There are several ways you could write this sentence, with and without the "with", for example: However, by the beginning of 2009, both prices had dropped remarkably (markedly?) with the oil price dropping by $90 and the food price index by 80 points. However, by the beginning of 2009, both prices had dropped markedly, the price of oil by $90 and the food price index by 80 points.

You are correct that you don't need "with" here, but notice these several other changes: "However, by the beginning of 2009, both prices had dropped remarkably: oil by $90, and the food price index by 80 points."