A dynamic analysis of nash equilibria in search models with fiat money

Abstract

We analyze the rise in the acceptability fiat money in a Kiyotaki–Wright economy by developing a method that can determine dynamic Nash equilibria for a class of search models with genuine heterogeneous agents. We also address open issues regarding the stability properties of pure strategy equilibria and the presence of multiple equilibria, numerical experiments illustrate the liquidity conditions that favor the transition from partial to full acceptance of fiat money, and the effects of inflationary shocks on production, liquidity, and trade.

Introduction

One central question of monetary economics is how an object that does not bring utility per se becomes accepted as a means of payment. It is well understood that the emergence of money depends on both trust and coordination of beliefs. While some recognize this observation and simply assume that money is part of the economic system, others have tried to explain the acceptance of money as the result of individuals’ interactions in trade and production activities.¹ Among the best-known attempts to formalize the emergence of money in decentralized exchanges is Kiyotaki and Wright (1989, henceforth, KW). The static analysis in KW provides important insights on how the specialization in production, the technology of matching, and the cost of holding commodities condition the emergence of monetary equilibria. Nevertheless, important issues are yet to be resolved. First, one would like to know if and how convergence to a particular long run equilibrium occurs from an arbitrary initial state of the economy. Historical accounts describe the different patterns that societies have followed in adopting objects as a means of payment.² What are the dynamic conditions that lead individuals in a KW economy to accept either commodity or fiat money? Second, static analysis gives little guidance about the short run consequences of a shock that causes, for instance, a sudden rise in inflation. How does the degree of acceptability of commodity and fiat money change with inflation?

The determination of dynamic equilibria in a KW environment is challenging. In an effort to improve its tractability, new classes of monetary search models have been proposed. These have incorporated some features of centralized exchanges but have also eliminated others, most notably the genuine heterogeneity across individuals and goods, and the storability of goods (see Lagos et al. (2017), for a recent review). Restoring these features turns out to be a useful exercise for characterizing the rise of money as a dynamic phenomenon.

The study of money acceptance in a KW environment requires a departure from the conventional set of tools employed to characterize the dynamics of an economy with centralized markets. Our method combines Nash’s (1950) definition of equilibrium with Perron’s iterative approach to prove the stable manifold theorem (see, among others, Robinson (1995)). This is the first work, to our knowledge, that shows how to determine pure strategy dynamic Nash equilibria in a KW search environment with fiat money. Previous works on the subject considered economies without fiat money, and often assumed bounded rationality.³ An exception is Iacopetta (2019) that also studies dynamic Nash equilibria in a KW environment, but does not consider fiat money. The work presented here extends the KW environment of Iacopetta (2019) in that it introduces fiat money and also considers seignorage (Li, 1994, Li, 1995). In the present environment it is possible to explicitly address how the distribution of individuals’ characteristics affects the emergence of a partial or full monetary equilibrium.

Steady-state results echo those of the inventory-theoreticmodels of money (e.g., Baumol (1952), Tobin (1956), and Jovanovic (1982)). For instance, higher levels of seigniorage may induce some to keep commodities in the inventory instead of accepting money, as a way to minimize the odds of being hit by seignorage tax. The dynamic analysis, however, generates novel results: it shows how changes in the liquidity of assets other than money can alter the proportion of individuals who accept fiat money in transactions. For instance, it reveals that an economy that converges to a long run equilibrium in which all prefer fiat money to all types of commodities (full acceptance) may go through a phase in which only a fraction of individuals do so (partial acceptance).⁴

The remainder of the paper is organized as follows: Section 2 describes the economic environment, characterizes the evolution of the distribution of inventories and money and defines a Nash equilibrium. Section 3 overviews steady-state Nash equilibria for some specifications of the model. Section 4 presents a methodology to determine Nash equilibria. Section 5 illustrates the acceptability of money and discusses multiple steady states through numerical experiments. Section 6 contains welfare considerations. Section 7 comments on future research. Appendix A contains proofs and mathematical details that are omitted in the main text. Appendix B explains how the stable manifold theorem is related to our solution algorithm.