Question

Coyote, Inc. sells in two markets, Market A and Market B.  Demand in Market A is Q_A = 80 - 2P_A and demand in Market B is Q_B = 120 - 4P_B.   For any given output Q, Coyote's cost of production C(Q) = 50 + 10Q + 0.125Q². If Coyote is able to practice 3rd degree price discrimination between these markets its maximum profit will equal ________.

Answer 1

Total output=Q=QA+QB

C(Q)=50+10*Q+0.125*Q^2

Marginal Cost=MC=dC(Q)/dQ=10+0.25Q=10+0.25QA+0.25QB

In case of market A

Demand is given by

QA=80-2PA

On rearranging we get

2PA=80-QA

PA=40-0.5QA

TRA=PA*QA=40*QA-0.5QA^2

Marginal Revenue from market A=MRA=dTRA/dQA=40-QA

Set MRA=MC for profit maximization

40-QA=10+0.25QA+0.25QB

30-1.25QA=0.25QB

QB=120-5QA

In case of market B

Demand is given by

QB=120-4PB

On rearranging we get

4PB=120-QB

PB=30-0.25QB

TRB=PB*QB=30*QB-0.25QB^2

Marginal Revenue from market B=MRB=dTRB/dQB=30-0.5*QB

Set MRB=MC for profit maximization

30-0.5*QB=10+0.25QA+0.25QB

20-0.75QB=0.25QA

Set QB=120-5QA

20-0.75*(120-5QA)=0.25QA

20-90+3.75QA=0.25QA

3.5QA=70

QA=20

QB=120-5*QA=120-5*20=20

PA=40-0.5QA=40-0.5*20=$30

PB=30-0.25QB=30-0.25*20=$25

Q=QA+QB=20+20=40

Total Revenue=TR=PA*QA+PB*QB=30*20+25*20=$1100

Total Cost=TC=50+10*Q+0.125*Q^2=50+10*40+0.125*40^2=$650

Profit=TR-TC=1100-650=$450