Stackelberg Duopoly - May 20

[Amit Goyal]

May20_Stackelberg.png

[AJ]

I got...

The firm that goes first will produce 8 units..
Other firm will produce 4 units...
price= 4

profits are.. 23 and 7 respectively

[aditi5000]

I'm getting
q of leader = 15/2
q of follower = 15/4
total Q = 45/4
P = 19/4
profit of leader = 225/8
profit of follower = 209/16

[anon_econ]

I'm getting the same answer as AJ's..
aditi how have u done this?

[aditi5000]

I may be wrong but this is the procedure I followed
- the follower decides his output keeping the leaders in mind so we get q2=(15-q1) /2 (simply MR=MC)
- now the leader maximises profits
- Pi 1 = TR - TC = 16q1 - q1 squared -q1q2 - q1
now put the value of q2 into this equation, take derivative , put it =0
then you'll get the values I got...

[Chocolate Frog]

Aditi, the marginal cost is not one; it's zero. So q2 should be (16 - q1)/2. If you follow through, you'll get the same answer as mine, Vasudha's and AJ's. :)

[aditi5000]

OH GOD!! I read that 9 as a q!!! Hahaha, got the same answer finally.. Thanks Chocolate frog :)

[Chocolate Frog]

[Amit Goyal]

Are you sure the best response of firm 2 is (16 - q1)/2 always?
What if firm 1 produces 12 units, what is firm 2's best response? Is it (16 - 12)/2 = 2? Or something else?

[aditi5000]

if firm 1 produces 12 units,
firm 2's best response would be to produce 0 and exit the market..
because 4*2=8<9

[Amit Goyal]

Thats right. But there is a minor error in what you wrote. Maximum profits from producing positive level of output is 2*2 - 9 = -5 < 0 (and not 4*2 - 9). So firm 2's best response at q1 = 12 is not (16 - q1)/2 but 0. Now rethink about this problem. Ok first tell me what will be the best response function of firm 2?

[aditi5000]

firm 2's BR fn:
q2= (16- q1)/2 if q1 <12
   = 0              if q1 >=12

[anon_econ]

q2=(16-q1)/2 if q1<10. q2=0 if q1>=10.
if we put q2=(16-q1)/2 in firm 1's profit function, the optimal output is 8 which is <10. if we put q2=0 we still get q1=8 but that is not >=10. so the final outcome is only q1=8 and q2=4. that's how i did it. i could be making a mistake.

[anon_econ]

ohh this is wrong. if q2=0 and q1=10 then firm 1 makes a profit of 51. so firm 1 would produce 10 and firm 2 would produce 0.

[Amit Goyal]

Thats right Vasudha.
Stackelberg or Subgame perfect equilibrium of this game is:

Firm 2's strategy:
q(2) = (16 - q(1))/2 if q(1) < 10,
and 0 if q(1) ≥ 10

Firm 1's strategy:
q(1) = 10

And hence the outcome is (10, 0).

[anon_econ]

Thanks a lot. These questions r cool :)

[AJ]

OMG....
I didn't considered the second part of best response function....

This was cool....!!

[Chocolate Frog]

I'll join the chorus and say that this question is too cool.

[vaibhavR]

seconded

its so cool that its even below 0 K

[tim]

cn someone pls clarify this...??
wen v r asked to find out the outcome..we actually need to find out the best response of both the firms ?