1)
Since the Bertrand duopoly model with differentiated products has been
analyzed. The case of homogeneous products yields a stark conclusion.
Suppose that the quantity that consumers demand from firm i is a-pi
when pi<pj (i&j are small subscript here), 0 when pi>pj, and (a-pi)/2
when pi=pj. Suppose also that there are no fixed costs and that
marginal costs are constant at c, where c<a. Show that if the firms
choose prices simultaneously, then the unique Nash equilibrium is that
both firms charge the price c.

2)
Recall the static Bertrand duopoly model(with homogeneous products)
from above question,1): the firms name prices simultaneously; demand
for firm i's product is a-pi if pi<pj, is 0 if pi>pj, and is (a-pi)/2
if pi=pj; marginal costs are c<a. Consider the infinitely repeated
game based on this stage game. Show that the firms can use trigger
strategies(that switch forever to the stage-game Nash equilibrium
after any deviation) to sustain the monopoly price level in a
subgame-perfect Nash equilibrium if and only if Gamma>=1/2.

vitaminc-ga,

This question is going to be tough without the aid of diagrams. But
I'll still give it a go.

Question 1:

The demand function for firm i is as follows :

         { a - pi        pi < pj
Di(pi) = { (a-pi)/2      pi = pj
         { 0             pi > pj

the profit function (which is maximized by firm i) is :

(pi - c)(a - pi)      if pi < pj
(pi - c)(a - pi)/2    if pi = pj
0                     if pi > pj

If pj > a then the profit function looks like an inverted parabola
intersecting the x axis at pi = c and again at pi = a > c.

 ^ Profit
 |   ___
 |  /   \
-+-/-----\---->
  /c     a\   pi

If this were the case firm i would choose pi = (a + c)/2

(Impressive graphics, huh?)

If c < pj <= a the diagram looks like :

 ^ Profit
 |   __
 |  /  | 
-+-/---+------>
  /c   |  a   pi
       pj

In this case, the firm would choose pi just below pj.

If pj <= c then the profit is 0 always and any price is optimal.

This holds symmetrically for firm j.

Given that firm i will always play pi < pj if pj > c, it is never an
equilibrium strategy for the firm to play pj > c. Firm j will be
enticed to play pj < pi. etc.

The only potential equilibrium that can then hold would be for pj = c,
in which case the optimal strategy for firm i is to use any price.
However by symmetry, pi = pj = c gives neither firm the incentive to
change.

The definition of a Nash equilibrium being one where neither party has
a positive incentive to unilaterally change his price confirms that by
the above analysis pi = pj = c is the unique equilibrium.

Question 2 :

Consider the infinite stage repeated game with payoff function :
Pit = (pit - c)(a - pit)      if pit < pjt
Pit = (pit - c)(a - pit)/2    if pit = pjt
Pit = 0                       if pit > pjt

at each stage t.

Consider the following symmetric stragegy :

Play pit = pjt = x > c in every period
If pit/pjt is not played then play pjs/pis = c for s = t+1, t+2 ,...

Assume that the cumulated payoff to each firm is :

sum{t=0 to infinity} of Pit*gamma^t

On the equilibrium path this is:
.5(x-c)(a-x) * 1/(1-gamma) > 0

Suppose there is a deviation by player i at time s. Instead of playing
x, they play y < x.

Their cumulated payoff is :
sum(t=0 to s-1) of .5(x-c)(a-x) * gamma^t + (y-c)(a-y) * gamma^s +
sum(t=s+1 to infinity) of 0
= .5(x-c)(a-x)*((1 - gamma^s)/(1-gamma)) + (y-c)(a-y) * gamma^s

For this strategy to be an equilibrium, we must have that :
.5(x-c)(a-x) * 1/(1-gamma) > .5(x-c)(a-x)*((1 - gamma^s)/(1-gamma)) +
(y-c)(a-y) * gamma^s for all pis < x and for all s.

Rearranging :

(gamma^s/(1-gamma))*.5(x-c)(a-x) > (gamma^s)*(y-c)(a-y)

or .5(x-c)(a-x) > (1-gamma)*(y-c)(a-y)

This is only true for all x,y if 1-gamma < .5 ie. if gamma >= .5

QED

Let me know if that makes sense.

calebu2-ga

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Clarification of Answer by calebu2-ga on 06 Nov 2002 14:31 PST

vitaminc-ga,

By the way - I have no clue about your other game theory questions
that you have asked! Just thought I'd mention in case you were hoping
that I was working on the others.

Regards

calebu2-ga

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Request for Answer Clarification by vitaminc-ga on 06 Nov 2002 20:09 PST

hi, calebu2-ga,

To work on the others......sure, why not?
But what do you mean that you have no clue about my other game theory
questions I have asked? Which one?

And after a quick scanning, i have a quesiton here.
Is ^ an exponent?(eg.gamma^t)
To see if I have further question, just give me some time to absorb
these.
 
Cheers :)

vitaminc-ga

---

Clarification of Answer by calebu2-ga on 07 Nov 2002 04:43 PST

Yes, ^ means exponent, or "to the power of".

What I meant by not being able to answer the other questions was your
questions with id numbers : 97923, 100006 and 100008. I'm not sure
that I can answer those ones (I'm better at the questions with actual
numbers in them :), so you'll have to hope that rbnn-ga or one of the
other researchers can answer them.

But you're all set with this one. Good Luck

Regards

calebu2-ga

P.S. If anything else is unclear feel free to request clarification

Thanks a lot!!!
By the way, the graph is really impressive. It is cute though and does
make me laugh. Unbelievable one. Hahaa :D