what is the equilibrium price? what is the monopolist's profit?

Question

A monopolist faces a cost function equal to  

C(Q) = 6Q

and a demand function of 

Q^D(P) = 200 - P/3

what is the equilibrium price?

what is the monopolist's profit?

Answer 1

It's really not an equilibrium you are looking for, unless it's the competitive equilibrium price/quantity when this isn't a monopoly.

Usually monopolies (or monopolists, same thing) look for the profit maximizing quantity and price, which is found by finding the quantity where MR = MC, then using that quantity to find the corresponding price along the demand curve. So here its:

C(Q) = 6Q

Use this cost function to find the marginal cost function, by taking the partial derivative of this function with respect to q:

MC = 6

Q^D(P) = 200 - P/3

This is a demand function, to find MR, which is the partial derivative of total revenue (TR) with respect to Q. Before finding TR, which is P * Q, you must rearrange this function so its inverse, meaning the P is isolated like P = ....

After doing this:

1/3 P = 200-Q

P = 600 - 3Q

TR = 600Q - 3Q²

MR = 600-3Q

Now set MR = MC

600-3Q = 6

594 = 3Q

Q = 198, which is the profit maximizing quantity for the monopoly. Now plug that into demand to find price:

P = 600 - 3(198) = 6

Now to find profit, its TR - TC

TR = 6*198 = 1188

TC = 6(198) = 1188

Since TR and TC are the same, profit is zero.