# arc elasticity of demand

1. Use any figures for prices and quantities to calculate and analyze the arc elasticity of demand relative to price for a product, and extend the analysis to showi ts implications on a product decision.

2. Using fully explained indifference curve analysis, derive a demand curve for a product. As part of your answer, explain verbally and show mathematically that consumer equilibrium in the ordinal and cardinal approaches to consumer equilibrium are equivalent.

3. If a firm enjoys economies of scale up to a certain output level, and cost then increases proportionately with ouput, what can you say about the shape of long-run average cost curves. Illustrate graphically the shape of this LRAC curve, and explain what kind of firm is likely to have this type of LRAC (a small firm with more labor and less capital or a large firm with more capital and less labor).

#### Solution Preview

1. Use any figures for prices and quantities to calculate and analyze the arc elasticity of demand relative to price for a product, and extend the analysis to showi ts implications on a product decision.

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.

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]

Implications. If the arc price elasticity of demand were less than 1 it would imply that the product demand was inelastic, that is the sales revenue would increase if there were an increase in prices and would decrease if the prices were lowered. Similarly if the price elasticity of ...

#### Solution Summary

Help to calculate and analyze the arc elasticity of demand is offered.