TC = 10 + 2q

(TC_{x} = 10 + 2q_{x}; TC_{y} = 10 + 2q_{y})

P_{x} = 16q_{x} - 2q_{}^{2}_{x}

P_{y} = 8q_{y} - q^{2}_{y}

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.

TR_{x} = 16q_{x} - 2q^{2}_{x}

TR_{y} = 8q_{y} - q^{2}_{y}

Therefore we can calculate the Total Revenue for both markets (X+Y) by simply adding the 2 functions.

Therefore TR_{Total} = 16q_{x} - 2q^{2}_{x} + 8q_{y} - q^{2}_{y}

Total Profit is equal to TR-TC:

(16q_{x} - 2q^{2}_{x} + 8q_{y} - q^{2}y) - (10 + 2(q_{x} + q_{y}))

=> 14q_{x} - 2q^{2}_{x} + 6q_{y} - q^{2}_{y} - 10

To find the profit maximising output in each market (like normal) we will differentiate TR_{x} and TR_{y} and set these equal to the differential of TC_{x} and TC_{y} respectively.

d(TR_{x})/dq = 16 - 4q_{x}

d(TC_{x})/dq = 2

16 - 4q_{x} = 2

14 = 4q_{x}

14/4 = q_{x}

3.5 = q_{x}

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(TR_{y})/dq = 8 - 2q_{y}

d(TC_{y})/dq = 2

8 - 2q_{y} = 2

6 = 2q_{y}

q_{y} = 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 q_{y} and q_{x} back into our Total Profit equation to work out the maximum profit made:

Total Profit = 14q_{x} - 2q^{2}_{x} + 6q_{y} - q^{2}_{y} - 10

Total Profit = (14*3.5) - (2*(3.5)^{2}) + (6*3) - (3^{2}) - 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 P_{x} and to P_{y}

P = 16 - 2q_{x}

q_{x} = 8 - 0.5P (Re-arrange equation P above)

P = 8 - q_{y }

q_{y} = 8 - P (Re-arrange equation P above)

q = q_{x} + q_{y}

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*q^{2}

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*q^{2} - 10 - 2q

Profit = (16/1.5*6.5) - (2/3*6.5^{2}) - 10 - (2*6.5)

Profit = £18.17

£18.17 > £23.50 therefore by undertaking Price Discrimination the firm makes £5.33 more in profit.