If Demand Varies Continuously with Price, Then Even If There
If demand varies continuously with price, then even if there are thousands of goods, there will be at least one set of prices such that demand equals supply in every market.
If allocation x is a competitive equilibrium at prices p and if everybody likes his bundle in allocation y better than his bundle in allocation x, then the total value of allocation y at prices p exceeds the total value of allocation x at prices p.
If the initial endowment is on the contract curve, then there must always be a competitive equilibrium in which no trade takes place.
Jack Spratt's utility function is U(F, L) = L. His wife's utility function is U(F, L) = F. If Jack's initial endowment is 10 units of F and 5 units of L and if Jack's wife's initial endowment is 6 units of F and 10 units of L, then in an Edgeworth box for Jack and his wife, an allocation of F and L will be Pareto optimal only if it is at a corner of the box.