Computing Price Elasticity
(Warning: This section involves some simple algebra. If you are math-challenged, take it slowly and it will probably be OK.)
When we try to use the equation in the first section to calculate elasticity coefficients, we run into a problem. If we look at an increase in price from $3.00 to $4.00, we have an increase of 33%. However, if we have a price reduction from $4.00 to $3.00, we have a reduction of 25%. Thus, there are two ways of viewing what happens between $3.00 and $4.00.
Suppose that when price of a product is $3.00, people will buy 60, but when price is $4.00, they will buy only 50. What is the elasticity over this segment of the demand curve, between prices $3.00 and $4.00? Should one start at the price of $3.00 and compute a price rise of 33 1/3% and a quantity decline of 16 2/3%, or
e =(16.67%)/(33.33%) = .5.
Or, should one start at the $4.00 price, and compute a price decline of 25% and a quantity increase of 20%, or
e = (20%)/(25%) = .8.
The formula mentioned in the previous section does not tell us which way to proceed, and it matters. To get around the problem of deciding which starting point to use, economists compute elasticity based on the midpoint, or in the example above, at a price of $3.50. The formula that does this is
e = (Change in quantity divided by average quantity) / (Change in price divided by average price)
or
e = ((Q1 - Q2) / ((Q1 + Q2)/2 )) / ((P1 - P2)/((P1 + P2)/2)).
Putting the numbers from the previous example into this equation yields:
e = (60-50)/ (60+50)/2 divided by ($4-$3) / ($4+$3)/2
or
e = (10/55)/(1.00 /3.50) = (10/55)x(35/10) = 7/11 = .6363...
This formula is the formula for arc elasticity, or the elasticity between two points on the demand curve. As the two points get closer together, arc elasticity approaches point elasticity, the measure of elasticity preferred by professional economists.
With a bit of algebra, one can show that the equation for elasticity above can be rewritten as:
e = (1 / (Slope of Demand Curve)) multiplied by ((Average Price)/(Average Quantity))
Using this last equation, consider what happens when the slope gets steeper, which means that the slope becomes a bigger number.1 Elasticity becomes smaller, which means that consumers are less responsive. As the demand curve approaches a vertical line, the slope approaches infinity and elasticity approaches zero. As the demand curve approaches a horizontal line, the slope approaches zero and elasticity approaches infinity.
One can also see what happens when the slope is constant, which means that the demand curve is a straight line. As one moves along the line, elasticity changes because average price and average quantity change. At the top of the demand curve, price is high and quantity is low, so elasticity is high. At the bottom price is low and quantity is high, so elasticity is low.
Wow. We made it through all that technical stuff. Now, we can relax a bit and look at some other things that can be measured with the elasticity concept.
1 Recall that we are making price elasticity a positive number, which means we are looking at the absolute value of the slope. Also note that as the segment of the demand curve we are using becomes very small, average price becomes price and average quantity becomes quantity, and the formula becomes the formula for point elasticity.