Arc Elasticity

One of the problems with the standard formulas for elasticity that are in many freshman texts is the elasticity figure you come up with is different depending on what you use as the start point and what you use as the end point. An example will help illustrate this.
When we looked at Price Elasticity of Demand we calculated the price elasticity of demand when price went from $9 to $10 and demand went from 150 to 110 was 2.4005. But what if we calculated what the price elasticity of demand when we started at $10 and went to $9? So we'd have:

Price(OLD)=10
Price(NEW)=9
QDemand(OLD)=110
QDemand(NEW)=150

First we'd calculate the percentage change in quantity demanded: [QDemand(NEW) - QDemand(OLD)] / QDemand(OLD)

By filling in the values we wrote down, we get:

[150 - 110] / 110 = (40/110) = 0.3636 (Again we leave this in decimal form)

Then we'd calculate the percentage change in price:

[Price(NEW) - Price(OLD)] / Price(OLD)

By filling in the values we wrote down, we get:

[9 - 10] / 10 = (-1/10) = -0.1

We then use these figures to calculate the price-elasticity of demand:

PEoD = (% Change in Quantity Demanded)/(% Change in Price)

We can now fill in the two percentages in this equation using the figures we calculated earlier.

PEoD = (0.3636)/(-0.1) = -3.636

When calculating a price elasticity, we drop the negative sign, so our final value is 3.636. Obviously 3.6 is a lot different from 2.4, so we see that this way of measuring price elasticity is quite sensitive to which of your two points you choose as your new point, and which you choose as your old point. Arc elasticities are a way of removing this problem.

When calculating Arc Elasticities, the basic relationships stay the same. So when we're calculating Price Elasticity of Demand we still use the basic formula:

PEoD = (% Change in Quantity Demanded)/(% Change in Price)

However how we calculate the percentage changes differ. Before when we calculated Price Elasticity of Demand, Price Elasticity of Supply,Income Elasticity of Demand, or Cross-Price Elasticity of Demand we'd calculate the percentage change in Quantity Demand the following way:

[QDemand(NEW) - QDemand(OLD)] / QDemand(OLD)

To calculate an arc-elasticity, we use the following formula:

[[QDemand(NEW) - QDemand(OLD)] / [QDemand(OLD) + QDemand(NEW)]]*2

This formula takes an average of the old quantity demanded and the new quantity demanded on the denominator. By doing so, we will get the same answer (in absolute terms) by choosing $9 as old and $10 as new, as we would choosing $10 as old and $9 as new. When we use arc elasticities we do not need to worry about which point is the starting point and which point is the ending point. This benefit comes at the cost of a more difficult calculation.

If we take the example with:

Price(OLD)=9
Price(NEW)=10
QDemand(OLD)=150
QDemand(NEW)=110

We will get a percentage change of:

[[QDemand(NEW) - QDemand(OLD)] / [QDemand(OLD) + QDemand(NEW)]]*2

[[110 - 150] / [150 + 110]]*2 = [[-40]/[260]]*2 = -0.1538 * 2 = -0.3707

So we get a percentage change of -0.3707 (or -37% in percentage terms). If we swap the old and new values for old and new, the denominator will be the same, but we will get +40 in the numerator instead, giving us an answer of the 0.3707. When we calculate the percentage change in price, we will get the same values except one will be positive and the other negative. When we calculate our final answer, we will see that the elasticities will be the same and have the same sign. To conclude this piece, I'll include the formulas so you can calculate the arc versions of price elasticity of demand, price elasticity of supply, income elasticity, and cross-price demand elasticity. I recommend calculating each of the measures using the step-by-step fashion I detail in the previous articles.

New Formulas - Arc Price Elasticity of Demand

To calculate the Arc Price Elasticity of Demand, we use the formulas:

PEoD = (% Change in Quantity Demanded)/(% Change in Price)

(% Change in Quantity Demanded) = [[QDemand(NEW) - QDemand(OLD)] / [QDemand(OLD) + QDemand(NEW)]] *2]

(% Change in Price) = [[Price(NEW) - Price(OLD)] / [Price(OLD) + Price(NEW)]] *2]

New Formulas - Arc Price Elasticity of Supply

To calculate the Arc Price Elasticity of Supply, we use the formulas:

PEoS = (% Change in Quantity Supplied)/(% Change in Price)

(% Change in Quantity Supplied) = [[QSupply(NEW) - QSupply(OLD)] / [QSupply(OLD) + QSupply(NEW)]] *2]

(% Change in Price) = [[Price(NEW) - Price(OLD)] / [Price(OLD) + Price(NEW)]] *2]

New Formulas - Arc Income Elasticity of Demand

To calculate the Arc Income Elasticity of Demand, we use the formulas:

PEoD = (% Change in Quantity Demanded)/(% Change in Income)

(% Change in Quantity Demanded) = [[QDemand(NEW) - QDemand(OLD)] / [QDemand(OLD) + QDemand(NEW)]] *2]

(% Change in Income) = [[Income(NEW) - Income(OLD)] / [Income(OLD) + Income(NEW)]] *2]

New Formulas - Arc Cross-Price Elasticity of Demand of Good X

To calculate the Arc Cross-Price Elasticity of Demand, we use the formulas:

PEoD = (% Change in Quantity Demanded of X)/(% Change in Price of Y)

(% Change in Quantity Demanded) = [[QDemand(NEW) - QDemand(OLD)] / [QDemand(OLD) + QDemand(NEW)]] *2]

(% Change in Price) = [[Price(NEW) - Price(OLD)] / [Price(OLD) + Price(NEW)]] *2]

Notes and Conclusion

Keep in mind that for all over these formulas it doesn't matter what you use as the "old" and as the "new" value, just as long as the "old" price is the one associated with the "old" quantity. You could call the points A and B or 1 and 2 if you like, but old and new works just as well.

So now you can calculate elasticity using a simple formula as well as using the arc formula. In a future article, we will look at using calculus to compute elasticities.