Question

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

 Q_{2 =} 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?

 

Answer 1

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