Point Elasticity And Arc Elasticity

Why Two Elasticity Measures?

Elasticity describes how responsive quantity demanded is to a change in price, using percentage rather than unit changes. The right calculation depends on the information available and the question being asked. If a demand function is known and responsiveness is needed at one particular price, use point elasticity. If the information consists of two observations and the question concerns the movement between them, use arc elasticity.

The distinction matters because a finite movement from one observation to another is not the same object as responsiveness at a single point. A slope alone cannot settle the question: it measures changes in quantity per unit of price, while elasticity scales responsiveness by price and quantity. In this tutorial, demand elasticity is reported with its sign, so a negative result indicates that quantity and price move in opposite directions.

Definition: Point price elasticity of demand measures the percentage responsiveness of quantity demanded to price at one specified point on a demand function. Arc price elasticity of demand measures that responsiveness across two observed price-quantity combinations, using their midpoints as the percentage-change bases.

Point Elasticity: Responsiveness at One Price

Suppose quantity demanded is given by a function $Q(P)$, where $P$ is price and $Q(P)$ is the quantity consumers demand at that price, with other relevant factors held fixed. To calculate point elasticity, the function must be differentiable at the price of interest, and quantity and price must be positive there. The derivative $dQ/dP$ is the local change in quantity associated with a small change in price.

Formula: For a differentiable demand function with $P>0$ and $Q(P)>0$, signed point price elasticity of demand is

\varepsilon_D(P)=\frac{dQ}{dP}\frac{P}{Q(P)}.

The derivative is not itself elasticity: it has units such as units of quantity per dollar. Multiplying the derivative by $P/Q(P)$ converts the local response into a ratio of percentage changes. If demand is downward sloping at the point, then $dQ/dP<0$, so signed elasticity is negative. If a demand function is not differentiable at the point, this derivative formula does not provide a unique point elasticity there.

Worked Example: Point Elasticity From a Nonlinear Demand Function

A hypothetical market has demand $Q(P)=500-2P^2$, where price is measured in dollars and quantity in units. Calculate signed point elasticity at $P=\$10$, and verify that the formula is applicable.

Method and conditions. The demand function is differentiable for all prices. At $P=10$, its quantity must also be positive for the elasticity formula to apply.

First calculate the quantity:

$$ Q(10)=500-2(10^2)=500-200=300. $$

The price and quantity are both positive. Next differentiate the demand function:

\frac{dQ}{dP}=-4P, \qquad \left.\frac{dQ}{dP}\right|_{P=10}=-4(10)=-40.

Now substitute the derivative, price, and quantity into the point-elasticity formula:

\varepsilon_D(10)=(-40)\frac{10}{300} =\frac{-400}{300} =-\frac{4}{3} \approx -1.3333.

Check. The derivative has units of quantity per dollar, while $P/Q$ has dollars per unit, so their product is unit-free. The negative sign is consistent with the negative local slope. The common mistake this method avoids is treating the slope $-40$ itself as an elasticity.

Arc Elasticity: Responsiveness Across Two Observations

When only two observations are supplied, a derivative may be unavailable or unnecessary. Let the observations be $(P_1,Q_1)$ and $(P_2,Q_2)$, with positive prices and quantities and with $P_1\ne P_2$. Midpoint arc elasticity divides the change in quantity by the average of the two quantities, and divides the change in price by the average of the two prices. It then takes the ratio of those percentage changes.

Formula: The signed midpoint arc elasticity between two observations is

\varepsilon_{\text{arc}} = \frac{(Q_2-Q_1)/\left((Q_1+Q_2)/2\right)} {(P_2-P_1)/\left((P_1+P_2)/2\right)}.

The midpoint method uses the same base values whichever observation is treated as the starting point. By contrast, calculating each percentage change from the initial observation can produce a different answer when the direction of comparison is reversed. The midpoint measure is a summary of the interval; it should not be described as the derivative at either endpoint.

Worked Example: Midpoint Arc Elasticity for a Price Increase

In a hypothetical market, price rises from $\$10$ to $\$14$, while quantity demanded falls from 300 units to 240 units. Calculate midpoint arc elasticity, and show what happens if the observations are reversed.

Method. Use the midpoint percentage change in quantity divided by the midpoint percentage change in price. This choice avoids making the result depend on which observation is named first.

The average quantity and average price are

\bar Q=\frac{300+240}{2}=270, \qquad \bar P=\frac{10+14}{2}=12.

Quantity changes by $240-300=-60$ units, and price changes by $14-10=4$ dollars. The corresponding midpoint percentage changes are

\frac{\Delta Q}{\bar Q}=\frac{-60}{270}=-\frac{2}{9}, \qquad \frac{\Delta P}{\bar P}=\frac{4}{12}=\frac{1}{3}.

Therefore,

\varepsilon_{\text{arc}} = \frac{-2/9}{1/3} =-\frac{2}{3} \approx -0.6667.

If the observations are reversed, the quantity change is $300-240=60$, the price change is $10-14=-4$, and the averages remain 270 and 12. The two midpoint percentage changes become $60/270=2/9$ and $-4/12=-1/3$; their ratio is still $-2/3$. Check. Both calculations give $-0.6667$, as required. A common mistake is to divide by the initial values: $(-60/300)/(4/10)=-0.5000$. That is not the midpoint arc calculation, which uses the averages as its bases.

Why the Midpoint Formula Works

Both measures are ratios of percentage responsiveness, but they build those percentages differently. The point formula uses local changes around one point. The midpoint formula uses finite changes across an interval and gives each endpoint a symmetric role. The following derivations make those choices explicit.

Start with local percentage changes.
For a small price change $dP$ near a point with price $P>0$ and quantity $Q(P)>0$, the corresponding local quantity change is $dQ$. The ratio of the percentage changes is $(dQ/Q)/(dP/P)$.

Use the derivative.
By the definition of the derivative, $dQ/dP$ describes the local change in quantity per unit change in price. Thus $(dQ/Q)/(dP/P)=(dQ/dP)(P/Q)$, which is the point-elasticity formula.

Replace local changes with finite changes.
For two observations, set $\Delta Q=Q_2-Q_1$ and $\Delta P=P_2-P_1$. Dividing $\Delta Q$ by the average quantity gives the midpoint percentage change in quantity; dividing $\Delta P$ by the average price gives the midpoint percentage change in price.

Take the ratio.
Dividing those two midpoint percentage changes produces the arc formula. Reversing the observations changes the signs of both $\Delta Q$ and $\Delta P$, but not the averages. Both numerator and denominator of the ratio change sign, so arc elasticity is unchanged.

This derivation also clarifies a limitation: point elasticity is defined from local changes, whereas arc elasticity is constructed from finite changes and midpoint bases. An arc value is therefore an interval measure, not automatically the point elasticity at the interval's midpoint. For a sufficiently small interval on a smooth demand curve, an arc measure can serve as an approximation to local responsiveness; for a substantial interval, report it as the arc measure.

Selecting the Appropriate Measure

The wording of a question and the form of the data usually determine the method. Use point elasticity when a differentiable demand equation is supplied and the question asks about responsiveness at a specified price. Use midpoint arc elasticity when two price-quantity observations are supplied and the question asks for responsiveness over the movement between them. If only two observations are known, they do not, by themselves, determine a derivative at either observation.

Information and question	Measure to use	Reason
A differentiable demand function and a particular price	Point elasticity	The derivative describes local responsiveness at that price.
Two observed price-quantity combinations and a question about the interval	Midpoint arc elasticity	The finite changes and their midpoint bases summarize the observed movement.
Only a slope or a change in units per dollar	Not enough to call it elasticity	Elasticity also scales the response by price and quantity.
A nondifferentiable demand function at the requested price	The derivative formula does not apply there	A local derivative at that point is not defined.

A practical exam habit is to label the requested measure before substituting numbers: write “point” if a price and a differentiable function are involved, or “arc” if the problem compares two observations. Then write down the relevant quantity and price—or their averages—before calculating. This makes it easier to catch a mismatched method or a percentage change formed from the wrong base.

Worked Example: Choosing Between Point and Arc Elasticity

A hypothetical demand function is $Q(P)=900-5P$. Find point elasticity at $P=\$60$, then find midpoint arc elasticity between $P=\$40$ and $P=\$60$. State which measure answers a question specifically about the interval from $\$40$ to $\$60$.

Point measure at $\$60$. First calculate quantity and the derivative:

Q(60)=900-5(60)=900-300=600, \qquad \frac{dQ}{dP}=-5.

Substitute into the point formula:

\varepsilon_D(60)=(-5)\frac{60}{600} =\frac{-300}{600} =-0.5000.

Arc measure over the interval. Calculate the quantities at both endpoints:

Q(40)=900-5(40)=700, \qquad Q(60)=600.

The midpoint values are $\bar Q=(700+600)/2=650$ and $\bar P=(40+60)/2=50$. The changes are $\Delta Q=600-700=-100$ and $\Delta P=60-40=20$. Therefore,

\varepsilon_{\text{arc}} = \frac{-100/650}{20/50} = \frac{-2/13}{2/5} =-\frac{5}{13} \approx -0.3846.

Selection and check. The point value $-0.5000$ describes responsiveness at $\$60$; the arc value $-0.3846$ describes the interval. Thus a question about the interval calls for $-0.3846$. The frequent error is to calculate the correct endpoint elasticity and use it as though it described the whole interval.

Common Mistakes and Checks

Using a slope as an elasticity. A slope such as $-5$ has units; calculate the derivative times $P/Q(P)$ for point elasticity.
Using the wrong percentage-change base. For midpoint arc elasticity, divide changes by the averages of the two observations, not by the initial values.
Applying a point formula to two observations. Two endpoints support an arc calculation. They do not supply a derivative unless additional functional information is given.
Calling an arc value a point value. Arc elasticity summarizes a finite interval. It is not automatically the elasticity at either endpoint or at the midpoint.
Dropping the sign during calculation. Keep the signs of both changes until their ratio is formed; for ordinary downward-sloping demand, the resulting signed elasticity is negative.

Key takeaway: Point elasticity uses the derivative at one price: $(dQ/dP)(P/Q)$. Midpoint arc elasticity uses two observations and divides their midpoint percentage change in quantity by their midpoint percentage change in price. Match the method to the question.

Check Your Understanding

For each problem, identify the requested measure before calculating. Keep the signed elasticity and round numerical results to four decimal places where needed.

A hypothetical demand function is $Q(P)=420-3P$. Calculate point elasticity at $P=\$20$. Verify that the quantity at this price is positive. (Answer: $-0.1667$)
Price moves from $\$8$ to $\$12$, while quantity demanded falls from 200 units to 160 units. Calculate midpoint arc elasticity and state whether reversing the observations changes it. (Answer: $-0.5556$; reversing the observations does not change it.)
Two observations are $(P_1,Q_1)=(\$25,500)$ and $(P_2,Q_2)=(\$35,400)$. Find midpoint arc elasticity, showing both midpoint percentage changes. (Answer: $-0.6667$)
A hypothetical demand function is $Q(P)=800-2P^2$. Calculate point elasticity at $P=\$15$. (Answer: $-2.5714$)
For the same demand function $Q(P)=800-2P^2$, calculate midpoint arc elasticity between $P=\$10$ and $P=\$15$, then state which measure should answer a question about responsiveness over that interval. (Answer: $-1.3158$; use arc elasticity for the interval.)
Two observations are $(P_1,Q_1)=(\$30,240)$ and $(P_2,Q_2)=(\$50,160)$. Calculate midpoint arc elasticity, then recalculate after reversing the observations to check the result. (Answer: $-0.8000$ in either order.)

What You'll Learn