Third degree price discrimination output of goods

Hi there
I was studying third degree price discrimination and encountered an issue which I can't seem to get my head around.
On page 283 of the textbook (which contains the example of how third degree price discrimination operates), it says that:

"Because the demand curves are linear, the total output sold is the same under third-degree price discrimination as it is under uniform pricing: i.e. Q*=Qh+Ql."

What is the relevance of the demand curves being linear in the above statement? What would happen if one or both demand curves were not?
Thanks for any help!

 

Furthermore, I tried testing this out with real numbers and it didn't work.
I set Qh = 12 - 0.5P and Ql = 8 - 2P
Therefore MRh = 12 - P and MRl = 8 - 4P

I then set MC = 2

If the firm can price discriminate:
Firm maxes profits re H where MRh = MC
12 - P = 2
P = 10
Qh = 12 - 0.5(10) = 7

Firm maxes profits re L where MRl = MC
8 - 4P = 2
P = 1.5
Ql = 8 - 2(1.5) = 5

If the firm can't price discriminate:
Q*= Qh + Ql (P =< 4 because we can't have Ql < 0)
= (12 - 0.5P) + (8 - 2P) = 20 - 2.5P
=> MR* = 20 - 5P

Firm maxes profits where MR* = MC
20 - 5P = 2
P = 3.6
Q* = 20 - 2.5(3.6) = 11

But Qh + Ql = 12
12 <> 11 ???

 

Hi

If the demand curves are linear, there is a simple relationship between the demand curves and the MR curves, ie the MR curves are twice as steep, so the effect of a change in the MC will have a simple and predictable effect on the quantities demanded. For example, if every decrease in price by £1 led to an increase in demand in Market X of 2 units and in Market Y of 3 units, it would lead to an increase in the total demand by 5 units; also a decrease in MC by £1 would lead to an increase in the profit-maximising level of output of 1 unit in Market X, 1.5 units in Market Y and 2.5 units in the total market.

In your example, you need to rewrite the demand function as a function of Q, so that you can work out the MR function (which is the derivative of TR wrt Q).

So you had Qh = 12-0.5P and Ql = 8-2P.
Rewrite as P=24-2Q and P=4-0.5Q
Then TR = 24Q-2Q^2 and TR = 4Q-0.5Q^2
So MRh =24-4Q and MRl=4-Q

In this case, there is a problem in that the low-priced customers aren't interested until the price falls below 4, but the MR of the high-priced group alone is negative by that point, and bringing in the low-priced market isn't going to shift it back into positive, so I don't think this example will work.

However you might want to try this one, which was on a specimen exam paper:
Qx = 11-P and Qy=12.5-0.5P.
The MC = 5.

Good luck!

 

Thanks freddie! I think my error was that I took the MR curve as simply twice the slope of the demand curve without going through the motions of multiplying out. Thumbs up from me!