Walrasian Equilibrium intuition given prices and some initial allocation

Question

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?

Answer 1

The "trick" of this question is that the fact that agents do not want to trade at the given prices does not mean the allocation is Pareto. The only thing you know is that if there is an allocation that is not a CE, there exists a vector of prices at which agents want to trade. Let me give an example:

You have 2 apples and I have 2 bananas. I'm willing to give you half a banana for an apple and you are willing to exchange a whole apple for a bite of a banana. Clearly, the initial allocation is not Pareto Efficient, nor a CE since it is profitable to trade and trade will occur if prices are somewhere between a bite of a banana and half a banana for an apple. The problem, however, assumes that prices are given. So in this toy example, if the price of an apple is set to one banana, there will be no trade (since I won't be willing to accept the prices).

From the lack of trades, it cannot be assumed that you are in a CE since the prices are not allowed to adjust to allow for mutually beneficial trading.