Answer to Question #231753 in Economics of Enterprise for MUFEES

Question #231753

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?

 


1
Expert's answer
2021-09-01T11:37:42-0400

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-.2P– 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 - 0.5P– 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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