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### D

#### duopoly equilibrium

The following explanation of how the duopoly equilibrium quantities were derived is for those interested – learning all the computations is not necessary to complete the course.

The duopoly price and quantities are based on insights from the so-called Cournot duopoly model. According to this model, the profit-maximising equilibrium outcome for two identical firms – which we assume Jane and Emma to run because they have identical cost structures and sell a homogeneous product – is to produce equal quantities. These two quantities jointly clear the market, meaning that the price is lowered until all that is produced is sold.

The following explains how the equilibrium price and quantity were determined. Emma’s profit (Π)function is as follows:

ΠEmma = Revenue – Total cost =

P*QEmma– TCEmma =

(100 – QEmma– QJane)QEmma– (800 + 10QEmma) =

100QEmma– QEmma2– QEmma*QJane– 800 – 10QEmma =

90QEmma– QEmma2– QEmma*QJane– 800

Emma maximises her profit by (we treat QJane as a constant):

ΠEmma/QEmma = 90 – 2QEmma– QJane

90 – 2QEmma– QJane = 0

-2QEmma = -90 + QJane

QEmma = 45 – 0.5QJane

The final equation is Emma’s reaction function. It gives Emma’s best response to whatever quantity Jane chooses to produce (QJane). Because Jane’s firm is identical to Emma’s, her reaction function is QJane = 45 – 0.5QEmma.

Based on the reaction functions, we can solve the market clearing (equilibrium) quantity as follows:

QEmma = 45 – 0.5(45 – 0.5QEmma)

QEmma = 45 – 22.5 – 0.25QEmma

0.75QEmma = 22.5

QEmma = 30

Because Jane’s reaction function is a mirror image of Emma’s, the resulting quantity would be the same. Thus, in a Cournot duopoly equilibrium, Emma and Jane would choose to produce 30 casks of beer each which results in a market price of (P = 100 – 30 – 30) of \$40.

 Keyword(s): duopoly equilibrium"duopoly equilibrium".

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