Question

Demand for propane is given by D(x) = 6.5 − 0.25x, and supply is given by S(x) = 2.1 + 0.15x, where x is in gallons per month customer and D(x) and S(x) are dollars per gallon. Find the followings:  

(a) Equilibrium point (X_e, P_e)  

(b) Find the consumer surplus and the producer surplus at the equilibrium point.

(c) Assume a price ceiling of $3 per gallon of propane is imposed. Find the point (X_c, P_c)  

(d) Find the new producer surplus and the new consumer surplus at (X_c, P_c).

(e) Find the deadweight loss.

Answer 1

Let us plot the given Demand and Supply equations:-

Plotting these on a graph we have:-

Thus the equilibrium point is

(b) Now, we draw a horizontal line passing through the equilibrium point, and then

The area of red triangle is the consumer surplus, and the area of the blue triangle is called the producer surplus.

So, for consumer surplus, the height of the triangle is 6.5 - 3.75 = 2.75 and the base is 11, so the area is given by

Now, for the producer surplus, the height of the triangle is 3.75 - 2.1 = 1.65, and the base is again, 11, so the area is given by

(c) Now, we introduce a price ceiling of $3 per gallon. This is equivalent to drawing a horizontal line passing through (0,3). So we have, drawing this line in green

Thus, as the price ceiling is lower than our equilibrium price, we choose the former point of intersection, (6,3). That is, the required values are

(d) and (e)

Now, the consumer surplus, deadweight loss, and the producer surplus after the introduction of a price ceiling is given by the yellow, red and orange areas

So, the trapezium giving us the consumer surplus has distance between the parallel lines = 6, and the two parallel lines have lengths 6.5 - 3 = 3.5 and 5 - 3 = 2, respectively. Hence

and the triangle giving us the producer surplus has base = 6 and height = 3 - 2.1 = 0.9, so

and the triangle giving us the dead-weight loss has base = 5 - 3 = 2, and height = 11 - 6 = 5, Hence

thus we have all the answers.