ms-09 mba assignment july dec 2010 Question 2

2. Given a firm’s demand function, P = 24 - 0.5Q and the average cost function, AC = Q² – 8Q + 36  + 3/Q, calculate the level of output Q which

a)  Maximizes total revenue

Since demand function is P = 24-0.5 Q

The total revenue will be TR =PQ=( 24-0.5Q) Q= 24-0.5Q²

To maximize TR , we find the derivative and set it to 0.

Hence  first order condition dR/dQ =24 – 2 ( 0.5 ) Q

                                                         = 24 – Q = 0

                                                      Q = 24.

The second order derivation d²R/dQ² to be negative .

Since dR/dQ² = -1

Which is negative, hence total revenue is maximized when output is 24 units .

b) From profit function.

P = TR – TC

TC    = AC X Q

        = ( Q2 – 8 Q + 36 + 3 / Q ) X Q

        = Q^{3 -} 8 Q² + 36 Q + 3

TR = (24 – O.5 Q ) Q

       = 24Q – 0.5 Q² after substituting TR & TC  we get

      P    =  ( 24 Q – 0.5 Q² ) – ( Q³- 8 Q² + 36 Q + 3 )

dP/dQ = ( 24 – Q – 3 Q² + 16 Q – 36 )

            = - 3Q^{2  _15 Q} -12

Now set = dπ /dQ = 0

-3Q^{2 +} 15 Q – 12 = 0

Dividing by 3 we get q2 +5 q – 4 = 0

(Q - 4 ) (Q – 1) = Q = 4 or 1