Uniqueness of utilities in competitive equilibrium

Question

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.

Answer 1

A sufficient but perhaps not necessary condition would be if the demand functions of all consumers were decreasing in price. (Price ratio to be exact.) This would cause the aggregate demand to be decreasing in price, and since the aggregate supply is constant there could only be one equilibrium price. If there is only one equilibrim price then for each consumer there is only one utility maximization problem and it has a maximum.
Denote by $\omega_m$ the initial endowment of good $m$. Let $e$ denote the income derived from the initial endowments given the prices, so $e = p_x\cdot \omega_x + p_y\cdot \omega_y$ Then the condition would be $$ \frac{d \ x(p_x,p_y,e)}{d \ p_x} = \frac{\partial x(p_x,p_y,e)}{\partial p_x} + \frac{\partial x(p_x,p_y,e)}{\partial e} \cdot \frac{d \ (p_x\cdot \omega_x + p_y\cdot \omega_y)}{d \ p_x} < 0 $$ This looks like the Slutsky equation so there are probably some theorems on what this implies, but I cannot readily say what this says about the underlying utility function. Most common examples (perfect substitutes, Cobb-Douglas, quasilinear) fulfill this condition.

EDIT: I first said the perfect complement utility functions always fulfill this condition but this is not true.

Answer 2

If you look at this manuscript by Jonathan Levin (2006), pages 25-26 provide an example where there is multiple equilibria, and calculating payoffs in this case (if I am not making any calculation mistakes) gives different levels of utilities in these different equilibria.