Question 4: [35 points] There are two groups of consumers who have inverse deman

Question

Question 4: [35 points] There are two groups of consumers who have inverse demands p1 = 150 q1 and p2 = 30 0.5q2. The monopolist has cost function C(q) = 20q.

(a) [5 points] If the monopolist can choose a separate p1 and p2 for each group, what will they be?

(b) [10 points] If the monopolist can only charge a single price, what will it be? Do both groups purchase? If so, how much does each group buy? [Hint: Add up demands, not inverse demands.]

(c) [5 points] How do the prices p1 and p2 compare to the “uniform” price you found in part b?

(d) [5 points] Does the monopolist prefer to set a separate p1 and p2, or does the monopolist prefer to set a single price?

(e) [10 points] How much would group 1 consumers pay to have a ban on price discrimination? How much would group 2 pay? (Might be negative)

Explanation / Answer

aI For this section we can use inverse demand curve. So it is given that 1st group's demand is P1= 150-q1

Now revenue R1= P.Q= (150-q1)q1= 150q1-q21

Hence MR1= 150-2q1 (derivation of R)

the monopolist's cost function is given by C(q)= 20q1

Now m arginal cost, MR= 20 (derivation of cost)

Now equates MR1 and Mc

150-2q1=20

or, q1= 65

Now monopolist's price for 1st group is found by substituting q1=65 into demand price equation

P1=150-q1=85

Hence P1 for 1st group be 85

Inverse demand function of second group is given by, P2= 30-0.5q2

therefore total revenue is, R2= q2*P2= (30-0.5q2)q2=30q2-0.5(q2)2

MR2= 30-q2.

Equating MR2 and MC we obtain

30-q2=20

or, q2=10

we can get p2 by substituting q2=10 in demand equation of second group.

hence p2=30-0.5q2=25

Therefore p2 of second group be 25.

b) we have two demands for two seperate groups. If we consider that monopolist charges a uniform price then both demand include p as their price. Moreover monopolist considers these two demands as a whole. Thus we have to sum these two demands in order to get marginal revenue.

inverse demand of first group is p1=150-q1

or, q1=150-p1

Similarly for second group , q2=60-2p2

Now adding q1 and q2 we have

q=q1+q2=150-p1+60-2p2

then assume that p1=p2=p

now,

q=210-2p

Again solve for p we have p=105-0.5q

Now R=(105-0.5q)q= 105q-0.5q2

MR=105-q

We know that MC= 20

Now quating MR and MC we have

105-q=20

or, q=85

Now, p=105-0.5*85 (substituting the value q=85 in demand price equation)

Hence p=62.5

Therefore monipolist's uniform price be 62.5.

At this uniform price second group can not buy, but 1st group obviously buy becasue they will pay lesser than the seperate price.

at this price 1st group will buy the amount of q1= 150-62.5=87.5

d) Monopolist price determination depends on its profit

If p1=85 and q1= 65

Then total revenue R1= p1*q1=5525, and total cost= 20*65=1300

Then profit from 1st group= 5525-1300=4225

in the other hand if p2=25 and q2=10 then total revenue R2=250, and total cosst=20*10=200

Then profits from second group= 250-200=50

Therefore total profits from both groups would be (4225+50=) 4275.

Now consider uniform price, p= 62.5 and quantity, q=85. but only first group will buyand quantity demanded =87.5

Now total revenue= p*q=62.5*87.5=5731.25

and total cost=20*q=20*87.5=1750

Now profts= 5731.25-1750=3981.25

Hence it is seen that the profit is lower at uniform price. So monopolist always prefer to set a seperate price for both customers.

e) seperately first group pays 85 but the uniform price is 62.5. Now first group will pay (62.5-85=)22.5 to ban price discrimination.

on the other hand seperately second group pays 25 and uniform price is 62.5. Now second group will pay (62.5-25=) 37.5.

Related Questions

Navigate

• Browse (All)
• Browse Q
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.