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I hope a question like this is fair game on this website!

I'm doing some research for my thesis, and have come across what seems to be a pretty simple model - two countries, A (representing the USA) and B (representing China), with the typical household optimization/BC, firm profit maximization, and goods clearing. There is a single good produced by each country, and consumers in each country can choose to consume either or both of the goods.

The paper can be found here: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1008734

However, this paper is sorely lacking in any sort of derivations or explanations, and there's one in particular that I've been trying to figure out all day that I just can't:

Using the assumptions made so far, and the definition of the price index, we may write the log deviation in $\frac{P_{BBT}}{P_{Bt}}$, the ratio of country B’s good price to the country B CPI, as $-(1-\alpha)\tau_t$.

(this is on page 8 of the PDF, by the way)

Previously (top of page 8), they have defined $\tau_t$ to be the log deviation of the country B terms of trade; the terms of trade is defined as $\frac{S_{t}P_{AAt}}{P_{BBt}}$ (where $S_{t}$ is the exchange rate) and the whole fraction is assumed to equal to 1 initially.

I'm guessing we can use this to assume the steady state log linearized terms of trade is 0 (since $\ln(1) = 0$), so the log deviation of the above fraction is just $\tau_t = \hat{s}_t + \hat{p}_{AAt} - \hat{p}_{BBt}$

Then, the price index is defined as

$$ P_i = [\alpha P_{ii}^{1-\theta} + (1-\alpha)P_{ij}^{1-\theta}]^{\frac{1}{1-\theta}} $$

but I can't figure out how to incorporate that into the log linearization calculation of $\frac{P_{BBT}}{P_{Bt}}$ to get $-(1-\alpha)\tau_t$. On page 7 there are some equilibrium conditions, which he could be using to arrive at this equality, but based on the wording I don't think that's the case.

I've tagged (and titled) this question as a log-linearization question, but it might be the case that I'm not understanding some aspect of the paper. Thanks a ton to anyone who takes a look at this!

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