# Economics: two rival firms in two countries

I am currently working on a paper in macroeconomics, where I found a result that I cannot manage to understand. Since we don't have a macroeconomic site yet, and this is mostly game theory, I will post it here.

On pages 787/788, they are talking about the situation in an asymmetrical market.

On page 787 (beginning of chapter "market size asymmetries"), they investigate whether the firms will stay in their countries as the market A grows. It is said that "because $\Pi_{AA}<\Pi_{AB}$ for all $n\geq1$, it is clear that the firm in B will not deviate to join its rival in country A."

But according to the introduced notation on page 781 (Where $\Pi_{ij}$ denotes the firm's profit when it is located in $i$ and the competitor in $j$), this statement would have to be expressed as $\Pi_{AA}<\Pi_{BA}$. On the other hand, the inequality in the text describes the decreasing profit of the firm in $A$ as the rival moves from $B$ to $A$, but since the rival's profits are irrelevant to each firm, and they cannot influence the rival's decision, this should not be relevant.

Similarly on the next page, we want to consider if the company in $A$ will move and join its rival in $B$, hence I think the left-hand side of (13) should read $\Pi_{BB}<\Pi_{AB}$ instead of the given $\Pi_{BB}<\Pi_{BA}$.

I know that this is quite an extensive thing to ask, and it requires quite some effort from any of you to get into the topic, but I hope that maybe one of you has the time to shortly explain what I am doing wrong with the notation, because I then also cannot follow the counter-intuitive conclusion that follows, according to which both firms "will optimally co-locate in country B" when $n$ is sufficiently large. Why are not both firms attracted to $A$ when the sufficiently large market size and reduced trade cost makes up for the increased competition? From (6), we can see that $\Pi_{AB}>\Pi_{BB}$ if $n>1$, so the company in $A$ should stay in $A$.

Best regards, M.

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From the point of view of the foreign firm you need to compare $\Pi_{BA}$ and $\Pi_{BB}$ indeed. Staying in B with a tax $T_B= \Pi_{BA}$ yields no profit.
Suppose a deviation to country A, then the firm earns $\Pi_{AA}$ which is inferior to $\Pi_{AB}$ by prop (6). Hence moving to country A for a profit $\Pi_{AA} \leq \Pi_{AB}=T_A$ will lead to a negative overall profit.
For the symetrical argument: you are firm A and consider a move from $\Pi_{AB}$ --> $\Pi_{BB}$; in the first case, you have $\Pi_{tot}= \Pi_{AB} - T_A= 0$ in the second $\Pi_{BB}- T_B \leq 0$ iff $\Pi_{BB} \leq \Pi_{BA}$ does hold for any n.
ah, so we are in fact comparing $\Pi_{BB}<T_B=\Pi_{BA}$ and $\Pi_{AA}<T_A<\Pi_{AB}$, i.e. the profit with the taxes, and not any rival firm's movement. now it's clear. thanks so much for the effort! – Marie. P. Oct 17 '12 at 11:11