# competitive equilibrium, Walrasian equilibrium, Walrasian auction

There are several sellers holding some indivisible goods, and several potential buyers with different valuations for these goods. I need to calculate the Walrasian equilibrium in this scenario, but first I need to understand what exactly is a Walrasian equilibrium. So I am looking for a short, formal, operational definition of this term.

Wikipedia just links to a page about General equilibrium theory, which is very long and verbal, but I couldn't find there a formal definition of a Walrasian equilibrium.

This page also hints that a Walrasian equilibrium is similar to Competitive equilibrium, but I didn't understand whether they are identical or different, and if they are different - what is the relation between them?

There is also a page about a Walrasian auction, but again, I am not sure what is the relation between this and a Walrasian equilibrium?

The definition in the about.com dictionary looks promising: "An allocation vector pair (x,p), where x are the quantities held of each good by each agent, and p is a vector of prices for each good, is a Walrasian equilibrium if (a) it is feasible, and (b) each agent is choosing optimally, given that agent's budget. In a Walrasian equilibrium, if an agent prefers another combination of goods, the agent can't afford it." But, they do not define what they mean by "feasible".

I would very much appreciate an orderly explanation of the relation between all these different terms.

Yes, Walrasian equilibrium and competitive equilibrium are used interchangably. Both refer to a set of allocations and prices such that

1. Taking prices as given, every agent weakly prefers their allocation to any other possible allocation they might receive (i.e., in an indivisible goods market, they do not want to buy some goods that they are not allocated in equilibrium, nor do they wish they didn't have to buy some goods that they are allocated in equilibrium).

2. The market clears: the entire endowment of goods is allocated, and no good is allocated more than once. (This is the feasibility requirement).

Note that in the case where the market has distinct seller and buyer roles (e.g. an auction), 1. and 2. jointly require that any unsold goods have a price of zero.

The term Walrasian auction is something rather unrelated. The standard classical model of supply and demand in a perfectly competitive market (i.e. the first thing you see in Econ 101) predicts price and quantity are set when supply is equal to demand. But how do we get there? Leon Walras proposed a thought experiment in which a hypothetical auctioneer called out prices and people made offers until the market converged on the prefect competition equilibrium. This hypothetical process is known as a Walrasian auction or, sometimes, Walrasian tâtonnement.

• Is it correct to say that a Walrasian auction converges to a Walrasian equilibrium? Jan 16, 2015 at 10:29
• Yes, this is true. But it only works because the Walrasian auctioneer is thought of as being able to either increase or reduce prices until he finds the equilibrium. If you constrain an auctioneer to only increase prices and if you have goods that are substitutes then tâtonnement won't work and you need to be much cleverer about how you run the auction. This insight is due to Gul and Stachetti (2001): princeton.edu/~fgul/english.pdf. Jan 16, 2015 at 10:36

The definition from about.com is what you are looking for. Feasible means that total consumption of each good does not exceed total endowment (initial holding) of that good in the economy. Terms Walrasian and Competitive equilibrium can be used interchangeably here and Walrasian auction is entirely different concept.

• So a Walrasian equilibrium is just another name to a competitive equilibrium? Jan 16, 2015 at 9:50
• It generally is. Sometimes competitive equilibrium may refer to other concepts, namely Arrow-Debreu equilibrium. Jan 16, 2015 at 10:33

Walrasrian Equilibrium is just basically the same as Market Equilibrium or Competitive Equilibrium. According to the books I am reading, market clears in equilibrium; demand = initial endowment.