Answer to Question #223359 in Microeconomics for Hafizh

Question

A monopolist sells its product in two isolated markets with demand functions
P1 = 32 − Q1 and P2 = 40 − 2Q2
The total cost function is TC = 4(Q1 + Q2).
(a) Show that the profit function is given by
π = 28Q1 + 36Q2 − Q12 − 2Q22
(b) Find the values of Q1 and Q2 which maximise profit and calculate the value of the
maximum profit. Verify that the second-order conditions for a maximum are satisfied

Answer 1

Solution:

a.). Profit function = TR – TC

TR = P $\times$ Q

TR for product 1 = (32 – Q1) Q1 = 32Q1 – Q1²

TR for product 2 = (40 – 2Q2) Q2 = 40Q2 – 2Q2²

TC = 4(Q1 + Q2) = 4Q1 + 4Q2

Profit function = TR – TC

Profit function = (32Q1 – Q1² + 40Q2 – 2Q2²) – (4Q1 + 4Q2))

Profit function = 28Q1 – Q1² + 36Q2 – 2Q2²

Profit function = 28Q1 + 36Q2 – Q1² – 2Q2²

 

b.). Profit maximizing function = 28Q1 + 36Q2 – Q1² – 2Q2²

Derive profit maximizing output for Q1 and Q2:

$\frac{\partial \pi } {\partial Q1}$ = πQ1 = 28 – 2Q1= 0

$\frac{\partial \pi } {\partial Q2}$ = πQ2 = πQ2 = 36 – 4Q2 = 0

Value for Q1:

28 – 2Q1= 0

28 = 2Q1

Q1 = 14

Value for Q2:

36 – 4Q2= 0

36 = 4Q2

     

 

Check second order conditions for maximum:

πQ1 = 28 – 2Q1

πQ2 = 36 – 4Q2

πQ1Q2 = 2

 

The second partial derivatives have to be negative:

πQ1Q1 = -2 < 0

πQ2Q2 = -4 < 0

πQ1Q2 = 2

 

The multiplication of the two-second partial derivatives must be greater than cross partial derivatives squared.

πQ1Q1πQ2Q2 > (πQ1Q2)²

(-2)(-4) > (2)²

8 > 4