Slope and elasticity

The post on Wed. (How Much Do Tariffs Raise Prices?) mentioned that slope and elasticity are not the same thing. Here I will give a couple of examples of demand lines that show this.

In the first graph, we can clearly see that one demand line is steeper than the other. I will calculate the price elasticity of demand for two different cases. In the first case I will drop the price from 8 to 4 for each line and calculate elasticity for each line using the mid-point formula (the steeper line will have a lower elasticity). Then I will calculate the elasticity for all of each line (the answer will be the same number in each case).

Here is the mid-point formula:

(Q2 - Q1)/(Q2 + Q1) times (P2 + P1)/(P2 - P1)

Calculations after the graph.

For the steeper line this is 

(8 - 6)/(8 + 6) times (4 + 8)/(4 - 8) = (2/14)*(12/4) = (1/7)*(3/1) = 3/7 (we ignore negative signs)

For the flatter line this is 

(12 - 4)/(12 + 4) times (4 + 8)/(4 - 8) = (8/16)*(12/4) = (1/2)*(3/1) = 3/2

Since 3/2 > 3/7, the steeper line is less elastic between those two prices (8 & 4).

But if we calculate the elasticity for all of each line, they would come out the same even though the lines have different slopes.

For the steep line it is (if we lower the price from 20 to 0)

(20 - 0)/(20 + 0) times (0 + 10)/(0 - 10) = (20/20)*(10/10) = 1

For the flat line it is (if we lower the price from 10 to 0)

(10 - 0)/(10 + 0) times (0 + 20)/(0 - 20) = (10/10)*(20/20) = 1

So the elasticity for all of each line is the same (1) even though it is clear that they have different slopes.

As long as we calculate the elasticity between two prices that are not the end points of either lines it looks like the steeper line will give us a lower elasticity. That is why economists sometimes say the steep line has inelastic demand and the flat line has elastic demand. 

But the next graph is different and I give an example of the elasticity being the same for two prices that are not the end points.

For the steeper line elasticity is 

(3 - 2)/(3 + 2) times (8 + 12)/(8 - 12) = (1/5)*(20/8) = (1/5)*(5/2) = 5/10 = 1/2

For the flatter line elasticity is 

(6 - 4)/(6 + 4) times (8 + 12)/(8 - 12) = (2/10)*(20/8) = (1/5)*(5/2) = 5/10 = 1/2

So notice that we use prices that are both between the end points of both demand lines yet the elasticity is the same on each line for a given price decrease. This is different than in the first graph. Maybe that is because these two lines don't cross.