Suppose we have two agents who are each assigned some initial allocation of two different goods, where the prices of each good are given. Also, suppose the utility functions for each agent are weakly concave and strictly increasing.
If, after receiving their allocations, no trades occur between the two agents, can we infer that this initial allocation represents a competitive equilibrium (CE)?
My intuition says that it may not be a CE, because perhaps agent 1 has an optimal allocation, but maybe agent 2 wishes that he could trade with agent 1 to change his (agent 2's) allocation. Although this would not be a competitive equilibrium, this scenario would still be Pareto efficient. Is my analysis correct?