Question

Suppose a stream is discovered whose water has remarkable healing powers. You decide to bottle the liquid and sell it. The market demand curve is linear and is given as follows: P = 30 – Q. The marginal cost to produce this new drink is $3. Answer the following questions:
a. What price would this new drink sell for if it is sold in a perfectly competitive market?
b. What is the monopoly price of this new drink?
c. What will be the price of this new drink if the industry is a Cournot duopoly?
d. What will be the price of this new drink if the industry is a Stackelberg duopoly?
e. What will be the price of this new drink if the industry is a Bertrand duopoly?

Answer 1

Answer:

Please find the detailed answer as follows:

Explanation:

1. Under perfect competition model, the profit maximising level of otuput is set where the P = MC

MC = 3 and P = 30 - Q

P = MC

30 - Q = 3.

Q = 27 units.

Thus P = 30 - 27 = $3.

2. Monopoly profit maximising level of output is where the MR = MC

P = 30 - Q

TR = P*Q

TR = (30 - Q)*Q

TR = 30Q - Q2

MR = \deltaTR/\deltaQ

MR = 30 - 2Q

MR = MC

30 - 2Q = 3

27 = 2Q

Q = 13.5 units.

P = 30 - 13.5 = $16.5

3. Under duopoly, Q = Qa + Qb

where Qa and Qb are the output share of firm A and firm B respectively.

Then, P = 30 - Qa - Qb.

for firm A :

TR = P*Q

TR = (30 - Qa - Qb)*Qa

TR = 30Qa - Qa2 - Qa*Qb

MR = \deltaTR/\deltaQa

MR = 30 - 2Qa - Qb

MR = MC

30 -2Qa - Qb = 3

or 27 - 2Qa = Qb..........................equation (i).

For firm B :

TR = P*Q

TR = (30 - Qa - Qb)*Qb

TR = 30Qb - Qa*Qb - Qb2

MR = \deltaTR/\deltaQ

MR = 30 - Qa - 2Qb

MR = MC

30 - Qa - 2Qb = 3

or 27 - 2Qb = Qa...............equation (ii).

substituting the value of equation (i) into equation (ii), we get,

27 - 2(27 - 2Qa) = Qa

27 - 54 + 4Qa = Qa

-27 = -3Qa

Qa = 9 units.

Plugging in this value to get Qb,

27 - 2*9 = Qb

Qb = 9 units.

Q = Qa + Qb = 9 + 9 = 18 units.

P = 30 - 18 = $12.

4. The standard reaction function of firm 2 is given as = a-Cb/2b - 1/2*Qa

P = 30 - (Qa + Qb)

where a= 30 b = 1 and C = 3.

Leader's output = (a + Cb - 2Ca)/2b

Leader's output = (30+3 - 3*2)/2 = 13.5 units.

Reaction function of firm B,

Qb = 30 - 3/2 = 13.5 - 1/2*13.5

Qb = 6.75 units.

P = 30 - (13.5 - 6.75) = $9.75