<p>Your answer for part (a) is correct. </p>

<p>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 <span class="math-container">$Q_r$</span>. </p>

<p>Marginal cost for the competitive fringe is:
<span class="math-container">$$MC_f=\frac{d}{dQ_f} C_f = \frac{Q_f}{2}.$$</span>
To find the supply function for the competitive fringe set <span class="math-container">$MC_f=P$</span>
<span class="math-container">$$Q_f=2P.$$</span></p>

<p>To find residual demand, subtract fringe supply from <strong>direct</strong> market demand (not inverse market demand). Direct market demand is given by 
<span class="math-container">$$Q=1005-P$$</span>
so residual demand is
<span class="math-container">$$Q_r=Q-Q_f=1005-P-2P$$</span>
<span class="math-container">$$Q_r=1005-3P.$$</span></p>

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

<p>Finally, we can find the quantity supplied by the fringe by plugging the leader's price into the fringe supply function
<span class="math-container">$$Q_f=2P=2*170=340.$$</span></p>

CC BY-SA 4.0
2019-03-17T21:08:19.793Z
2019-03-17T21:08:19.793Z
dlnB
<p>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. </p>

<p>(a) Find the profit maximizing P and q1 and profit. </p>

<p>(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.</p>

<p>My attempt for (a):</p>

<p>P = 1005 - Q. </p>

<p>MR = 1005 - 2Q. </p>

<p>MR = MC. </p>

<p>1005 - 2Q = 5</p>

<p>Q = 500</p>

<p>P = 505</p>

<p>Profit = 505 x 500 - 5 x 505 = 249975</p>

<p>Hopefully I'm right.</p>

<p>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</p>

CC BY-SA 4.0
2019-03-18T00:41:15.353Z
Let me know if you have any questions about my explanation below.
dlnB
CC BY-SA 4.0
2019-03-21T22:49:32.427Z
It was perfect. Thank you very much. I understood it!
Lily Turner
<microeconomics>
Profit maximizing, Quantity and Price with Dominant Firm and Competitive Fringe
Lily Turner
# Profit maximizing, Quantity and Price with Dominant Firm and Competitive Fringe

(Asked by: username_0 on 2019-03-17)

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

**Tags:** microeconomics
> **Comments:**
> **username_1**: Let me know if you have any questions about my explanation below.  
> **username_0**: It was perfect. Thank you very much. I understood it!  

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## 1 answers

### Answer by username_1 on 2019-03-17. Score: 1 | Accepted answer

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
$$Q\_f=2P=2\*170=340.$$


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