Question

Sarabeth Tucek is monopoly singer from New York City and has identified two types of customers that attend her concerts: adults (A); and non-adults (N). Adult demand is pA = 100 – qA and non-adult demand is pN= 50 – (1/2)qN, where p is the price per concert ticket, and q is the quantity of tickets demanded. Her marginal cost (MC = AC) of playing music to an additional patron is 10. Assume that Sarabeth practices third-degree price discrimination.

. Given the information above, now assume that Sarabeth must practice uniform pricing. 1.What is the profit-maximizing price and quantity? 2. What is the producer surplus in this market?

Answer 1

For calculating a uniform price, a combined demand curve and MR curve must be calculated first.

Combined demand curve can be calculated by horizontally summing up the quantities (qA & qN) for different price levels. It is shown in the following image. Case (A) shows the demand curve and MR curve for adults, (B) shows the same for non-adults and (C) shows the combined demand and MR curves.

The equilibrium quantity is calculated at the point where MR=MC, which is 85 units at which the equilibrium price is 38.33. This is calculated by estimating the equation for the lower segment of combined Demand curve which can be shown as follows:

After the kink at point (50,50), the equation for the segment can be calculated by using the points (50,50) & (200,0) which lie on the line. Equation for a line is given by y=c+mx, where c is the intercept and m is the slope. In our case the equation is p=c+mq. For calculating an equation, the slope is calculated first by using the formula,

m=(p1-p2)/(q2-q1), which in our case is, m = (50-0)/(50-200) = -1/3

For the equation, p-50= m.(q-50)

=> p-50= -1/3(q - 50)

=> 3p= 200 - q

As can be observed from the diagram that at MC=MR=10, q=85. Therefore, the price can now be calculated from the above derived equation which comes out to be 38.33.

Producer Surplus, which refers to the total benefit accruing to the producer by selling a specific amount of quantity, is shown by the shaded rectangle, highlighting the area above MC and below P* till q*.

Therefore, PS= (p*-MC)q* = 28.33 x 85 = 2408.05.