Answer to Question #210551 in Microeconomics for Sedy

Question #210551

Two dairy farmers produce milk for a local town with local milk demand given by Q = 100 - 1/3P (P denotes price measured in Rands, Q denotes the quantity measured in liters). Both farmers have the same cost function given by TC = 150 + 2Q (where Q denotes output).


 (a) Determine the reaction function of each farmer.  


1
Expert's answer
2021-06-28T03:10:02-0400

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. 

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