Question

1. A monopolist produces good y at a cost c(y) = 10y, so that marginal cost is a constant €10 per unit. Two distinct groups of consumers, A and B, have demands for y as follows:

            y_A(p_A) = 120 - p_A

            y_B(p_B) = 200 - p_B

1. First assume that the firm can practice third-degree price discrimination and so can set a different price for each group, p_A and p_B. Write the inverse demand functions for each group of consumers. Find the firm's optimal choice of y_A and y_B, and the associated prices for each group, p_A and p_B.
2. Now assume that the firm is not able to practice price discrimination and must set one price p for the whole market. Write total demand for the market as a whole. Write the inverse demand curve for the market as a whole. Find the firm's optimal choice of y, and the associated price p. Show that the firm does better under price discrimination as in 3.1 than under a single price as in 3.2.

Answer 1

A.... From question    TC =10Y MC =10 ( Y is quantity of output) YA= 120—PA or PA=120—YA and YB =200—PB or PB =200—YB TRA( total revenue from market segment A) =PA.YA Or TR​​​​​A = 120YA–YA2 or MRA =d(TRA)/YA =120 —2YA   

And TRB = PB . YB   = 200 YB —YB2 OR MRB = 200—2YB

Now, at equilibrium in' A' market segment MRA= MC or 120—2YA=10 or YA=55 so optimal quantity =55    Here optimal price =120 —55=65 (put value in A's demand.fun.)  

Now, at equilibrium in the market segment B MRB=MC or 200—2YB=10 or YB = 95 (optimal qu.) Here optimal price P= 200—95 =105   

B.......Now without price discrimination

Demand function (YA+YB)=(120–P)+(200—P) ; PA=PB Y=320—2P   

Inverse demand function 2P=320—Y or P=160 —0.5Y

Total revenue (TR) =P × Y = 160Y —0.5Y2 Marginal revenue (MR) = d(TR)/dy = 160 —Y Now at equilibrium of the firm

MR =MC or 160—Y = 10 or =150(equilibrium q) now,   equilibrium price P= 160—0.5×150 =85