Cournot-Bertrand Nash Equilibrium - I have found the Cournot but I don't understand how to find the Bertrand.

Question

There are two firms producing gizmos, the demand for gizmos is:

p = 120 - y

They have different marginal costs

Firm 1: MC1 = 15 and Firm 2: MC2 = 30

Therefore, because of the different MCs, the best-response functions will not be symmetric.

I have calculated the best-response functions using the Cournot method to be:

y1* = 105/2 - y2/2 and y2* = 45 - y1/2

Solving the two equations yields:

y1* = 40 and y2* = 25

Now the part I don't understand.

Suppose that the firms choose prices instead of quantities. What are the Bertrand equilibrium prices? How much does each firm earn in the Bertrand equilibrium?

I would really like to understand how the prices are determined. All of the examples assume symmetric best-response functions, but that is not the case here.

Thank you very much for any help you can offer.

Answer 1

Cournot (asymetric because MC1≠MC2):

P=120-(A+B)

MC1=15 ; MC2=30

TR1=P*A=120A-A²-BA ; TR2=P*B=120B-B²-BA

MR1=(TR1)' = 120-2A-B ; MR2=(TR2)' = 120-2B-A

MR=MC ; MR1=MC1 ; MR2=MC2

120-2A-B=15 ; 120-2B-A=30

105-2A=B ; 90-2B=A

105-180+4B=B

3B=75 ; B=25 ; A=40 ; P=55

Bertrand (asymetric because MC1≠MC2):

No capacity (output) limitations for firms.

Q=120-P

Profit¹=(p¹-MC¹)*q¹

Profit²=(p²-MC¹)*q²

....................| 0  ;  if P¹>P² (because second firm will take whole market)

Profit¹(p¹;p²)= | (P¹-15)*(60-P¹/2)  ;  if P¹=P²

....................|  (P¹-15)*(120-P¹)  ;  if P¹<P²

....................| 0  ;  if P²>P¹ (because first firm will take whole market)

Profit²(p¹;p²)= | (P²-30)*(60-P²/2)  ;  if P¹=P²

....................|  (P²-30)*(120-P²)  ;  if P²<P¹

Profit→MAX => (Profit)'=0

P¹=P²

(P¹-15)*(60-P¹/2)=60P¹-900-P¹^2/2+7.5P...

(Profit¹)'=67.5-P¹=0

67.5=P¹=P²

P¹=P²

(P²-30)*(60-P²/2)

(Profit²)'= 75 - P²=0

P²=P¹=75

P¹<P²

(Profit¹)'=135 - 2 P

P¹=67.5

P²<P¹

(Profit²)'=150 - 2 P

P²=75

Actually second firm will loose Bertrand competition because of higher marginal costs, so it will be pressed to follow first firm for price P¹=P²=67.5 . Market quantity will be 52.5 and each firm's Q¹=Q²=26.25

Profit²=(67.5-30)*26.25=984.375

Profit¹=(67.5-15)*26.25=1378.125

But I'm not so sure that I remember Bertrand competition model right way, so better double-check because my solution may be wrong.