<p>Note the revenue is the quantity sold in each country times its respective price.  Write out the profit equation
$$
\pi= Q_a*P_a+Q_b*P_b - TC(Q_a+Q_b)
$$
You already have expressions for $P_a$ and $P_b$ in terms of $Q_a$ and $Q_b$.  Substitute these in.  The result will be a profit function in terms of only $Q_a$ and $Q_b$.  This is your profit function.</p>

<p>Take the derivative of the profit function with respect to $Q_a$.  Set equal to 0 and solve for $Q_a$.  Repeat for $Q_b$.</p>

<p>If you do it right, you will get $Q_a=450$ and $Q_b=4750$.  Substitute these into the profit equation to get the total profit.</p>

<p>All these types of questions are solved the same way.</p>

<ol>
<li><p>Write out the correct profit function.  </p></li>
<li><p>Substitute everything in until the only remaining variables are the things you are able to control (in this case, the quantities).</p></li>
<li><p>Take derivatives with respect to each choice variable and solve the resulting first order conditions (which sometimes may be a system of equations, but not here).  </p></li>
<li><p>Substitute your optimal quantities back in to get the resulting profit.</p></li>
<li><p>In a more complex case you should also take the second derivative to ensure that you are getting the maximums, not the minimums.</p></li>
</ol>

<p>=================== EDIT: Additional Help =============</p>

<p>Profit function:</p>

<p>$$
\pi=1000Q_a - Q_a^2+2000Q_b-0.2Q_b^2-100(Q_a+Q_b)-1000000
$$</p>

<p>First order conditions
$$
\frac{d\pi}{dQ_a}=1000-2Q_a - 100
$$
$$
\frac{d\pi}{dQ_b}=2000-0.4Q_b - 100
$$</p>

<p>Note that from the second first order equation
$$
Q_b=\frac{2000-100}{0.4} = 4750
$$</p>

<p>Substituting 450 and 4750 into the profit function gives exactly 3,715,000.</p>

<p>Please don't say that right answers are wrong when you are unable to replicate them on your own.  It causes those who don't know any better to downvote.</p>

CC BY-SA 3.0
2017-08-04T13:08:28.247Z
Thank you so much for the answer! I learned a lot from it, and it seems right to me also. However, I think it is not right that Qb = 4750, I got Qb = 250, but the same thing fro Qa as you. 
Unfortunately, both yours and my answer is wrong when I substitute the quantities into the profit function. It should be "3 715 000"
Nora
CC BY-SA 3.0
2017-08-04T14:42:47.957Z
@user3257736 Actually my answer is correct and gives the correct profit.  Since you are still having problems I will add more info to the answer.
farnsy
CC BY-SA 3.0
2017-08-04T15:18:52.967Z
@farnsy Actually I downvoted your answer because it [answers an off-topic question](https://economics.meta.stackexchange.com/questions/1662/should-i-downvote-valid-answers-to-off-topic-questions).
Giskard
CC BY-SA 3.0
2017-08-04T15:39:24.453Z
@denesp Not a very good practice in my opinion.  Econ stack exchange is filled with this type of question and it's Ph.D.-level equivalent (which is no better).  The community hasn't chosen to close the question nor the many other questions like it, nor to downvote their answers.  If you downvote every answer to a homework-style question, you end up being little more than the on-topic police.  Up to you how you spend your time, of course.
farnsy
CC BY-SA 3.0
2017-08-04T16:08:41.887Z
Feel free to vote on the meta question, we value your opinion. The consensus did not come about lightly, I don't think anyone thinks fully banning or fully allowing these types of questions is ideal.
Giskard
CC BY-SA 3.0
2017-08-04T16:15:34.557Z
I will.  But note that you accepted the answer (that had 4 votes--that's no consensus), giving it legitimacy it doesn't deserve.  I answer plenty of questions, including poor questions, on StackExchange and this is the only time I've seen an answer downvoted because someone thought the question was beneath them.  Note that my answer provided a general approach to answering all this type of question, which suggests that the answer (and question) have value beyond this person's homework.  The purpose of StackExchange is to help people, and this question and answer accomplished that.
farnsy
CC BY-SA 3.0
2017-08-04T16:45:34.460Z
I am happy that you like helping people. If you feel strongly about this I encourage you to read the arguments on meta (not just in the question I linked, there is a lot more on this subject). Once you did that if you still disagree you can post on meta and challenge the current consensus. (If you feel like 4 votes is not a consensus just post an opposing answer and see if you get more votes: Economics is a small SE.)
Giskard
CC BY-SA 3.0
2017-08-06T06:51:14.613Z
@farnsy: Thank you so much! I see that it is correct now, and sorry for saying that it was wrong when it was not. I will be much more careful next time :). I really really appreciate your help!
Nora
CC BY-SA 3.0
2017-08-04T02:54:52.023Z
2017-08-04T15:05:10.520Z
2017-08-04T15:05:10.520Z
farnsy
<p>I am really struggling with the following question: </p>

<p><strong>Question:</strong> </p>

<p>A company produces a product that they sell in two different countries. Because of transport costs, customs etc. it is not profitable for someone to buy the product in one country, to sell it in the other. </p>

<p>The demand in one country (country A), is given by P<sub>A</sub> = 1000 - Q<sub>A</sub>, where P<sub>A</sub> is the price, and Q<sub>A</sub> is the quantity. </p>

<p>In the other country (country B) demand is given by P<sub>B</sub> = 2000 - 0.2Q<sub>b</sub>. For the company, the cost per unit sold is the same in both markets. The companies cost can be expressed as: TC(Q) = 100(Q<sub>A</sub> + Q<sub>B</sub>) + 1 000 000. </p>

<p>What is the optimal profit for the company?</p>

<p>The answer is apparently 3 715 000, but I am not sure why. </p>

<p><strong>My reasoning so far:</strong></p>

<p>I know that profit is equal to TR(Q) - TC(Q), and optimal profit can probably be found by taking the derivative of the expression, and setting the derivative to 0. We are given TC(Q), from the expression above, but I am not sure how to find TR(Q). Also, I am not sure if I am on the right track at all. I also don't understand why the first piece of information (that it is not profitable to buy the product in one country, and sell it in the other) was given. </p>

<p>Any hints, or more detailed solutions, would be greatly appreciated! </p>

<microeconomics>
What is optimal profit for this company?
Nora
# What is optimal profit for this company?

(Asked by: username_0 on 2017-08-01)

I am really struggling with the following question:

**Question:**

A company produces a product that they sell in two different countries. Because of transport costs, customs etc. it is not profitable for someone to buy the product in one country, to sell it in the other.

The demand in one country (country A), is given by PA = 1000 - QA, where PA is the price, and QA is the quantity.

In the other country (country B) demand is given by PB = 2000 - 0.2Qb. For the company, the cost per unit sold is the same in both markets. The companies cost can be expressed as: TC(Q) = 100(QA + QB) + 1 000 000.

What is the optimal profit for the company?

The answer is apparently 3 715 000, but I am not sure why.

**My reasoning so far:**

I know that profit is equal to TR(Q) - TC(Q), and optimal profit can probably be found by taking the derivative of the expression, and setting the derivative to 0. We are given TC(Q), from the expression above, but I am not sure how to find TR(Q). Also, I am not sure if I am on the right track at all. I also don't understand why the first piece of information (that it is not profitable to buy the product in one country, and sell it in the other) was given.

Any hints, or more detailed solutions, would be greatly appreciated!

**Tags:** microeconomics

---

## 1 answers

### Answer by username_1 on 2017-08-04. Score: 0 | Accepted answer

Note the revenue is the quantity sold in each country times its respective price. Write out the profit equation
$$
\pi= Q\_a\*P\_a+Q\_b\*P\_b - TC(Q\_a+Q\_b)
$$
You already have expressions for $P\_a$ and $P\_b$ in terms of $Q\_a$ and $Q\_b$. Substitute these in. The result will be a profit function in terms of only $Q\_a$ and $Q\_b$. This is your profit function.

Take the derivative of the profit function with respect to $Q\_a$. Set equal to 0 and solve for $Q\_a$. Repeat for $Q\_b$.

If you do it right, you will get $Q\_a=450$ and $Q\_b=4750$. Substitute these into the profit equation to get the total profit.

All these types of questions are solved the same way.

1. Write out the correct profit function.
2. Substitute everything in until the only remaining variables are the things you are able to control (in this case, the quantities).
3. Take derivatives with respect to each choice variable and solve the resulting first order conditions (which sometimes may be a system of equations, but not here).
4. Substitute your optimal quantities back in to get the resulting profit.
5. In a more complex case you should also take the second derivative to ensure that you are getting the maximums, not the minimums.

=================== EDIT: Additional Help =============

Profit function:

$$
\pi=1000Q\_a - Q\_a^2+2000Q\_b-0.2Q\_b^2-100(Q\_a+Q\_b)-1000000
$$

First order conditions
$$
\frac{d\pi}{dQ\_a}=1000-2Q\_a - 100
$$
$$
\frac{d\pi}{dQ\_b}=2000-0.4Q\_b - 100
$$

Note that from the second first order equation
$$
Q\_b=\frac{2000-100}{0.4} = 4750
$$

Substituting 450 and 4750 into the profit function gives exactly 3,715,000.

Please don't say that right answers are wrong when you are unable to replicate them on your own. It causes those who don't know any better to downvote.

> **Comments:**
> **username_0**: Thank you so much for the answer! I learned a lot from it, and it seems right to me also. However, I think it is not right that Qb = 4750, I got Qb = 250, but the same thing fro Qa as you.  Unfortunately, both yours and my answer is wrong when I substitute the quantities into the profit function. It should be "3 715 000"  
> **username_1**: @user3257736 Actually my answer is correct and gives the correct profit.  Since you are still having problems I will add more info to the answer.  
> **username_2**: @farnsy Actually I downvoted your answer because it [answers an off-topic question](https://economics.meta.stackexchange.com/questions/1662/should-i-downvote-valid-answers-to-off-topic-questions).  
> **username_1**: @denesp Not a very good practice in my opinion.  Econ stack exchange is filled with this type of question and it's Ph.D.-level equivalent (which is no better).  The community hasn't chosen to close the question nor the many other questions like it, nor to downvote their answers.  If you downvote every answer to a homework-style question, you end up being little more than the on-topic police.  Up to you how you spend your time, of course.  
> **username_2**: Feel free to vote on the meta question, we value your opinion. The consensus did not come about lightly, I don't think anyone thinks fully banning or fully allowing these types of questions is ideal.  
> **username_1**: I will.  But note that you accepted the answer (that had 4 votes--that's no consensus), giving it legitimacy it doesn't deserve.  I answer plenty of questions, including poor questions, on StackExchange and this is the only time I've seen an answer downvoted because someone thought the question was beneath them.  Note that my answer provided a general approach to answering all this type of question, which suggests that the answer (and question) have value beyond this person's homework.  The purpose of StackExchange is to help people, and this question and answer accomplished that.  
> **username_2**: I am happy that you like helping people. If you feel strongly about this I encourage you to read the arguments on meta (not just in the question I linked, there is a lot more on this subject). Once you did that if you still disagree you can post on meta and challenge the current consensus. (If you feel like 4 votes is not a consensus just post an opposing answer and see if you get more votes: Economics is a small SE.)  
> **username_0**: @farnsy: Thank you so much! I see that it is correct now, and sorry for saying that it was wrong when it was not. I will be much more careful next time :). I really really appreciate your help!  

---

