Suppose several traders are given some initial endowment of goods. Then, a free market opens and they trade until the market is at a competitive equilibrium. Each trader now has a final utility which is at least as high as his initial utility.
MY QUESTION: Is the vector of final utilities unique? I.e, is this possible that, with the same initial endowments and the same utility functions, there will be two different equilibria in which the final utilities are different?
EXAMPLE: Suppose there are two goods and two traders, Alice and Bob, with the same utility function: $u(x,y)=x+y$. The initial endowment is $(10,0)$ for Alice and $(0,10)$ for Bob. In competitive equilibrium, the price vector has $p_x=p_y$. There are many different equilibrium allocations, for example: $(10,0),(0,10)$, $(9,1),(1,9)$ and $(5,5),(5,5)$ are all equilibrium allocations. But, in all these allocations, the utilities of both traders are the same: $(10,10)$. I would like to know in what cases this uniqueness happens.