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Maximising Profits under 3rd Degree Price Discrimination

We can use differentiation and similar methods as used in Maximising Maths to calculate the output levels firms should set in separate markets in order to maximise profits from Price Discrimination. We will also analyse the profit it makes from conducting price discrimination and compare this with what it would have made had it not conducted price discrimination. Note, when referring to price discrimination in this lesson we are referring to 3rd Degree, unless stated. To see the lesson on differentiation please click here.

If given the demand functions for both market x and market y we can calculate the Total Revenue functions (by multiplying the demand function by q), we can then add both of these functions to give the Total Revenue function for the whole market. If also given the total cost (which remember must be the same for both markets, otherwise price discrimination isn't occurring but the firm has different costs of production) then we can differentiate this to find the marginal cost, and we can also differentiate the Total Revenue function to find the marginal revenue. To find the profit-maximising output we would set MR equal to MC and rearrange to find q. We can then substitute this value of q into TR-TC to calculate the profit obtained at that output, which is the maximum output.

Example
TC = 10 + 2q 
(TCx = 10 + 2qx; TCy = 10 + 2qy)
Px = 16qx - 2q2x
Py = 8qy - q2y
The above is the demand function, to calculate the Total Revenue function we will multiply the above equations by q. We can see for the demand functions that the demand slope in Market X is twice as steep as in Market Y, making Market X the more inelastic market.
TRx = 16qx - 2q2x
TRy = 8qy - q2y
Therefore we can calculate the Total Revenue for both markets (X+Y) by simply adding the 2 functions.
Therefore TRTotal = 16qx - 2q2x + 8qy - q2y
Total Profit is equal to TR-TC:
(16qx - 2q2x + 8qy - q2y) - (10 + 2(qx + qy))
=> 14qx - 2q2x + 6qy - q2y - 10
To find the profit maximising output in each market (like normal) we will differentiate TRx and TRy and set these equal to the differential of TCx and TCy respectively.
d(TRx)/dq = 16 - 4qx
d(TCx)/dq = 2
16 - 4qx = 2
14 = 4qx
14/4 = qx
3.5 = qx
Therefore the optimal output level for Market X is 3.5 units
To calculate the price we substitute 3.5 back into the demand function:
P = 16 - (2*3.5)
P = £9
 
Therefore products should be sold at £9 in Market X and 3.5 units will be sold.

We will do the same for Market Y:
d(TRy)/dq = 8 - 2qy
d(TCy)/dq = 2
8 - 2qy = 2
6 = 2qy
qy = 3
Therefore the optimal output level for Market X is 3 units
To calculate the price we substitute 3 back into the demand function:
P = 8 - 3
P = 5
 
Therefore products should be sold at £5 in the more elastic Market Y and 3 units will be sold.
We can substitute qy and qx back into our Total Profit equation to work out the maximum profit made:
Total Profit = 14qx - 2q2x + 6qy - q2y - 10
Total Profit = (14*3.5) - (2*(3.5)2) + (6*3) - (32) - 10
Total Profit = £23.50

We can compare this total profit with the total profit if price discrimination hadn't been conducted.
To do this we have to set P equal to Px and to Py
P = 16 - 2qx
qx = 8 - 0.5P (Re-arrange equation P above)
P = 8 - qy 
qy = 8 - P (Re-arrange equation P above)

q = qx + qy
q = 8 - 0.5P + 8 - P
q = 16 - 1.5P

We can re-arrange this to find an equation in terms of Q
P = 16/1.5 - q/1.5
Total revenue is therefore P multiplied by Q
TR = 16/1.5q - 2/3*q2
TC = 10 + 2q
To find the maximum profit we can differentiate TR and TC to find the marginals:
dTR/dq = 16/1.5 - 4/3*q
dTC/dq = 2
And set them equal to calculate the optimal output:
16/1.5 - 4/3*q = 2
16 - 2q = 3
13 = 2q
q = 6.5
Therefore the optimal output is 6.5 units, to find the price we can substitute 6.5 into the demand function:
P = 16/1.5 - 6.5/1.5
P = £6.33
To find the maximum profit we would find TR-TC and substitute q into this:
Profit = TR - TC
Profit = 16/1.5q - 2/3*q2 - 10 - 2q
Profit = (16/1.5*6.5) - (2/3*6.52) - 10 - (2*6.5)
Profit = £18.17
£18.17 > £23.50 therefore by undertaking Price Discrimination the firm makes £5.33 more in profit.

Page last updated on 25/05/13
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