Solution 2

There are two ways to do this problem. One approach is to figure out what one person's CS will be and then to multiply that by 10 to account for the fact that there are 10 people involved. The first step down that road is to calculate each person's Q:

W2P = 72 - 6*Q
W2P = P
72 - 6*Q = 12
60 = 6*Q
Q = 10

Their CS will be the area of the triangle between their W2P curve and the price:

CS = (1/2)*10*(72-12)
CS = 300

Multiplying by 10 to scale up to the whole market would give a CS of 3,000.

The alternative approach is to derive the market demand curve and then calculate the area between it and the $12 price. To build the market demand, start by rearranging the W2P equation to give the quantity demanded by an individual, Qi, as a function of P:

W2P = P = 72 - 6*Qi
6*Qi = 72 - P
Qi = (1/6)*(72 - P)

Since there are 10 people, the market demand, Qm, will be 10 times the individual demand:

Qm = Q1 + Q2 + Q3 + ... + Q10 = 10*Qi
Qm = 10*(1/6)*(72-P)
Qm = 120 - (10/6)*P

Solving for Qm:

Qm = 120 - (10/6)*12
Qm = 100

To find the total consumer surplus, we need to know where the demand curve hits the vertical axis. We can find that by looking for the price at which Qm would be driven to zero:

Qm = 120 - (10/6)*P
0 = 120 - (10/6)*Pmax
120 =(10/6)*Pmax
72 = Pmax

With luck, that's not surprising: if each person's maximum W2P is $72, then the overall market's maximum W2P should be $72 as well (since that's the maximum any individual is willing to pay).

Calculating the area under the market demand curve:

CS = (1/2)*100*(72-12)
CS = (1/2)*100*60
CS = 3,000

The methods are equivalent: they give identical results. Which one is easier to use will depend on the context of the problem.