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Microeconomics 2
John Hey
The remaining lectures
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We have three weeks left this term:
This week chapters 31 and 32.
Next week chapters 33 and 34.
Final week?
I propose looking at the second half of the first specimen
paper, and general revision (questions in advance please).
Next term we have two slots: 13.15 Tuesday the 30th of
April and 13.15 Tuesday the 7th of May.
I propose to use them for the second specimen paper...
... and for telling you in what way I have strengthened the
examination for this year.
OK? Comments?
Game Theory
• In Chapter 29 we talked about games, in which two
players have to choose the value of some decision
variables, the values of which affect both players.
• We introduced the idea of a Nash Equilibrium in which
each is optimising given the decision of the other.
• We had our doubts about NE in general but in some
cases it seems reasonable.
• Do note that the NE depends heavily on the ‘rules of the
game’, particularly about sequentiality and repetition.
• This week’s tutorial reinforces these results.
• Today we will apply game theory in Duopoly.
Duopoly
• We deliberately keep things simple, but
we do not lose anything of interest.
• Consider a market in which there are
two identical sellers – two firms – Firm 1
and Firm 2, selling an identical good.
• Suppose the demand curve in the
market is given by:
• p = a – b(q1 + q2)
The Cournot Model
• Each firm chooses independently its
output.
• The price is determined by the demand
curve (recall that their output is identical).
• What outputs do the firms choose?
• Nash: each firm choosing profitmaximising output given the output of the
other firm.
Profits
• Let us denote the profit of firm 1 by π1.
• Suppose that its total cost function is given by:
C(q1) = cq1 (Note that this is linear and so constant returns to scale.)
• Hence its profits are given by:
• π1 = pq1 - cq1
= [a – b(q1 + q2)]q1 - cq1
• Assuming the same cost function for firm 2 its
profits are given by
• π2 = pq2 – cq2
= [a – b(q1 + q2)]q2 – cq2
Isoprofit Curves
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For firm 1 an isoprofit curve is given by:
π1.= constant
Hence
[a – b(q1 + q2)]q1 - cq1 = constant
Note that this is quadratic in q1 for given q2.
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For firm 2 an isoprofit curve is given by:
π2.= constant
Hence
[a – b(q1 + q2)]q2 – cq2 = constant
Note that this is quadratic in q2 for given q1.
Parameters in Maple file
• C(q) = 10q
(Note linear so Constant Returns to Scale).
• and hence the Marginal Cost is 10.
• Demand is p = 110 – q.
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So a = 110, b = 1 and c = 10.
• Hence Revenue = pq = 110q – q2
• and so Marginal Revenue = 110 – 2q.
• Hence the monopoly output (given by
MR=MC) is 50, and the monopoly price is 60.
• Competitive output (given by Price=MC) is
100, and the competitive price) is10.
Reaction Curve
(showing how a firm should optimally react to the decision of the other)
• If Firm 1 chooses its output to maximise its
profits given a level of output of Firm 2, we get:
• q1 = (a-c-bq2 )/2b
• …the reaction curve of Firm 1. Note that its
slope is negative. (We will see this graphically in Maple.)
• (To find, differentiate π1 = [a – b(q1 + q2)]q1 cq1 with respect to q1 and put the derivative
equal to zero.)
• Let’s go to Maple...
The Nash Equilibrium with quantity-setting
• The NE is given by the intersection of the two
reaction curves…
• Total output = 2(a-c)/3b
• With monopoly (marginal cost – marginal
revenue) output = (a-c)/2b
• With perfect competition (price equal to marginal
cost) output = (a-c)/b
• The output with a duopoly is between the
monopoly output and the competitive output.
• In an oligopoly with n identical firms, the NE has
a total output of n(a-c)/(n+1)b. Rises with n.
The Bertrand Model
• Each firm independently chooses its
price.
• The demand all goes to the firm with the
lowest price (recall that the firms
produce an identical product).
• What prices will the firms choose?
• Nash: each firm choosing optimal price
given the price of the other firm.
What happens with a duopoly
• Is very sensitive to the rules of the
game…
Summary
• With quantity-setting rules in the Nash Equilibrium the total
output with a duopoly is between the monopoly output and the
competitive output.
• A collusive outcome is better for both firms – but is unstable.
• For a firm it is better to be the leader.
• With price-setting the Nash Equilibrium has price equal to
marginal cost (and therefore like competition).
• Surplus is lost with quantity-setting but not with price-setting.
• After this week’s tutorial you might like to ask what happens if
the ‘game’ is repeated many times.
• Might some kind of (illicit?) agreement be reached?
• Who would enforce it?
• Legality of such agreements?
• Government regulation of duopolies/oligopolies?
Chapter 31
• Goodbye!