Uniqueness of utilities in competitive equilibrium

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.

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.

• What exactly is meant by "decreasing in price"? If $p_x$ increases and all other prices remain equal, then naturally the demand for $x$ decreases. But what if $p_x$ increases while $p_y$ decreases? Then (if $x$ and $y$ are complementary goods), it is not clear whether the demand for any of them increases or decreases. – Erel Segal-Halevi Oct 10 '15 at 20:07
• @ErelSegal-Halevi this is why the next sentence reads "(Price ratio to be exact.)" – Giskard Oct 10 '15 at 22:13
• If $p_x$ increases and $p_y$ decreases, then the price ratio between them strictly increases. – Erel Segal-Halevi Oct 10 '15 at 22:23
• @ErelSegal-Halevi In this case, when the income is derived solely from the value of the initial endowment, so $p_x \cdot \omega_x + p_y \cdot \omega_y$, it would just mean that the price ratio increased. Price levels are irrelevant. – Giskard Oct 10 '15 at 23:14
• @ErelSegal-Halevi I also say at the end of my answer that perfect complement utility functions do not fulfill this condition. – Giskard Oct 11 '15 at 12:47

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.

• Thanks. It seems from this paper that, if the goods are gross-substitutes, then there is a unique equilibrium. – Erel Segal-Halevi Oct 10 '15 at 20:09