<p>Firm 1's objective is to maximize profits :
$$\max_{q_1\geq 0} \ \ \left(a - b(q_1+q_2)\right)q_1 - cq_1$$</p>

<p>Solving the above problem, we get the best response function of firm 1 as
$$q_1 = \dfrac{a - c - bq_2}{2b}$$</p>

<p>Firm 2's objective is to maximize revenue subject to the constraint that its profits are non-negative :
$$\max_{q_2\geq 0} \ \ \left(a - b(q_1+q_2)\right)q_2$$ $$\text{s.t. }\left(a - b(q_1+q_2)\right)q_2 - cq_2 \geq 0$$</p>

<p>Solving the above problem, we get the best response function of firm 2 as
\begin{eqnarray*} q_2 =\begin{cases}  \dfrac{a - bq_1}{2b} &amp; \text{if } a - bq_1 \geq 2c \\ \dfrac{a - c - bq_1}{b} &amp; \text{if } a -  bq_1 &lt; 2c \end{cases}\end{eqnarray*}</p>

<p>Solving the best response functions for $q_1$ and $q_2$ yields
\begin{eqnarray*} (q_1^*, q_2^*) = \begin{cases} \left(\dfrac{a-2c}{3b}, \dfrac{a+c}{3b}\right) &amp; \text{if } a \geq 2c\\
 \left( 0, \dfrac{a-c}{b}\right) &amp; \text{if } a &lt; 2c \end{cases} \end{eqnarray*}</p>

CC BY-SA 4.0
2018-05-13T07:12:12.563Z
Thank you so much dear @Amit I couldn’t think how to write revenue maximization problem. Since you explain really great and understandable, please also can you explain this question? (By the way, these are not exactly homework, I ‘m preparing an exam. And since you tell how I should solve such interesting and difficult questions that I am not familiar with, I instantly ask you. Thanks a lot:) https://economics.stackexchange.com/questions/21967/question-on-consumer-theory
studentp
CC BY-SA 4.0
2018-05-15T03:31:31.573Z
when you Find time, will you look my question?:)
studentp
CC BY-SA 4.0
2018-05-12T23:36:46.900Z
2018-05-12T23:36:46.900Z
Amit
<p>We have two firms with identical cost structure compete in a market </p>

<p>Demand function = $p=a-bq$ </p>

<p>And $q=q_1+q_2$</p>

<p>They are identical in every way. However, firm 1 maximizes profit and firm 2 maximizes revenue as long as shareholders are satisfied, which he achieves by keeping profits nonnegative. </p>

<p>Both firms have constant and equal marginal cost c. So I want to find the quantities that they will choose. </p>

<p>———</p>

<p><strong>What I did is...</strong></p>

<p>For firm 1, </p>

<p>$$\pi_1=max[(a-b(q_1+q_2))q_1-cq_1]$$</p>

<p>FOCs for $q_1$</p>

<p>$$a-2bq_1-bq_2-c=0$$</p>

<p>So $$q_1={a-bq_2-c\over 2b}$$</p>

<p>For firm 2,</p>

<p>$$max [(a-b(q_1+q_2))q_2]$$</p>

<p>FOCs $$a-bq_1-2bq_2=0$$</p>

<p>$$q_2={a-bq_1\over 2b}$$</p>

<p>So, 
$$q_1={a-b({a-bq_1\over 2b})-c\over 2b}$$</p>

<p>$$q_1^*={a-2c\over 3b}$$</p>

<p>And $$q^*_2={5a+2b\over 6b}$$</p>

<p>—-</p>

<p>The question says that “* firm 1 maximizes profit and firm 2 maximizes revenue as long as shareholders are satisfied, which he achieves by keeping profits nonnegative.*”</p>

<p>Because of this sentence, I am exactly not sure about my solution. Especially for firm 2. </p>

<p>I’m confused at this point. Please tell my mistakes. Thank you. </p>

CC BY-SA 4.0
2018-05-12T17:58:21.123Z
You have to check whether $q_2^*$ is such that profits for firm 2 are non-negative.
Alecos Papadopoulos
CC BY-SA 4.0
2018-05-12T18:53:45.253Z
Non-negative _profits_. $q^*_2$ does not represent profits.
Alecos Papadopoulos
<microeconomics><mathematical-economics><self-study><profit-maximization><theory-of-the-firm>
Revenue maximization
studentp
# Revenue maximization

(Asked by: username_0 on 2018-05-11)

We have two firms with identical cost structure compete in a market

Demand function = $p=a-bq$

And $q=q\_1+q\_2$

They are identical in every way. However, firm 1 maximizes profit and firm 2 maximizes revenue as long as shareholders are satisfied, which he achieves by keeping profits nonnegative.

Both firms have constant and equal marginal cost c. So I want to find the quantities that they will choose.

———

**What I did is...**

For firm 1,

$$\pi\_1=max[(a-b(q\_1+q\_2))q\_1-cq\_1]$$

FOCs for $q\_1$

$$a-2bq\_1-bq\_2-c=0$$

So $$q\_1={a-bq\_2-c\over 2b}$$

For firm 2,

$$max [(a-b(q\_1+q\_2))q\_2]$$

FOCs $$a-bq\_1-2bq\_2=0$$

$$q\_2={a-bq\_1\over 2b}$$

So,
$$q\_1={a-b({a-bq\_1\over 2b})-c\over 2b}$$

$$q\_1^\*={a-2c\over 3b}$$

And $$q^\*\_2={5a+2b\over 6b}$$

—-

The question says that “\* firm 1 maximizes profit and firm 2 maximizes revenue as long as shareholders are satisfied, which he achieves by keeping profits nonnegative.\*”

Because of this sentence, I am exactly not sure about my solution. Especially for firm 2.

I’m confused at this point. Please tell my mistakes. Thank you.

**Tags:** microeconomics, mathematical-economics, self-study, profit-maximization, theory-of-the-firm

> **Comments:**
> **username_1**: You have to check whether $q_2^*$ is such that profits for firm 2 are non-negative.  
> **username_1**: Non-negative _profits_. $q^*_2$ does not represent profits.  

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## 1 answers

### Answer by username_2 on 2018-05-12. Score: 1 | Accepted answer

Firm 1's objective is to maximize profits :
$$\max\_{q\_1\geq 0} \ \ \left(a - b(q\_1+q\_2)\right)q\_1 - cq\_1$$

Solving the above problem, we get the best response function of firm 1 as
$$q\_1 = \dfrac{a - c - bq\_2}{2b}$$

Firm 2's objective is to maximize revenue subject to the constraint that its profits are non-negative :
$$\max\_{q\_2\geq 0} \ \ \left(a - b(q\_1+q\_2)\right)q\_2$$ $$\text{s.t. }\left(a - b(q\_1+q\_2)\right)q\_2 - cq\_2 \geq 0$$

Solving the above problem, we get the best response function of firm 2 as
\begin{eqnarray\*} q\_2 =\begin{cases} \dfrac{a - bq\_1}{2b} & \text{if } a - bq\_1 \geq 2c \\ \dfrac{a - c - bq\_1}{b} & \text{if } a - bq\_1 < 2c \end{cases}\end{eqnarray\*}

Solving the best response functions for $q\_1$ and $q\_2$ yields
\begin{eqnarray\*} (q\_1^\*, q\_2^\*) = \begin{cases} \left(\dfrac{a-2c}{3b}, \dfrac{a+c}{3b}\right) & \text{if } a \geq 2c\\
\left( 0, \dfrac{a-c}{b}\right) & \text{if } a < 2c \end{cases} \end{eqnarray\*}

> **Comments:**
> **username_0**: Thank you so much dear @username_2 I couldn’t think how to write revenue maximization problem. Since you explain really great and understandable, please also can you explain this question? (By the way, these are not exactly homework, I ‘m preparing an exam. And since you tell how I should solve such interesting and difficult questions that I am not familiar with, I instantly ask you. Thanks a lot:) https://economics.stackexchange.com/questions/21967/question-on-consumer-theory  
> **username_0**: when you Find time, will you look my question?:)  

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