# Optimal tax level that raises revenue R < e given Hicksian demand elasticities

I have to find a uniform tax rate that raises revenue R < e and I am given a Hicksian cross price elasticity demand matrix below. I know I want to use Ramsey's first-best but I am not sure, how to derive $$\tau=\frac{R}{R+wT}$$ given these elasticities. I have scoured the internet trying to understand it better but the only thing I am seeing is that Ramsey's first best assumes cross-price elasticities to be zero, and from there you can you inverse elasticity rule where the tax rate is proportional to its own price elasticity. I am not really understanding this: if we have 3 different goods, each with its own price elasticity, how can we determine a uniform tax rate? Completely not sure how to start thinking about this. Any help would be much appreciated.

• I think that the first thing to clear up is that the problem is looking for a single tax rate that optimizes revenues, e.g., minimizes deadweight loss, given this market in which various recreational drugs are partial substitutes for one another. It is not asking you to find the tax rate that will reduce deadweight loss to zero (which would be a so-called "first-best" outcome). – heh Feb 19 at 15:34