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.