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I hope that this is the right stackexchange-site for my question, if not, please move it! sorry!!! :) So, I have a problem with a paper I've got to read for one of my classes, and I think you guys can help me!

Assume there are two countries, and two firms that want to sell products in both countries. Each firm chooses a country to be located in, and sales in the other country are taxed.

First, assume we have one firm $A$:

Quantity demanded is given by $Q_A=\alpha - p_A$, $\alpha$ is some constant and $p_A$ is the selling price. This is the monopolist demand function. (right?) The cost of production is consisting of marginal cost $\omega$, and the trade cost $\tau$ if the products have to be brought from one country into another (direction doesn't matter).

The profit of firm $A$ is then: $\pi_A(p_A)=Q_A*M_A$, where the last factor is the marginal profit per unit, which is $M_A=p_A-\omega-\tau$, $\tau$ only being substracted if there is international trade, so:

\begin{eqnarray} \pi_A^L(p_A)&=&(\alpha-p_A)*(p_A-\omega) \text{ if the firm is local (implied by the index $L$};\\ \pi_A^F(p_A)&=&(\alpha-p_A)*(p_A-\omega-\tau) \text{ if the firm is foreign and has to import the goods.} \end{eqnarray}

Assuming still that $A$ is monopolist, its profits are maximized where the derivative of the profit function is 0, hence we solve (if we leave taxes away now): \begin{eqnarray} (\alpha-p_A)-(p_A-\omega)&=&0 \text{ or, if the firm imports and pays taxes:}\\ (\alpha-p_A)-(p_A-\omega- \tau)&=&0 \end{eqnarray}

and the result is:

\begin{eqnarray} p_A^L&=&\frac{\alpha+\omega}{2} \text{ or}\\ p_A^F&=&\frac{\alpha+\omega+\tau}{2} \end{eqnarray}

and if we plug this into $\pi(p_A)$ to get the maximal profit, which will be

$(\alpha-\frac{\alpha + \omega}{2})*(\frac{\alpha+\omega}{2}-\omega)$ if there is no tax (analogously with $\tau$ and tax), and finally

\begin{eqnarray} \pi_A^L &=&\frac{1}{4}(a-\omega)^2 \\ \pi_A^F &=&\frac{1}{4}(\alpha-\omega-\tau)^2 \end{eqnarray} where the indices $F,L$ indicate if the firm is serving the market from the other side of the interstate border (foreign), paying taxes, of if it is the local firm. Do you follow me until here? I hope the notation is okay.

Now, the thing is that the formulas are not given in the paper, except for the demand function and the final profits, $\pi_A^L, \pi_A^F$. But since the final results are given and coincide, I assume strongly that my calculations and ideas are fine.

Now the problem: assume we have two firms, $A,B$, that serve one country. The paper then, without any calculations, gives the following table:

$\pi_A^{LF} = \frac{1}{9} (\alpha-\omega+\tau)^2$ if the firm $A$ is producing locally and the rival firm abroad, and similarly:

$\pi_A^{LL}=\frac{1}{9}(\alpha-\omega)^2$ if both are local;

$\pi_A^{FF}=\frac{1}{9}(\alpha-\omega-\tau)^2$ if both are foreign and importing;

$\pi_A^{FL}=\frac{1}{9}(\alpha-\omega-2\tau)^2$ if our firm is importing and the rival is local.

How do I get there???

It is clear that the marginal gain is still $M_A=p_A-\omega-\tau$ (with $\tau$ only if we are abroad, again...), but what is the demand curve? The Duopoly demand function? does it look like $Q_A=\alpha-p_A-p_B$? I tried it first, and got close (!), but did not quite receive the profits I'm supposed to receive.

Does anybody still follow me? :) I hope someone can help

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Does noone have any idea? :( I just now thought that I can give details on the article: "Competing for a duopoly international trade and tax competition", by Ben Ferrett, Ian Wooton... – Marie. P. Sep 12 '12 at 20:07
Please make questions self-contained. You don't introduce many of the symbols you use; people shouldn't have to refer to an external article to understand the question, especially not if you don't provide a link to a free online version of the article. – joriki Sep 13 '12 at 9:34
okay, is it better now? I tried to give it more structure and explanaition! :) – Marie. P. Sep 13 '12 at 10:09
up vote 2 down vote accepted

Your exposition is mostly fine. You could state that the costs $\omega$ and $\tau$ are costs per unit, and I think you should state that you assume they are constant and do not depend on the level of production.

Moving to the case of two producers, you must understand that the market does not care who is supplying them. The demand will only depend on the price, and it will be impossible for the two producers to charge different prices since the goods are perfect substitutes. Hence the demand function is $$Q=\alpha - p$$ where the price $p$ is common for the two producers. Since the quantity $Q$ is supplied by the two producers, we have $$Q=Q_A+Q_B$$ You now need to describe your assumptions on how the producers behave, and I believe what you are looking at is what is called the Cournot solution. Then each producer will take the production level of the other producer as given, and try to find an optimal level of its own production. It is easiest to do this by eliminating $p$ and differentiating with respect to the quantity, instead of the other way around as you do in the monopoly case above.

In the case where both producers are local (I'll leave the other cases to you), we get $$\pi _A=q_A(p-\omega)=q_A(\alpha-q_A-q_B-\omega)$$ $$\pi _B=q_B(p-\omega)=q_B(\alpha-q_A-q_B-\omega)$$ (or $\pi_A^{LL}$ and $\pi_B^{LL}$ as you write in this case). The first order conditions are $${\partial \pi_A\over \partial q_A}=\alpha-2q_A-q_B-\omega=0$$ $${\partial \pi_B\over \partial q_B}=\alpha-q_A-2q_B-\omega=0$$ Solving this system with respect to $q_A$ and $q_B$ you get the solution $q_A=q_B={1\over 3}(\alpha-\omega)$. Substituting this into the profit functions, we get $\pi_A=\pi_B={1\over 9}(\alpha-\omega)^2$, which is the answer given in your paper.

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