Question

2. Suppose a monopoly firm is allowed to price discriminate in 3 markets where the prices for the good in each market are given by: P1 = 63 - 401 P2 = 105-502 P3 = 75 - 6Q3 The cost of the output is (Q) = 20 + 15Q+Q? where: Q = Q1 + Q2 + Q3 a) Give the profit function for the firm. b) Find the FOC's and find the p*'s and Qo's that maximize profit c) Find the SOC's.

Answer 1

2.a) P₁=63-4Q₁

Then, total revenue TR₁=P₁*Q₁ = 63Q₁-4Q₁²

and MR₁=dTR₁/dQ₁ = 63-8Q₁

Similarly, P₂=105-5Q₂

Then, total revenue TR₂=P₂*Q₂ = 105Q₂-5Q₂²

and MR₂=dTR₂/dQ₂ = 105-10Q₂

Again, P₃=75-6Q₃

Then, total revenue TR₃=P₃*Q₃ = 75Q₃-6Q₃²

and MR₃=dTR₃/dQ₃ = 75-12Q₃

Given : TC = 20+15Q+Q²

Then, Marginal cost MC = dTC/dQ = 15+2(Q₁+Q₂+Q₃)

Now, profit function $\pi$ = TR₁+TR₂+TR₃-TC

b) Then, FOC's for profit maximization are d$\pi$/dQ₁ = 0 , d$\pi$/dQ₂ =0 and d$\pi$/dQ₃=0

or, d/dQ₁(TR₁-TC) = 0 , d/dQ₂(TR₂-TC) = 0 and d/dQ₃(TR₃-TC) = 0

or, MR₁=MC , MR₂=MC and MR₃=MC

Then, 63-8Q₁= 15+2Q₁

or, Q₁=4.8 and P₁=63-(4*4.8)=$43.8

Again, 105-10Q₂=15+2Q₂

or, Q₂=7.5 and P₂=105-(5*7.5)=$67.5

and 75-12Q₃=15+2Q₃

or, Q₃=4.3 and P₃=75-(6*4.3) = $49.2

c) SOC's are d²$\pi$/dQ₁² < 0, d²$\pi$/dQ₂² < 0 and d²$\pi$/dQ₃² < 0

or, d/dQ₁(dTR₁/dQ₁-dTC/dQ₁) < 0 , d/dQ₁(dTR₂/dQ₂-dTC/dQ₂) < 0 and d/dQ₃(dTR₃/dQ₃-dTC/dQ₃) < 0

or, d/dQ₁(MR₁-MC) < 0 , d/dQ₂(MR₂-MC) < 0 and d/dQ₃(MR₃-MC) < 0

Then, d/dQ₁(63-8Q₁-15-2Q₁)<0

or, -10<0 (is valid)

Again, d/dQ₂(105-10Q₂-15-2Q₂)<0

or, -12<0 (is valid)

and d/dQ₃(75-12Q₃-15-2Q₃)<0

or, -14<0 (is valid)

Thus, SOC's are fulfilled.