optimal level of output

[gayathri]

the monopolists faces  a demand curve D(p)= 100-2p and C(y)=2y
find the optimal price and level of output.

can someone explain with steps ?

[Homer Simpson]

given, q = 100 - 2p
=>  p = 50 - (1/2)q - i have adjusted this because i need to find what is the quantity monopolist produces.

find R = pq
      R = (50 - 0.5q)q = 50q - 0.5q^2

Differentiate R wrt q to get MR = 50 - q

Now monopolist produces at point where MR = MC
Here, MC = 2 (y taken here as q and Cost is differentiated wrt q as well)

Thus, 50 - q = 2 (after equating)
               q = 48

He has decided optimal q here. However, since he is subjected to the market demand curve, we use that to find his optimal price at decided output q=48

thus, p = 26

[gayathri]

thank you