Question

Problem 2: Bertrand or Monopoly? How can you tell?

After your time in Collegeville, you move on to Adultville - a much bigger town. You are again the economic advisor to the mayor of Adultville. You see that in Adultville there is only one firm selling widgets, and that

Q = 25, P= 20, and ϵ = −2.

You also know (from your predecessor) that demand is linear,

Q = A − BP,

and that the technology for producing widgets in Adultville is linear, or that marginal costs are constant,

mc = c.

Also, there are no other costs of production, such as licensing fees. You are naturally concerned!

Part 1: Suppose you believe that the widgets seller is a monopolist. What is her markup? Specifically, what is P − mc?

Part 2: Suppose you believe that the widgets seller is a Bertrand competitor. Can you say something about the seller’s markup (P - mc)?

a. No, this gives no information on her markup

b. Yes, her markup is at least $6

c. Yes, her markup is $0

d. Yes, her markup is at most $12

e. Yes, her markup is at most $20

f. Yes, her markup is $10

Answer 1

Solution:

Elasticity = (dQ/dP)*(P/Q) where Q is quantity and P is price. Given the linear demand function: Q = A - B*P, we have

dQ/dP = -B

Elasticity = -B*(P/Q)

With P = 20, Q = 25, and elasticity = -2, this becomes: -2 = -B*(20/25)

So, B = 25*2/20 = 2.5

Then, Q = A - B*P

25 = A - 2.5*20

A = 25 + 2.5*20 = 75

Then, demand function becomes: Q = 75 - 2.5*P

a) Inverse demand function becomes: P = 30 - 0.4*Q

Total revenue, TR = P*Q = (30 - 0.4*Q)*Q

TR = 30Q - 0.4Q^2

Then, marginal revenue, MR = dTR/dQ

MR = 30 - 2*0.4Q

MR = 30 - 0.8Q

For a monopoly, profit maximization occurs where the marginal revenue equals the marginal cost. So, MC = MR = 30 - 0.8*25 = 10

Markup P - MC = 20 - 10 = 10

Shortcut: in a monopoly, we have the markup formula as follows: (P - MC)/P = 1/|e|

P - MC = P/|e|

P - MC = 20/|-2| = 20/2 = 10

b) If the seller were a Bertrand competitor, that is in case of price competition, the bertrnad competitors end up charging price same as their marginal cost (due to such price cut competition). So, with P = MC, P - MC = 0.

Thus, the correct option is (C).