THE PHILLIPS CURVE IN A MATCHING MODEL

• Author: Tai‐Wei Hu,Neil Wallace
• Published date: 01 November 2019
• DOI: http://doi.org/10.1111/iere.12393
• Date: 01 November 2019

INTERNATIONAL

ECONOMIC

REVIEW

November 2019

Vol. 60, No. 4

DOI: 10.1111/iere.12393

THE PHILLIPS CURVE IN A MATCHING MODEL∗

BYTAI-WEI HUAND NEIL WALLACE1

University of Bristol, U.K.; Penn State University, U.S.A.

Following ideas in Hume, monetary shocks are embedded in the Lagos-Wright model in a new way: There

are only nominal shocks accomplished by individual transfers that are sufficiently noisy so that realizations

of those transfers do not permit the agents to deduce much about the aggregate realization. Assuming that

the distribution of aggregate shocks is almost degenerate, aggregate output increases with the growth rate of

the stock of money—our definition of the Phillips curve. This almost degeneracy assumption is far from being

necessary; under some mild conditions, the Phillips curve result holds for a large class of distributions.

1. INTRODUCTION

In his Nobel lecture (see “Monetary Neutrality,” Lucas, 1996), Lucas begins by describing

Hume’s (1970) views about the effects of changes in the money supply. Lucas emphasizes that

Hume’s views were dependent on how changes in the quantity of money come about. In order to

get neutrality, Hume set out very special conceptual experiments which, when correct, amount

to changes in monetary units. For some other kinds of changes, Hume says that there is a

short-run Phillips curve:

Accordingly we find that, in every kingdom into which money begins to flow in greater abundance

than formerly, everything takes a new face: labour and industry gain life, the merchant becomes more

enterprising . . .

To account, then, for this phenomenon, we must consider, that though the high price of commodities

be a necessary consequence of the encrease of gold and silver, yet it follows not immediately upon that

encrease, but some time is required before the money circulates through the whole state, and makes its

effect be felt on all ranks of people. At first, no alteration is perceived; by degrees the price rises, first of

one commodity, then of another, till the whole at last reaches a just proportion with the new quantity

of specie in the kingdom. In my opinion, it is only in this interval or intermediate situation, between the

acquisition and rise of prices, that the encreasing quantity of gold and silver is favorable to industry.

When any quantity of money is imported into a nation, it is not at first dispersed into many hands but is

confined to the coffers of a few persons, who immediately seek to employ it to advantage. (Hume 1970,

p. 37)

∗Manuscript received February 2018; revised September 2018.

1We are grateful to seminar participants at National Taiwan University, University of Bath, Penn State–Cornell

Macro workshop, Richmond Fed, and UNSW. We are also grateful to the referees for helpful comments on an earlier

draft of this article. Please address correspondence to: Tai-Wei Hu, Department of Economics, University of Bristol,

Priory Road Complex, Priory Road, Bristol BS81TU, United Kingdom of Great Britain and Northern Ireland (GB).

E-mail: taiwei.hu@bristol.ac.uk.

1469

C

(2019) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

1470 HU AND WALLACE

Hume asserts that there is a positive association between the changes in the stock of money

and real economic activity, which is our definition of the Phillips curve.2He also offers what

may at some time have been regarded as an explanation of it. A modern economist would not

treat his discussion as an explanation, but might look to it for hints about modeling ingredients

that, when rigorously analyzed, could conceivably constitute an explanation.

The passage from which the above excerpt comes contains at least two hints about modeling

ingredients. First, changes in the quantity of money come about in a way that gives rise to

changes in relative money holdings among people. In particular, the changes for individuals

are not uniformly proportional to initial holdings as is required for neutrality. Second, trade

seems to be occurring within small groups, rather than in a centralized market. That suggests

the use of some sort of search/matching model. Given those ingredients, the passage hints at

two conjectures that might be studied. One is that a change in relative money holdings has

Phillips-curve-type effects that dissipate over time through the effects of subsequent trades on

those relative holdings. The other is that the change occurs in a way that is not seen by everyone

when it occurs and that the Phillips-curve-type effects dissipate when people learn about it.

Although these are not mutually exclusive conjectures, we pursue only the second here.

In order to study it, we embed monetary shocks in the Lagos and Wright (2005; LW) model and

assume that the aggregate monetary shocks (i) are observed with a lag and (ii) are accomplished

by way of individual transfers in such a way that those transfers are imperfect signals about

the aggregate shock. The information imperfection is modeled in the usual way: There is a

fixed support for individual transfers, and the aggregate shock determines the distribution over

that fixed support. Our main contribution is to show that when these transfers are relatively

uninformative about the aggregate shock, there is a Phillips curve. For general parameter values,

we show this when the distribution of the aggregate shock is close to a degenerate distribution—

that is, when shocks are rare. But this near degeneracy is far from being necessary: Under some

mild conditions, we obtain the Phillips curve for a large class of distributions of the aggregate

shock. We also provide a counterexample that illustrates how things could go wrong.

We are not the first to study the Phillips curve in a model of small-group trade. Wallace

(1997) and Katzman et al. (2003) do so in a model in which money holdings are limited to be

in the set {0,1}. However, Phillips curve results in both papers depend on the assumption that

less than half the population has money, an assumption that is troublesome because it has no

analog when money is divisible. Nor are we the first to use some version of LW to study the

Phillips curve. Faig and Li (2009) embed a version of the signal-extraction problem in Lucas

(1972) into that model. That signal-extraction problem involves a delicate confounding of real

and nominal shocks, and the sign of the slope of the Phillips curve depends on preferences.

In contrast, our Phillips curve relies on the assumption that individual transfers are relatively

uninformative about the aggregate shock. Moreover, our specification is closer to Hume, is

conceptually simpler, and is strategic.3In it, people meet in pairs and do not see the transfers

received or the trades in other meetings. Therefore, they can only use what they experience in

their meeting to draw inferences about the aggregate shock.

2. THE MODEL

The background model is that in LW. Time is discrete, and there are two stages at each date.

In the first stage, the decentralized market (the DM), production and consumption occur in

pairwise meetings that occur at random in the following way. Just prior to such meetings, each

person looks forward to being a consumer (buyer) who meets a random producer (seller) with

probability σ, looks forward to being a producer who meets a random consumer with probability

σ, and looks forward to a no-coincidence meeting with probability 1 −2σ, where σ≤1/2. In

2To be precise, we use the term Phillips curve to mean that total output is strictly increasing in the growth rate of

the stock of money. As is true of many models of the relationship between money and economic activity, there is no

unemployment in our model.

3Lucas (1972) and Faig and Li (2009) use rational-expectations competitive-equilibrium as a solution concept and

have agents learn from the prices they see. Such learning has not been given a strategic foundation.