Question

Q5. A monopolist has the cost function C(Q) = m*Q +k*Q2 where m and k are parameters. It faces two types of consumers, A and B, with the following demand curves for its product: PA =60–3*QA PB =80–2*QB For (a)—(c) below, assume Assume m=30 and k=0.

(a) [4] What price will the monopolist charge under uniform pricing?

(b) [4] What prices will the monopolist charge if it can price discriminate?

(c) [4] How much higher is the monopolist’s profit in (b) than in (a)? Now, for part (d), assume m=30 and k=3.

(d) [10] What prices and quantities will the monopolist choose under price discrimination, and under uniform pricing?

Answer 1

a) Under uniform pricing,

Pa = Pb

and Q = total output = Qa + Qb

We have,

Qa = 20 - 1/3 P

and

Qb = 40 - 1/2 P

=> Q = Qa + Qb = 60 - 5/6 P

or P = 72 - 6/5 Q

=> Total Revenue, TR = P*Q = 72Q - 6/5 Q2

=> Marginal Revenue, MR = dTR/dQ = 72 - 12/5 Q

We know,

Total Cost = mQ + k Q2

or Total Cost, C = 30 Q (at m=30 and k=0)

=> Marginal Cost, MC = dC/dQ = 30

At Equilibrium,

MR = MC

=> 72 - 12/5 Q = 30

=> Q = 17.5 units

From the demand function,

P = $51

Profit = TR - TC = $367.5

b) For Type A consumers

Pa = 60 - 3Qa

=> Total Revenue, TRa = Pa*Qa = 60Qa - 3Qa2

=> Marginal Revenue,MRa = dTRa/dQa = 60 - 6Qa

and Total Cost, TCa = 30Qa

=> Marginal Cost, MCa = dTCa/dQa = 30

At Equilibrium,

MRa = MCa

=> 60 - 6Qa = 30

=> Qa = 5units

From the demand function

Pa = $45

Profita = TRa - TRb = $75

For type B

Pb = 80 - 2Qb

=> Total Revenue, TRb = Pb*Qb = 80Qb - 2Qb2

=> Marginal Revenue,MRb = dTRb/dQb= 8b - 4Qb

and Total Cost, TCb = 30Qb

=> Marginal Cost, MCb = dTCb/dQb = 30

At Equilibrium,

MRb = MCb

=> 80 - 4Qb = 30

=> Qb= 12.5units

From the demand function

Pb= $55

Profitb = TRb - TCb = $312.5

c) Profit when it sets a uniform price = $367.5

Profit when it price discriminates = Profita + Profitb = 312.5 +75 = $387.5

Therefore, the monopolist earns $20 more profit if it uses price discrimination

d)Under uniform pricing,

Pa = Pb

and Q = total output = Qa + Qb

We have,

Qa = 20 - 1/3 P

and

Qb = 40 - 1/2 P

=> Q = Qa + Qb = 60 - 5/6 P

or P = 72 - 6/5 Q

=> Total Revenue, TR = P*Q = 72Q - 6/5 Q2

=> Marginal Revenue, MR = dTR/dQ = 72 - 12/5 Q

We know,

Total Cost = mQ + k Q2

or Total Cost, C = 30 Q + 3Q2 (at m=30 and k=3)

=> Marginal Cost, MC = dC/dQ = 30 + 6Q

At Equilibrium,

MR = MC

=> 72 - 12/5 Q = 30 + 6Q

=> Q = 8.75 units

From the demand function,

P = $61.5

Profit = TR - TC = 8.75*61.5 - 3*8.75*(10+8.75)

=> Profit = $45.9375

For type A

Pa = 60 - 3Qa

=> Total Revenue, TRa = Pa*Qa = 60Qa - 3Qa2

=> Marginal Revenue,MRa = dTRa/dQa = 60 - 6Qa

and Total Cost, TCa = 30Qa + 3Qa2

=> Marginal Cost, MCa = dTCa/dQa = 30 + 6Qa

At Equilibrium,

MRa = MCa

=> 60 - 6Qa = 30 + 6Qa

=> Qa = 2.5 units

From the demand function

Pa = $52.5

Profita = TRa - TRb = 2.5*52.5 - 3*2.5*(10+2.5) = $37.5

For type B

Pb = 80 - 2Qb

=> Total Revenue, TRb = Pb*Qb = 80Qb - 2Qb2

=> Marginal Revenue,MRb = dTRb/dQb= 8b - 4Qb

and Total Cost, TCb = 30Qb + 3Qb2

=> Marginal Cost, MCb = dTCb/dQb = 30 + 6Qb

At Equilibrium,

MRb = MCb

=> 80 - 4Qb = 30 + 6Qb

=> Qb= 5units

From the demand function

Pb= $70

Profitb = TRb - TCb = 70*5 - 3*5*(10+5) = $225

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