When price elasticity is calculated using two points, it is known as **arch elasticity**. The midpoint method involves using the average of the initial and ending values.

Let's use our previous example, where:

Price (P) |
Quantity demanded (Q_{d}) |
---|---|

8 | 100 |

10 | 90 |

The calculation of price elasticity using the midpoint method involves the following:

**Step 1: Calculate the midpoints for P and (Qd)**

$$\text{Midpoint of P} = {(8 + 10) \over 2}$$

$$\text{ } = 9$$

$$\text{Midpoint of Q} = {(90 + 100) \over 2}$$

$$\text{ } = 95$$

**Step 2: Calculate the percentage change in P and Qd if price increases from R8 to R10.**

$$\%∆P = {(10 - 8) \over 9} \times 100$$

$$\text{ } = {2 \over 9} \times 100$$

$$\text{ } = 22,2\%$$

$$\%∆Q_d = {(90 - 100) \over 95} \times 100$$

$$\text{ } = {10 \over 95} \times 100$$

$$\text{ } = 10,5\%$$

**Step 3: Use the formula for price elasticity to calculate the price elasticity coefficient**

$$e_p = {\%∆Q_d \over \%∆P}$$

$$\text{ } = {10,5\% \over 22,2\%}$$

$$\text{ } = 0,47$$

Assuming that the price decreases from R10 to R8 will give the same result as an increase from R8 to R10, we have eliminated the "increase versus decrease" issue.

If we add all the above steps together, we end up with the following formula for calculating arch elasticity:

$$

e_p = {

\begin{array}{c|c}

(Q_2 - Q_1) \\

\hline

Q_1 + Q_2 \\

\hline

2 \\

\hline

(P_2 - P_1) \\

\hline

P_1 + P_2 \\

\hline

2

\end{array}}

$$

This looks rather intimidating, but it is nothing more than a combination of the various steps. The (Q_{2} – Q_{1}) is the difference between the end value and the initial value of the quantity demanded (90 – 100).

The [(Q_{1} + Q_{2} )/2] is the midpoint value for the quantity demanded [(100 + 90)/2]. The (P_{2} – P_{1}) is the difference between the end value and initial value for the price (R10 – R8).

The [(P_{1} + P_{2} )/2] is the midpoint value for the price [(R8 + R10)/2]. We have left out the multiplication by 100 since it occurs above and below the line, which means that each cancels out the other.

The (Q_{2} – Q_{1} )/ [(Q_{1} + Q_{2} )/2] gives us the percentage change in quantity demanded (10,5%) while (P_{2} – P_{1} )/[(P_{1} + P_{2} )/2] gives us the percentage change in price (22,2%).

We can even further simplify the formula by getting rid of the 2 since it occurs above and below the line, and end up with the following formula for arch elasticity using the midpoint method:

$$

e_1 = {

\begin{array}{c|c}

(Q_2 - Q_1) \\

\hline

Q_1 + Q_2 \\

\hline

(P_2 - P_1) \\

\hline

P_1 + P_2 \\

\end{array}}

$$

Using the data in our example, the arch elasticity can be calculated as follows:

$$

e_p = {

\begin{array}{c|c}

(90 - 100) \\

\hline

(100 + 90) \\

\hline

(10 - 8) \\

\hline

8 + 10 \\

\end{array}}

$$

$$

\text{ } = {

\begin{array}{c|c}

10 \\

\hline

190 \\

\hline

2 \\

\hline

18 \\

\end{array}}

$$

$$

\text{ } = {0,052 \over 0,111}

$$

$$

\text{ } = {0,47}

$$

This price elasticity coefficient of 0,47 is the ratio between the percentage change in price and the percentage change in quantity demanded, and tells us that a 1% increase in price will cause a 0,47% decrease in quantity demanded, and that a 1% decrease in price will cause a 0,47% increase in quantity demanded.

#### Activity

**Do the following activity about the calculation of price elasticity of demand.**

The following diagram shows what happens if the supply of say, restaurant meals, decreases because of an increase in the cost of producing restaurant meals. The equilibrium price increases and the equilibrium quantity declines. What you need to do is to calculate the price elasticity coefficient for restaurant meals for the price increase range R100 to R110.

The data in the diagram is reproduced in the following table:

P | Q_{d} |
---|---|

R100 | 1 000 |

R110 | 750 |

In the above diagram, we see that as the price increases from R100 to R110, the quantity demanded decreases from 1 000 to 750. We now have sufficient data to calculate the price elasticity coefficient for restaurant meals for the price range R100 to R110. We do this by applying the formula for price elasticity.

$$

e_p = {

\begin{array}{c|c}

(Q_2 - Q_1) \\

\hline

Q_1 + Q_2 \\

\hline

2 \\

\hline

(P_2 - P_1) \\

\hline

P_1 + P_2 \\

\hline

2

\end{array}}

$$

$$

\text{ } = {

\begin{array}{c|c}

(1000 - 750) \\

\hline

(750 + 1000)) \\

\hline

(110 - 100) \\

\hline

(100 + 110) \\

\end{array}}

$$

$$

\text{ } = {

\begin{array}{c|c}

250 \\

\hline

1750 \\

\hline

10 \\

\hline

210

\end{array}}

$$

$$

\text{ } = 3

$$