium, rejecting any candidate outside its interval. (Answer: $p^*=22,\ Q^*=44$.)

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