Bertrand Duopoly

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Bertrand Duopoly
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We have seen that Oligopoly is a situation where
there are just a few firms. In this situation
each firm understands that the outcomes of its
actions are also influenced by the actions of the
other firms. In a Cournot duopoly the firms
competed on quantity. There we saw that the
output level and price of the firms fell
somewhere between the monopoly level and the
competitive level. In the Bertrand model firms
will compete on price. It seems to me firms in
the computer business compete on price. In other
situations firms may compete on features of the
product. Lets turn to the Bertrand model.
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Bertrand Duopoly Some assumptions of the
model If the two firms charge the same price
each will get half of the market demand at that
price. If one firm charges more than the other,
even just a little bit, then the one with the
higher price will sell nothing and the one with
the lower price will have all the demand at that
price. Each firm wants to maximize its
profit. Lets say the demand in a market is Q
100 5P And say the marginal cost for each firm
is 2.
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Lets see what the result would be if there was
only one firm in this market. With Q 100 5P P
20 - .2Q MR 20 -.4Q and with MR MC for
profit maximization 20 - .4Q 2, so Q 18/.4
45 and then P 11. On the next slide lets
explore the demand from firm 2s perspective. We
will want to remember the assumptions we made on
a previous slide.
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Firm 2s perspective on the demand it will
have Q2 0 if P2 gt P1 and profit 0. Q2
.5(100 5P2) if P2 P1 and profit P2Q2
MCQ2 (P2 MC)Q2
(P2 MC).5(100 5P2) Q2 (100 5P2) if P2 lt
P1 and profit P2Q2 MCQ2 (P2
MC)Q2 (P2 MC)(100 5P2) Lets
say both charge the monopoly price of 11. Firm
two would then have Q2 .5(100 5(11)) 22.5
and profit (11-2)22.5 202.5 If
firm 2 charged just a little less than 11, say
10.99, when firm 1 charges 11, then firm 2 will
make Q2 100 5(10.99) 45.05 and profit will
be (10.99 2)45.05 404.9995
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Firm two finds it irresistible to not charge a
lower price here, when the other firm is charging
a monopoly price. Now, imagine that firm 1
charges less than its marginal cost of
production. Firm 1 would lose money. If firm 2
charged an even lower price firm 2 would lose
money. Firm 2 would be better off not producing
at all. So, when firm 1 has price lower than
marginal cost, firm 2 does not want to have a
lower price. If firm 1 has price at marginal
cost then firm 2 doesnt want to have a lower
price because it would lose money and it doesnt
want to have a higher price because it wont sell
anything. On the next slide I will have a summary
of firm 2s reactions given what firm 1 does we
will see the best price response for firm 2.
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Firm 2s best price response If P1 gt monopoly
price of 11, set P2 11 and sell monopoly
Q, 2(mc) lt P1 lt 11, set P2 P1 small amount,
sell all of mkt at P2, P1 2(mc), set P2
2(mc) and sell half of market at P2 2, 2(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 11, set P1 11 and sell
monopoly Q, 2(mc) lt P2lt 11, set P1 P2 small
amount, sell all of mkt at P1, P2 2(mc), set
P1 2(mc) and sell half of market at P1
2, 2(mc) gt P2 gt 0, set P1 gt P2 and sell
nothing.
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Lets check some possible solutions to see if
they qualify as a Nash Equilibrium. Is P1 11,
P2 10.99 a Nash Equilibrium? Firm 2 does not
want to change from P2 10.99 if firm 1 has P1
11, but firm 1 would want to change (what does
firm 1s best response from the previous slide
suggest?). Firm 1 would want P1 10.98 in this
case. But then firm 2 would want 10.97. This
would spiral downward. What about P1 1.5 and P2
1.6 as a Nash equilibrium? Firm 2 would not
want to change, but firm 1 would want something
like 1.61. But then firm two would want P21.62.
This would spiral upward.
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Is P12 and P22 a Nash equilibrium? Both would
not want to change so it is a Nash
Equilibrium. Wow, here with only two firms if
they compete on price the price gets driven to
MC. This is the competitive solution!