A Note on Simple Strategic Bargaining for Models of Money or Credit

• Published date: 01 March 2019
• Author: YU ZHU
• Date: 01 March 2019
• DOI: http://doi.org/10.1111/jmcb.12508

DOI: 10.1111/jmcb.12508

YU ZHU

A Note on Simple Strategic Bargaining for Models

of Money or Credit

Many papers on liquidity have bilateral trade with buyers constrained by

their money holdings or debt limits. Axiomatic bargaining, typically Nash,

determines the terms of trade. However,there are reasons to prefer strategic

bargaining. I analyze a bargaining game that is useful in models of liquidity.

Advantages include (i) it has simple microfoundations, both in and out of

steady state; (ii) it is more tractable than Nash, but the outcomes share

interesting features; (iii) the benchmark version is consistent with axiomatic

approaches in the literature, while another version is not, but can still be

used; and (iv) it is arguably realistic.

JEL codes: E42, E51

Keywords: strategic bargaining, liquidity constraint, NewMonetarist

model.

ALARGE RECENT LITERATURE ANALYZING money, credit, and

liquidity builds on the New Monetarist model introduced in Lagos and Wright (2005)

and Rocheteau and Wright (2005) (see surveys by Nosal and Rocheteau 2011 and

Lagos, Rocheteau, and Wright 2017). An important ingredient in the framework is a

frictional market in which money or credit facilitates trade. In this market, buyers and

sellers trade bilaterally, and the former are constrained by their liquidity positions—

that is, their money holdings or debt limits. Bargaining is typically used to determine

the bilateral terms of trade (although there are other options, as discussed in the

above-mentioned surveys). Usually, axiomatic solution concepts such as Nash or

Kalai bargaining are used. However, in many scenarios, there are reasons to prefer

I am very grateful to Randall Wright for his valuable suggestions and comments. I thank the editor and

two anonymous referees for their helpful comments. I also thank Chao He and Han Han for their input.

I acknowledge the support from the Department of Economics at the University of Leicester. The views

expressed in this paper are solely those of the author and may differ from official Bank of Canada views.

No responsibility for them should be attributed to the bank.

YUZHU is a Senior Economist, Funds Management and Banking Department, Bank of Canada (E-mail:

zhuy@bankofcanada.ca).

Received November 18, 2016; and accepted in revised form March 23, 2018.

Journal of Money, Credit and Banking, Vol. 51, Nos. 2–3 (March–April 2019)

C

2018 The Ohio State University

740 :MONEY,CREDIT AND BANKING

strategic bargaining. This note analyzes a particular strategic bargaining game that is

useful in this context and, more generally, in other frameworks such as labor search

and matching models, where liquidity issue matters.1

The game has two stages. First, a seller proposes terms of trade. If accepted,

the game ends with trade. If rejected, they proceed to the second stage, where one

agent is randomly given the chance to make a take-it-or-leave-it offer. This simple

game works well in monetary models and has advantages over other bargaining so-

lutions. Versions of this game appear elsewhere (e.g., see Marchesiani and Nosal

2017 and Wright and Wong 2014, for recent examples and discussions) but not

in models where liquidity is important. A related game appearing in some search-

and-bargaining models (e.g., Gale 1987 or Mortensen and Wright 2002) is similar,

except it does not have the first stage—the agents simply flip a coin to see who

makes a take-it-or-leave-it offer. Since agents in these models have linear utility,

this does not matter. But here, with nonlinear utility, agents generally prefer to

use the first stage to avoid the risk associated with the coin flip. This is an ad-

vantage of the version studied here. There are several other advantages, as I now

discuss.

First, it has simple microfoundations in discrete time, which is the natural way

to formulate the monetary environment. Taking the standard approach in Rubinstein

(1982) and Binmore, Rubinstein, and Wolinsky (1986), and deriving Nash as the

limit of an alternating offer game as the time between offers goes to 0, is awkward in

Lagos–Wright, where bargaining should not go on forever because in finite time the

frictional market closes. Of course there are options, including working in “imaginary

time,” but this does not solve the whole problem. Consider a nonstationary equilib-

rium and suppose agents do not have linear utility, both of which are important in

monetary applications. Coles and Wright (1998) and Coles and Muthoo (2003) show

the approach in Binmore, Rubinstein, and Wolinsky (1986) delivers a path for the

terms of trade solving a differential equation that has the Nash solution as a steady

state, but is different from and more complicated than Nash outside of steady state.

Indeed, the authors argue that using Nash outside of steady state is tantamount to

having agents bargain strategically but myopically. The game used here has no such

problem.

Second, while the game is much simpler than Nash, the outcome shares some

interesting features in models of money and credit. One is that monetary economies

generally cannot achieve the efficient allocation at the optimal monetary policy with

Nash and, as I show, also with the game considered here. This is not true of all so-

lution concepts (e.g., Kalai bargaining always gives efficiencyat the optimal policy).

A related feature of both Nash and the solution here is that a buyer’s surplus is not

generally monotone in his liquidity position, which can give rise to complicated equi-

librium dynamics in models with endogenous debt limits (Gu et al. 2013, Bethune,

Hu, and Rocheteau Forthcoming). This does not happen with all solution concepts

(e.g., again it cannot happen with Kalai). Also, while in either case one can prove

1. A straightfoward example is provided in Section 3.