Bargaining Under Liquidity Constraints:
Experimental Evidence∗John Duffy†

Lucie Lebeau‡

Daniela Puzzello§

September 16, 2022

Abstract
Bargaining is widely used in monetary, labor and finance models to determine termsof trade. The chosen bargaining solution can matter for welfare analysis, for example
when agents are liquidity constrained. Here we report on an experiment in which buyers
and sellers engage in semi-structured bargaining to determine the terms of trade withthe aim of evaluating the empirical relevance of two bargaining solutions, the generalized
Nash bargaining solution and Kalai’s proportional bargaining solution. These bargaining
solutions predict different outcomes when buyers are constrained in their money holdings.
We first use the case when the buyer is not liquidity constrained to estimate the bargaining
power parameter, which we find to be equal to 1/2. Then, imposing liquidity constraints
on buyers, we find strong evidence in support of the Kalai proportional solution. Ourfindings have policy implications, e.g., for the welfare cost of inflation in search-theoretic
models of money.
JEL Codes: C78, C92, D83.
Keywords: Bargaining, Monetary Economics, Experimental Economics.
∗

We gratefully acknowledge funding for this project from the National Science Foundation under grants

SES#1529272 and SES#1530820. The experimental protocol used to collect data for this study was approvedby the UC Irvine Institutional Review Board, HS# 2015-2082. The views expressed in this paper are thoseof the authors and are not necessarily reflective of views at the Federal Reserve Bank of Dallas or the Federal
Reserve System.
†
Department of Economics, University of California, Irvine (duffy@uci.edu)‡
Federal Reserve Bank of Dallas (lucie.lebeau@dal.frb.org)§
Department of Economics, Indiana University, Bloomington (dpuzzell@indiana.edu)

1Introduction

Decentralized markets are a key feature of many workhorse models in monetary and labor economics and finance (e.g., Aruoba et al. (2007), Duffie et al. (2005), Lagos and Wright (2005),
Lagos and Zhang (2020), and Weill (2020)). These settings typically involve bilateral meetings where determination of the terms of trade (quantity and payment) is not obvious. One
common approach is to suppose that traders bargain over the terms of trade. Importantly, theproperties of the bargaining solution matter both qualitatively and quantitatively, particularly
when agents face constraints. In this paper we focus on liquidity constraints but our approach
would also apply to other constraints such as on production capacity or hours worked. If
agents are liquidity constrained, then the solution that characterizes bargaining outcomesmatters for welfare analyses in New Monetarist models (Craig and Rocheteau (2008)). Twomain solutions are considered in this literature, the generalized Nash solution (Nash (1950),
Nash (1953)) and Kalai’s proportional bargaining solution (Kalai (1977)). In this paper, weseek to understand which bargaining solution is more empirically relevant in such models. We
address this question by placing subjects in the typical bargaining setting faced in the New
Monetarist models and observing the allocations that they agree upon. We find strong support
for the Kalai bargaining solution and we further explore the implications of this finding for
the welfare cost of inflation.
Nash bargaining solution follows uniquely from several axioms: solutions are assumed to be
individually rational, Pareto efficient, independent of irrelevant alternatives and invariant toscale changes in utility representations. By contrast, Kalai’s approach replaces the last axiom,
invariance to scale transformations of utility, with a strong monotonicity axiom, which implies
that the resulting solution exhibits proportionality; if there are higher gains from trade, thenboth parties must gain from that expansion proportionally. A third solution possibility, due
to Kalai and Smorodinsky (1975) replaces the independence of irrelevant alternatives axiomof Nash with a weaker version of Kalai’s monotonicity axiom. As the applied literature mainly
uses the Nash or Kalai solutions and since the Kalai-Smorodinsky solution responds similarly
to the Nash solution as liquidity constraints are varied, we chose to focus our attention on acomparison between the Nash and Kalai solutions only.
Relative to the existing experimental bargaining literature (for surveys, see, e.g., Roth1

(1995), Camerer (2003) and Güth and Kocher (2014)) we consider an environment that differsin two important aspects. First, the payoff functions of the buyer and seller are nonlinear andas already noted, there are two dimensions to the bargaining problem, over quantities andmoney. Second, we consider the case where buyers may be liquidity constrained.
We show that in the case where buyers are not liquidity constrained, the Nash and Kalai
approaches yield the same solution. However, in the case where buyers are liquidity constrained
and the payoff functions are nonlinear, we show that the Nash solution will generally differ fromthe Kalai solution, enabling a direct test as to which bargaining solution best characterizes
the experimental data.
An alternative approach to the one we pursue here would be to directly test the axioms
that Nash and Kalai rely upon to determine the bargaining solution. However, as we candirectly observe trading behavior in our experiment, it seems less interesting to evaluate moregeneral axioms, e.g., Pareto optimality, than to consider the relevance of different bargaining
solutions for predicting exchange outcomes. Further, to test the axioms that underlie thetwo different bargaining solutions would require testing of different sets of axioms for eachsolution that need not overlap. Such an exercise has been attempted already by Nydegger andOwen (1974) who find support for some (but not all) of the axioms found in Nash (1950) andfor some (but not all) of the different axioms found in Kalai and Smorodinsky (1975). Morerecently, Navarro and Veszteg (2020) confirm that one axiom assumed in the Nash (1950)
and Kalai and Smorodinsky (1975) solutions – scale invariance – finds little support in theirexperimental unstructured bargaining experiments.
Our experiment consists of three treatments. In the first treatment, buyers and sellers areunconstrained in their ability to achieve the first best allocation. In the other two treatments,
buyers are constrained by their money endowments from implementing the first best allocation.
The first, unconstrained treatment, enables us to estimate the distribution of bargaining power
between the buyer and the seller. Given these bargaining weights, the two bargaining solutions
predict different allocations in the two constrained treatments.
To preview our results, in the unconstrained case we estimate that the bargaining weight isequal to 1/2. Using that bargaining weight we compare the predictions of the Nash and Kalai
bargaining solutions in the constrained case and we find strong evidence favoring the Kalai
proportional solution over the generalized Nash solution. We further discuss the implications2

of our findings for applied work. Specifically, we show that our estimates have important
quantitative implications for the welfare costs of inflation in search-theoretic models of money.

2

Related Literature

Our paper adds to the literature in three ways: 1) We consider a two-dimensional bargaining
game where players endogenously and simultaneously determine both the size of the pie, i.e.,the quantity to be produced, and how to divide that pie, i.e., the amount of money thebuyer offers the seller. 2) We further consider the role of liquidity constraints where one
party, the buyer, comes to the bargaining table with constraints on the amount of money thats/he can offer; these constraints enable us to differentiate between two axiomatic bargaining
solutions due to Nash and Kalai. 3) We provide evidence on the appropriate bargaining
solution and weights in such a setting that will be useful to researchers working with suchmodels; specifically, we provide an application to the welfare cost of inflation in money-searchmodels.
We start by discussing how our paper contributes to monetary economics. New monetarist
economists emphasize the importance of micro-foundations. Given that bargaining solutions
and weights matter both qualitatively and quantitatively, and that theory is silent on thebargaining solution/weight selection, empirical investigations are crucial to shed light on theseissues. There is a relevant applied theoretical literature employing different price formation
mechanisms—including generalized Nash and Kalai bargaining—in monetary economics, andstudying their implications for monetary equilibria and the transmission of monetary policy,e.g., Lagos and Wright (2005), Molico (2006), Aruoba et al. (2007), Craig and Rocheteau(2008), Rocheteau and Wright (2005). These studies show that the efficiency of the monetary
equilibrium, the welfare costs of inflation, and the impact of monetary policy depend on the
trading protocol and the bargaining weights. A common approach in this literature is to
impose a bargaining solution (e.g., Nash or Kalai), fix the bargaining parameter (e.g., 1 or
0.5), or calibrate it to target retail markups (the ratio of price to marginal cost) observed in
the data in the United States. Using the latter method in a model where the bargaining setup
is virtually similar to the one built into our experiment, and imposing the Nash bargaining
solution, Lagos and Wright (2005) estimate the consumer’s bargaining power to range between

30.315 and 0.404. In a closely related but slightly different model, Aruoba et al. (2011) obtaina consumer bargaining weight of 0.92, also using Nash bargaining. Using Kalai bargaining,
Bethune et al. (2019) obtain an estimate of 0.72, while Venkateswaran and Wright (2013)and Davoodalhosseini (2021) estimate the consumer’s bargaining power to range, respectively,
between 0.68 and 0.86 and between 0.75 and 0.87. The range of estimates is due to variationin the targeted markup value as well as preference parameters and specifics of the model. In
our study, we do not impose a bargaining solution. We find that the experimental evidence
supports the use of the Kalai solution, and provides an additional data point for estimates ofconsumers’ bargaining power, suggesting that the assumption of symmetric bargaining powers
is appropriate in settings close to Lagos and Wright (2005).
Our paper also contributes to the experimental literature on bargaining. As Karagözoğlu(2019) notes, prior to 1982, the experimental literature on bargaining generally employed
unstructured designs. For example, Nydegger and Owen (1974) study two-player unstructured,face-to-face bargaining over chips where they vary the utility of value of chips between players
and introduce irrelevant constraints to test the axioms of Nash and Kalai and Smorodinsky.
See also Roth and Malouf (1979), Roth and Murnighan (1982), Hoffman and Spitzer (1982)
and Hoffman and Spitzer (1985).
Since 1982 with Rubinstein’s alternating offers bargaining model (Rubinstein (1982)) andGüth et al.’s ultimatum game experiments (Güth et al. (1982)), much research has been
conducted using more structured models of bargaining albeit mainly in the one-dimensional,
divide a pie framework. For instance, Binmore et al. (1989), Binmore et al. (1998) lookedat the role played by outside options and compared a “split the surplus” solution with a“deal-me-out” solution. In the former solution, both players get their outside option and split
equally the remaining pie net of those outside option values. In deal-me-out, the players agree
to split the pie in half unless one player is worse off than under her outside option, in whichcase she gets her outside option and the other player gets the remainder of the pie. The
evidence here seems to favor the deal-me-out solution. Our game, described in Section 3, does
not allow us to distinguish between deal-me-out, split the surplus or the Kalai solution withsymmetric weights.1
1

Both deal-me-out and split-the-surplus coincide because the disagreement values are normalized to zero, as

is usual in the search-theoretic literature that makes use of bargaining. In this setup, both of these solutions,4More recently there has been a revival of interest in unstructured bargaining experiments.
For instance, Feltovich and Swierzbinski (2011), Anbarci and Feltovich (2013) and Anbarci and
Feltovich (2018) study the role of outside options and disagreement values in one dimensional,
unstructured bargaining games as well as in structured games, such as the Nash demandgame (Nash (1953)). While outside options have to be forgone if parties enter into bargaining,
disagreement values are payoffs earned in the event that a bargain is not reached. They too
find that disagreement values do not matter as much as theory would predict, with subjects
often dividing the pie down the middle regardless of the disagreement values. Bolton andKaragözoğlu (2016) study the role of hard leverage (ultimatum game proposal rights) versus
soft leverage (appeal to a focal precedent) for outcomes in an unstructured bargaining gameand find that focal precedents play an important role in bargaining outcomes. Dufwenberg
et al. (2017) consider unstructured pre-play negotiation in a “lost wallet game” and varywhether agreed-upon bargaining outcomes are binding or informal (reneging is allowed). Theyfind that regardless of whether negotiated outcomes are binding or informal, equal splits are themost commonly agreed upon outcome. Camerer et al. (2019) study unstructured bargaining
with one-sided private information. They find that the incidence of bargaining failures isdecreasing in the pie size. They use a machine learning approach to show that features of thebargaining process play an important role in the determination of agreements. Korenock andMunro (2021) study unstructured wage bargaining between firms and workers in a dynamic
labor-search model under the assumption that the exogenously determined match surplus is
split equally. They find no effects on wages from changes in the unemployment rate and anunder-reaction of wages to changes in unemployment benefits.
Like us, Galeotti et al. (2019) use an unstructured bargaining setting to explore equityefficiency trade-offs. However, in their study the bargaining process is over two or three predefined allocations where an equal split allocation may or may not be efficient and bargaining
only involves online chat between the two parties as to which pre-defined allocation they willagree upon. Using this design, they find evidence inconsistent with both the Nash (1950) andwhen Pareto efficient, also coincide with the Kalai proportional solution in the special case where the bargaining
weights are symmetric, so that focusing on Kalai bargaining is without loss of generality in our context. The
addition of non-zero disagreement values would only shift the paths of the Nash and Kalai solutions but notalter the characteristics we test for in our experiment.

5Kalai and Smorodinsky (1975) solutions, but they do not address the Kalai (1977) proportional
solution as we do in this paper. Further, they show that focality on equal earnings outcomes is
not universal. Indeed, in decision settings where, unlike in fixed-pie settings, there are tradeoffs between equality and efficiency, a substantial proportion of subjects choose an unequaland total welfare maximizing allocation over equal and Pareto efficient ones.
As noted earlier, Navarro and Veszteg (2020) construct unstructured bargaining situations
with the goal of testing the axioms of several bargaining solutions. They provide evidence
against the scale invariance axiom, and show that the Nash bargaining solution and the KalaiSmorodinsky solution are poor predictors of bargaining outcomes.
In sum, in all of these unstructured bargaining studies, there is typically only a time limit
to bargaining and bargaining typically takes place in a single dimension, e.g., how to divide
a pie. By contrast, we study unstructured bargaining in a two-dimensional setting where
subjects simultaneously decide on both the size of the pie and how to divide it.
There is a related experimental literature on “joint production” where non-cooperative
bargaining occurs among players who have jointly produced the pie, often via some real effort
task. After production has occurred, subjects subsequently bargain over how to divide thatpie. For references see the survey by Karagözoğlu (2012). Research questions in that literature
concern whether and how heterogeneities in efforts/abilities to produce the pie, in interaction
with equality, equity and other fairness norms, matter for the subsequent bargaining and division of the pie. While our bargaining task is related to this “joint production” literature,
there are important differences between our set-up and joint production bargaining experiments. First, as noted earlier, subjects in our experiment decide on the size of the pie and thedivision of that pie simultaneously, that is, there is no sequential structure to the game that
our subjects play.2 Second, subjects in our setup have distinct roles as buyer/consumers or
seller/producers. If a bargaining agreement is reached, then it is only the seller who actually
produces the good (the buyer supplies money in exchange for this production). Thus, production is not joint and in such settings it would be unnatural for the seller to produce first,
before there was any agreement on the terms of trade. Finally, the second stage bargaining
2

Taking seriously the sequential structure of joint production bargaining games, once the size of the pie isdetermined, the costs are sunk and so it should not matter for bargaining outcomes whether and how the pie
was produced by the bargainers or not. The experimental literature, however, shows otherwise.

6

game in the joint-production literature is typically a structured bargaining game, taking theform of an ultimatum game or the Nash demand game; by contrast we consider a less structured bargaining game. Still, we see our approach as complementary to the literature on joint
production.

3

Theoretical Framework

We focus on a bargaining problem where a buyer and a seller need to determine the termsof trade in their match. The bargaining problem is inspired by the “bargaining stage” of themonetary model proposed by Lagos and Wright (2005) and captures many interesting bargaining situations where the outcome of the bargaining problem includes the size of the surplus, inaddition to the surplus division between the two parties. Examples of such bargaining problems with endogenous surplus determination include labor-management negotiations, politicalparties during coalition-building processes following elections and family negotiations.
There are two agents: a buyer (consumer) and a seller (producer). The buyer gets utilityu(q) from consuming quantity, q, but he cannot produce for himself. The buyer is endowedwith m tokens, which can be interpreted as money. He can offer y ≤ m tokens in exchange forsome amount q, produced by the seller. The seller incurs a cost c(q) from producing quantity qof the good. Production is thus made to order (it does not occur in advance) and is conditionalon the two parties reaching an agreement.
If an exchange of y tokens for an amount q of the consumption good is agreed to, then thebuyer’s payoff is S b = u(q) − y and the producer’s payoff is S s = y − c(q), such that the total
surplus is equal to S = u(q) − c(q). If no agreement is reached, then the buyer and the seller
get a payoff of zero. Note that the choice of q determines the joint gains from trade (size ofthe pie or total surplus), while y determines how these gains are split between the buyer andthe seller. Units of the good and tokens are perfectly divisible.3 Information about utility
and cost functions is complete. Tokens have a redemption value in terms of points only in theevent that an exchange is agreed to, and then only in the amount of tokens actually exchanged;
otherwise, tokens are worthless. This is equivalent to normalizing the buyer’s threat point to3

While the theory assumes perfect divisibility, in the experiment we limited units of q and y to incrementsof size 0.01.

7

Figure 1: Seller’s production cost and buyer’s consumption utility as parametrized in the experiment.
Gains from trade are maximized for q = 4.

0.4 Note that in Appendix B, we show that the predictions that we test in our experimentremain unchanged when allowing for non-zero disagreement values.
The utility and cost functions satisfy the following assumptions: u0 > 0, u00 < 0, c0 > 0, c00 >0, u(0) = c(0) = 0. There exists q ∗ such that u0 (q ∗ ) = c0 (q ∗ ). That is, q ∗ is the amount thatmaximizes the joint surplus in a pair.
Figure 1 shows the utility function of the buyer against the cost function of the seller using
the parameterization of our experiment (see also Section 4). The bargaining problem is tochoose q ∈ [0, q], where q > 0 s.t. u(q) = c(q) and to choose y ≤ m. A proposal is a (q, y) pairoffered by either the buyer or the seller. The first best solution is given by q ∗ (q ∗ = 4 in theexperiment).

3.1

Solutions to the Bargaining Problem

In this section, we apply the axiomatic approach to characterize the solution to the bargaining
problem. We focus on the generalized Nash bargaining and the Kalai proportional solutions.
According to the axiomatic approach, “One states as axioms several properties that it would
seem natural for the solution to have and then one discovers that the axioms actually determine
4

If we allowed tokens to have a redemption value, then, in the event of no exchange, the threat point of thebuyer would be m. In that case, the surplus of the buyer would remain the same (u(q) + m − y) − m = u(q) − y.

8the solution uniquely” (Nash (1953)). The main difference between the generalized Nashbargaining and Kalai proportional solutions hinges upon one of the axioms. Specifically Kalai’ssolution satisfies strong monotonicity (i.e., as the bargaining set expands, each player gets ahigher surplus), while the Nash solution does not. This difference generates distinct bargaining
outcomes when the buyer is liquidity-constrained. Recall that an agreement consists of a pair(q, y) where q is the amount produced by the seller, and y ≤ m is the amount of tokens thebuyer transfers to the seller. If the buyer and seller strike an agreement, the buyer’s surplus isgiven by S b = u(q) − y and the seller’s surplus is given by S s = −c(q) + y, such that the total
surplus is equal to S = u(q) − c(q). In case of no agreement, both parties’ payoff equals 0, i.e.,the disagreement point is (0, 0). The set of feasible utility levels for this bargaining problem
is given by
S(m) = {(u(q) − y, −c(q) + y) : 0 ≤ y ≤ m and q ≥ 0}.
The left panel of Figure 2 depicts the bargaining set for three values of m (30, 60 and 171)
associated with the parameterization of our experiment, restricting our attention to the region
where both players’ surpluses are positive (i.e., their participation constraint is satisfied). Notethat as m increases, the bargaining set expands, which implies that as the buyer’s money
holdings m increase, more outcomes can be attained. The Pareto frontier of the bargaining
set is linear when the buyer is not liquidity constrained, and it is concave when the liquidity
constraints bind (see also Aruoba et al. (2007) for details). Next, we describe what bargaining
outcome is selected under the generalized Nash bargaining solution and the Kalai bargaining
solution.
3.1.1

Generalized Nash Bargaining

The Nash (1950) bargaining solution satisfies the axioms of symmetry, Pareto optimality, scale
invariance, and independence of irrelevant alternatives. The generalized Nash bargaining solution that we study in this paper drops the symmetry axiom. The axiom of Pareto optimality
implies that the solution is such that there is no attainable outcome that makes one player
better off without making the other player worse off. Scale invariance implies that the solution is invariant to affine transformations of the buyer or seller’s surplus. Independence of
irrelevant alternatives means that, if some outcomes are removed from the bargaining set and9the solution is not among them, then the solution must remain the same. The generalized
Nash solution is given by the following optimization problem:max [u(q) − y]θN [y − c(q)]1−θNq,ysubject to
0 ≤ y ≤ m,where θN ∈ [0, 1] denotes the buyer’s bargaining power (which in the generalized version candiffer from the seller). Let m∗N = (1 − θN )u(q ∗ ) + θN c(q ∗ ). Then, the solution is given by:
If m ≥ m∗N ,q = q∗,(1)y = m∗N = (1 − θN )u(q ∗ ) + θN c(q ∗ ).

(2)

If m < m∗N , the consumer is money constrained, so y = m and q is the implicit solution
to:
m=
3.1.2

(1 − θN )c0 (q)u(q) + θN u0 (q)c(q).
θN u0 (q) + (1 − θN )c0 (q)(3)

Kalai or Proportional bargaining

The Kalai (1977) solution satisfies the axioms of Pareto optimality, independence of irrelevantalternatives and strong monotonicity. Strong monotonicity implies that players cannot bemade worse off as the bargaining set expands and greater surplus levels become attainable.
The Kalai solution does not satisfy the axiom of scale invariance. It is given by:
max [u(q) − y]q,ysubject toθK[y − c(q)]1 − θKy ≤ m,u(q) − y =where θK ∈ [0, 1] denotes the buyer’s bargaining power. This can be rewritten as:
max θK [u(q) − c(q)]q10subject to:(1 − θK )u(q) + θK c(q) ≤ m.
Let m∗K = (1 − θK )u(q ∗ ) + θK c(q ∗ ). If m ≥ m∗K , the Kalai solution has the same functional
form as in the Nash solution:q = q∗,(4)y = m∗K = (1 − θK )u(q ∗ ) + θK c(q ∗ ).

(5)

If m < m∗K , the buyer is money constrained, so y = m and q is the implicit solution to:m = (1 − θK )u(q) + θK c(q).

3.2

(6)

Comparison of Nash and Kalai

When the buyer is not liquidity constrained, the bargaining parameters θK and θN have thesame interpretation: they pin down the fraction of the total surplus assigned to the buyer.
The two solutions are observationally equivalent in the unconstrained case. Therefore, in what
follows, we set θK = θN = θ.5 Both the Nash and Kalai solutions are the same when m is large
enough, in the unconstrained case. That is, q = q ∗ , and y = m∗ = (1 − θ)u(q ∗ ) + θc(q ∗ ), i.e.,
the first best is achieved and θ determines how the total surplus u(q ∗ ) − c(q ∗ ) is distributed
between the buyer and seller. However, when the buyer is liquidity constrained, the twosolutions differ.
Under Nash bargaining, the buyer spends all her/his money, i.e., y = m and so q is pinned
down by:m = [1 − Θ(q)]u(q) + Θ(q)c(q), where Θ(q) =θu0 (q).
θu0 (q) + (1 − θ)c0 (q)Under Kalai bargaining, we also have y = m, but q is determined by

m = (1 − θ)u(q) + θc(q).
The two solutions differ so long as q < q ∗ . Importantly, Θ(q) depends on q while θ is aconstant.
5

Following the applied literature from which our game is derived, we take θ as a primitive of the model andassume that it is exogenously fixed. We discuss in Section 5.4 the possibility of allowing θ to vary as a functionof liquidity constraints.

11

This implies that, under Kalai, the buyer’s surplus is S b ≡ u(q) − y = θ[u(q) − c(q)], thatis, the buyer’s surplus share, S b /S, stays constant and is equal to θ. Further, the buyer’s
surplus increases in q up to q ∗ , since [u(q) − c(q)] is increasing for q ≤ q ∗ . Under Nash, on theother hand, the buyer’s surplus is given by S b ≡ u(q) − y = Θ(q)[u(q) − c(q)], and the buyer’s
surplus share, Θ(q), depends on q. While [u(q) − c(q)] is increasing for q ≤ q ∗ , it is easy toshow that Θ(q) is decreasing in q. We can show that the buyer’s surplus is non-monotonic
in q, first increasing as q increases but then decreasing as q gets closer to q ∗ . Also, note that
Θ(q) ≥ θ for q ≤ q ∗ . This implies that the buyer obtains a higher share of the surplus under
Nash as long as the liquidity constraint is binding. The seller’s surplus is increasing underboth solutions.6
We illustrate the differences between the two solutions in the case where the bargaining
weight is θ = 1/2. The Nash solution is given by the tangency points of the Nash productlevel curves with the bargaining set, while the Kalai solution is given by the intersection of
the 45 degree line with the bargaining set. The left panel of Figure 2 compares the buyers’
and sellers’ surplus allocations as the buyers’ endowment of tokens, m, increases. The figure
shows the bargaining sets for the three cases we study in the experiment, labeled m = 30,m = 60 and m = 315. Notice first that liquidity constraints (here, when m < 171) produce
asymmetries in the bargaining set that the two players face; the maximum buyer’s surplus(vertical intercept) is always greater than the maximum seller’s surplus (horizontal intercept).
Notice further that under the Nash solution, the buyer’s surplus increases and then decreases
as m increases, and the buyer’s surplus is always greater than or equal to the seller’s surplus.
By contrast, under the Kalai solution, when θ = 1/2, the buyer’s and seller’s surpluses arealways monotonically increasing and are equal. These differences between the two bargaining
solutions are the main focus of our experiment.7
6Note that here, we compare the Nash and Kalai outcomes for a given bargaining weight, θ. However, anotherway to compare the two solutions would be to start from a given allocation (S b , S s ) on the Pareto frontier (orequivalently, a given quantity and payment pair generating such an allocation) and compute the bargaining
weights implied by both solutions. Under Kalai, the implied bargaining weight would be θK = S b /(S b + S s ),i.e., simply the buyer’s share of the total surplus. One can show that the bargaining weight implied by the Nashsolution, θN , would necessarily be lower, θN < θK , as long as q < q ∗ . Also, the difference with θK increases asliquidity constraints are tightened (i.e., θN decreases).
7
These differences remain if we allow for a non-zero, and possibly asymmetric, disagreement point. See
Appendix B for details. The right panel of Figure 2 can also be used to compare the bargaining path under12m = 30m = 60m > 171

Figure 2: Surplus predictions (left panel) and terms of trade predictions (right panel) under Nash andKalai bargaining for θ = 1/2. The circle and triangle markers depict outcomes when m ∈ {30, 60, 315},from left to right.

4

Experimental Design and Hypotheses

We employ a 3×1 experimental design, where the treatment variable is the buyer’s endowmentof money (or “tokens”), m. We adopt a between-subjects design, i.e., each subject participatedin only one environment, either m = 30, m = 60 or m = 315. Our main outcome variables are
the quantities q and amounts of tokens (money) y that buyers and sellers bargain over.

4.1

Model Parameterization

In parameterizing the model, we had several objectives. First, we did not want to make thefirst best choice too focal, so we chose to make q ∗ off-center between 0 and q̄. Second, we
made q ∗ and u(q ∗ ) − c(q ∗ ) integer values and we sought to have a significant slope on bothsides of u(q ∗ ) − c(q ∗ ) so that the first best was sufficiently salient. Third, we wanted there to
be large, significant differences between the Nash and Kalai solutions so that we would havesome chance of detecting those solutions in our data.
With these considerations in mind, we chose:u(q) = 74.1752q 0.6
Kalai and Nash in the (q, y) space instead of the surplus space. For m = 30 and m = 60, the Nash solution
predicts a higher quantity traded.

13c(q) = 8.23249q 1.51678 .
It follows that:q̄ = 11, q ∗ = 4, u(q ∗ ) = 170.41, c(q ∗ ) = 67.41, u(q ∗ ) − c(q ∗ ) = 103.

4.2

Treatments

The maximum m needed to achieve the first best satisfies m̂ − u(q ∗ ) = 0, i.e., it corresponds
to the transfer of tokens required to compensate the seller assuming he/she has all of thebargaining power, in which case the buyer’s gains from trade are equal to 0. Given ourmodel parameterization, m̂ = 170.41, so for our “unconstrained” treatment we set m to bemuch higher than m̂ and equal to m = 315. For our two “constrained” treatments, we set
m < 170.41 ≡ m̂. Specifically, we chose m = 60 and m = 30. Both of these values areless than c(q ∗ ) = 67.41 to ensure that liquidity constraints were binding regardless of thebargaining weight θ. Further, these two values for the constrained treatment nicely capture
the non-monotonicity in the buyer’s surplus as m is varied as illustrated in the left panel of
Figure 2.
In all other respects, the environment was held constant. Each session consisted of 10
subjects who participated in 30 rounds of bargaining as either a buyer or a seller. At the
beginning of the session, subjects were randomly assigned a role as a buyer or seller (5 ofeach) and they maintained the same role in all 30 rounds.8
In each round, buyers and sellers were randomly and anonymously paired and were tasked
with bargaining over q and y within T = 2 minutes without communication. In all treatments,
0 ≤ q ≤ 11, while y was restricted by m = 30, 60 or 315, depending on the treatment. Asidefrom the time limit and the requirement that proposals consist of (q,y) pairs, bargaining waslargely unstructured and tacit. Still, because of the time limit and the restriction on proposals,
we refer to our experimental design as involving semi-structured bargaining.9
Figures 3 and 4 show screenshots of the top and bottom parts, respectively, of the bargaining interface for a buyer in the unconstrained treatment where m = 315 (the seller’s interface
8
9

We chose this design to enable subjects to gain experience with a particular role.
See also Camerer et al. (2019).

14

Figure 3: Top part of decision screen for a buyer, unconstrained treatment

is similar).10 On the top part of this screen (Figure 3), subjects had two slider bars, one forquantity, q, and one for tokens, y. By moving the position of one or the other of the twoslider bars to different values for q or y, subjects were informed of the payoff to themselves
and to the other player if the implied proposal of (q,y) was accepted. Thus, the sliders also
worked as calculators for the subjects, avoiding the need for them to directly calculate payoffs.
The buyer’s payoff was calculated as S b = u(q) − y and the seller’s payoff was calculated as
S s = y − c(q), where u(q) and c(q) were parameterized as discussed above and illustrated
on the decision screen. The calculators showed both the buyer’s and the seller’s payoff fromany given proposal. All payoffs were denoted in “points” and subjects understood that theirmonetary payoffs were increasing in their points earned. Note that tokens, y, only had value
in points to sellers or a cost in points to buyers if those tokens were part of an agreed upon
trade. If an agreement was not reached, the buyer’s token endowment was declared worthless.
10

The constrained case is also similar, with the only difference being that the slider for tokens was restricted

to [0, 60] or [0, 30].

15

Figure 4: Bottom part of decision screen for a buyer, unconstrained treatment

If an agreement was reached, tokens in excess of the agreed upon exchange amount, y, werealso declared worthless. In this manner, we capture the fiat money nature of the token object;
the tokens only have value/cost if they are used as part of an agreed upon exchange.11
Once a subject found a proposal she would like to make, she clicked on a Submit button to
make that proposal “live.” We restricted subjects from making proposals that would result innegative payoffs to themselves or to the other player. Once a proposal was made live, it could
not be withdrawn and the other player in the match could accept it any time by clicking on
the Accept button next to that proposal (See illustration in Figure 4).12 Additional proposals
11This is one interpretation. Other interpretations are that tokens are commodity money or that liquidity

constraints are technological constraints on transfers. This has also the advantage that the buyer’s disagreement
value is normalized to zero and equal to the seller’s disagreement value. Theoretically, this is without loss ofgenerality, see footnote 6.
12This design choice was made for two reasons. First, we wanted to ensure that players think seriously about
the proposals they make. Second, we wanted to prevent players using “fake” proposals to take advantage ofthe 2-minute time limit (for example, withdrawing with only a few seconds lefts to constrain the other player
to agree to less favorable terms of trade).

16

made by either player within the 2 minute round were added to the existing set of proposals
and did not replace any prior proposal.
Buyers and sellers could see each others’ submitted proposals at all times under columns
for buyer and seller proposals. If there were many proposals made in a round, then a scrollbar appeared allowing subjects to review all live proposals. Proposals were live only for theduration of each 2 minute round; the set of proposals was cleared out at the start of each newround.
A round ended when either a proposal was agreed to or the 2 minute time limit had expired,
whichever came first. Importantly, our experiment was implemented in two phases. In thefirst phase, we explored the unconstrained treatment (where m = 315) in order to estimate
the bargaining weight, θ. As noted earlier, we picked m = 315, as it was well above u(q ∗ ). In
the unconstrained treatment, we found strong evidence that θ = 1/2, as shown in Section 5.2.
With that knowledge, we designed the second phase of the experiment, where we studied
environments where the buyer’s money holdings were sufficiently low that the first best could
not be achieved. Specifically, in the constrained treatments, m = 60 and m = 30.13
As Figure 2 shows in these constrained cases, the buyer’s surplus under the Nash solution
is first increasing and then decreasing, whereas under the Kalai solution, the buyer’s surplus
is strictly increasing and is equal to the seller’s surplus in all three treatments. Note that in
the unconstrained case, the Nash and Kalai solutions coincide.
Summarizing, our three treatments involve three different values for m:
1. Unconstrained: m = 315 ≥ u(q ∗ )2. Constrained-High: m = 60 < c(q ∗ )3. Constrained-Low: m = 30 < c(q ∗ ).

4.3

Hypotheses

Table 1 provides predictions for q, y, the per unit price y/q, the seller and buyer’s surpluses,b

S s and S b , the total surplus, S, and the ratio of the buyer’s surplus to the total surplus, SS ,13

While we conducted our experiment in two stages, we employed a between subjects design. Each subject

participated in only a single treatment, either m = 315, m = 60 or m = 30.

17

Table 1: Theoretical predictions, Nash vs. Kalai, θ = 1/2.

m = 315

q

y

y/q

Ss

Sb

S

Sb
S

Nash

4

118.91

29.73

51.5

51.5

103

.5

Kalai

4

118.91

29.73

51.5

51.5

103

.5

m = 60

q

y

y/q

Ss

Sb

S

Sb
S

Nash

2.17

60

27.65

33.27

58.19

91.46

.64

Kalai

1.69

60

35.5

41.71

41.71

83.42

.5

m = 30

q

y

y/q

Ss

Sb

S

Sb
S

Nash

1.24

30

24.19

19.59

54.4

73.99

.74

Kalai

0.625

30

48

25.96

25.96

51.92

.5

under the Nash and Kalai bargaining solutions in the θ = 1/2 case for all three treatments.14
Based on the theoretical predictions, we have the following hypotheses.
Hypothesis 1. In the unconstrained case, subjects achieve the first best.
This hypothesis checks whether there are any inefficiencies in the unconstrained case.
Hypothesis 2. As m increases, the agreed upon q, the amount of tokens spent y, thetotal surplus S, and the seller’s surplus S s all increase.
Hypotheses 1 and 2 are valid under both bargaining solutions and are independent of θ.
However, the buyer’s surplus, S b , may increase or decrease in m depending on the bargaining
solution.
Hypothesis 3a (Nash) As m increases, the buyer’s surplus, S b , is increasing and thendecreasing as q increases toward q ∗ .
Hypothesis 3b (Kalai) As m increases, the buyer’s surplus, S b , is monotonically increasing as q increases toward q ∗ .
14

We use predictions for the θ = 1/2 case because, as we show below in Section 5.2, this is the empiricalestimate for θ that emerges in the unconstrained case.

18

Hypothesis 3a is the prediction of the Nash bargaining solution, while hypothesis 3b is theprediction of the Kalai bargaining solution.

5

Results

In this section we first report on the number of sessions and results relevant to the questionof the appropriate bargaining weight and solution. Then, we report on an analysis of thebargaining process data.

5.1

Sessions, Subjects and Payments

The experiment was programmed in oTree (Chen et al. (2016)) and was conducted over networked computers in the Experimental Social Science Laboratory at UC Irvine. Subjects were
undergraduate students from a variety of different majors and had no prior experience with
our study. They were recruited using Sona Systems software. At the start of each 90 minutesession, subjects were given written instructions which were read out loud. All participants
had to successfully complete comprehension test questions before proceeding to the bargainingtask. Sample instructions and comprehension test questions are found in Appendix A.
Following the instructions and test which took about 30 minutes, the remaining hour was
devoted to the 30 bargaining rounds (maximum of 2 minutes each).
As noted earlier, subjects earned points in each round depending on bargaining outcomes.
At the end of the session two rounds were randomly chosen. The sum of subjects’ point totals
from those two rounds were multiplied by 0.25 to determine subjects’ monetary earnings fromthe bargaining task. In addition, subjects earned a $7 show up payment. Subjects’ average
total earnings varied by treatment: $31.08 for the m = 315 treatment, $26.61 for the m = 60
treatment and $19.03 for the m = 30 treatment.

5.2

Estimation of the Bargaining Weight

Our first result concerns the buyer’s share of the total surplus. We examine the buyer’s
share in the unconstrained treatment where m = 315 because under both the Nash and Kalai
solutions, the buyer’s share gives us an estimate of the bargaining weight θ.

19

Table 2: Random-effects estimation of the buyer’s share of the surplus among accepted offers.

Buyer’s surplus

Total surplus, S

Observations

(1)

(2)

(3)

(4)

0.50120.49940.49870.4989(0.0032)(0.0018)(0.0019)(0.0019)

698

574

412

348

Standard errors clustered at the session level in parentheses

To estimate the bargaining weight, we focus on accepted offers and regress the buyer’s
surplus Sib on the total surplus achieved by each pair, Si in the unconstrained treatment.
Specifically we run the regression
Sib = θSi + i ,using a random effects regression estimator where i indexes an individual buyer.15 We consider
several specifications that vary in terms of how close the quantity is to the first best predictionof q ∗ = 4. Specifically, the sample we use consists of accepted offers in the unconstrained
treatment for four different neighborhoods of q ∗ = 4:
(1) all(2) s.t. |q − 4| < 0.5(3) s.t. |q − 4| < 0.1(4) s.t. |q − 4| < 0.05.

The results are reported in Table 2. We see that regardless of any restrictions placed on traded
quantities q, the estimate θ̂ is not statistically different from 0.5.

Finding 1. The buyer’s share of the surplus in the unconstrained case is equal to 0.5.
Finding 1 suggests that we can use θ = 1/2 to evaluate our theoretical predictions in
the constrained treatments where bargaining outcomes differ between the two solutions. The
predictions for the case where θ = 1/2 were reported earlier in Table 1.
15

Regression results obtained when random effects are indexed at the individual seller level are reported inTable D3 in Appendix D.

20

Table 3: Average agreed-upon outcomes by treatment.

5.3

q

y

y
q

Ss

Sb

S

Sb
S

m = 315

4.03

119.67

29.82

50.99

51.09

102.08.50m = 60

1.69

59.05

35.88

40.65

42.14

82.79

.50

m = 30

0.67

29.70

47.19

25.09

28.11

53.21

.52

Which solution?

Figure 5 shows the distribution of traded quantities and tokens over all five sessions of eachof the three treatments. Mean values are reported in Table 3.16 Figure 5 and Table 3, in
conjunction with Table 1, which reports the theoretical predictions for θ = 1/2, indicate that
there is overall support for aspects of the theoretical predictions captured in Hypotheses 1
and 2, and more support for the Kalai solution, as formalized in Hypotheses 3b. We provide
a more rigorous analysis of the data next.
We focus first on the unconstrained case, where we see that in case of agreement, the mean
traded quantity is 4.03 and the mean traded tokens are 119.69. Using a Wilcoxon signed-rank
test on session-level averages for agreed upon trades, we find that we cannot reject the null
hypothesis that in the unconstrained treatment, q = 4 and y = 118.91, i.e., the first best isachieved (p-values=.626 and .4375, respectively).
Finding 2. Consistent with Hypothesis 1, in the unconstrained case, we cannot reject the nullthat subjects achieve the first best.
We next consider traded quantities and tokens across all three treatments. In addition to
the averages displayed in Table 3, histograms show the distributions of traded quantities andtokens in Figure 5.
Finding 3. Consistent with Hypothesis 2, as m increases, the traded quantity q, the amountof tokens spent y, the total surplus S and the seller’s surplus S s all increase.
16

Means by session and for each first/last 15 rounds are provided in Appendix D in Table D2.

21

Figure 5: Distribution of traded quantities and tokens by treatment.

Support for Finding 3 is found in rows 1-4 of Table 4, which reports results from nonparametric Jonckheere tests for ordered alternatives using session-level average data over all
periods, and the first and second halves of each session (periods 1-15 and 16-30). Specifically,
we test the null hypothesis that population medians for each treatment value for m (x̃m ) for
the variables q, y, and S, S s are the same, i.e., H0 : x̃30 = x̃60 = x̃315 , against the orderedalternative hypothesis predicted by theory: HA : x̃30 ≤ x̃60 ≤ x̃315 , with at least one strict
inequality. We find that we can easily reject the null in favor of the alternative in all cases.
Next, we study the impact of varying the liquidity constraint, m, on the buyer’s surplus,
so as to discriminate between Hypotheses 3a and 3b.
Finding 4. Consistent with Hypothesis 3b (Kalai) but counter to Hypothesis 3a (Nash) the
buyer’s surplus is monotonically increasing as m increases.
Support for this finding is found in Figures 6 and 7 and row 5 of Table 4, where we test
the null of no difference in the buyer’s surplus across the three treatments against the orderedalternative predicted by the Kalai (but not by the Nash) solution, using session-level average22m = 30m > 171m = 60Figure 6: Experimental bargaining agreements in the surplus space (left panel) and in the (quantity,
payment) space (right panel). Each colored circled corresponds to an agreement between a buyer anda seller. Treatments are color-coded as follows: m = 30 in red, m = 60 in orange, m = 315 in green.
Note that a region becomes darker as more data points overlap within it.
Mean Buyer's Surplus

60
Nash

51.09
50
42.14
40
Kalai
30

28.11

20

10

0
Constrained, m = 30

Constrained, m = 60

Unconstrained, m = 135

Figure 7: Mean buyer’s surplus and 95% confidence intervals across the three treatments along withNash and Kalai predictions.

data. We find that we can reject the null of no difference in favor of the alternative that thebuyer’s surplus S b , increases as m increases from 30 to 60 to 315, which is consistent with the
Kalai solution.17
17

We also tested the null hypothesis that the buyer’s surplus was equal in pairwise comparisons between the23

Table 4: Results and p-values from Jonckheere tests of ordered alternatives for the variables,
q, y, S, S s and S b , all periods, first half (periods 1-15) and second half (periods 16-30), using
session-level average data.

Row No.,Hypotheses:

All

First Half

Second Half

Variable

H0 vs. HA

periods

Periods 1-15

Periods 16-30

1

H0 : q30 = q60 = q315

Reject H0

Reject H0

Reject H0q

HA : q30 ≤ q60 ≤ q315

p = .0000

p = .0000

p = .0000

2

H0 : y30 = y60 = y315

Reject H0

Reject H0

Reject H0y

HA : y30 ≤ y60 ≤ y315

p = .0000

p = .0000

p = .0000

3

H0 : S30 = S60 = S315

Reject H0

Reject H0

Reject H0

S

HA : S30 ≤ S60 ≤ S315

p = .0000

p = .0000

p = .0000

4s = Ss = Ss
H0 : S30
60
315

Reject H0

Reject H0

Reject H0Ss

s ≤ Ss ≤ Ss
HA : S30
60
315

p = .0000

p = .0000

p = .0000

5b = Sb = Sb
H0 : S30
60
315

Reject H0

Reject H0

Reject H0

Sbb ≤ Sb ≤ Sb
HA : S30
60
315

p = .0000

p = .0000

p = .0000

To quantify these effects, we also run the regression
Sib = β1 1{m=30} + β2 1{m=315} + εi ,(7)using a random effects regression estimator where i indexes an individual buyer. We focus on
accepted offers, and report results for two different specifications:(1) all accepted offers(2) accepted offers s.t. |q − 4| < 0.5 when m = 315, y − 60 < 0.5 when m = 60, andy − 30 < 0.5 when m = 30.18bbb
three treatments using non-parametric Mann Whitney tests on session-level average data, i.e., S30
vs. S60; S60
bbb
vs. S315; S30
vs. S315; We are always able to reject the null of no difference in favor of the alternative that thebuyer’s surplus is higher when m is higher (p = .0079 for all three two-sided tests), which again favors the Kalai
solution and runs counter to the non-monotonic prediction for the buyer’s surplus under the Nash solution.
18
These conditions ensure that offers are close to the Pareto frontier, i.e., players are close to behaving

24

Table 5: Random-effects estimation of the impact of varying the liquidity constraint, m, on
the buyer’s surplus.

Buyer’s surplus

m = 30m = 315

Constant (m = 60)

Observations

(1)

(2)

-13.9488-14.2499(1.0885)(1.1902)8.9191

9.0271(0.8067)(0.8106)

42.199342.3712(0.7856)(0.7943)

2028

1801

Standard errors clustered at the session level in parentheses

The results are reported in Table 5.19 For both specifications, consistent with Kalai’s predictions, the buyer’s surplus significantly decreases when m is reduced from 60 to 30, andsignificantly rises when m is increased from 60 to 315.

5.4

Discussion

Given the estimated bargaining weight of θ = 0.5, the Kalai solution coincides with the“efficient equal split” outcome, or the case where the buyer and seller surpluses are maximized
subject to the constraint that both surpluses are equal. Efficient equal split is not a generalproperty of the Kalai solution. For values of θ different from 0.5, one can show that 1) buyer
and seller’s surpluses will not be equal under the Kalai solution; 2) in the presence of liquidity
constraints, total earnings efficiency remains lower under the Kalai solution as compared with
the Nash solution.
rationally by proposing a Pareto-efficient joint production.
19
Regression results obtained when random effects are indexed at the individual seller level are reported inTable D4 in Appendix D.

25

We do not pursue here the exercise of varying θ experimentally, as that would involve
adding more structure to our bargaining game and complicate interpretations across treatments; in the present setup, θ does not enter explicitly as a parameter affecting choices anywhere in our game as it is a model primitive. Instead, θ is estimated using data for the casewhere buyers are not liquidity constrained. Further, we have no reason to think that θ wouldchange as liquidity constraints become binding, as existing theory is silent on this issue.
As a thought exercise, however, we can reverse our approach to analyzing the data: insteadof assuming that bargaining powers are a constant primitive of the experiment and checking
which solution fits the data best, we can impose a bargaining solution and solve for thebargaining power implied by the allocations we observe. First, suppose we fixed the buyer’s
share to its empirical average, S b /S = 0.5. We know that this is theoretically consistent with
the Kalai solution for all treatments when θK = 0.5 and for the Nash solution with θN = 0.5
in the unconstrained treatment (m = 315). We can then ask how the bargaining weight
would have to change as the liquidity constraint becomes binding so as to make the allocationconsistent with the Nash solution. This is equivalent to solving for θN such that Θ(q) = 0.5.
We obtainθN =c0 (q)
u0 (q) + c0 (q).

(8)

Plugging in the values for q predicted by the Kalai solution for m ∈ {30, 60, 315} into (8),we respectively obtain θN = 0.15, θN = 0.31, and θN = 0.5. That is, large changes in θN
across environments would be needed in order for the Nash solution to fit the data. Second, inFigure 8, we go further and plot the empirical distributions of bargaining weights implied by
all agreements, assuming that these agreements implement the Kalai solution (in orange) or
the Nash solution (in white with dashed outline). That is, for each agreement in the data, we
compute θK and θN . The distribution under Kalai bargaining is unimodal, centered around
0.5, while the distribution under Nash is trimodal, reflecting the three treatments. The figure
confirms that in order to be consistent with Nash bargaining, the data we observe requires
significant shifts in the bargaining weights of players across treatments.

26

80
60

θN

0

20

Percent
40

θK

0

.2

.4

.6

.8

1

Figure 8: Empirical distribution of bargaining weights assuming that agreements follow the Kalai
bargaining rule (in orange) v.s. the Nash bargaining rule (in white with dashed outline)

6

The Bargaining Process

In this section we consider the process by which buyers and sellers reached a trade agreementin each two minute round.
Overall, 91% of negotiations ended with an agreement. More specifically, the looser theconstraint on tokens, the higher the agreement rate: 89% for m = 30, 91% for m = 60 and
94% for m = 315. Table D1 in Appendix D reports agreement rates for the first 15 and thelast 15 rounds, by session.
Figure 9 shows the share of the surplus that buyers assign to themselves as part of theiroffers, as the negotiation unfolds. The three panels correspond to the three treatments. The
sample is restricted to negotiations that ended with an agreement during sessions 1 to 6.20 In
each panel, offers are ranked by their order relative to the last offer made. For example, inthe left panel, the yellow dots represent all the fourth-to-last offers made by buyers (i.e., thethird subsequent offer was the agreed-upon offer). The black square represents the median20

The sample here is restricted to the first six sessions because timestamps tracking the order in which

proposals were made were not recorded for subsequent sessions.

27

0.65

m = 300.65

m = 600.65

m = 315
Median
Mean

0.55

0.5

0.45

0.4

0.6

Buyer's share of surplus

0.6

Buyer's share of surplus

Buyer's share of surplus

0.6

0.55

0.5

0.45

0.4

0.35
4th to last 3rd to last 2nd to last

Buyer's Offer

Last

0.55

0.5

0.45

0.4

0.35
4th to last 3rd to last 2nd to last

Buyer's Offer

Last

0.35
4th to last 3rd to last 2nd to last

Last

Buyer's Offer

Figure 9: Buyers’ share of the surplus in their own offers, by rank of the offer relative to the accepted
offer and by treatment. Colored circles represent all offers. Median offers are represented with a blacksquare, and mean offers are represented by a black cross.

offer in this sample, and the black cross the average. We can see that buyers start out making
proposals that significantly favor themselves (above the 0.5 equal split line), but they adjustthese proposals downward as the negotiation proceeds. The bargaining process is similar forsellers, who also start out making proposals that greatly favor themselves before agreeing toapproximately equal splits as the bargaining time limit approaches (see Figure E1 in Appendix
E).
This process data analysis reveals that subjects do not immediately jump to the final
outcome where the total surplus is split equally as reported in Section 5. That is, the equal
split final outcome need not be due to an inherent desire for fairness on the part of players.
Rather, it seems that subjects initially try to extract more surplus for themselves and only
converge to an equal split as a result of the back-and-forth bargaining process.
To obtain an estimate of the compromises made by players as the negotiation unfolds, westudy a new variable,∆ player’s share = player’s share in first offer round − agreed-upon player’s share.
28

Table 6: Random-effect estimation of the change in a player’s share of surplus between thefirst offer and the accepted offer, dependent on having made the first offer. Random effects
are indexed at the individual player level. Sample is restricted to buyers.

∆ Player’s share
Player made the first offer-0.1351(0.0222)

Constant

0.0480(0.0089)

Observations

835

Standard errors clustered at the session level in parentheses

We then regress this new variable on a dummy variable indicating whether the player or her
negotiation partner made the first offer. Observations include all rounds that ended withan agreement in sessions 1 to 6. Since each recorded offer corresponds to two observations(one recorded for the buyer and one recorded for the seller), we run two separate regressions,splitting the sample by roles. Results for the sample restricted to buyers are presented inTable 6 while results for the sample restricted to sellers are available in Appendix D in Table
D5.
When the player makes the first offer, she initially proposes to allocate to herself a share of
the total surplus that is, on average, 8.71 percentage points higher than is eventually agreedupon. On the other hand, if her trade partner made the first offer, the player eventuallyobtains a share of the total surplus that is, on average, 4.80 percentage points higher than
was originally proposed. Finding 5 summarizes this analysis. Similar conclusions can be madewhen the sample is restricted to sellers.
Finding 5. If more than one proposal is made, the initial proposal is more favorable to theplayer making the proposal than is the final agreed upon proposal.
Insights into the bargaining process can also be obtained by looking into the opening bids
made by players (that is, the first bid made by either of the two players during a bargaining
29Buyer's share in opening bids

0.8
0.75

Nash prediction
Kalai prediction

0.7

Bid made by buyer
All rounds
Round 1

0.65

Bid made by seller
All rounds
Round 1

0.6
0.55
0.5
0.45
0.4
m = 30m = 60m = 135

Figure 10: Average buyer’s share across opening bids, split by treatment and player identity. Opening
bids made by sellers are displayed in orange and marked by crosses, while opening bids made by buyers
are displayed in green and marked by plus signs. The black dots represent the Nash solution and theblack triangles the Kalai solution.

round). Average opening bids split by treatment and by the identity of the payer that madethe offer (buyer or seller) are displayed in Figure 10. Looking at averages across all rounds(represented with the dot-dashed lines), we can see that the buyer’s share proposed by sellers
in their opening bids is relatively constant across treatments, reaching approximately 45% of
the total surplus. Buyers’ behavior differs, as the share they offer to themselves when they
open negotiations is non-monotone as constraints are relaxed—increasing from m = 30 tom = 60, then decreasing from m = 60 to m = 315. This is consistent neither with the Kalai
solution, which predicts constant shares, neither with the Nash solution, which predicts astrictly decreasing share of surplus for buyers as liquidity constraints are relaxed. Opening
offers made by players when they were not yet experienced, in the first bargaining round, arealso plotted in the Figure 10 with dotted lines. It is interesting to note that both inexperienced
buyers and sellers made opening offers that were relatively more beneficial to buyers in theconstrained treatments, which is consistent with the Nash prediction. Thus, players seemed
to converge towards the Kalai split with equal weights along two margins: first, as a given
negotiation was unfolding, and second, as rounds progressed and players were gaining more
and more experience.
30

Table 7: Average number of proposals made by a pair of players during one round.

Type of negotiation

All treatments

m = 30m = 60m = 315

All

Agreement

No agreement

All rounds

7.22

6.84

11.15

1-15

6.65

6.21

10.47

16-30

7.79

7.44

12.11

All rounds

11.37

10.98

14.31

1-15

9.98

9.49

13.29

16-30

12.75

12.43

15.63

All rounds

5.44

5.23

7.55

1-15

5.05

4.8

7.24

16-30

5.84

5.66

7.93

All

4.85

4.48

10.43

1-15

4.93

4.47

9.91

16-30

4.77

4.49

11.53

31

Note that the bargaining process delineated above, with both players starting by proposing higher shares of surplus to themselves before converging towards a potential agreement,
generates delays. Table 7 shows the average number of proposals made by a pair of players during one round (over the entire sample), broken down across treatments, rounds, andwhether bargaining ended with an agreement. On average, players made 7.22 proposals perround. This number is markedly higher for rounds that did not end up with an agreement.
While we do not have timed evidence, this finding suggests that players who agreed on a tradewere typically not constrained by the time limit, since they typically made fewer offers thantheir counterparts who did not agree. It is also interesting to note that the more stringent is
the liquidity constraint, the greater is the number of offers made on average by the players(and thus the larger the delays): it took fewer than 5 proposals, on average, for players in theunconstrained treatment to agree, compared to almost 11 proposals for players in the mostconstrained treatment.

7

Welfare Cost of Inflation

The results we reported on in the previous section provide strong support for the Kalai bargaining solution over the Nash solution and for equal bargaining weights for buyers and sellers.
In this section we show how these findings can be of use to researchers working in the searchmoney literature, who make use of the bargaining setting that our experiment studies. In
particular, we show how our findings on bargaining weights and bargaining solution haveimportant implications for the estimation of the welfare costs of inflation.21
Consider the inverse demand curve for real money balances, with money demand on thex-axis and the nominal interest rate on the y-axis. The empirical money demand curve isrepresented by the circles in Figure 11 for the years 1900-2000, where each observation corresponds to a year. Money demand corresponds to the aggregate balance of M1 divided bynominal GDP, while the nominal interest rate corresponds to the rate on short-term commercial paper. The original method developed to estimate the welfare cost of inflation is due to21

We recognize that this section may not appeal to all readers, but the question of the appropriate bargaining

solution to use in search-money models and the implications for the welfare cost of inflation did serve as an
impetus for this project, and so we feel we would be remiss not to include this discussion here. Readers who
are not interested in this topic can skip to the concluding section 8.

32

Bailey (1956) and was later expanded upon by Lucas (2000). It consists in first estimating
the demand curve and then measuring the area under the curve between the relevant nominalinterest rates, where the latter are mapped into inflation rates (Π) using the Fisher equation,i = r + Π.
0.15
Data
Fitted Curve

Nominal interest rate

0.13
0.11
0.09
0.07
0.05
0.03
0.01
0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Money demand
Figure 11: Money demand curve in the US, 1900-2000. Empirical observations in black circles (source:
Craig and Rocheteau, 2006). Non-linear least square curve fit for θ = 0.5 under the Kalai bargaining
solution in orange solid curve.

Craig and Rocheteau (2006, 2008) highlight that the Bailey-Lucas approach is only correct
if the private benefits of real money balances to money holders are equal to their social benefits.
Indeed, the money demand curve only captures the benefits of money to its holders, not tosociety. Since any transaction is two-sided, conditional on sellers extracting some surplus fromtransactions, the welfare triangle approach underestimates the benefits of real money balances(and thus underestimates the cost of inflation). Obtaining a correct estimate therefore requires
accounting for the surplus obtained by sellers in transactions. For a given surplus received
by a buyer (money holder), the surplus received by the seller depends entirely on the way inwhich terms of trade are determined. For example, in a transaction where the buyer receives
a surplus S b , the seller would receive a surplus S s = (1 − θ) S b /θ, assuming Kalai bargaining
33

with a buyer’s bargaining weight of θ. A different bargaining solution or a different distributionof bargaining powers would lead to inferring a different surplus for the seller.
Using a typical search-theoretical model of money where competitive markets alternatewith decentralized markets with bilateral trades that make money essential, Craig and Rocheteau (2006, 2008) are then able to estimate the welfare cost of inflation for four different
trading protocols (fixed markup, Nash bargaining, Kalai bargaining, and take-it-or-leave-it
offers) and a variety of distributions of bargaining powers.22 They calibrate their model to
the US economy between 1900 and 2000.
We follow a similar method but calibrate the decentralized market to reflect our experimental setup. Specifically, we estimate the money demand curve and the cost of inflation using
the parameters from the experiment, varying both the distribution of bargaining powers andthe bargaining protocol assumed.23 Following the literature, we focus on estimating the costof a 10% inflation regime compared to a no-inflation regime. Given a discount rate of 3%, thisis equivalent to comparing an economy with a i = 13% nominal interest rate to an economy
with a i = 3% nominal rate. Note that a nominal interest rate of 0% (thus an inflation rateof -3%) corresponds to the Friedman rule.
Table 8 shows our results. Considering both the Kalai and Nash bargaining solutions, theTable 8: Estimates of the quantities traded bilaterally and of the welfare cost of inflation.
Buyer’s bargaining

Kalai

Nash

Quantity traded, qWelfare cost (% of GDP),

power, θi = 0% (Friedman rule)i = 3% (No inflation)i = 13% (10% inflation)from i = 3% to i = 13%

0.33

4.00

2.84

0.32

5.70

0.50

4.00

2.70

0.43

3.35

0.66

4.00

2.70

0.74

2.13

1.00

4.00

3.06

1.68

1.20

0.33

1.54

1.04

0.51

3.67

0.50

2.17

1.53

0.76

2.86

0.66

2.74

1.98

1.02

2.26

1.00

4.00

3.06

1.68

1.20

table reports the quantity traded in bilateral meetings as well as the cost of 10% inflation as
a function of the buyer’s bargaining weight.
First notice that the higher is the inflation rate, the higher is the nominal interest rate,
22

The alternation of competitive and decentralized markets is also designed to capture that, in field settings,

some trades take place in frictionless markets, while other trading activities take place in markets with frictions,e.g., where it may be more difficult to find counterparties. See also Lagos et al. (2017).
23
See Appendix C for details about the model and the estimation.

34

and the lower is the quantity that is traded, q. Indeed, the higher the inflation rate, themore costly it is to hold real money balances from one period to the next (in other words, thenominal interest rate is the opportunity cost of holding real money balances). As this cost
increases, buyers economize by carrying fewer real balances, and thus they cannot purchaseas many units of the consumption good from sellers. This phenomenon is partially due toa hold up problem: as long as θ < 1, buyers do not extract all of the surplus generated bycarrying real balances, which leads them to “underinvest” in real balances. This disappears
when θ = 1, in which case the first-best quantity of goods is traded both under the Nash andKalai bargaining solutions, q = q ∗ = 4. Interestingly, note that under the Nash solution, evenwhen the Friedman rule is in place so that carrying real money balances is costless (i = 0),buyers may not carry the optimal amount of real balances, leading to suboptimal trade sizes.
This is due to the non-monotonicity of the Nash solution, as depicted in Figure 2. At somepoint, carrying additional real balances worsens the buyer’s bargaining position under the
Nash solution as it reduces the amount of the surplus that she can extract. Even if real money
balances are costless, this is another reason for buyers to limit their money holdings.24
Estimates obtained under the specification in line with our experimental results are highlighted in grey. The corresponding estimated money demand curve is represented in Figure 11.
Finding 6. Under Kalai bargaining and with a buyer’s bargaining power of 0.5, the cost of a
10% inflation amounts to 3.35% of GDP. Assuming instead that buyers have all the bargaining
power (θ = 1) leads to underestimating the welfare cost of inflation by a factor of 2.79 (fora cost of 1.20% of GDP). Setting the bargaining power equal to θ = 0.5 but using the Nashbargaining solution instead of Kalai leads to underestimating the cost of inflation by a factorof 1.17 (for a cost of 2.86% of GDP).

8

Conclusion

We have studied a bargaining setting where players simultaneously determine both the sizeof the gains from trade and the division of those gains. We have further considered the case24

An interesting result is that while in partial equilibrium, for a given amount of money holdings, the Nashbargaining leads to a higher total surplus shared between a buyer and a seller, in general equilibrium, Nashbargaining leads to lower real money balances holdings and therefore lower surpluses.

35where one party, the buyer, is constrained in terms of the amount of money that they canbring to the bargaining table. Such liquidity constraints are a common phenomenon andmake the bargaining set asymmetric between the two players. Under the Nash bargaining
solution, the presence of liquidity constraints gives rise to a larger surplus going to the buyer
relative to the seller and larger total gains from trade than under the Kalai solution, where
the surpluses are predicted to be equal (given the bargaining weight of 1/2) regardless ofliquidity constraints. These two different solutions are used in the money search literature tounderstand such questions as the welfare cost of inflation.
The evidence from our experiment clearly favors the Kalai bargaining solution over the
Nash solution. Still, this is just a first step in understanding how liquidity constrained players
approach the bargaining problem.
It is interesting to note that by favoring a solution close to the proportional Kalai bargaining solution rather than Nash bargaining, players effectively agree to share a smaller pie,decreasing total welfare, in order to achieve more equality. The Nash solution would allow fora larger joint production, albeit at the expense of the seller. Theory predicts that this welfare
result would be overturned were the liquidity constraints endogenized through a costly ex-antechoice of real balances by buyers (see, e.g., Lebeau (2020), Rocheteau et al. (2020)). In thatcase, playing according to the Nash solution rather than the Kalai solution would lead thebuyer to carry fewer real balances, making the negotiation more liquidity-constrained, eventually resulting in lower trade volumes and total welfare. In that case, it would also be to theadvantage of the buyer to implement Kalai bargaining. This suggests that our findings would
be strengthened were liquidity constraints endogenized rather than imposed upon subjects.
A promising avenue to test this hypothesis in the lab would be to consider the following
two stage game. First buyers decide how much to borrow in terms of money. Then, in thesecond stage, bargaining takes place. Finally, in the payoff stage, buyers have to repay theirborrowings with interest and realize their payoffs. Higher interest rates would also capture
higher inflation rates, allowing us to explore the welfare costs of inflation more directly. Other
promising investigations include the implementation of bargaining settings that provide noncooperative foundations to Kalai’s solution (see, e.g., Dutta (2012, 2021), Hu and Rocheteau(2020)) as well as the incorporation of unstructured bargaining in fully dynamic settings (e.g.,
Duffy and Puzzello (2021), Jiang et al. (2021)). We leave these extensions to future research.
36

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Appendix For Online Publication OnlyA

Experimental Instructions

Note: The following instructions were used in the constrained m = 60 treatment. Other
instructions are similar.
Welcome to this experiment in the economics of decision-making. Please read these instructions carefully as the cash payment you earn at the end of today’s session may dependon how well you understand these instructions.

If you have a question at any time, pleasefeel free to ask the experimenter. There is no talking for the duration of this 2-hour session.
Please turn off your cell phone and any other electronic devices.
Your Role and Matching
There are 10 participants in today’s session. You will participate in 30 rounds of decisionmaking using the networked computer workstations of the laboratory. Prior to the first round,
one-half of participants will be randomly assigned the role of Buyer and the other half therole of Seller. You will remain in the same role in all 30 rounds.
In each and every round, the computer program randomly and anonymously matches
Buyers and Sellers in pairs. All possible pairings of one Buyer with one Seller are equally
likely in each round. Thus, while the player you are paired with will always be of the opposite
type, they are likely to change from round to round. You will not know the identity of any ofthe players you are paired with, nor will they know your identity, even after the experimentis over.
Your Decision in Each RoundIn each round, Sellers can produce a good that is valuable to Buyers only. In each round
Buyers have an endowment of 60 tokens that can be used to purchase units of the good that
Sellers produce. A round involves bargaining between the Buyer and Seller over the quantityof the good the Seller will produce, q, and the number of tokens, τ , the Buyer will give theSeller in exchange.
A proposal thus consists of a pair, (q, τ ). Quantities, q, can range from 0 to 11, inclusive,1

in increments of up to 2 decimal places. The number of tokens, τ , can range from 0 to 60,
inclusive, in increments of up to 2 decimal places. Proposals can be made by either the Buyeror the Seller at any time during each 2-minute bargaining round. Any number of proposals
can be made by either player and in any order over the duration of the bargaining round.
Once a proposal is made, it is shown on both the Buyer and the Seller’s decision screens andis considered “live”. If the player not making the proposal chooses to accept that proposal,
the bargaining round is declared over and the proposed exchange of q units of the good for τ
tokens is implemented. The payoffs from such an agreed upon exchange are explained below.
Proposals submitted by one player can be accepted by the other player at any time during each
2-minute round. If no proposal is made or there is no agreement on any proposal by players
within each 2-minute bargaining round, then the round is declared over and no exchange takes
place.
Payoffs From Exchange Outcomes
1. If a proposal is accepted within the 2-minute bargaining round, then an exchange takes
place. In that case:• The Buyer’s payoff is given by:
Buyer’s payoff in points = u(q) − τ.

(A1)

Here u(q) is an increasing function of the quantity q representing the buyer’s value
from consuming q units of the good. This function is illustrated on your decision screen, and Table A1 provides the actual function for u(q) along with a nonexhaustive list of possible values for u(q). Note that while u(q) is increasing in q,
these increases are lower with higher values for q. Notice also that each token a
Buyer gives to the Seller reduces the Buyer’s payoff by one point. Importantly,
tokens that are not offered as part of an agreed upon proposal have no payoff inpoints to the Buyer. That is, if an agreed upon proposal involves an exchange ofτ tokens, then the Buyer loses τ points. Any remaining tokens held by the buyer,
60 − τ , have 0 value.

2• The Seller’s payoff is given by:
Seller’s payoff in points = τ − c(q)(A2)

Each token a Seller receives from a Buyer increases the Seller’s payoff by one point.
Recall that tokens only have value to the Seller in terms of points (and a cost to
the Buyer in terms of points) if a proposal is accepted. Notice also that the Seller’s
payoff is decreased by c(q), which represents the Seller’s cost of producing q units.
This cost function, c(q), is illustrated on your decision screen. Table A1 provides
the actual function for c(q) along with a non-exhaustive list of possible values forc(q). Note that while c(q) is increasing in q these increases are higher with higher
values for q.
2. If no proposal is made or is accepted within a 2-minute bargaining round then no exchange takes place. In that case:• Both the Buyer and the Seller payoffs are 0 for that round.
The Decision Screen
The decision screen you face is illustrated in Figures 1-2. The top part of this screen, as
shown in Figure 1, illustrates the functions u(q) for Buyers and c(q) for Sellers. Below thisillustration are two slider bars, one for the quantity, q, and the other for tokens, τ , which arerepeated in Figure 2, which illustrates the bottom part of the decision screen from the Buyer’s
perspective (the Seller’s perspective is similar). By moving the slider bars for q and τ you willsee, to the right of these slider bars, how the Buyer’s value, u(q), and the Seller’s cost, c(q),will be affected by the choice of q as well as the payoff to you and the other player from yourproposal of (q, τ ). You can move both slider bars as often as you wish to experiment withvarious proposals. When you find a proposal that you would like to submit to the other player,press the blue Submit button, which makes your proposal ‘live’. This means that the otherplayer can accept your proposal at any time during the two-minute round. Once submitted,
proposals cannot be withdrawn.

Note that proposals that would result in negative payoffs

to either you or the other player are not allowed. The computer program will check whether

3Figure A1: Top Part of Decision Screen (Buyer View)

your proposal would result in negative payoffs to you or the other player. If you try to submit
such a proposal you will receive an error message and that proposal will not be made live.
The proposals, in the form, q units for τ tokens, submitted by the Buyer or by the Seller
are shown at the bottom of your decision screen under the headings “Buyer proposals” and“Seller proposals”. Next to each proposal is the number of points the Buyer would earn andthe number of points the Seller would earn if the proposal were accepted. A scroll bar willappear if the number of proposals is large so that you can review all proposals made in thecurrent round.
To accept a proposal, simply click on the Accept button next to a proposal at the bottomof your decision screen. Note: you cannot accept your own submitted proposal; it is assumed
that you agree to abide by any proposal that you submit. The bargaining round ends if one
player chooses to accept a proposal made by the other player or the 2-minute time limit to
the round has been reached, whichever comes first.
Payoffs
If you complete this experiment, you are guaranteed a $7 participation payment. In addition,at the end of the experiment, the computer program will randomly choose 2 rounds from all4

Figure A2: Bottom Part of Decision Screen (Buyer View)

30 rounds played. Your payoff in points from those 2 rounds will be converted into dollars at
the rate of 1 point = $0.25. Your dollar payoff from the chosen rounds will be added to yourparticipation payment. Your total payment will be made to you in cash and in private.
Summary of the Experiment
1. Prior to the first round, you will be randomly assigned a role as a Buyer or a Seller. You
will remain in the same role in all 30 rounds of the experiment.
2. In each round, Buyers and Sellers are randomly paired and have 2 minutes to bargain
over an amount, q, of the good the Seller will produce for the Buyer and the amount oftokens, τ , the Buyer will give the Seller in exchange.
3. Bargaining consists of making proposal pairs of (q, τ ). Proposals can be made by either
player, in any order. Proposals that would result in negative payoffs to either player are
not allowed.
4. Once submitted, proposals are considered live and can be accepted at any time.
5. If a proposal, (q, τ ), is accepted by the other player, the bargaining round is over. In
that case, the Buyer’s payoff in points for the round is u(q) − τ . The Seller’s payoff in5

points is τ − c(q).
6. The Buyer’s tokens have no value or cost (i.e., they are worthless) unless a proposal isaccepted. In that case, the τ tokens offered yield τ points to the Seller and cost the
Buyer τ points. Any excess tokens held by the Buyer, 60 − τ , have 0 value or cost.
7. If no proposal is made or is accepted within the 2-minute bargaining round, then bothplayers earn 0 points.
8. At the end of the experiment, 2 rounds, from all 30 rounds, will be randomly chosen for
payment. Your points from those 2 rounds will be converted into dollars at the rate of
1 point = $0.25 cents.
Questions?
Now is the time for questions. If you have a question, please raise your hand and your question
will be answered in private.
Quiz
To check your understanding of the instructions, we ask you to complete the following quiz
before we move on to the experiment. The numbers in these quiz questions are for illustration
purposes only. The actual numbers in the experiment may be quite different. When you havecompleted the quiz, please raise your hand. An experimenter will check your answers. If there
are any wrong answers we will go over the relevant part of the instructions again.
1. My role as a Buyer or as a Seller will (circle choice):
a. change every round.
b. remain the same in all rounds.
2. Suppose a player makes a proposal of q = 3.25 and τ = 50, and this proposal is acceptedby the other player. What is the payoff in points to the Buyer from this proposal? (UseEquation (A1) and Table A1).

What is the payoff in points to the Seller fromthis proposal? (Use Equation (A2) and Table A1).

6

3. Suppose a player makes a proposal of q = 1 and τ = 60, and this proposal is acceptedby the other player. What is the payoff in points to the Buyer from this proposal? (UseEquation (A1) and Table A1).

What is the payoff in points to the Seller fromthis proposal? (Use Equation (A2) and Table A1).
4. If several proposals are made but none are accepted by the end of a 2-minute trading
round, then (circle choice):a. the last proposal made will be implemented.
b. both players will earn 0 for the round.
5. True or false: If I make a proposal in a round, I can later withdraw it and make aproposal that is better for me. Circle one:

True

False

6. True or false: At the end of the experiment my payoff in points from two randomly
chosen rounds will be converted into dollars at the rate of 1 point = $0.25. Circle one:
True

B

False

Non-zero disagreement values

Figure B1: Pareto frontiers and predicted allocations under Nash and Kalai solutions with a (10,5)
disagreement point compared to allocations with a (0,0) disagreement point.

In our experiment, the disagreement point was normalized to (0, 0). This section shows7that the theoretical predictions that we test for in Hypotheses 1 to 3 remain identical if thedisagreement point is different from zero.
Assume that if the two players do not agree on a trade, the buyer obtains db > 0 and theseller obtains ds > 0. The Nash bargaining problem ismax [u(q) − y − db ]θ [y − c(q) − ds ]1−θ s.t. y ≤ m,
q,y

(B1)

while the Kalai bargaining problem ismax u(q) − y − db s.t. u(q) − y − db =q,yθ[y − c(q) − ds ] and y ≤ m.
1−θ

(B2)

After redefining S b ≡ u(q) − y − db , S s ≡ y − c(q) − ds , and S ≡ u(q) − c(q) − db − ds , it iseasy to show that we still obtain S b = θS and S s = (1 − θ)S under Kalai. Under Nash, we
also still have S b = Θ(q)S and S s = [1 − Θ(q)]S, where q is given by
θu0 (q)[c(q) + ds ] + (1 − θ)c0 (q)[u(q) − db ]∗.
min q , q :θu0 (q) + (1 − θ)c0 (q)(B3)

Because the Nash and the Kalai solution predict how the net pie (i.e., net of disagreement
payoffs) is split, changing disagreement values only has an impact on the size of what can beshared, not on how it is shared between the two players. More precisely, the buyer still earns ashare θ of the net pie according to the Kalai solution, and a share Θ(q) according to the Nashsolution. In addition, the buyer’s surplus under Nash is still non-monotone, first increasing
then decreasing as the liquidity constraint is relaxed.
Figure B1 provides a graphical illustration using the same parameters as those used in
the laboratory and imposing θ = 0.5, comparing the paths and allocations as m increases
when (ds , db ) = (0, 0) and when (ds , db ) = (10, 5). Going from the former to the latter
disagreement point, the origin translates to (10, 5), with the new axes now represented inblue. The bargaining path predicted by Kalai is then represented by the blue dashed line,
while the blue dotted line is the new path under Nash. Blue squares represent the newly
predicted allocations for m = 30, m = 60 and m = 315. Surpluses can be obtained by taking
the difference between the axis value corresponding to the new allocations and the new origin,(10, 5). Shares of the total surplus remain identical conditional on the quantity traded, but
gross payoffs generically differ.

8

C

Welfare cost of inflation

Model. We embed the model of bilateral trade used in our experiment into a general searchtheoretic model of monetary exchange. The resulting model is equivalent to the one by Lagos
and Wright (2005) and can be described as follows.
The economy is populated by a continuum of agents of measure one. Time is discrete
and infinite. In each period, there are two subperiods. In the first subperiod, agents may
meet bilaterally in a decentralized market. An agent meets with another player who likes
the good they can produce with probability σ. In this case, he becomes a buyer in thebilateral meeting. He may instead meet another player whose good they would like to consume,
also with probability σ. In that case, he becomes a seller. We assume that there are no
meetings where both agents like each other’s products (no double-coincidence of wants), suchthat with probability 1 − 2σ, an agent does not get the opportunity to trade in the first
subperiod. In that decentralized market, the production of q units of good costs c(q), while
their consumption provides u(q) utils. In addition, payments require money, available insupply M in the economy. In the second subperiod, there is a centralized and frictionlesscompetitive market where all costs and utilities are linear.
In the decentralized market, when an agent gets the opportunity to buy from anotheragent, he can purchase a quantity q of output against a payment z(q), for a payoff u(q) − z(q).
Choosing money holdings is thus equivalent to choosing a trade quantity q. Thus agents pick q
to maximize their expected trade surplus net of the cost of holding the real balances required
to purchase that amount of goods:
q = arg max {−iz(q) + σ[u(q) − z(q)]} .

(C1)

Solving for this program determines the quantity of goods purchased in bilateral meetings, q(i),as a function of the nominal interest rate, i. In addition, we can solve for a payment functionz(q), which depends on the specifics of the bargaining solution. With Kalai bargaining, z(q) =(1 − θ)u(q) + θu(q). With Nash bargaining, z(q) = u(q) − Θ(q)[u(q) − c(q)]. We can thenobtain individual money demand as a function of real balances,z̃(i) = z ◦ q(i).

(C2)

Aggregate money demand is equal to M/(σM + pA), where A is the real output and p is the9price in the centralized market. Using z = M/p, we can rewrite aggregate money demand as
L(i) =z̃(i).
σz̃(i) + A(C3)

Calibration and money demand curve estimation. Consistent with the calibration
used in our experiment, we assume u(q) = 74.1752q 0.6 and c(q) = 8.23249q 1.51678 . Regarding
the determination of the terms of trade, we vary both the bargaining protocol (Nash and Kalai
bargaining), and the distribution of bargaining powers (θ ∈ {0.33, 0.50, 0.66, 1}), leading to
8 different specifications. For each specification, we estimate A and σ by fitting (C3) to ourempirical money demand curve by non-linear least squares.
Welfare cost estimations.

To estimate the welfare cost of a 13% nominal interest rate

(10% inflation) compared to a 3% nominal interest rate (0% inflation), we start from theno-inflation steady state and compute the drop in consumption that would make a plannerindifferent with a 10% inflation steady state. Mathematically, the cost ∆ must satisfy
σ {u[q(0.03)(1 − ∆)] − c[q(0.03)]} − A∆ = σ {u[q(0.13)] − c[q(0.13)]} .
Table 8 reports 100∆ for the 8 specifications we consider.

D

Additional Tables

In this section we provide additional tables.

10

(C4)

Table A1: Buyer values u(q) = (74.1752)q 0.6 and Seller costs c(q) = (8.23249)q 1.51678 in points.

Difference u(q) − c(q) pts.

qu(q) in pointsc(q) in points

0

0.00

0.00

0.00

0.25

32.29

1.01

31.28

0.5

48.94

2.88

46.06

0.75

62.42

5.32

57.09

1

74.18

8.23

65.94

1.25

84.80

11.55

73.25

1.5

94.60

15.23

79.38

1.75

103.77

19.24

84.53

2

112.43

23.56

88.87

2.25

120.66

28.17

92.50

2.5

128.54

33.05

95.49

2.75

136.10

38.19

97.91

3

143.39

43.57

99.82

3.25

150.4549.20

101.25

3.5

157.2955.05

102.243.75

163.9461.12

102.814

170.41

67.41

103.004.25

176.7273.90

102.824.5

182.8980.60

102.294.75

188.92

87.48

101.44

5

194.82

94.56

100.26

5.25

200.61101.83

98.79

5.5

206.29109.27

97.02

5.75

211.87116.8994.97

6217.34124.69

92.66

6.25

222.73132.6590.08

6.5

228.04140.78

87.26

6.75

233.26149.0784.19

7238.41157.53

80.88

7.25

243.48166.1477.34

7.5

248.48174.91

73.58

7.75

253.42183.8369.59

8258.29192.89

65.40

8.25

263.11

202.11

61.00

8.5

267.86211.47

56.39

8.75

272.56220.9851.58

9277.21230.63

46.58

9.25

281.80240.41

41.39

9.5

286.35250.34

36.01

9.75

290.84260.4030.45

10295.30270.59

24.71

10.25

299.70280.92

18.79

10.5

304.07291.37

12.70

10.75

308.39301.966.43

11

312.68312.680.00

11

Table D1: Agreement rates for all treatments and sessions.

m = 30m = 60m = 315

Session \ Round

1-15

16-30

All

1-15

16-30

All

1-15

16-30

All

1

0.91

0.90

0.90

0.93

0.89

0.91

0.89

0.92

0.91

2

0.91

0.91

0.91

0.93

0.97

0.95

0.99

0.99

0.99

3

0.84

0.93

0.89

0.87

0.93

0.90

0.8

0.92

0.86

4

0.85

0.91

0.88

0.92

0.93

0.93

0.97

1.00

0.99

5

0.84

0.87

0.85

0.85

0.86

0.86

0.92

0.97

0.95

All

0.87

0.90

0.89

0.90

0.92

0.91

0.91

0.96

0.94

12

0.66
0.63
0.72
0.65
0.69
0.67

1

2

3

4

5

All

0.67

0.69

0.66

0.73

0.63

0.66

1-15

0.67

0.69

0.64

0.71

0.63

0.66

16-30

13
1.78
1.64
1.66
1.68
1.68
1.69

1

2

3

4

5

All

1.67

1.64

1.68

1.63

1.61

1.78

1-15

1.70

1.72

1.68

1.69

1.66

1.78

16-30

All
3.99
4.08
4.06
3.83
4.19
4.03

Session \Round

1

2

3

4

5

All

4.10

4.35

3.74

4.11

4.19

4.11

1-15

3.96

4.03

3.92

4.02

3.97

3.87

16-30

Mean Quantity

All

Session \Round

Mean Quantity

All

Session \Round

Mean Quantity

119.67123.87114.51

119.65121.27119.14

All

29.89

29.80

29.92

30.00

29.80

29.93

16-30

58.35

58.01

59.35

57.81

57.88

58.67

1-15

59.74

59.84

59.96

59.01

59.93

59.95

16-30

121.38128.22112.07120.52124.12

122.231-15

118.01119.70116.93118.87118.41116.1016-30

Mean Tokens

59.05

58.93

59.66

58.43

58.93

59.28

All

29.51

29.50

29.68

29.64

29.03

29.70

1-15

Mean Tokens

29.70

29.65

29.80

29.83

29.41

29.82

All

Mean Tokens

28.47

29.42

27.52

31.26

27.10

27.27

1-15

27.75

29.41

25.73

29.99

26.17

27.32

16-30

42.15

41.56

41.78

41.27

40.17

45.80

1-15

42.13

42.42

41.08

42.50

40.59

44.32

16-30

51.09

50.93

51.21

51.69

51.06

50.58

All

50.80

50.39

50.93

51.55

50.80

50.40

1-15

51.37

51.46

51.49

51.81

51.32

50.76

16-30

Mean Buyer Surplus

Treatment m = 315

42.14

41.99

41.43

41.90

40.39

45.09

All

Mean Buyer Surplus

Treatment m = 60

28.11

29.42

26.62

30.59

26.64

27.30

All

Mean Buyer Surplus

Treatment m = 30

53.33

54.18

52.74

55.61

51.97

52.34

1-15

53.08

54.50

51.23

55.02

51.91

52.61

16-30

82.36

81.85

83.00

81.58

80.86

84.38

1-15

83.22

83.27

83.02

83.19

82.68

84.04

16-30

102.08102.06102.10101.45102.52102.19All

101.38101.19101.27100.36102.08101.82

1-15

102.76102.90102.93102.42102.96102.5516-30

Mean Total Surplus

82.79

82.57

83.01

82.41

81.79

84.22

All

Mean Total Surplus

53.20

54.34

51.98

55.30

51.94

52.47

All

Mean Total Surplus

0.52

0.53

0.51

0.55

0.51

0.51

1-15

0.51

0.54

0.47

0.54

0.50

0.50

16-30

0.51

0.50

0.50

0.50

0.49

0.54

1-15

0.50

0.50

0.49

0.51

0.49

0.52

16-30

0.50

0.50

0.50

0.51

0.50

0.49

All

0.50

0.50

0.50

0.52

0.50

0.49

1-15

0.50

0.50

0.50

0.51

0.50

0.49

16-30

Mean Buyer Share

0.50

0.50

0.50

0.51

0.49

0.53

All

Mean Buyer Share

0.52

0.54

0.49

0.54

0.51

0.51

All

Mean Buyer Share

Table D2: Mean outcomes for all treatments and all sessions.

Table D3: Random-effect estimation of the buyer’s share of surplus in accepted offers. Random
effects are indexed at the individual seller level. The sample in column (1) includes all accepted
offers in the unconstrained treatment. Column (2) includes the subsample s.t. |q − 4| < 0.5.
Column (3) includes the subsample s.t. |q − 4| < 0.1. Column (4) includes the subsample s.t.
|q − 4| < 0.05.

Buyer’s surplus

Total surplus, S

Observations

(1)

(2)

(3)

(4)

0.50060.49970.50000.5005(0.0027)(0.0014)(0.0017)(0.0019)

698

574

412

348

Standard errors clustered at the session level in parentheses

Table D4: Random-effect estimation of the impact of varying the liquidity constraint, m, on
the buyer’s surplus. Random effects are indexed at the individual seller level. The sample incolumn (1) includes all accepted offers, and the sample in column (2) includes accepted offers
s.t. |q − 4| < 0.5 when m = 315, y − 60 < 0.5 when m = 60, and y − 30 < 0.5 when m = 30.

Buyer’s surplus

m = 30m = 315

Constant (m = 60)

Observations

(1)

(2)

-14.0669-14.3074(1.0303)(1.1053)

8.94729.1411(0.7480)(0.7373)42.1275

42.2828(0.7304)(0.7264)

2028

1801

Standard errors clustered at the session level in parentheses

14

Table D5: Random-effect estimation of the change in a player’s share of surplus between thefirst offer and the accepted offer, depending on having made the first offer. Random effects
are indexed at the individual player level. The sample is restricted to sellers.

∆ Player’s share
Player made the first offer-0.1348(0.0219)

Constant

0.0861(0.0186)

Observations

835

Standard errors clustered at the session level in parentheses

15

E

Figures0.65

m = 300.65

m = 600.65

m = 315
Median
Mean

0.55

0.5

0.45

0.4

0.6

Seller's share of surplus

0.6

Seller's share of surplus

Seller's share of surplus

0.6

0.55

0.5

0.45

0.4

0.35
4th to last 3rd to last 2nd to last

Seller's Offer

Last

0.55

0.5

0.45

0.4

0.35
4th to last 3rd to last 2nd to last

Seller's Offer

Last

0.35
4th to last 3rd to last 2nd to last

Last

Seller's Offer

Figure E1: Seller’ share of the surplus in their own offers, by rank of the offer relative to the accepted
offer and by treatment. Colored circles represent all offers. Median offers are represented with a blacksquare, and mean offers are represented by a black cross.

16