I would like to know whether a weighted average can be defined as the product of the different values, to the power of their weights?
Basically, I have this formula and I have to describe it in one or two sentances:
$$E_{i,t}=\prod_{j=1}^{n}(S_{i,j,t}P_{i,t}/P_{jt})^{w_{i,j}}$$
E is the real effective exchange rate of country i at time t
S is the bilateral exchange rate between country i and country j
The two P's represent the price level in the relevant countries
Lastly, Wij is the weight of country j in the overall trade activities in country i.
I believe this can be described as: the Real effective exchange rate is the trade weighted average of bilateral real exchange rates between the home country and a basket of other countries. But, I am uncertain, since until now, I have seen weighted averages represented differently.
So does this formula correspond to a weighted average?
Yes, it is similar to the Weighted geometric mean.
If $\{x_1, x_2, ... x_n\}$ are the values, and $\{w_1, w_2, ... w_n\}$ are the weights, the weighted geometric mean is defined as
$$(\prod_{i=1}^{n}x_i^{w_i})^{1/\sum_{i=1}^{n}w_i}$$
Note that this version accounts for the sum of the $w_i$'s not being 1.