Question

Two dairy farmers produce milk for a local town with local milk demand given by (P denotes price measured in Rands, Q denotes the quantity measured in liters). Both farmers have the same cost function given by (where denotes output). (a) Determine the reaction function of each farmer. [8] (b) Find the Cournot-Nash equilibrium. [2] (c) Calculate profits for each farmer [2] (d) Suppose that both farmers decide to form a cartel, determine profits for each farmer under the cartel 

Answer 1

a)

Milk demand is given as $Q = 100 - \frac{1}{3P}$

Inverse function can be written as $P = 300 - 3Q$

Q = output of farmer 1 (Q₁) + output of farmer 2 (Q₂) 

Therefore $P = 300 - 3(Q_1+ Q_2 )$

$P = 300 - 3Q_1 - 3Q_2$

For farmer 1,

$TR = P × Q$ gives

$TR _1 = (300 - 3Q_)1 - 3Q_2) Q_1\\ TR _1 = 300Q_1 - 3Q1^2 - 3Q_1Q_2\\ MR _1 =\frac {\delta TR _1} { \delta Q_1} = 300 - 6Q_1 - 3Q_2\\ TC = 150 + 2Q \\$

$MC =\frac {\delta TC} {\delta Q} = \frac {\delta (150+2Q)} {\delta Q}=2$

MC = 2 for both farmers because total cost is same for both. 

Equating $MR _1 = MC$

$300 - 6Q_1 - 3Q_2 = 2\\ 6Q_1 = 298 - 3Q_2$

$Q_1 = 49.67 - \frac{1}{2} Q_2$ is farmer 1's reaction function. 

For farmer 2,

$TR _2 = (300 - 3Q_1 - 3Q_2) Q_2\\ TR |_2 = 300Q_1 - 3Q_1Q_2 - 3Q_2^2\\ MR_ 2 = \frac{\delta TR_ 2 /}{\delta Q_2 }= 300 - 3Q_1 - 6Q_2\\ MC = 2$

Equating

$MR_ 2 = MC\\ 300 - 3Q_1 - 6Q_2 = 2\\ 6Q_2 = 298 - 3Q_1\\ Q_2 = 49.67 - \frac{1}{2} Q_1$ is farmer 2's reaction function. 

Therefore,

$Q_1 = 49.67 - \frac{1}{2}Q_2$ is Farmer 1's reaction function

$Q_2= 49.67 - \frac{1}{2}Q_1$ is Farmer 2's reaction function. 

b)

To find the equilibrium let us substitute the reaction function of firm 1 in the reaction function of firm 2:

substitute equition 1 into equition 2

$Q_2=49.67-\frac{1}{2}(49.67-\frac{1}{2})\\Q_2=49.67-24.835+0.25Q_2\\Q_2-0.25Q_2=24.835\\Q_2=33.11$

$Q_1=49.67-\frac{1}{2}(33.11)\\Q_1=49.67-16.555\\Q_1=33.11$

The nash-equilibrium outputs are

$Q_1=33.11\\Q_2=33.11$

c)

$profit=TR-TC\\ First \space farmer = 300Q_1 - 3Q1^2 - 3Q_1Q_2-150-2Q\\ second \space farmer\\= 300Q_1 - 3Q_2Q_2- 3Q1^2 -150-2Q$

d)

$Q=Q_1+Q_2\\MC=2\\P=300-3Q=300-3(50)=150$

$Profit=TR-TC\\=(150\times 50)-(150\times 300)=7050$

profit per firm$=7050\div2=3525$