Solution Set-Up

The following table lists what we know and what we want to determine:

KNOWN
DETERMINE
 Producer's surplus = 108,900 Consumer's surplus = 108,900/5 = 21,780

Supply curve

Demand curve, D(q), is linear.

1) Equilibrium point

2) Demand Curve Equation

Since the demand curve is linear, then it has the form

and because this linear curve will be true for all q and p, then it will, in particular, hold for the point (q', p'):

 (1)

where m is the slope and b is the y-intercept of the line.

Since the supply curve S(q) is known for all q, we have

 (2)

The consumer's surplus (21,780) is the area of the triangle shown shaded in the graph below
.

Since the area of a triangle is (base x height)/2, we obtain

 (3)

The producer's surplus of 108,900 is the area shown shaded in the graph below

and it results in the equation

 (4)

Now we are ready to solve Sam's problem. Equations (1) through (4) give you 4 equations in the 4 unknowns m, b, p' and q'. After we have solved this system of equations, the equation of the linear demand curve can be determined using the values for m and b. In addition the equilibrium point will be specified by (q', p').