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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!

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closed as off-topic by Herr K., luchonacho, Kitsune Cavalry, optimal control, Martin Van der Linden Aug 23 '17 at 16:27

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question does not meet the standards for homework questions as spelled out in the relevant meta posts. For more information, see our policy on homework question and the general FAQ." – Herr K., luchonacho, Kitsune Cavalry, optimal control, Martin Van der Linden

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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.

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  • $\begingroup$ 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" $\endgroup$ – Nora Aug 4 '17 at 13:08
  • $\begingroup$ @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. $\endgroup$ – farnsy Aug 4 '17 at 14:42
  • $\begingroup$ @farnsy Actually I downvoted your answer because it answers an off-topic question. $\endgroup$ – Giskard Aug 4 '17 at 15:18
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    $\begingroup$ @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. $\endgroup$ – farnsy Aug 4 '17 at 15:39
  • $\begingroup$ 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. $\endgroup$ – Giskard Aug 4 '17 at 16:08

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