Duo p Bertrand

7/30/2019 Duo p Bertrand

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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 thelower 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 = 1005P

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 = 1005P

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 madeon a previous slide.


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Firm 2s perspective on the demand it will have

Q2 = 0 if P2 > P1 and profit = 0.

Q2 = .5(1005P2) if P2 = P1 and profit = P2Q2MCQ2

=(P2MC)Q2

=(P2MC).5(1005P2)

Q2 = (1005P2) if P2 < P1 and profit = P2Q2MCQ2=(P2MC)Q2

=(P2MC)(1005P2)

Lets say both charge the monopoly price of 11. Firm two would

then have Q2 = .5(1005(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 = 1005(10.99) = 45.05 and

profit will be (10.992)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 ofproduction. 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 reactionsgiven what firm 1 doeswe will see the best price response for

firm 2.


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Firm 2s best price response

If

P1 > monopoly price of 11, set P2 = 11 and sell monopoly Q,

2(=mc) < P1 P1 >= 0, set P2 > P1 and sell nothing.

Firm 1s best price response is similar

If

P2 > monopoly price of 11, set P1 = 11 and sell monopoly Q,2(=mc) < P2 P2 >= 0, set P1 > 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. Thiswould 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 P2=1.62. This would spiralupward.


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Is P1=2 and P2=2 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 pricegets driven to MC. This is the competitive solution!

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Duo p Bertrand