Estimating Equilibrium From Rounded Or Plotted Data

Why an Estimated Equilibrium Needs a Range

Market schedules are not always available as exact equations. A table may round quantities to the nearest ten units, or a graph may only allow approximate readings from its axes. In either case, a calculated equilibrium price is an estimate, not an exact value. Reporting a range as well as a central estimate makes clear how much confidence the data’s precision supports.

Let $Q_d(P)$ be the quantity demanded at price $P$, and $Q_s(P)$ the quantity supplied. The excess-demand gap is $G(P)=Q_d(P)-Q_s(P)$. Equilibrium occurs when the quantities are equal, or equivalently when $G(P)=0$. A positive gap indicates excess demand; a negative gap indicates excess supply.

Definition: An estimated equilibrium price is an approximate price at which estimated quantity demanded equals estimated quantity supplied. A reasonable equilibrium-price range is an interval that reflects the stated precision of the quantity and price readings, under an explicitly stated model of how the gap changes between observations.

The method in this tutorial uses two observations that bracket a crossing: the gap is positive at the lower price and negative at the higher price. Before interpolating, assume the gap changes linearly between those observations. This is a local approximation; it does not assert that demand and supply are linear at every price. If the observed gaps do not have opposite signs, these two observations alone do not bracket an equilibrium.

Linear Interpolation and Its Justification

Suppose the observed gaps are $g_1>0$ at price $P_1$ and $g_2<0$ at price $P_2$, with $P_1<P_2$. Linear interpolation estimates the price where the gap reaches zero. The estimated price is closer to the observation whose gap is closer to zero.

\widehat{P}^{\,*} = P_1+(P_2-P_1)\frac{g_1}{g_1-g_2}

To see why, let $t$ be the fraction of the distance from $P_1$ to $P_2$. Under the linear-gap assumption, the gap at that fraction is $g_1+t(g_2-g_1)$. At the estimated equilibrium the gap is zero. Solving $0=g_1+t(g_2-g_1)$ gives $t=g_1/(g_1-g_2)$; the price is then $P_1+t(P_2-P_1)$, which is the formula above. Because $g_1>0$ and $g_2<0$, $t$ lies between zero and one, so the estimate lies between the two observed prices.

Turning Rounded Quantities into a Price Range

Suppose each quantity is rounded to the nearest $h$ units. A reported quantity can differ from its unrounded value by as much as $h/2$. Since the gap subtracts a supply quantity from a demand quantity, the two rounding errors can act in opposite directions: the true gap can differ from the reported gap by as much as $h$.

Thus, if the reported gap at a price is $g$, a conservative interval for the true gap is $[g-h,\ g+h]$. For two observations, write the possible true gaps as $G_1$ and $G_2$. If the first remains positive and the second remains negative throughout their intervals, the crossing is bracketed even after accounting for rounding. Substituting the extreme possible gaps into the interpolation formula gives a conservative range for the equilibrium price.

Important condition: This range assumes the gap is linear between the two observed prices and that the stated rounding precision is accurate. If rounding uncertainty allows either endpoint gap to change sign, the observations may not guarantee a crossing strictly between the prices. Report that limitation rather than claiming unwarranted precision.

Worked Example: Rounded Quantities to the Nearest Ten

A market table reports the following quantities, rounded to the nearest 10 units. At a price of $20, reported demand is 100 and reported supply is 60. At a price of $28, reported demand is 80 and reported supply is 100. Assuming a linear gap between these prices, estimate the equilibrium price and give a conservative range.

Method. First calculate the reported gaps. At $20, $g_1=100-60=40$. At $28, $g_2=80-100=-20$. The positive and negative signs bracket a crossing.

Central estimate. The reported gap falls by $40-(-20)=60$ units across an $8 price increase. Its zero is $40/60=2/3$ of the way from the first observation to the second. Therefore,

\widehat{P}^{\,*} =20+8\left(\frac{40}{40-(-20)}\right) =20+8\left(\frac{2}{3}\right) =25.333\ldots

The central estimate is $25.33, rounded to the nearest cent.

Range from rounding. Each reported quantity is rounded to the nearest 10, so the true gap at either price can differ from the reported gap by up to 10 units. Thus $G_1$ is between 30 and 50, and $G_2$ is between $-30$ and $-10$. Both endpoint signs remain secure. For the lowest possible crossing fraction, use the smallest positive first gap and the most negative second gap: $30/(30-(-30))=1/2$. For the highest fraction, use the largest first gap and the second gap closest to zero: $50/(50-(-10))=5/6$. Hence,

20+8\left(\frac{1}{2}\right) \leq P^* \leq 20+8\left(\frac{5}{6}\right), \qquad 24 \leq P^* \leq 26.666\ldots

The conservative price range is $24.00 to $26.67, with endpoints rounded to the nearest cent. The central estimate $25.33 lies inside it. The reported gaps of 40 and $-20$ are multiples of 10, as they must be when each reported quantity is a multiple of 10.

Common mistake avoided. Using a rounding error of only 5 units for the gap would treat the difference as if only one quantity had been rounded. Both demand and supply are rounded, so their errors can combine to make the gap error as large as 10 units.

Reading an Approximate Crossing from a Graph

A graph introduces a related issue: the values read from its axes may themselves be approximate. If the price coordinates are uncertain as well as the quantity coordinates, account for both. One practical approach is to calculate a central interpolation from the readings, then find a conservative range by combining the extreme possible endpoint prices and gaps.

For two endpoint gaps $G_1>0$ and $G_2<0$, the interpolation fraction is $t=G_1/(G_1-G_2)$. It increases when either endpoint gap increases. Therefore, its lowest value uses the smallest possible positive first gap and the most negative second gap; its highest uses the largest first gap and the second gap closest to zero. The interpolated price is $(1-t)P_1+tP_2$, so its range must also reflect uncertainty in $P_1$ and $P_2$.

Worked Example: Graph Readings with Price and Quantity Uncertainty

A plotted demand-and-supply graph is read at approximately $P_1=\$30$ and $P_2=\$50$. At $30, the readings are $Q_d=90$ and $Q_s=70$; at $50, they are $Q_d=70$ and $Q_s=100$. Each quantity reading may be off by up to 5 units, and each price reading by up to $1. Assume the gap is linear between the actual endpoint readings. Find the central estimate and a conservative price range.

Method. The central gaps are $g_1=90-70=20$ and $g_2=70-100=-30$. The central interpolation fraction is $20/(20-(-30))=20/50=0.4$. Thus the estimated equilibrium price is

\widehat{P}^{\,*}=30+(50-30)(0.4)=\$38.00.

Range from quantity readings. Each gap subtracts two quantity readings, each with a possible error of 5 units. The gap uncertainty is therefore up to 10 units. The first gap lies between 10 and 30; the second lies between $-40$ and $-20$. The first remains positive and the second remains negative. The smallest interpolation fraction is $10/(10-(-40))=0.2$; the largest is $30/(30-(-20))=0.6$.

Include price-reading uncertainty. The actual first price lies between $29 and $31, and the actual second price between $49 and $51. The smallest possible crossing uses the lowest endpoint prices and the smallest fraction: $29+(49-29)(0.2)=\$33$. The largest uses the highest endpoint prices and the largest fraction: $31+(51-31)(0.6)=\$43$. The conservative range is therefore $33.00 to $43.00. The estimate and range are consistent with the stated graph-reading precision.

Common mistake avoided. Reporting $38.00 as though it were exact ignores uncertainty in all four graph readings. The central value is useful, but the range communicates what the graph can actually support under the stated assumptions.

When the Data Do Not Secure a Crossing

A range is meaningful only if the observations and assumptions support it. If rounding is so coarse that a supposedly positive gap might actually be negative, or a negative gap might actually be positive, the endpoint signs are not guaranteed. The central interpolation may still be a descriptive estimate based on reported values, but the data do not establish that the actual equilibrium is bracketed by those prices. In that case, state the uncertainty clearly; do not treat the central estimate as a precise solution.

The same caution applies when the gap may curve substantially between observations. Linear interpolation is a local approximation, not a result that follows from observing two points alone. If the curve’s shape is known to be nonlinear, or if the interval is wide enough that linearity is implausible, two-point interpolation may be a poor estimate. The appropriate range should reflect both measurement precision and the model’s limitations.

Common Mistakes and a Practical Check

Ignoring rounding in both quantities. When both demand and supply are rounded to the nearest $h$, the gap uncertainty can be $h$, not $h/2$.
Using gaps that cannot come from the stated data. If quantities are rounded to the nearest 10, each reported quantity is a multiple of 10, so their difference must also be a multiple of 10. A reported gap of 45 is incompatible with that premise.
Interpolating without a sign change. A positive gap at both prices, or a negative gap at both, does not bracket a zero. Do not apply the crossing formula as if it did.
Reporting false precision. A calculated value with many decimal places does not make approximate graph readings exact. Round sensibly and state a range.
Forgetting the model assumption. The range calculated from two points relies on a linear gap between them. It is not a guarantee when the gap’s shape is unknown.

Calculate the reported gaps.
At each observed price, subtract reported supply from reported demand. Check that the two gaps have opposite signs.

State the approximation.
Use a linear gap between the observations, and record the rounding or graph-reading precision for both price and quantity.

Interpolate the central value.
Use the reported gaps and prices to estimate the zero of the gap.

Build and check the range.
Translate measurement precision into possible endpoint gaps and prices. Confirm whether the endpoint signs remain opposite, and check that the central estimate lies inside the resulting range.

Key takeaway: Interpolation gives a central equilibrium estimate, while measurement precision determines how wide a defensible range should be. Check that rounded data are internally consistent, that the gap signs truly bracket equilibrium, and that the linear-gap assumption is stated.

Check Your Understanding

For each problem, use linear interpolation between the stated observations. Give a central estimate and, where possible, a conservative range that reflects the stated precision.

Quantities are rounded to the nearest 10. At prices $10 and $22, the reported demand-supply gaps are 30 and $-20$ units. Find the central estimate and range. (Answer: central estimate $17.20; range $14.80 to $19.60.)
Quantities are rounded to the nearest 2 units. At price $8, reported demand is 44 and supply is 30; at price $14, demand is 32 and supply is 40. Find the central estimate and conservative range. (Answer: central estimate approximately $11.82; range approximately $11.27 to $12.36.)
A graph gives gap readings of $+18$ units at price $25 and $-22$ units at price $45. Each quantity reading may be off by up to 3 units, while prices are exact. Find the central estimate and range. (Answer: central estimate $34; range $31 to $37.)
Quantities are rounded to the nearest whole unit. The reported gaps are $+6$ at price $40 and $-4$ at price $52. Find the central estimate and conservative range. (Answer: central estimate $47.20; range $46.00 to $48.40.)
Quantities are rounded to the nearest 10 units. The reported gaps are $+10$ at price $5 and $-10$ at price $15. Under a linear-gap assumption, what central estimate and range are supported by the readings, and are the endpoint signs guaranteed? (Answer: central estimate $10; possible crossing range $5 to $15; the endpoint signs are not guaranteed because either gap may be zero.)

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