More 3rd degree discrimination and elasticity calculation

Q

P

5

20

3

8

If we have a point on a demand curve like Q = 3 and P = 5, then we can calculate the elasticity with the following method as seen on the next few slides.

Some folks who are really slick found out when you have a straight line demand curve you can calculate the elasticity at a point on the demand curve relatively easily. You need the P and Q values of the point for which you want to calculate the elasticity and you need the slope of the line.

Let’s look at the slope on the next slide.

Q

P

5

20

3

8

run

rise

We talk about the slope of the line as the rise over the run. This is a fraction rise/run.

One way to get the rise is to go from 0 to 20 and thus the rise is 20.

Similarly, to get the run go from 0 to 5 to get 5. The slope is then 20/5.

(The slope is really -20/5, but we ignore the minus sign)

The definition of the elasticity of demand as the percent change in Qd divided by the percent change in P simplifies along a straight line demand curve at a point to

P/Q times run/rise.

You will note I have run/rise and the slope is the rise over the run. So we need the inverse of the slope.

In our example we have P = 8, Q = 3, run = 5, and rise = 20, so

Elasticity at the point P=8, Q = 3 is

(8/3)(5/20) = 40/60 = 2/3

Before we had a monopolist that had two markets to sell its product in. The demands in each market were

P = 6 – Q and P = 8 – Q.

If the monopolist acts as a single price monopolist it would add the two demands together and have 1 market demand. To get this demand we first switch each separate demand to the “Q” form, like so:

Q = 6 – P and Q = 8 – P and so added together we get

Q = 14 – 2P, or P = 7 - .5Q. Then MR = 7 – Q and with MR = MC (and the MC before was a constant at 4), we have

7 – Q = 4, or Q = 3 and the single price is 5.5

Review

Demands separately Demands together

P = 6 – Q (MKT1) P = 7 - .5Q

P = 8 – Q (MKT2)

If single price monopoly P = 5.5 and Q = 3. Note at this price the separate quantity demanded amounts are 0.5 and 2.5 for a total of three. The revenue is 5.5(3) = 16.5

If the firm discriminates we (as we saw before) have in

MKT 1 Q = 1 and P = 5 for revenue = 5, and in

MKT2 Q= 2 and P = 6 for revenue = 12. Thus, on the 3 units sold the firm is better off selling in the separate markets at separate prices because revenue is 17 total. (Would this always be the case?)

P

6

Q6

5.5

0.5

P

Q8

8

5.5

2.5

MKT1 MKT2

Here I show the two separate markets, but after we have treated them as 1 and found the market price of 5.5.

The elasticity in MKT1 is (5.5/.5)(6/6) = 11 and the elasticity in MKT2 = (5.5/2.5)(8/8) = 2.2.

The demand in both markets is elastic, but in MKT2 it is relatively more inelastic. Note when the firm acts as a 3rd degree price discriminator it raises the price in the relatively inelastic market and lowers the price in the relatively elastic market.