Question

In the small village in Tirol there are only two farmers, Herbert and Wolfgang, sellingthe milk to consumers. The cows in each herd are from the same ancestors and areeating the grass on the same fields, so the farmers are producing a homogeneous product.However, Herbert has chronic back pain, so the production of each additional litre ofmilk is more costly for him than for Wolfgang:CH= 20 + 5QH+Q2HandCW= 20 + 3QW+ 0.5Q2W,whereQHis the output level (in litres) of Herbert andQWis the production level ofWolfgang. The total demand for milk is:P= 250?8Q, whereQ=QH+QW.

a) Calculate the equilibrium output, price and profits if the farmers compete onthe basis of output (Cournot competition) and make the production decisionssimultaneously.

b) Draw the reaction functions of the firms. Label the equilibrium point and theresulting output levels.

Answer 1

given,

P=250-8Q Q=QH+QW

P=250-8(QH+QW)

TRh=250Qh-8QH²-8 QHQW

MRh=250-16QH-8Qw

given Ch=20+5QH+Q²_H

MCh=5+2Q_H

for profit maximisation, MR=MC

250-16QH-8Qw=5+2QH

=> 245-8Qw=18QH

=>QH=13.611-0.44Qw ----------------reaction curve for Herbert

now , TRh={250-8(Qh+Qw)}Qw

MRh=250-8Qh-16qw

given Cw=20+3Qw+0.5Q_w²

MCw=3+Qw

for profit maximisation MR=MC

250-8Qh-16qw=3+Qw

=> 17Qw=247-8Qh

=>Qw=14.53-0.47Qh-----------reaction curve for Wolfgang

now plugging the value of Qw in the Qh, we get

Qh=9.10 & Qw=10.25

thus q = qw + qh=19.35

putting this values in demand equation

P=250-8*19.35

=95.2