Question

10,000 where p is the price per 12) The demand equation for a monopolist's product is p q225 unit (in dollars) when q units are demanded. (a) Determine the value of q for which revenue is maximum. (b) What is the maximum revenue? 13) A manufacturer found that the total cost c of producing q units of a product is given by c 0.02q22800. At what level of production will average cost be a minimum?

Answer 1

Answer

12)

(a)

Demand is given by :

p = 10,000/(q² + 25)

Revenue(R) = pq = (10,000/(q² + 25))*q = 10,000q/(q² + 25)

First order condition :

d(R)/dq = 0 => 10,000[(q² + 25)*1 - q(2q)]/(q² + 25)² = 0

=> (q² + 25)*1 - q(2q) = 0

=> (q² + 25)*1 - 2q² = 0

=> q² = 25

=> q = 5

Hence, q for which Revenue is maximum is q = 5 units

(b)

Thus Maximum revenue(R) = 10,000q/(q² + 25) = 10,000*5/(5² + 25) = 1000

Hence, Maximum Revenue = $1000

(13)

Average Cost(AC) = c/q = (0.02q² + 2q + 800)/q = 0.02q + 2 + 800/q

First order condition :

d(AC)/dq = 0 => 0.02 - 800/q² = 0

=> 0.02q² = 800

=> q² = 40000

=> q = 200

Hence, Level of production at which Average Cost will be minimized is q = 200 units.