Bertrand Again

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Bertrand Again

• Here we study a situation known as capacity
  constraints.

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Say the demand in the market is Q 6000-60P.
Say we have two firms who will compete on price
as in the Bertrand duopoly situation and each has
a constant marginal cost 10. Lets see what the
result would be if there was only one firm in
this market. With Q 6000 - 60P P 6000/60
(1/60)Q MR 6000/60 (2/60)Q and with MR MC
for profit maximization 6000/60 (2/60)Q 10,
so Q (6000-600)/2 2700 and then P
55. Next, lets look at the best price response
for each firm in a Bertrand duopoly situation.
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Firm 2s best price response If P1 gt monopoly
price of 55, set P2 55 and sell monopoly
Q, 10(mc) lt P1 lt 55, set P2 P1 small
amount, sell all of mkt at P2, P1 10(mc), set
P2 10(mc) and sell half of market at P2
10, 10(mc) gt P1 gt 0, set P2 gt P1 and sell
nothing. Firm 1s best price response is
similar If P2 gt monopoly price of 55, set P1 55
and sell monopoly Q, 10(mc) lt P2lt 55, set P1
P2 small amount, sell all of mkt at P1, P2
10(mc), set P1 10(mc) and sell half of market
at P1 10, 10(mc) gt P2 gt 0, set P1 gt P2 and
sell nothing.
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Now, although we really havent mentioned it
explicitly, both firms recognize each others
best price response. So, each recognizes what
are the options for the other. For example, firm
2 recognizes that firm 1 has a best price
response. Plus firm 2 believes as a credible
position that firm 1 will want to have a lower
price than firm 2s if firm 2 makes the price,
say, 45 in our example. In the context of the
economic literature, it is said that firm 1
threatens to lower the price if firm 2 makes its
price 45, and firm 2 sees it as a credible
threat. Firm 1 see similar credible
threats. Economic actors must deal with credible
threats.
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So, we havent done anything new here yet except
introduce the notion of credible threats. We
just looked at a new problem. The Nash
equilibrium price for each will be 10 and in the
market Q will be 5400. Each will take half of
5400, for a Q of 2700 for each. Up to this point
we have assumed that each firm had the capacity
to meet the demand. But, lets now say that firm
1 can only make 1000 units and firm 2 can only
make 1400. When you look at the market demand,
2400 units could be sold at a price of 2400
6000 60P, or P (6000 2400)/60 60. How
does each firm view each others best price
response in the context of capacity constraints?
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Here is a new wrinkle in firm 2s thinking. If I,
firm 2, put a price of 60 on the product, then
from firm 1s best price response function I see
firm 1 will set a price of 55. But I do not view
it as credible because at a price of 55 it will
have too much demand. In other words, if you can
charge 1000 people 60 because they will pay it,
why charge them 55? So, in the face of capacity
constraints firm 2 does not see firm 1s best
price response function as credible. Firm 1 has
a similar view of firm 2s best price response
function. The authors suggest that in the face of
capacity constraints the Bertrand duopoly model
does not result in the competitive
solution. Next, lets look at how the authors add
to the discussion by considering residual demand
curves.
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The total demand we saw before was Q 6000
60P. If P 60 the Q 2400 and the firms could
each get their capacity amount. Another way to
see things from firm 2s perspective If firm 1
makes P 60 and takes 1000 customers, then my
residual demand will be Q 6000 1000 60P
5000 60P and if I, firm 2, make P60 I will
have Q 5000 3600 1400, my capacity
constraint. Lets see if firm 2 can do better by
taking the residual demand and act like a
monopolist with that demand. With Q 5000 60P,
P 5000/60 (1/60)Q and MR 5000/60
(2/60)Q. Then with MR MC, the best Q in a
perfect world with no capacity problems would be
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5000/60 (2/60)Q 10, so Q (5000 600)/2
2200 and the price would be (5000 2200)/60
46.67. So, firm 2 says, hey, if firm 1 does a
price of 60 I might as well do 60 as well because
if I try a price of 46.67 I would attract 2200
but can only serve 1400 so I might as well charge
those 1400 peopel 60 instead of 46.67. Firm 1
will similarly see its residual demand if firm 2
has P 60 as Q 6000 1400 60P. Thus P
4600/60 (1/60)Q and MR 4600/60 (2/60)Q and
with MR MC Q (4600 600)/2 2000 and P
(4600 2000)/60 43.33. Firm 1 will say why
charge 2000 people 43.33 when I can only serve
1000 of them. Just charge the 1000 people the
60. In the face of capacity constraints in
Bertrand model the competitive solution will not
result because it is not a credible solution.