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As a part of self-study for entrance exams I was trying to solve the following problem: enter image description here

The problem with this question is that I feel like the equation for $TB$ is incorrect, because we are given that $e$ has to be adjusted for $TB=0$. How can the trade balance be independent of the real exchange rate, and under such a scenario, how does $e$ adjust to balance the trade if it is not even part of $TB$?

I feel like the expression for trade balance should be $TB=\bar T + \frac{\beta eP^*}{P}-mY=\bar T +\beta \epsilon -mY$ where $\epsilon=\frac{eP^*}{P}$ is the real exchange rate This formulation of $TB$ also ensures that the Marshall-Lerner conditions hold as a real depreciation (increase in $\epsilon$) increases the trade balance. Please let me know your thoughts on this

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