Profit maximizing, Quantity and Price with Dominant Firm and Competitive Fringe A dominant firm operates in various markets. In one of these markets, it is a monopolist and produces with the following cost function: C(q1) = 5q1. The market demand is P = 1005 - Q. 

(a) Find the profit maximizing P and q1 and profit. 

(b) A new fringe firm with C = 28,900 + (qf^2)/4 enters the market. The fringe firm acts as a price taker and maximizes its profits. The dominant firm, after the entry , subtracts the fringe firm's supply and behaves as a monopolist for the residual demand. Find the new profit maximizing P, q1, qf and profits as well.

My attempt for (a):

P = 1005 - Q. 

MR = 1005 - 2Q. 

MR = MC. 

1005 - 2Q = 5

Q = 500

P = 505

Profit = 505 x 500 - 5 x 505 = 249975

Hopefully I'm right.

My real problem is I'm stuck on (b). I'm not familiar with price taker. I did read my course book but it wasn't that helpful. I'd appreciate if anyone can help me, thank you. Please let me know if (a) was right as well
Your answer for part (a) is correct. 

For part (b) you need to first determine what's called 'residual demand', which can intuitively be thought of as 'leftover' demand given how much the competitive fringe will produce. Denote residual quantity demanded as $Q_r$. 

Marginal cost for the competitive fringe is:
$$MC_f=\frac{d}{dQ_f} C_f = \frac{Q_f}{2}.$$
To find the supply function for the competitive fringe set $MC_f=P$
$$Q_f=2P.$$

To find residual demand, subtract fringe supply from direct market demand (not inverse market demand). Direct market demand is given by 
$$Q=1005-P$$
so residual demand is
$$Q_r=Q-Q_f=1005-P-2P$$
$$Q_r=1005-3P.$$

Rearranging residual demand gives inverse residual demand of 
$$P=335-\frac{Q_r}{3},$$
which gives residual marginal revenue of
$$MR_r=335-\frac{2Q_r}{3}.$$
First find the profit-maximizing output for the leader firm by setting $MR_r=MC$:
$$335-\frac{2Q_r}{3}=5$$
$$Q_r=495.$$
To find the leader's profit-maximizing price, plug $Q_r$ into inverse residual demand to get
$$P=335-\frac{Q_r}{3}=335-165=170.$$

Finally, we can find the quantity supplied by the fringe by plugging the leader's price into the fringe supply function
$$Q_f=2P=2*170=340.$$