Gizmo brothers, manufactures two types of hi-tech yo-yo: The Exterminator and the Eliminator. Denoting Exterminator output as Q1 and Eliminator output as Q2, the company has estimated the following demand equation for its yo-yos:
Q1= 10-.2P1 – 0.4Q2
Q2 = 20 -0.5P2 – 2Q1
The total cost equation for producing Exterminators and Eliminators are
TC1 = 4+2Q12
TC2 = 8+6Q22
a) If Gizmo Brothers is a profit-maximizing firm, how much should it charge for Exterminators and Eliminators?
b) What is the profit maximizing level of output for Exterminators and Eliminators?
c) What is Gizmo Brother’s profit?
Solution:
a.). Amount to be charged for Exterminators and Eliminators:
A profit-maximizing firm maximizes profits at the point where MR = MC.
Derive MR from TR:
TR = P "\\times" Q
Derive inverse demand function for Exterminator:
Q1= 10-.2P1 – 0.4Q2
P1 = 50 – 5Q1 – 2Q2
TR1 = (50 – 5Q1 – 2Q2) x Q1 = 50Q – 5Q12 – 2Q2Q1
TR1 = 50Q1 – 5Q12 – 2Q2Q1
MR1 = ∂TR1/∂Q1 = 50 – 10Q1 – 2Q2
MR1 = 50 – 10Q1 – 2Q2
Derive MC from TC for Exterminator:
TC1 = 4+2Q12
MC = "\\frac{\\partial TC_{1} } {\\partial Q_{1} }" = 4Q1
MC = 4Q1
Set MR1 = MC1:
50 – 10Q1 – 2Q2 = 4Q1
50 – 2Q2 =14Q1
Q1 = 25/7 – Q2/7
Derive inverse demand function for Eliminator:
Q2= 20 - 0.5P2 – 2Q1
P2 = 40 – 2Q2 – 4Q1
TR2 = (40 – 2Q2 – 4Q1) x Q2 = 40Q2 – 2Q22 – 4Q1Q2
TR2 = 40Q2 – 2Q22 – 4Q1Q2
MR2 = "\\frac{\\partial TR_{2} } {\\partial Q_{2} }" = 40 – 4Q2 – 4Q1
MR2 = 40 – 4Q2 – 4Q1
Derive MC from TC for Eliminator:
TC2 = 8+ 6Q22
MC2 = "\\frac{\\partial TC_{2} } {\\partial Q_{2} }" ∂TC2/∂Q2 = 12Q2
MC2 = 12Q2
Set MR2 = MC2:
40 – 4Q2 – 4Q1 = 12Q2
40 – 4Q1 =16Q2
Q2 = 2.5 – Q1/4 = 2.5 – 0.25Q1
Substitute to get their values:
Q1 = 3.6 – "\\frac{Q_{2} }{7}"
Q1 = 3.6 – "\\frac{2.5 - 0.25Q_{1}}{7}"
Q1 = 3.6 – 0.36 – 0.036Q1
Q1 = 3.1
Substitute to get Q2:
Q2 = 2.5 – 0.25Q1 = 2.5 – 0.25(3.1) = 2.5 – 0.78 = 1.72
Q2 = 1.72
Substitute in the price inverse demand function to determine price charged:
Price charged for Exterminator:
P1 = 50 – 5Q1 – 2Q2
P1 = 50 – 5(3.1) – 2(1.72) = 50 – 15.5 – 3.44 = 41.06
P1 = 41.06
Price charged for Exterminator = 41.06
Price charged for Eliminator:
P2 = 40 – 2Q2 – 4Q1
P1 = 40 – 2(1,72) – 4(3.1) = 40 – 3.44 – 12.4 = 24.16
P1 = 24.16
Price charged for Eliminator = 24.16
b.). Profit maximizing level of output for Exterminator = 3.1
Profit maximizing level of output for Eliminator = 1.72
c.). Profit = TR – TC
Profit for Exterminator = (41.06 "\\times"3.1) – (4 + 2(3.12))
= 127.29 – 23.22 = 104.07
Profit for Eliminator = (24.16 "\\times"1.72) – (8 + 6(1.722))
= 41.56 – 25.75 = 15.81
Total profit for Gizmo Brother’s = 104.07 + 15.81 = 119.88
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