Answer to Question #284473 in Microeconomics for beki

Question

Given the demand function for the two markets and total cost function that:

Q₁ = 55 – P₁; Q₂ = 70 – 2P₂ and TC=5Q + 20 where Q = Q₁ + Q₂

A.   Calculate the equilibrium price, equilibrium quantity in each market with price discrimination.

B.    Calculate the maximum profit using equilibrium price and quantity with price discrimination.

C.    Calculate the price elasticity of demand in each market using MR=P(1+1/Ɛ_p).

D.   Calculate the equilibrium price and quantity without price discrimination.

E.    Calculate the maximum profit without price discrimination.

F.     Is the Monopolist better off with or without price discrimination.

Answer 1

Solution:

A.). Derive the inverse demands for both markets:

Market 1:

Q₁ = 55 – P₁

P₁ = 55 – Q₁

TR₁ = P₁ $\times$ Q₁ = (55 – Q₁) x Q₁ = 55Q₁ – Q₁²

Derive MR₁ from TR₁:

MR₁ = $\frac{\partial TR_{1} } {\partial Q_{1} }$ = 55 – 2Q₁

Market 2:

Q₂ = 70 – 2P₂

P₂ = 35 – 0.5Q₂

TR₂ = P₂ $\times$ Q₂ = (70 – Q₂) $\times$ Q₂ = 70Q₂ – Q₂²

Derive MR₂ from TR₂:

MR₂ = $\frac{\partial TR_{2} } {\partial Q_{2} }$ = 70 – 2Q₂

 

Derive MC:

MC = $\frac{\partial TC } {\partial Q }$ = 5

 

Set MR₁ = MC

55 – 2Q₁ = 5

50 = 2Q₁

Q₁ = 25

Substitute to get P₁:

P₁ = 55 – Q₁ = 55 – 25 = 30

Market 1 equilibrium quantity (Q₁) = 30

Market 1 equilibrium Price (P₁) = 25

 

Set MR₂ = MC

70 – 2Q₂ = 5

65 = 2Q₂

Q₂ = 32.5

Substitute to get P₂:

P₂ = 35 – 0.5Q₂ = 35 – 0.5(32.5) = 35 – 16.25 = 18.75

Market 2 equilibrium quantity (Q₁) = 32.5

Market 2 equilibrium Price (P₁) = 18.75

 

B.). Profit Market 1 = TR₁ – TC

TR₁ = P₁ $\times$ Q₁ = 30 $\times$ 25 = 750

TC = 5Q + 20 = 5(30 + 32.5) + 20 = 312.5 + 20 = 332.5

Profit Market 1 = 750 – 332.5 = 437.5

 

Profit Market 2 = TR₂ – TC

TR₂ = P₂ $\times$ Q₂ = 32.5 $\times$ 18.75 = 609.38

TC = 5Q + 20 = 5(30 + 32.5) + 20 = 312.5 + 20 = 332.5

Profit Market 2 = 609.38 – 332.5 = 276.88

 

C.). Price elasticity Market 1 = $\frac{\triangle Q_{1} }{\triangle P_{1}} \times \frac{ Q_{1} }{P_{1}}$

$\frac{\triangle Q_{1} }{\triangle P_{1}}$ = -1

= -1 $\times \frac{25}{30} = -0.83$

Price elasticity Market 1 = -0.83

 

Price elasticity Market 2 = $\frac{\triangle Q_{2} }{\triangle P_{2}} \times \frac{ Q_{2} }{P_{2}}$

$\frac{\triangle Q_{2} }{\triangle P_{2}}$ = -2

= -2 $\times$ $\frac{18.75}{32.5} = -1.15$

Price elasticity Market 2 = -1.15