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
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 \u00d7 Q" gives
"TR _1 = (300 - 3Q_)1 - 3Q_2) Q_1\\\\\n\nTR _1 = 300Q_1 - 3Q1^2 - 3Q_1Q_2\\\\\n\nMR _1 =\\frac {\\delta TR _1} { \\delta Q_1} = 300 - 6Q_1 - 3Q_2\\\\\n\nTC = 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\\\\\n\n6Q_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\\\\\n\nTR |_2 = 300Q_1 - 3Q_1Q_2 - 3Q_2^2\\\\\n\nMR_ 2 = \\frac{\\delta TR_ 2 \/}{\\delta Q_2 }= 300 - 3Q_1 - 6Q_2\\\\\n\nMC = 2"
Equating
"MR_ 2 = MC\\\\\n\n300 - 3Q_1 - 6Q_2 = 2\\\\\n\n6Q_2 = 298 - 3Q_1\\\\\n\nQ_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"
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