Approximation of a sum within an exponential

Question

There is a well-known equation in international economics, called the gravity equation. This equation expresses imports from a country as a function of importer size, exporter size, and distance. In particular, $$ X_{ij}=\frac{Y_{i}Y_{j}}{\tau_{ij}} $$ Here, the LHS measures the value of imports by $i$ from~$,$$\tau_{ij}$ represents distance between $i$ and $j,$$Y_{i}$ represents $i's$ size and~$Y_{j}$ represents $j's$ size., Now, in a regression framework, one estimates this equation in additive form: $$ lnX_{ij}=lnY_{i}+lnY_{j}-ln\tau_{ij}+\epsilon_{ij} $$ where $\epsilon_{ij}$ is a random measurement regression error term. The OLS predicted values one obtains are defined as $\widehat{lnX_{ij}}$. As always, the hats denote the orthogonal projection of $X_{ij}$ onto the space spanned by $Y_{i},Y_{j}$ and $\tau_{ij}.$Without getting into the details of the procedure, one wishes to add over all the trading partners for a country $i,$ after exponentiating this expression. In other words, total\emph{ predicted} imports by $i$ are $$ \sum_{j\neq i}exp\left(\hat{\beta}_{1}lnY_{i}+\hat{\beta}_{2}lnY_{j}+ln\tau_{ij}\right) $$ The expression above is quite complicated. If there were no exponential, this would be easily simplified. Can this be approximated by a linear function, or any function that allows me to say collect terms that don't involve $j$ , and just make this expression more tractable? I feel that this question is not apt for the economics forum, as the question is not about the economics.