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Wize AP Microeconomics Textbook > Elasticity

Arc Price Elasticity of Demand

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Arc Price Elasticity of Demand

Arc price elasticity of demand is used when you want to calculate the average elasticity between two prices
Arc Price Elasticity of Demand= Average %ΔQAverage %ΔP  =  ΔQΔP ⋅ P(average)Q (average)\boxed{\text{Arc\ Price\ Elasticity\ of\ Demand} =\ \frac{Average\ \%ΔQ}{Average\ \%ΔP}\ \ =\ \ \frac{ΔQ}{ΔP}\ \cdot\ \frac{P\left(average\right)}{Q\ \left(average\right)}}

How do you Calculate Average Percentage Change?

Average Percentage Change= (New − Old)Average⋅100\boxed{\text{Average\ Percentage\ Change} =\ \frac{\left(New\ -\ Old\right)}{Average}\cdot100}


Examples:

Between points A and B in the diagram above, the average percentage change in price is

(50−40)45×100%=22.22%\displaystyle \frac{\left(50-40\right)}{45}\times100\%=22.22\%

Note: We calculate the average price using (25+30)2=27.5\frac{\left(25+30\right)}{2}=27.5

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Between points A and B in the diagram above, the average percentage change in quantity is

(25−30)27.5×100%=−18.18%\displaystyle \frac{\left(25-30\right)}{27.5}\times100\%=-18.18\%

Note: We calculate the average quantity using (25+30)2=27.5\frac{\left(25+30\right)}{2}=27.5


Between points A and B in the diagram above, the arc price elasticity of demand is
Arc Price Elasticity of Demand (Arc Ed) = (Average % Change in Q)(Average % Change in P) = −18.18%22.22%  = −0.82\frac{\left(Average\ \%\ Change\ in\ Q\right)}{\left(Average\ \%\ Change\ in\ P\right)}\ =\ \frac{-18.18\%}{22.22\%\ }\ =\ -0.82

This is inelastic because the absolute value of -0.82 is 0.82 which is less than 1.
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Example: Arc Price Elasticity of Demand

a) Suppose the price of textbooks increase from $20 to $30 and the quantity demanded drops from 100 units to 90 units. What is the arc price elasticity of demand (midpoint method)? Is demand elastic or inelastic in this range?

Arc price elasticity of demand (Arc Ed) =  Average % Change in QAverage % Change in P\frac{\ Average\ \%\ Change\ in\ Q}{Average\ \%\ Change\ in\ P}


Average % Change in P = (New−Old)Average⋅100 =(30−20)25⋅100 = 40%\frac{\left(New-Old\right)}{Average}\cdot100\ =\frac{\left(30-20\right)}{25}\cdot100\ =\ 40\%


Note: we calculated average price using (40+50)2=45\frac{\left(40+50\right)}{2}=45


Average % Change in Q = (New−Old)Average⋅100 = (90−100)95⋅100 = −10.53%\frac{\left(New-Old\right)}{Average}\cdot100\ =\ \frac{\left(90-100\right)}{95}\cdot100\ =\ -10.53\%


Note: we calculated average quantity using (90+100)2=95\frac{\left(90+100\right)}{2}=95


Arc Ed = −10.53%40% = −0.26\frac{-10.53\%}{40\%}\ =\ -0.26


This is inelastic because absolute value is less than 1.



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b) If the demand equation is P = 70 - 2Q, what would be the elasticity of demand between $40 and $50?

First we plug in 40 and 50 in to the demand equation as the price to get our two quantities:

40 = 70 - 2Q
2Q = 30
Q = 15


50 = 70 - 2Q
2Q = 20
Q = 10

Now we have 2 prices and 2 quantities so we can find the percentage changes:


% Change in P = (New−Old)Average⋅100 =(50−40)45⋅100 = 22.22%\frac{\left(New-Old\right)}{Average}\cdot100\ =\frac{\left(50-40\right)}{45}\cdot100\ =\ 22.22\%


Note: we calculated average price using (40+50)2=45\frac{\left(40+50\right)}{2}=45


% Change in Q = (New−Old)Average⋅100 =(10−15)12.5⋅100 = −40%\frac{\left(New-Old\right)}{Average}\cdot100\ =\frac{\left(10-15\right)}{12.5}\cdot100\ =\ -40\%


Note: we calculated average quantity using (40+50)2=45\frac{\left(40+50\right)}{2}=45


Arc Ed = (Average % Change in Q)(Average % Change in P) = −40%22.22 = −1.8\frac{\left(Average\ \%\ Change\ in\ Q\right)}{\left(Average\ \%\ Change\ in\ P\right)}\ =\ \frac{-40\%}{22.22}\ =\ -1.8


This is elastic because the absolute value is greater than 1.

Note: In this example we took $40 as the old price and $50 as the new price. You can do it the opposite way also (putting $50 as the old price) but then you would also have to switch the quantities (put 15 as the old quantity) and you would get the exact same final answer.