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