# From Walras’ auctioneer to continuous time double auctions: A general dynamic theory of supply and demand

In standard Walrasian auctions, the price of a good is defined as the point where the supply and demand curves intersect. Since both curves are generically regular, the response to small perturbations is linearly small. However, a crucial ingredient is absent of the theory, namely transactions themselves. What happens after they occur? To answer the question, we develop a dynamic theory for supply and demand based on agents with heterogeneous beliefs. When the inter-auction time is infinitely long, the Walrasian mechanism is recovered. When transactions are allowed to happen in continuous time, a peculiar property emerges: close to the price, supply and demand vanish quadratically, which we empirically confirm on the Bitcoin. This explains why price impact in financial markets is universally observed to behave as the square root of the excess volume. The consequences are important, as they imply that the very fact of clearing the market makes prices hypersensitive to small fluctuations.

# Introduction

One of the most time-worn statement of economic science is that “prices are such that supply matches demand”. In order to explain how this really comes about, one usually invokes a Walras auctioneer, who attempts to measure the supply and demand curves $$S(p)$$ and $$D(p)$$, that give the total amount of supply/demand for a given good (or asset), would the price be set to $$p$$. The equilibrium price $$p^*$$ is then such that $$D(p^*) = S(p^*)$$, which maximiz