Question

Q₁ = (4310/22) - (P₁/22)

Q₂ = (5021/17) - (P₂/17)

TC = 760 + 14Q₁ + 32Q₂

Form the profit equation, π and solve for the critical values.

(a) What is the critical value of Q₁?

(b) What is the critical value of Q₂?

(c) What is the value of |H₂|?

(d) What is the value of maximum profit?

( (Give your answer to two decimal places, if necessary)

Answer 1

The demand functions are given as,

Q1 = (4310/22) - (P1/22)

or, P1 = 4310 - 22.Q1.......(1)

Q2 = (5021/17) - (P2/17)

or, P2 = 5021 - 17.Q2.........(2)

Revenue from type 1 demand is,

R1 = P1.Q1 = (4310 - 22.Q1).Q1

or, R1 = 4310.Q1 - 22.(Q1)²

Revenue from type 2 demand is,

R2 = P2.Q2 = (5021 - 17.Q2).Q2

or, R2 = 5021.Q2 - 17.(Q2)²

Total Cost is,

TC = 760 + 14Q1 + 32Q2

Profit of the firm is,

π = R1 + R2 - TC

or, π = 4310.Q1 - 22.(Q1)² + 5021.Q2 - 17.(Q2)² - (760 + 14Q1 + 32Q2)

Hence,

π1 = dπ/dQ1 = 4310 - 44.Q1 - 14

or, π1 = 4296 - 44.Q1

And,

π2 = dπ/dQ2 = 5021 - 34.Q2 - 32

or, π2 = 4989 - 34.Q2

When profit is maximized, we can write,

π1 = 0

or, 4296 - 44.Q1 = 0

or, Q1* = 97.6

(a) Critical value of Q1 is 97.6.

Also, when profit is maximized, we can write,

π2 = 0

or, 4989 - 34.Q2 = 0

or, Q2* = 146.7

(b) Critical value of Q2 is 146.7.

Now, from π1 and π2 we calculate the followings.

π11 = dπ1/dQ1 = -44

π12 = dπ1/dQ2 = 0

π21 = dπ2/dQ1 = 0

π22 = dπ2/dQ2 = -34

The determinant |H2| is,

(c) The value of |H2| is 1496.

Now, putting Q1* and Q2* in the drmand functions we get the corresponding prices P1* and P2*. Hence,

P1* = 4310 - 22.Q1* = 4310 - 22×97.6

or, P1* = 2162.8

And, P2* = 5021 - 17.Q2* = 5021 - 17×146.7

or, P2* = 2527

And, total cost is,

TC = 760 + 14Q1* + 32Q2*

or, TC = 760 + 14×97.6 + 32×146.7

or, TC = 6820.8

Profit is,

π = R1 + R2 - TC = P1*.Q1* + P2*.Q2* - TC

or, π = (2162.8)×97.6 + (2527)×146.7 - 6820.8

or, π = 574979.38

(d) The maximum value of profit is 574979.38.