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FEDERAL RESERVE BANK OF CLEVELAND

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papers

NUMBER 12

by Ben Craig and Guillaume Rocheteau

POLICY DISCUSSION PAPER

Inflation and Welfare:
A Search Approach

JANUARY 2006

POLICY DISCUSSION PAPERS

FEDERAL RESERVE BANK OF CLEVELAND

Inflation and Welfare:
A Search Approach
By Ben Craig and Guillaume Rocheteau

This paper extends recent findings in the search-theoretic literature on monetary exchange regarding
the welfare costs of inflation. We present first some estimates of the welfare cost of inflation using the
“welfare triangle” methodology of Bailey (1956) and Lucas (2000). We then derive a money demand
function from the search-theoretic model of Lagos and Wright (2005) and we estimate it from U.S. data
over the period 1900–2000. We show that the welfare cost of inflation predicted by the model accords
with the welfare-triangle measure when pricing mechanisms are such that buyers appropriate the social
marginal benefit of their real balances. For other mechanisms, welfare triangles underestimate the true
welfare cost of inflation because of a rent-sharing externality. We also point out other inefficiencies
associated with noncompetitive pricing, which matter for estimating the cost of inflation. We then
illustrate how endogenous participation decisions can mitigate or exacerbate the cost of inflation, and
we provide calibrated examples in which a deviation from the Friedman rule is optimal. Finally, we
discuss distributional effects of inflation.

Ben Craig is an economic
advisor at the Federal
Reserve Bank of Cleveland,
and Guillaume Rocheteau
is an economist at the
Bank.The authors have
benefited from the comments
of Ray Batina, Chuck
Carlstrom, Richard Dutu,
Sebastien Lotz, Ed Nosal,
and Christopher Waller. They
thank Patrick Higgins for his
research assistance and
Monica Crabtree-Reusser for
editorial assistance.

Materials may be reprinted,
provided that the source is
credited. Please send copies
of reprinted materials to the
editor.

POLICY DISCUSSION PAPERS

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ISSN 1528-4344

FEDERAL RESERVE BANK OF CLEVELAND

Introduction
Assessing the welfare costs of inflation requires a sound understanding of the benefits of monetary
exchange. The search theory of money, developed in the last 15 years from the pioneering works of
Kiyotaki and Wright (1989, 1991, 1993), offers such a framework. However, the first generation of
search models of money were based on assumptions that were too restrictive to be able to deliver
useful insights for monetary policy (goods and money were indivisible, for example, individuals’ portfolios were limited to one unit of one object, and so forth).These severe restrictions have been relaxed
in several recent extensions of the theory, by Shi (1997, 1999), Molico (1999), and Lagos and Wright
(2005).The extensions have opened up the perspectives for a better understanding of the costs, and
also maybe benefits, associated with inflationary finance.As an example, Lagos and Wright provide estimates for the cost of 10 percent inflation ranging from 1.4 percent of GDP to 4.6 percent of GDP.
Interestingly, these numbers are significantly larger than estimates based on the traditional method
developed by Bailey (1956), which consists of computing the area underneath a money demand function. For instance, Lucas (2000), using Bailey’s approach, estimates the cost of 10 percent inflation at
slightly less than 1 percent of GDP.1
In this paper we clarify and extend recent findings provided by models of monetary exchange to
the evaluation of the cost of inflation for society. Our approach consists of relating the measures of the
welfare cost of inflation obtained from different versions of the search-theoretic model of Lagos and
Wright (2005) with the traditional measures based on the area underneath the money demand func-

1. Based on this methodology,
Fischer (1981) and Lucas (1981)
obtained estimates for the cost
of 10 percent inflation ranging
from 0.3 percent of GDP to 0.45
percent of GDP.

tion. We show the conditions under which the two measures are consistent, and those under which
they differ. We also disentangle the different effects of inflation in search models of monetary exchange: a real-balance effect, an effect on participation decisions, and a distributional effect.We show
that the estimates for the welfare cost of inflation provided by the basic version of the search model of
Lagos and Wright coincide with those provided by the Bailey method whenever money holders can
appropriate the marginal social return of their real balances. This condition is satisfied when buyers
have all the bargaining power to set prices in bilateral trades, or when pricing is competitive. If this
condition does not hold, then the welfare cost of inflation is larger than what traditional estimates
predict.This discrepancy arises because of a rent-sharing externality associated with noncompetitive
pricing mechanisms.We establish a simple relationship between the cost of inflation, the area underneath the money demand function, and the buyer’s share in the surplus of a trade. We also discuss
various inefficiencies associated with different bargaining solutions.
We also extend the Lagos-Wright model by introducing participation decisions and trading frictions.We show that the measures of the cost of inflation based on the Bailey methodology are in general misleading since they do not take into account the effects of inflation on participation decisions.
We also illustrate how the presence of search frictions can mitigate or exacerbate the welfare cost of
inflation.We provide calibrated examples in which the Friedman rule, where the interest rate is set to
zero, is not the optimal policy.
The Lagos-Wright model is based on assumptions that yield a degenerate distribution of money
balances in equilibrium.2 While these assumptions make the model tractable, they prevent an analysis
of the distributional effects of inflation.We reintroduce such effects by considering a simple extension
of the Lagos-Wright model in which agents receive idiosyncratic productivity shocks. We discuss the
distributional effects of monetary policy and the insurance role of inflation.

2. Shi (1997, 1999) constructs a
different model, which also yields
a degenerate distribution of
money balances. The economy
is populated by households
composed of a large number of
members, who pool their money
balances at the end of each
period. See also Faig (2004).

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POLICY DISCUSSION PAPERS

NUMBER 12, JANUARY 2006

The paper is organized as follows. First, we present the methodology Bailey (1956) developed
to compute the welfare cost of inflation. Next, we use a search-theoretic model to derive a money
demand function, and we fit this function to the data. This allows us to compute an alternative measure of the cost of inflation and to compare it to the Bailey estimates.Then, the benchmark model is
extended to discuss the importance of pricing mechanisms. Next, we consider participation decisions
and search frictions. Finally, we introduce the distributional effects of inflation.3

The “Welfare Triangle”

3. A more detailed presentation of
the model occurs in appendix 1,
and the data are given in
appendix 2.

The traditional approach to measuring the cost of inflation was developed by Bailey (1956).The estimates provided by this approach will be useful in explaining some findings of the search model. Bailey (1956) measures the welfare cost of inflation by calculating the area underneath a money demand
curve over an appropriate interval. We plot in figure 1 the (inverse) demand for real balances, where
the cost of holding real balances, the nominal interest rate, is represented on the vertical axis.The demand for real balances is downward-sloping since individuals reduce their money balances and resort
to alternative payment arrangements, such as credit or barter, as the interest rate increases. The area
underneath the money demand relationship over the interval [m1, m*], the “triangle” ABC in figure 1,
measures the welfare cost of having a positive interest rate r1 instead of zero. (In this analysis, the interest rate is assumed to vary one to one with the inflation rate.) Obviously, the welfare cost of inflation is minimized when the nominal interest rate is zero.4 This corresponds to the Friedman (1969)
rule for optimal monetary policy. In the following, we will measure the cost of inflation as the cost of
raising the interest rate from r0 = 3 percent, interpreted as the interest rate consistent with zero inflation, to r1, say, the interest rate associated with 10 percent inflation. Graphically, this cost is measured
by the area ABDE.
We define monetary assets according to the monetary aggregate M1, that is, currency and demand
deposits.5 Money demand is then defined by the aggregate money balances M1 divided by nominal
gross domestic product.6 The nominal interest rate, r, is measured by the short-term commercial
paper rate. In figure 2, we represent each observation (r,m) by a circle for the period 1900–2000.
FIGURE 1

THE WELFARE TRIANGLE

B

r0

0

2

5. Alternatively, several authors,
including Fischer (1981), define
money as high-powered money.
The operational definition of
monetary assets is somewhat
arbitrary. For a discussion, see
Lucas (1981) and Marty (1999).

6. By measuring real balances as a
fraction of domestic output, the
area of the money triangle can
be interpreted as the fraction
of income that is needed to
compensate individuals for an
interest rate of r1 instead of zero
(Lucas, 2000).

Nominal interest rate

r1

4. Since the interest rate is approximately the sum of a constant
real interest rate and the inflation
rate, the Friedman rule would
imply that the inflation rate is
negative and approximately equal
to the opposite of the real interest
rate.

D
A
m1

E
m0

C
m*

Money real balances

FEDERAL RESERVE BANK OF CLEVELAND

To measure the welfare triangle, one estimates a curve that fits the observations in figure 2, and
then computes the appropriate area underneath the implied money demand curve. Lucas (2000)
considers two specifications for money demand: the log–log specification, m( r ) = Ar −η , where m
is aggregate real balances divided by output, r is the interest rate, and A and η are two estimated
parameters; and the semilog specification, where m( r ) = Ae −η r. In order to estimate the parameters A
and η we use nonlinear least squares.7 We also estimate the money demand curve by using a kernel
regression.8
It can be seen in figure 3 that the welfare cost associated with an interest rate of 13 percent
(10 percent inflation, approximately) is quite different across specifications for the money demand

7. This method is different from the
one used by Lucas (2000), who
constrains the curves to pass
through the geometric means of
the data and who uses a visual
test to identify the best fit.

function, from slightly more than 0.5 percent to slightly less than 1.5 percent. These differences
simply reflect different ways to fit the data.According to the R-squared criterion, the best fit is obtained
for the kernel regression which evaluates the cost of 10 percent inflation at about 1 percent of GDP.9
This number is similar to Lucas’s measure.The estimate from the semilog specification, about 1.5 percent of GDP, is comparable to Lagos and Wright’s smallest estimate of the welfare cost of inflation.

Search for a Money Demand Curve
The Bailey approach does not identify explicitly the benefits of monetary exchange for society.10 An
alternative approach consists of constructing a microfounded model economy in which money has
an essential role in trades, so that there is a well-specified demand for real balances and a natural measure of welfare. A theory that emphasizes the transactional role of money is the search approach of
monetary exchange pioneered by Kiyotaki and Wright (1989, 1991, 1993).The recent extension proposed by Lagos and Wright (2005) describes an economy in which trades take place under different
market structures. Some trades occur in a decentralized (or search) market, where buyers and sellers
are matched bilaterally, and other trades occur in a centralized market. Money is useful because of a
standard double-coincidence-of-wants problem in the decentralized market:The buyer does not pro-

8. The kernel was estimated with a
local bandwidth computed using
plug-in techniques, modified at
each boundary. See Brockman et
al. (1993).
9. For the kernel regression,
R2=0.6795; for the log–log
specification, R2=0.6238; for the
semilog specification, R2=0.6750.
10. The Bailey analysis is subject
to the following caveats. It is a
partial equilibrium analysis that
assumes away externalities,
general equilibrium effects, and
distributional effects. Also, an
underlying assumption is that
the government has access to
nondistorting taxes, so that a
change in seigniorage revenue
has no welfare consequences.

duce a good that the seller wants to consume.

FIGURE 2

FITTING MONEY DEMAND
Real balances
0.6

0.5

Nonparametric
0.4

0.3

Log-Log

0.2

Semilog
0.1
0

2

4

6
8
10
Interest rate, percent

12

14

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POLICY DISCUSSION PAPERS

NUMBER 12, JANUARY 2006

The instantaneous utility of an agent is u( q b ) − c ( q s ) + x, where qb is the consumption and qs is
the production in a bilateral match, and x is the net consumption in the centralized market (x is negative if an agent produces more at night than he consumes).11 The probability of a single-coincidenceof-wants meeting in which an agent meets someone who produces a good he likes is σ ≤ 1 2 .There
are no double-coincidence-of-wants meetings in which agents could use barter. So with probability σ
an agent is a buyer in a bilateral match, with probability σ he is a seller, and with probability 1 − 2σ
he is unmatched.A social planner, who would dictate the quantities to produce and consume, would
choose q b = q s = q *, where q* satisfies u′( q *) = c ′( q *) .The socially efficient level of production and

11. The linearity of the preferences is
what guarantees that the distribution of wealth at the beginning
of each period is degenerate.
Intuitively, this linearity eliminates
wealth effects in the choice of
real balances and ensures that all
buyers carry the same amount of
real balances.

consumption in the centralized market is indeterminate.The quantity of fiat money in the economy is
growing at a constant rate, π , through lump-sum transfers in the centralized market.
Denote z(q) as the real balances that an agent must hold in order to buy the quantity q ∈ [ 0, q * ]
in a bilateral match. The specific form for z(q) will depend on the assumed pricing mechanism in
the decentralized market.The Lagos-Wright model can be reduced to one equation that specifies the
quantity q = q b = q s traded in bilateral matches.This equation says that an agent chooses the quantity
q to consume in the decentralized market so as to maximize the expected surplus he gets as a buyer,

σ [u( q ) − z ( q ) ] minus the cost of holding real balances, rz(q),
q = arg max {− rz ( q ) + σ [u( q ) − z ( q ) ]} .
In order to calibrate the model, we adopt the same functional forms as the ones used in LagosWright: u( q ) = q1−η / (1 − η ), where η ≥ 0 and c ( q ) = q. Furthermore, the matching probability σ
is set to 1/2, so that each agent trades with probability one. Half of the time an agent is a buyer, and
half of the time he is a seller.12 The money demand function that is estimated is defined as aggregate
money balances divided by aggregate nominal output. It is equal to L = z / (σ z + A ), where A is the
real output in the centralized market (this quantity is indeterminate in the model) and where z is a
function of the nominal interest rate, r.13

FIGURE 3

13. Since agents readjust their real
balances in the centralized
market, the output A must be at
least equal to σz.

WELFARE COST OF INFLATION USING THE BAILEY METHOD
Cost of inflation, percent
2.0
Semilog
1.5

Nonparametric
1.0

0.5

Log-Log

0

–0.5
0

4

2

4

6
8
Interest rate, percent

10

12. Alternatively, one could consider a
model in which half of the agents
are buyers in even periods and
the remaining half are buyers
in odd periods. This formulation
would give identical results for the
cost of inflation.

12

14

FEDERAL RESERVE BANK OF CLEVELAND

One needs to take a stand on how prices (or terms of trade) are determined in decentralized
markets in which buyers and sellers are matched bilaterally. We will assume here that the monetary
transfer from the buyer to the seller is such that the seller is exactly compensated for his production
cost, z ( q ) = c ( q ) .This bargaining solution, called the dictatorial solution, is the outcome of a game in
which the buyer makes an offer which the seller can accept or reject. If the offer is rejected, no trade
takes place.After some calculation, aggregate real balances satisfy
−1

1
⎡
⎤
η
⎢σ + A ⎛1 + r ⎞ ⎥ .
L=
⎜
⎟
⎢
⎝ σ⎠ ⎥
⎣
⎦

As in the previous section, the parameters A and η can then be estimated from the data for the U.S.
economy from 1900 to 2000.The parameter A could be interpreted as the extent of the tax base.The
parameter η represents the sensitivity of individual real balances to changes in the interest rate.
In order to measure the welfare cost associated with a given interest rate, r, relative to 3 percent
(the interest rate consistent with zero inflation), we ask the following question. What is the percentage of total consumption that individuals would be willing to sacrifice in order to be in the steady
state with an interest rate of 3 percent instead of the steady state associated with r?
In figure 4, we compare the two measures of the welfare cost of inflation, namely, the compensated
measure and the welfare triangle measure. The welfare triangle measure is the area underneath the
money demand function as estimated from the search model. Figure 4 shows that the two measures
are nearly identical. In order to understand this result, consider the individual demand for real balances given by
dq ⎤
⎡
−1 ,
r = σ ⎢u′( q )
dz ⎥
⎣
⎦

FIGURE 4

COMPENSATED WELFARE AND THE WELFARE TRIANGLE
Cost of inflation, percent
2.0

1.5

Compensated welfare
1.0
Welfare triangle
0.5

0

–0.5
0

2

4

8
6
Interest rate, percent

10

12

14

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POLICY DISCUSSION PAPERS

NUMBER 12, JANUARY 2006

where dq / dz = 1 / z ′( q ) . Compute the area underneath this money demand function over the interval [ z0 , z1 ],
z1

∫ r ( z )dz = σ {u [q ( z )] − z } − σ {u [q ( z )] − z }.
1

1

0

0

z0

Using the assumed dictatorial solution, where the seller is exactly compensated for production
costs, z = c ( q ), it is easy to see that the right-hand side of the previous expression is just the change
in steady-state welfare. So the area underneath the individual demand for real balances coincides with
the change in steady-state welfare. These two measures do not exactly coincide in figure 4 because
we express real balances as a fraction of aggregate output and we do not look at the change in steadystate welfare but at a compensated measure of welfare.
The welfare cost of a nominal interest rate of 13 percent relative to a 3 percent interest rate is
about 1.5 percent.14 This measure is bigger than the ones of Lucas (2000) but it accords with the nonlinear least square estimate based on the semilog specification.The difference between the numbers
simply stems from different strategies of fitting the points in the data. In all cases, the cost of inflation
corresponds to the area underneath a money demand function.15

14. Assuming a competitive pricing
mechanism, Rocheteau and
Wright (2004, 2005) and Reed
and Waller (2004) find similar
estimates for the welfare cost
of inflation, between 1 and 1.5
percent of GDP.

Pricing
The estimate for the welfare cost of inflation in the previous section has been obtained by assuming a special pricing mechanism: Buyers are able to extract the whole surplus from trade.Alternatively, one can model prices, and the way in which the surplus from a trade is shared between trading
partners, differently. In this section, we consider first a simple bargaining solution, called the proportional solution, which will illustrate the role played by the pricing mechanism in assessing the

15. Aiyagari, Braun, and Eckstein
(1998) also show, using a
cash-in-advance model with a
credit sector, that the welfare
cost of inflation using an
estimated money demand curve
is consistent with the prediction of
the model.

welfare cost of inflation. The proportional bargaining solution assumes that the buyer obtains a constant fraction, called the buyer’s share and denoted θ , of the surplus of a match defined as the difference between the buyer’s utility of consumption and the seller’s disutility of production. Formally,
u( q ) − z ( q ) = θ [u( q ) − c ( q ) ], and therefore z ( q ) = θ c ( q ) + (1 − θ ) u( q ). The dictatorial solution of
the previous section corresponds to θ = 1. From the agent’s choice of real balances, one can find a
simple relationship between z, individual real balances, and the nominal interest rate, r. Aggregate real
balances are defined as L = z / (σ z + A ) .
We use the same method as before to derive the cost of inflation: We estimate the parameters A
and η of money demand generated by the model, and we use compensated welfare to compute the
cost of inflation. In figure 5, we report the welfare cost of inflation for different values for the buyer’s
share (θ = 0.3, 0.5, 0.8, 1) .16 When the buyer’s share is less than 1, the cost of inflation is typically
larger than the measure given by the money triangle. In fact, the area of the welfare triangle is approximately equal to the buyer’s share multiplied by the compensated measure of the welfare cost of
inflation. For instance, if the buyer’s share is 50 percent, the welfare cost of inflation is always about
twice the size of the area of the money triangle (see figure 6).To understand this result, consider the
area underneath the individual demand for real balances. It satisfies
z1

∫ r ( z )dz = σ {u [q ( z )] − z } − σ {u [q ( z )] − z }
1

1

0

0

z0

= θσ {u [ q ( z1 ) ] − c [ q ( z1 ) ]} − θσ {u [ q ( z0 ) ] − c [ q ( z0 ) ]} .

6

16. The R2 for the case of θ = 1.0 is
0.6757, which is higher than the
log–log and semilog parametric
models above. Other values of

θ gave smaller values for the R2
than the semilog model.

FEDERAL RESERVE BANK OF CLEVELAND

It is equal to the change in steady-state welfare multiplied by θ. In figure 6, real balances are divided
by aggregate output, and the welfare metric of the model is a compensated measure, which explains
the slight discrepancy between the two lines.
As one varies the buyer’s share from 0.3 to 1, the cost of 10 percent inflation varies from slightly
more than 1 percent to 6 percent of GDP. In order to understand why the welfare triangle can underestimate the true welfare cost of inflation, consider the following example. Suppose that each unit of
good produced in a bilateral match is worth $1 for the buyer and costs $0.9 to produce.The marginal
surplus of a trade is then $0.1. Suppose that the price is $0.95, so that both the buyer and the seller get
a surplus of $0.05.The private return of money is equal to the buyer’s surplus divided by the amount
of money that the buyer must carry to buy the good, 0.05/0.95 = 5.2 percent. The social return of

FIGURE 5

WELFARE COST OF INFLATION UNDER PROPORTIONAL BARGAINING

Cost of inflation, percent
7.0

θ = 0.3

6.5
5.5
4.5

θ = 0.5

3.5
2.5

θ = 0.8

1.5

θ = 1.0

0.5
–0.5
–1.5
0

FIGURE 6

2

4

6
8
Interest rate, percent

10

12

14

WELFARE TRIANGLE VERSUS COMPENSATED WELFARE ( θ = 0.5)
Cost of inflation, percent
3.5
3.0
2.5
Compensated welfare
2.0
Welfare triangle x2
1.5
1.0
0.5
0
–0.5
0

2

4

6

8
Interest rate

10

12

14

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POLICY DISCUSSION PAPERS

NUMBER 12, JANUARY 2006

money is the total surplus divided by the price of the good, 0.1/0.95 = 10.5 percent. If the interest rate
is 10 percent, the cost of holding $0.95 for the buyer is larger than the marginal gain of $0.05. So the
buyer has no incentive to bring an additional dollar even though the return of this dollar to society is
larger than the opportunity cost incurred by the buyer.
This discrepancy between the private and social benefits of real balances arises from a rent-sharing externality. The marginal benefit of the real balances from the buyer’s point of view is smaller
than the marginal benefit from society’s point of view. Since the money demand estimated from the
data only captures the marginal benefit of money from the buyer’s side, the welfare triangle misses a
fraction of the welfare cost of inflation.This externality arises from any pricing rule that stipulates that
the buyer does not get the full marginal return of his real balances.17
This point is illustrated in figure 7. For a given stock of real balances m0, the marginal benefit that
money provides to the money holder, the length of segment AB, is smaller than the marginal societal
benefit of money, the length of segment AD. If prices are determined according to a proportional solution, the ratio BA/DA is the buyer’s share. As a consequence, when measuring the area underneath
the demand for real balances, the ABC area, one underestimates the social benefit of money, the ADC
area, by a factor equal to the inverse of the buyer’s share.
In figure 7, the two curves representing the private and social benefits of real balances intersect
the horizontal axis (r = 0) for the same value m* of real balances.At this point, the total benefits of real
balances are maximized for both buyers and society.This observation implies that the Friedman rule
yields the best allocation of resources for society for all values of the buyer’s share, θ .
The result in which the Friedman rule generates the first-best allocation does not hold for all
bargaining solutions. For instance, it does not hold for the Nash (1950) solution, according to which
z ( q ) = Θ( q )c ( q ) + [1 − Θ( q ) ] u( q ) with
Θ( q ) =

θ u′( q )
,
θ u′( q ) + (1 − θ )c ′( q )

FIGURE 7

THE RENT-SHARING EXTERNALITY
Fraction of a dollar

Private marginal benefit
of money = Money demand

D

Social marginal benefit of money

r0

B

A
0

8

m0

C
m*

Real money balances

17. This rent-sharing externality is
closely related to holdup problems
noted in the investment literature.

FEDERAL RESERVE BANK OF CLEVELAND

where θ is now the bargaining power of the buyer.18 The buyer’s share, Θ , depends on both θ and q,
and it is equal to θ when q = q*. In particular, the buyer’s share, Θ , decreases as q increases.As shown
by Lagos and Wright (2005), if prices in bilateral matches are determined according to the Nash solution, then the Friedman rule is optimal but the quantities traded in the decentralized market are too
low. This result is illustrated in figure 8. At the Friedman rule (r = 0) the economy’s real balances are
m, while the real balances that would maximize society’s welfare are m*. In other words, if the interest rate is zero, an individual’s demand for real balances is satiated even though the marginal benefit of
money to society is still positive.This inefficiency is called a “nonmonotonicity inefficiency” to reflect
the fact that the buyer’s surplus from a trade does not necessarily increase with the match surplus.19
Put differently, the buyer’s surplus u( q ) − z ( q ) reaches a maximum for a value of q smaller than q*.
This inefficiency has two important consequences for the welfare effects of inflation. First, a small increase of the interest rate above r = 0 will have a larger effect on welfare than in the case under proportional bargaining solutions. Indeed, the welfare cost of a small interest rate can be approximated
by the change in real balances multiplied by the social benefit of real balances at r = 0. This second
term, measured by the length of the segment EC in figure 8, is now positive at r = 0. Second, real balances are inefficiently low at r = 0, and, as a consequence, the quantities produced and consumed in
bilateral matches are also too low.

18. The bargaining power θ , which
varies from 0 to 1, is a measure
of the buyer’s strength in the
bargaining process. In an explicit
bargaining game with offers and
counteroffers, the bargaining
power of an individual depends,
among other things, on his
ability to threaten to terminate the
negotiation if his offer is rejected.
19. In order to illustrate this
inefficiency, consider a bargaining
problem where two individuals
must share a prize. If one of the
two individuals gets worse off as
the size of the prize increases,
then the bargaining solution is
said to be nonmonotonic. For a
detailed treatment of alternative bargaining solutions and
their properties in monetary
economies, see Rocheteau and
Waller (2004).

Figure 9 plots the welfare cost of inflation when prices are determined according to the Nash
solution for different values of the buyer’s bargaining power, θ . The comparison of figures 5 and 9
reveals that the welfare cost of 10 percent inflation under the Nash solution is of same magnitude as
the cost under the proportional bargaining solution. In both cases, there is a rent-sharing externality
at work, which amplifies the cost of inflation. However, under the Nash solution, the buyer’s share,

Θ, gets larger for higher inflation rates so that the rent-sharing externality gets smaller.The most noticeable difference between figures 5 and 9 is the gain associated with a reduction of the interest rate
from 3 percent to zero (which corresponds to the optimal deflation rate).This gain can be as high as
2.5 percent of GDP when the buyer’s bargaining power is 0.3. Under the proportional solution, this

FIGURE 8

BARGAINING INEFFICIENCIES
Fraction of a dollar

Money
demand

D

Social marginal benefit of money

r0

B
E

0

A
m0

C
~
m

F
m*

Real money balances

9

POLICY DISCUSSION PAPERS

NUMBER 12, JANUARY 2006

gain is about 0.5 percent of GDP. So the nonmonotonicity inefficiency is important in that it predicts
large welfare gains if inflation is reduced from zero to the optimal deflation rate.
To conclude this section, we present a pricing mechanism which exhibits the same type of inefficiencies as the Nash bargaining solution but which is based on the possibly more familiar idea that
prices are set as a markup over the cost incurred by sellers.20 More precisely, the transfer of money
from the buyer to the seller corresponds to the cost incurred by the seller in producing the amount
asked for by the buyer multiplied by a constant factor, 1 + μ, (where μ ≥ 0 ), which we interpret as the
“markup,” z ( q ) = (1 + μ )c ( q ). As shown in figure 10, the cost of inflation increases with the markup.
When the markup is 20 percent, the cost of 10 percent inflation is slightly more than 3 percent of
GDP, which is similar to the prediction of the model under the symmetric Nash solution.

FIGURE 9

COST OF INFLATION UNDER NASH BARGAINING
Cost of inflation, percent
6.0

θ = 0.3

5.0
4.0

θ = 0.5

3.0
2.0

θ = 0.8

1.0

θ = 1.0
0
–1.0
–2.0
–3.0
0

FIGURE 10

2

4

6
8
Interest rate, percent

10

12

14

12

14

COST OF INFLATION UNDER CONSTANT MARKUP
Cost of inflation, percent
3.5

μ = 20%

3.0
2.5

μ = 10%

2.0
1.5
1.0

μ=0

0.5
0
–0.5
–1.0
–1.5
0

10

2

4

8
6
Interest rate, percent

10

20. For a search model with price
posting by sellers, see Ennis
(2004). Ennis (2004) describes
an economy in which buyers
have private information about
their tastes, and sellers make
take-it-or-leave-it offers. The annual welfare cost of a 10 percent
inflation in this model is between
4 percent and 7 percent of GDP.

FEDERAL RESERVE BANK OF CLEVELAND

Figure 11 illustrates how a constant markup affects the assessment of the welfare cost of inflation.
The constant markup, μ , shifts the curve indicating the social return of real balances up by a constant
amount (BD = CE), which reflects the seller’s marginal benefit from buyers’ real balances. The larger
the markup, the larger the difference between private and social benefits of real balances. Also, it is
clear from figures 8 and 11 that both the Nash solution and the pricing with constant markup induce
qualitatively similar effects of inflation. In both cases, the quantities traded at the Friedman rule (r = 0)
are too low.

Participation Decisions
In the basic model described in the previous sections, the frequency of trade is assumed to be constant. So inflation affects the quantities traded in bilateral meetings, but it does not affect the number
of those meetings. In order to endogenize the number of trade matches, one can let buyers and sellers choose whether or not to participate in the market, or let them choose on which side of the market to participate in, or even let them choose the resources they will invest in the search for a trading
partner.21 By taking into account these participation decisions, one can introduce general equilibrium
effects of inflation, which are not captured by the Bailey methodology. Also, in environments with
search frictions, participation decisions tend to be inefficient. Consequently, the welfare effects of inflation are ambiguous and depend on the way in which inflation distorts participation decisions.
We consider in the following an extension of the Lagos-Wright model, which is based on an
assumption in Shi (1997).22 The economy is similar to the one previously described except that at the
beginning of each period, before matches are formed, individuals can choose to be buyers or sellers in
the decentralized market. For instance, agents in the labor market can choose to be buyers (entrepreneurs) or sellers (workers).An agent who chooses to be a buyer cannot produce during the day, while
an agent who chooses to be a seller cannot consume during the day.The composition of buyers and

21. Rocheteau and Wright (2005)
consider a model with free entry
of sellers. Shi (1997) describes
an economy in which individuals
can choose which side of the
market to participate in. See also
Rocheteau and Wright (2004)
and Faig (2004). Li (1995, 1997)
introduces endogenous search
intensities.
22. Our model is similar to the one
in Rocheteau and Wright (2004),
except that we consider different
pricing mechanisms.

sellers is then endogenous. Let n denote the fraction of sellers in the economy. Assume further that
the matching process is such that a buyer meets a seller with probability n, whereas a seller meets a

FIGURE 11

CONSTANT MARKUP
Fraction of a dollar

Money
demand

Social marginal benefit of money
D

r0

B
E

A
0

m0

C
~
m

Real money balances

11

POLICY DISCUSSION PAPERS

NUMBER 12, JANUARY 2006

buyer with probability 1 – n.23 The fraction of sellers in equilibrium is such that agents are indifferent
to being buyers or sellers,
−rz ( q ) + n [u( q ) − z ( q ) ] = (1 − n ) [ z ( q ) − c ( q ) ] .
The left-hand side of the previous equation is the expected utility of a buyer, the right-hand
side is the expected utility of a seller, and q is the equilibrium quantity traded in bilateral matches.24 We assume that prices are determined according to the proportional bargaining solution,
z ( q ) = θ c ( q ) + (1 − θ )u( q ).
The equilibrium allocation is socially efficient if θ = 0.5 and r = 0. The second requirement corresponds to the Friedman rule, and it guarantees that q = q*. The first requirement, θ = 0.5, is the

23. The specification for the matching
function is the same as the one
used in most monetary models,
including Kiyotaki and Wright
(1993). Obviously, it would be
desirable to estimate the transaction technology. This strategy has
been pursued successfully in the
labor literature. We leave this
extension for future investigation.
24. The money supply is growing
through lump-sum transfers. Such
transfers do not affect agents’
decisions to be buyers or sellers.

condition under which the number of trades is maximized; this maximization requires n = 1/2. It corresponds to the Hosios (1990) condition according to which the number of trades is efficient only in
the unlikely event that an agent’s share of the match surplus is equal to his marginal contribution to
the creation of trade matches.25 The best outcome for a search monetary economy requires that both

market, A,

25. The elasticity of the number of
trades with respect to the number
of buyers in the economy is n.
Therefore, the Hosios condition
requires θ = n. When r = 0,
n = 1 − θ , so that the optimal
allocation requires θ = 0.5.

(1 − n ) z .
L=
n (1 − n ) z + A

26. For an elaboration of this idea,
see Berentsen, Rocheteau, and
Shi (2004).

the Hosios condition and the Friedman rule hold.26
The definition of the aggregate demand for money is the money held by the 1 – n buyers divided
by the sum of the output in the decentralized market, n(1 – n)z, and the output in the centralized

We use the same strategy as before to estimate the parameters A and η of money demand. In
figure 12, we report the welfare cost of inflation. When the buyer’s share is less than 50 percent, the
number of buyers is too low and inflation lowers the fraction of buyers even further.The welfare cost
of low inflation is then larger than what would be obtained under the assumption in which the frequency of trades is constant. Reciprocally, the welfare gains from reducing the interest rate to zero are
also large.When the buyer’s share is above 50 percent, a deviation from the Friedman rule is optimal.
FIGURE 12

COST OF INFLATION WITH SEARCH EXTERNALITIES (PROPORTIONAL BARGAINING)
Cost of inflation, percent
6.0
5.5

θ = 0.3

5.0
4.5
4.0
3.5
3.0

θ = 0.5

2.5
2.0
1.5
1.0

θ = 0.8

0.5
0

θ = 0.9

–0.5
–1.0
0

12

2

4

8
6
Interest rate, percent

10

12

14

FEDERAL RESERVE BANK OF CLEVELAND

In this case, if r = 0, then the number of buyers is too high and the number of trades is too low. Since
inflation has a direct negative effect on buyers’ expected utility, the number of buyers falls, while the
number of sellers increases.27 The composition in the market becomes more even and the number of
trades increases.28 When the buyer’s share is 90 percent, then the cost of implementing the Friedman
rule is about 0.5 percent of GDP while the cost of 10 percent inflation is close to 0.
The results reported in figure 12 have been derived by assuming that terms of trade are set according to a proportional bargaining solution.As in the previous section, different bargaining solutions can
have very different properties in terms of the efficiency of the equilibrium allocation at the Friedman
rule. For example, in the basic Lagos-Wright model, the equilibrium allocation is efficient at the Friedman rule under the proportional bargaining solution, but it is inefficient under the Nash solution. Not
too surprisingly, the choice of the pricing mechanism will also matter considerably for the welfare
cost of inflation when participation decisions are endogenous.
Figure 13 reports the cost of inflation when the terms of trade are determined according to the
Nash solution. In sharp contrast to the results obtained under the proportional solution, there is no
positive effect of inflation on welfare. The optimal monetary policy is the Friedman rule. The “nonmonotonicity inefficiency” of the Nash solution changes the nature of the trade-off between the
negative effect of inflation on individual real balances and the effect of inflation on the composition
of the market. Indeed, since real balances are inefficiently low at the Friedman rule under the Nash
solution, a small increase of the interest rate reduces real balances, which has a first-order negative
effect on welfare. For the calibrated version of the model, this negative real balance effect dominates
any positive effect of inflation on the composition of the market.29 As a consequence, a deviation
from the Friedman rule is not optimal under the Nash solution. The comparison of figures 9 and 13
reveals that the presence of search externalities exacerbates the cost of inflation for large values of
the buyer’s bargaining power. If the buyer’s bargaining power is 0.5, results are largely similar to those
obtained in the absence of search externalities, because at this value of θ , the composition of the
market is similar to the market in which σ = 1 2. If the buyer’s bargaining power is 0.9, the cost of
10 percent inflation is about 4.5 percent of GDP in the presence of search frictions, whereas it is less
FIGURE 13

COST OF INFLATION WITH SEARCH EXTERNALITIES (NASH BARGAINING)
Cost of inflation, percent
6.0
5.5

27. This effect is sensitive to the
choice of the pricing mechanism.
For instance, under the Nash
solution the number of buyers can
increase with inflation because
the buyer’s share in the surplus of
a match gets bigger.
28. The result in which the Friedman
rule is not always optimal in the
presence of search externalities
could raise the following objection. If the government was able
to make transfers contingent on
agents’ participation decisions,
then it could take care of the
search externalities by an appropriate transfer scheme, while
the Friedman rule would be used
to reduce the monetary wedge
associated with the inflation tax.
In practice, however, it may be
difficult to implement transfers
contingent on agents’ participation
decisions since these decisions
may not be observable. Inflation,
which is a tax on market activities,
may be an effective instrument for
correcting inefficient participation
in the market.
29. For other parameterizations, a
deviation from the Friedman rule
could raise society’s welfare.
However, the result according
to which a deviation from the
Friedman rule is less likely under
Nash bargaining than under
proportional bargaining is robust.

θ = 0.3

5.0
4.5

θ = 0.9

4.0
3.5

θ = 0.8

3.0

θ = 0.5

2.5
2.0
1.5
1.0
0.5
0
–0.5
–1.0
–1.5
–2.0
–2.5
0

2

4

8
6
Interest rate, percent

10

12

14

13

POLICY DISCUSSION PAPERS

NUMBER 12, JANUARY 2006

than 2 percent of GDP when the frequency of trades is exogenous. Also, the welfare cost of inflation
is not a monotonic function of the buyer’s bargaining power. An increase in the buyer’s bargaining
power makes the rent-sharing externality less severe, but it also distorts the composition of the market toward too many buyers.
To summarize, the introduction of endogenous participation decisions has several implications.
First, the welfare triangle is a misleading measure since it does not capture the distortionary effects
of inflation on individuals’ participation decisions. Second, the presence of search externalities can
mitigate or exacerbate the welfare cost of inflation.Third, the Friedman rule may no longer be optimal,
as the positive effect of inflation on the composition of the market and the frequency of trades can
dominate the negative effect of inflation on real balances.

Distributional Effects
The Lagos-Wright model described in the previous sections has been designed in such a way that all
agents, despite different trading histories in the decentralized market, start each period with the same
money balances.This property of the model is what makes it tractable. However, because the distribution of money balances is degenerate, inflation does not have any distributional effect. Levine (1991),
Molico (1999), and Deviatov and Wallace (2001), among others, have provided examples in which
a policy that consists of increasing the money supply through lump-sum transfers induces some redistribution across individuals. For individuals with large stocks of nominal assets, the burden associated with the inflation tax is greater than the benefit of receiving a lump-sum transfer. On the other
hand, individuals with few nominal assets enjoy a net benefit from the lump-sum transfer. In some circumstances this redistribution can be beneficial to society. For example, if agents are subject to idiosyncratic shocks (on endowments, productivity, and so forth) that cannot be insured against, money
growth can provide an insurance mechanism.30
We capture the distributional effects of inflation through a simple extension of the Lagos- Wright
model. Recall that in the Lagos-Wright model individuals live forever, and, in each period of their lives,
they trade sequentially in a centralized market and a decentralized market. We depart from these assumptions by assuming that individuals live only two periods.They are born at the beginning of the
centralized market, and they die at the end of the following period after the centralized market has
closed. Also, agents do not discount utility across periods. When agents are born, they have access to
the centralized market in order to make their choice of money balances. In the second period of their
lives, they trade in the decentralized market and then have access to the centralized market before
they die. Only a fraction p of the newly-born agents are able to produce in the first period of their
lives. One can interpret this assumption as individuals receiving productivity shocks. In the absence
of money growth, agents who cannot produce cannot accumulate money and, therefore, cannot consume in the decentralized market in the second period of their lives. By inflating the money supply
the government can transfer money to all individuals, irrespective of their productivities, and therefore smooth consumption across all agents.
The demand for real balances from productive agents is the same as the one described in the previous sections.The main difference with respect to the previous models is the fact that the distribution
of real balances has two points, z, the real balances of productive agents, and z , the real balances of
unproductive agents.The aggregate demand for money balances is then

14

30. In Molico’s (1999) version of the
search-theoretic model, individuals trade only in a decentralized
market with bilateral random
matching. Since the matching
process is random, trading
opportunities arrive according
to a stochastic process. Some
individuals are lucky and can sell
their output often: They then have
a large stock of money balances.
Other individuals are less lucky
and have a low stock of real
balances. Molico shows that for
low inflation rates the redistributive effect of inflation, according to
which inflation acts as a subsidy
for the poor and a tax on the rich,
can dominate the real balance
effect of inflation, according to
which inflation reduces aggregate
real balances. In Craig and
Waller’s (2004) version of a
decentralized market with two
currencies, one of which inflates,
low rates of inflation can induce a
strong redistributive effect in the
inflating currency where buyers
are likely to hold either large
amounts of the currency or none
at all.

FEDERAL RESERVE BANK OF CLEVELAND

L=

pz + (1 − p ) z
.
σ ⎡ pz + (1 − p ) z ⎤ + A
⎣
⎦

The strategy to estimate the model is the same as before. Society’s welfare is measured by the sum
of the trade surpluses in all matches.The welfare cost corresponding to an interest rate, r, is the fraction by which total consumption at the steady state with a 3 percent interest rate must be reduced in
order to achieve the same welfare as the one that prevails at the steady state with r.31 In figure 14, we
plot the welfare cost of inflation when prices are determined according to the proportional bargaining solution with θ = 1/2. As in Molico (1999), society’s welfare is maximized for a positive inflation
rate. The welfare gains associated with a positive interest rate are rather small. When calibrating the

FIGURE 14

31. Note that we kept the benchmark
for the interest rate at 3 percent.
Since agents do not discount
future utility, this interest rate corresponds to a 3 percent inflation.

DISTRIBUTIONAL EFFECTS OF INFLATION
Cost of inflation, percent
3.5
3.0
2.5
2.0
1.5
1.0
p = 0.8
0.5

p = 0.5
p = 0.3

0

p = 1.0
–0.5
0

FIGURE 15

2

4

6
8
Interest rate, percent

10

12

14

DISTRIBUTIONAL EFFECTS OF INFLATION (HIGHER RISK AVERSION)

Cost of inflation, percent
3.5

p=1.0

3.0

p=0.8

2.5
p=0.5

2.0

p=0.3

1.5
1.0
0.5
0
–0.5
0

2

4

6

8

10

12

14

Interest rate, percent

15

POLICY DISCUSSION PAPERS

NUMBER 12, JANUARY 2006

model, the coefficient that describes the aversion of individuals toward risk is low (the coefficient of
relative risk aversion is close to 0.2), and it is, in fact, lower than what is usually thought realistic.
If one increases this coefficient to make it a little bit more realistic, the welfare gain corresponding
to the redistribution effect of inflation gets larger. For example, in figure 15 we plot the cost of inflation for the same value of the parameter A as the one used to draw figure 14 but for a coefficient of
relative risk aversion raised to 0.5.The benefits of inflation get significantly bigger and increase with
the probability that an agent receives a negative productivity shock.

Conclusion
We have presented some insights provided by the search theory of monetary exchange for the understanding of the welfare cost of inflation. Using different extensions of the model of Lagos and Wright
(2005), we have identified and quantified various effects of inflation on welfare. First, inflation has a
negative real balance effect.The inflation tax introduces a wedge in the decision to invest in real balances. The extent of this distortion depends crucially on the assumed pricing mechanism. If buyers
receive the full marginal benefit of their money balances, the cost of inflation is essentially the Bailey
measure, given by the area underneath the money demand function. In all other cases, the Bailey measure has to be scaled up by a function that depends on the sellers’ share in the surplus of a trade. We
have also provided examples of pricing mechanisms under which the Friedman rule fails to generate
the first-best allocation. In such cases, the benefit of implementing the optimal deflation is large. Second, inflation affects agents’ decisions to participate in the market and therefore it affects the number
of trades. However, since participation decisions tend to be inefficient in search environments, the
effect of inflation on participation choices can be welfare-enhancing or welfare-worsening. We have
provided calibrated examples in which a deviation from the Friedman rule is optimal.We have shown
once again that the welfare effects of inflation depend crucially on the pricing mechanism.Third, inflation can generate a redistribution across agents. If agents are subject to idiosyncratic shocks, this
redistribution can prove useful to society.
Additional extensions would be worth considering. As emphasized by Cooley and Hansen (1989,
1991) and Dotsey and Ireland (1996), the inflation tax can distort a variety of marginal decisions:
among others, the leisure-consumption choice and the accumulation of capital. Aruoba and Wright
(2003) and Aruoba, Waller, and Wright (2004) have extended the Lagos-Wright model to allow for
capital accumulation and have considered various assumptions regarding how capital enters the
economy. Also, as Kocherlakota (2004) pointed out, it is important to extend search models in order
to incorporate additional assets beside money, such as government bonds, and to take into account
distortionary taxes. Finally, one can introduce realistic nominal rigidities in the search model of money
along the lines suggested by Craig and Rocheteau (2004).

16

FEDERAL RESERVE BANK OF CLEVELAND

Appendix 1:The Model
The Lagos-Wright Model of Monetary Exchange
We present briefly the search-theoretic model of monetary exchange proposed by Lagos and Wright
(2005). Time is discrete and each period of time is divided into two subperiods: the day and night.
During the day trades take place in a decentralized market where agents are matched bilaterally.
There is a lack of double coincidence of wants in bilateral matches. The probability for an agent to
find someone who produces a good he likes is σ , and the probability to find someone who likes the
good he produces is σ . With probability 1 − 2σ an agent is unmatched. Night trades take place in a
competitive market. Money is introduced in the economy through lump-sum transfers in the centralized market and the supply of money is growing at the rate π . We only focus on steady-state equilibria where real balances are constant.
An agent’s utility function is
u( q b ) − c ( q s ) + x ,
where qb is the consumption and q s the production in a bilateral match and x is the net consumption in the centralized market (x is negative if an agent produces more at night than he consumes).
When we calibrate the money demand function we assume u( q ) = q1−η /(1 − η ) , where η ∈ ( 0,1)
and c ( q ) = q .The discount factor is β = (1 + ρ ) −1 ∈ ( 0,1) .
Let pt denote the price in the centralized market.The utility of an agent holding mt units of money
during the night of period t, denoted by W ( mt / pt ) , satisfies
⎧
⎛m ⎞
⎛ m ⎞⎫
ˆ
⎪
⎪
W ⎜ t ⎟ = max ⎨ xt + β V ⎜ t +1 ⎟ ⎬ ,
ˆ
xt mt +1 ⎪
⎝ pt ⎠
⎝ pt +1 ⎠ ⎪
⎩
⎭
ˆ
s.t. pt xt + mt +1 = mt + Tt
where V (⋅) is the expected utility of the individual in the decentralized market.
The individual receives a lump-sum transfer Tt and chooses his net consumption xt and his money
balances mt +1 in the next period. It is straightforward to check that the value function is linear and
ˆ
that the choice of mt +1 is independent of mt. Note that the absolute value of consumption and producˆ
tion in the centralized market is not determined within the model. The aggregate production in the
centralized market will be determined by calibration.
Terms of trade (q,d) in a bilateral match, where q is the output and d the real transfer of money, are
determined by a bargaining solution. In general, the terms of trade maximize a monotonic function of
the surpluses of the buyer and the seller in the match.Those surpluses are independent of the money
balances of the buyer and the seller.Also, for standard bargaining solutions, (q,d) only depends on the
real balances zt = mt / pt of the buyer.
The utility of an agent in the decentralized market satisfies

⎡
⎡
⎛m ⎞
⎛m ⎞
⎛m
⎞⎤
⎛m
⎞⎤
V ⎜ t ⎟ = σ ⎢u( q b ) + W ⎜ t − d b ⎟ ⎥ + σ ⎢ −c ( q s ) + W ⎜ t − d s ⎟ ⎥ + (1 − 2σ )W ⎜ t ⎟ ,
pt
pt
pt ⎠
⎥
⎥
⎢
⎢
⎝ pt ⎠
⎝
⎝
⎠⎦
⎝
⎠⎦
⎣
⎣

17

POLICY DISCUSSION PAPERS

NUMBER 12, JANUARY 2006

where (q b ,d b ) are the terms of trade when the agent is the buyer and (q s ,d s ) are the terms of trade
when the agent is the seller. In equilibrium, ( q b , d b ) = ( q s , d s ) since all agents hold the same money
balances. Substitute V(m t / pt) into the Bellman equation for W ( mt / pt ) and rearrange in order to
obtain the following problem

max {−r z ( q ) + σ [u( q ) − z ( q )]} ,
t

t

t

t

qt

where the nominal interest rate rt satisfies 1 + rt = (1 + π )(1 + ρ ) and where z(q) relates the buyer’s
real balances to the quantity of goods he can purchase from the seller. So the agent essentially maximizes the expected surplus he gets when he is a buyer minus the cost of holding real balances.

Pricing Mechanisms
The form of the function z(q) depends on the bargaining solution. If buyers have all the bargaining
power, then z(q) = c(q) and the first-order condition is
u′( q )
r
= 1+ .
c ′( q )
σ
Using the specifications for the utility and cost functions, this gives
−1

r ⎞η
⎛
q = z = ⎜1 + ⎟ .
⎝ σ⎠
If terms of trade are determined according to the proportional bargaining solution, then
the relationship between individual balances and the quantity traded in a match is given by
z ( q ) = θ c ( q ) + (1 − θ )u( q ) , where θ is the buyer’s share in the match surplus. Using the linear
specification for c(q) and the CRRA specification for u(q), and the first-order condition for the choice
of real balances, one obtains
⎛ 1 − θ ⎞ ⎛ ( r + σ )θ ⎞
z =⎜
⎟⎜
⎟
⎝ 1 − η ⎠ ⎝ σθ − r (1 − θ ) ⎠

− ( 1−η )
η

−1

⎛ ( r + σ )θ ⎞ η
+θ ⎜
⎟ .
⎝ σθ − r (1 − θ ) ⎠

If terms of trade are determined according to the generalized Nash bargaining solution where the
buyer’s bargaining power is θ , the pair (z,q) satisfies

θ u′( q )c ( q ) + (1 − θ )c ′( q )u( q )
,
θ u′( q ) + (1 − θ )c ′( q )
r
u′( q )
= 1+ .
z ′( q )
σ
z (q ) =

Finally, if prices are set as a constant markup over the cost incurred by sellers, the function z(q)
satisfies
z ( q ) = (1 + μ )c ( q ).

18

FEDERAL RESERVE BANK OF CLEVELAND

Calibration
The money demand function that is confronted with the data is defined as aggregate money balances
divided by aggregate nominal output. It is equal to
M
,
σ M + pA
where M is aggregate money balances, σ is the frequency of trades in the decentralized market, p
is the price in the centralized market, and A is the real output in the centralized market. Since z =
M/p, the money demand function is
z
.
σz+ A
If the pricing mechanism is “buyers take all” then the expression for money demand is
z
=
σZ + A

1
1

r ⎞η
⎛
σ + A ⎜1 + ⎟
⎝ σ⎠

.

The frequency of trades is set to σ = 0.5 so that each individual is matched and is either a buyer or
a seller.The parameters A and η are chosen to fit money demand in the United States over the period
1900–2000.The same procedure is used for other pricing mechanisms.

Social and Private Marginal Returns of Real Balances
A marginal unit of real balances allows a buyer to buy ∂q / ∂z units of goods in the event a
match occurs. The expected private marginal return of real balances, or equivalently, the expected increase of the buyer’s utility from holding an additional unit of real balances, is then

σ [u′( q ) − z ′( q ) ] ∂q / ∂z = σ [u′( q ) / z ′( q ) − 1] , which is precisely r from the first-order condition for the choice of real balances. The expected social marginal return of real balances is

σ [u′( q ) − c ′( q ) ] ∂q / ∂z = σ [u′( q ) − c ′( q ) ] / z ′( q ). Note that the private and social returns of real balances are equal when buyers have all the bargaining power since z(q) = c(q). If prices are determined
by the proportional bargaining solution then the social return of real balances is
⎡ u′( q ) − c ′( q ) ⎤ σ ⎡ u′( q ) − z ′( q ) ⎤ r
σ⎢
⎥= .
⎥= ⎢
z ′( q )
z ′( q )
⎦ θ
⎦ θ ⎣
⎣
So the social return of real balances is equal to the interest rate divided by the buyer’s bargaining power. Finally, if prices are determined according to a constant markup μ over the cost incurred by the
seller then the social return of real balances is
⎡ u′( q ) − z ′( q ) /(1 + μ ) ⎤
⎡ u′( q ) − c ′( q ) ⎤
μ
σ⎢
.
⎥ = r +σ
⎥ =σ ⎢
z ′( q )
z ′( q )
1+ μ
⎦
⎣
⎦
⎣
So the social return of real balances is equal to the interest rate plus a constant term that is increasing in μ .

19

POLICY DISCUSSION PAPERS

NUMBER 12, JANUARY 2006

Accurateness of the Welfare Triangle Measure
The inverse money demand function of the search model is given by
r = σ {u′ [ q ( x ) ] q′( x ) − 1} ,
where q(x) is the function that specifies the output traded in bilateral matches as a function of real
balances. Denote z0(zr) the real balances at the steady-state equilibrium with an interest rate of 3 percent (r percent). Integrate the inverse money demand function from zr to z0 in order to obtain the
welfare triangle measure (denoted WT),
z0

WT = ∫ σ {u′ [ q ( x ) ] q′( x ) − 1} dx
zr

= σ {u [ q ( z0 ) ] − z0 } − σ {u [ q ( z r ) ] − z r } .
When buyers make take-it-or-leave-it offers,
WT = σ {u [ q ( z0 ) ] − c [ q ( z0 ) ]} − σ {u [ q ( z r ) ] − c [ q ( z r ) ]} .
In this case, he welfare triangle coincides exactly with the change in society’s welfare. When prices
are determined according to the proportional bargaining solution,
WT = σθ {u [ q ( z0 ) ] − c [ q ( z0 ) ]} − σθ {u [ q ( z r ) ] − c [ q ( z r ) ]} .
Now the welfare triangle measure is equal to the buyer’s share times the change in society’s welfare.

Measuring the Welfare Cost of Inflation
Consider two steady states, one associated with an interest rate of 3 percent (the nominal interest rate
that is consistent with 0 inflation) and one associated with an interest r. The welfare cost of an interest
rate r is measured by the rate Δ , at which consumption in the steady state with an interest rate of 3
percent must be decreased to make agents indifferent between this steady state and the steady state
with a nominal interest rate r. Let qr denote the quantities traded in a steady state when the interest
rate is r.The cost of inflation Δ solves

σ {u ⎡q0.03 (1 − Δ ) ⎤ − c ( q0.03 )} − AΔ = σ [u( qr ) − c ( qr ) ] .
⎣
⎦
Using the functional forms for u(q) and c(q) this gives
( q0.03 )1−η
( q )1−η
A
(1 − Δ )1−η − Δ = r
+ q0.03 − qr .
1 −η
σ
1 −η

Endogenous Composition of Buyers and Sellers
We introduce an assumption used by Shi (1997), and subsequently by Rocheteau and Wright (2004),
Rocheteau and Waller (2005) and Faig (2004), to endogenize the frequency of trades. In the LagosWright model, each individual can be a buyer or a seller in the decentralized market depending on
whom he meets.The frequency of trades is then given by an exogenous matching probability σ . We

20

FEDERAL RESERVE BANK OF CLEVELAND

now allow each agent to choose on which side of the market to participate in: Each agent can choose
to be a buyer or a seller in the decentralized market. Let n denote the fraction of sellers.The matching
technology is such that a buyer meets a seller with probability n and a seller meets a buyer with probability 1 – n. So the aggregate number of trades is n(1 – n) and it is maximum when the composition
of the market is symmetric, n = 1 – n = 1/2.The number of sellers in equilibrium is such that an agent
is indifferent between being a buyer or a seller. Consequently, n satisfies
−rz ( q ) + n [u( q ) − z ( q ) ] = (1 − n ) [ z ( q ) − c ( q ) ] ,
where z(q) is the buyer’s real balances as a function of q. (The growth of money supply occurs
through lump-sum transfers. These transfers do not affect agents’ decisions to be buyers or sellers.)
The right hand-side is the expected surplus of a buyer in the decentralized market net of the cost of
holding real balances. The right hand-side is the expected utility of a seller in the decentralized market.The previous equation can be solved for n,
n=

(1 + r ) z ( q ) − c ( q )
.
u( q ) − c ( q )

The form taken by z(q) depends on the pricing mechanism that is assumed. If prices are determined
by the proportional bargaining solution then
z ( q ) = θ c ( q ) + (1 − θ )u( q ).
The choice of real balances satisfies an equation similar to the one in the basic Lagos-Wright model,
that is,
u′( q )
r
= 1+ .
z ′( q )
n
Substituting n by its expression as a function of q gives
r [u( q ) − c ( q ) ]
u′( q )
= 1+
.
z ′( q )
(1 + r ) z ( q ) − c ( q )
The aggregate money demand function that is fitted to the data is
(1 − n ) z
,
n(1 − n ) z + A
where both n and z are functions of the interest rate r. Since the number of trades is n(1 – n), steadystate welfare is measured by n(1 – n)[u(q) – c(q)]. The welfare cost of inflation is the value Δ that
solves
n0.03 (1 − n0.03 ) {u ⎡q0.03 (1 − Δ ) ⎤ − c ( q0.03 )} − AΔ = nr (1 − nr ) [u( qr ) − c ( qr ) ] ,
⎣
⎦
where nr and qr are the values for n and q at the steady state corresponding to the interest rate r.

21

POLICY DISCUSSION PAPERS

NUMBER 12, JANUARY 2006

Distributional Effects of Monetary Policy
We now consider a variant of the Lagos-Wright model where individuals only live for three periods.All
agents are born at the beginning of night before the centralized market opens, and they die at the end
of the subsequent period after the centralized market closes. So agents can trade in the first period
of their life in the centralized market and in the second period of their life in both the decentralized
market and the centralized market. For simplicity, we assume that agents do not discount future utility,

ρ = 0, so that r = π . In the second period of their lives, agents are identical to the agents in the LagosWright model. In order to introduce distributional effects of monetary policy we add the following
assumption. Only a fraction p of the newly-born agents are able to produce in the first period of their
lives.The remaining 1 – p cannot produce and therefore cannot obtain money balances except from
lump-sum transfers by the government. If p = 1 the model is analogous to the Lagos-Wright model.
The choice q (the quantity consumed in the decentralized market) by newly-born agents satisfies

max {−rz ( q ) + σ [u( q ) − z ( q )]} ,
q

where z(q) is given by the bargaining solution. Let Tt denote the lump-sum transfer at night in period
t. By definition Tt = Mt +1 − Mt = π Mt , where Mt is the quantity of money in the decentralized market
in period t. Let mt denote the nominal money balances held by agents who can produce.Agents who
cannot produce hold Tt–1.Then, pm t + (1 – p)T t–1 = M t and
Tt −1 =

pπ
mt .
1 + pπ

Let qt ( qt ) denote the quantity consumed in the decentralized market by those agents who can (cannot) produce in the first period of their lives. Since z ( qt ) = Tt −1 / pt and z ( qt ) = mt / pt then
z (q ) =

pπ
z ( q ).
1 + pπ

If π = 0, then q = 0 ; and if π = ∞ , then q = q . The aggregate money demand function that has to
be fitted to the data is
pz ( q ) + (1 − p ) z ( q )
z (q )
=
.
σ [ pz ( q ) + (1 − p ) z ( q ) ] + A σ z ( q ) + (1 + pπ ) A
p(1 + π )
Social welfare is measured simply by the sum of utilities across agents, that is,

σ p [u( q ) − c ( q ) ] + σ (1 − p ) [u( q ) − c ( q ) ] .
The welfare cost of inflation is measured by Δ that solves

σ p {u ⎡q0.03 (1 − Δ ) ⎤ − c ( q0.03 )} + σ (1 − p ) {u ⎡q0.03 (1 − Δ ) ⎤ − c ( q0.03 )} − AΔ
⎣
⎦
⎣
⎦
= σ p [u( qr ) − c ( qr ) ] + σ (1 − p ) [u( qr ) − c ( qr ) ] .

22

FEDERAL RESERVE BANK OF CLEVELAND

Appendix 2: Data Description
The interest rate is the short-term commercial paper rate. From 1900 to 1975, it is taken from Friedman and Schwartz (1982), Table 4.8, Column 6. From 1976 to 1994, it is from the Economic Report
of the President (1996),Table B-69. From 1995 to 1997, it is from Economic Report of the President
(2003), Table B-73. From 1998 to 2000 it is the short-term 90-day AA credit rate from the Federal Reserve Board, www.federalreserve.gov/releases/h15/data/m/fp3m.txt. Money supply is M1, as of December of each year, and is not seasonally adjusted. From 1900 to 1914, it is from the Historical Statistics of the United States (1960), Series X-267. From 1915 to 1960, it is from Friedman and Schwartz
(1963), pp. 708–744, col. 7. From 1961 to 2000, it is from the FRED II database of the St. Louis Fed.
Nominal GDP from 1900 to 1928 is taken from the Historical Statistics of the United States, Colonial Times to Present (1970, F-1 p. 224). From 1929 to 2000, it is from the GDPA series from the Citibase database.

DATA SET

Year

Interest rate

Money demand

1900

4.38

0.307540107

1901

4.28

0.318792271

1902

4.92

0.331018519

1903

5.47

0.327729258

1904

4.2

0.34139738

1905

4.4

0.346533865

1906

5.68

0.322752613

1907

6.34

0.314868421

1908

4.37

0.32833935

1909

3.98

0.283203593

1910

5.01

0.282691218

1911

4.03

0.289860335

1912

4.74

0.277106599

1913

5.58

0.277727273

1914

4.79

0.300906736

1915

3.45

0.334925

1916

3.42

0.323043478

1917

4.74

0.303228477

1918

5.87

0.272015707

1919

5.42

0.279345238

1920

7.37

0.252983607

1921

6.53

0.296623563

1922

4.42

0.308205128

1923

4.97

0.269529965

1924

3.9

0.288099174

1925

4

0.280204082

23

POLICY DISCUSSION PAPERS

NUMBER 12, JANUARY 2006

DATA SET (CONTINUED)

Year

1926

4.23

0.262257732

1927

4.02

0.271243414

1928

4.84

0.272536082

1929

5.78

0.25515444

1930

3.55

0.273267544

1931

2.63

0.286196078

1932

2.72

0.346524702

1933

1.67

0.350336879

1934

0.88

0.345060606

1935

0.75

0.368785812

1936

0.75

0.368162291

1937

0.94

0.316528836

1938

0.86

0.368513357

1939

0.72

0.390574837

1940

0.81

0.413323471

1941

0.7

0.380102605

1942

0.69

0.386689314

1943

0.72

0.402129909

1944

0.75

0.412843494

1945

0.75

0.459058718

1946

0.81

0.484030589

1947

1.03

0.459459459

1948

1.44

0.411218425

1949

1.49

0.411896745

1950

24

Interest rate

Money demand

1.45

0.392443839

FEDERAL RESERVE BANK OF CLEVELAND

DATA SET (CONTINUED)

Year

Interest rate

Money demand

1951

2.16

0.359563808

1952

2.33

0.352777003

1953

2.52

0.33790195

1954

1.58

0.346477392

1955

2.18

0.324493732

1956

3.31

0.312

1957

3.81

0.293862503

1958

2.46

0.301369863

1959

3.97

0.279313068

1960

3.85

0.268047112

1961

2.97

0.266935928

1962

3.26

0.259393784

1963

3.55

0.25504452

1964

3.97

0.248426763

1965

4.38

0.240084828

1966

5.55

0.224554455

1967

5.1

0.22630675

1968

5.9

0.222897802

1969

7.83

0.212624416

1970

7.71

0.211929706

1971

5.11

0.208094224

1972

4.69

0.206849713

1973

8.15

0.195440804

1974

9.84

0.18788

1975

6.32

0.180258805

25

POLICY DISCUSSION PAPERS

NUMBER 12, JANUARY 2006

DATA SET (CONTINUED)

Year

1976

5.34

0.172301539

1977

5.61

0.167401152

1978

7.99

0.160335556

1979

10.91

0.153380408

1980

12.29

0.150375336

1981

14.76

0.142873993

1982

11.89

0.149237481

1983

8.89

0.150777278

1984

10.16

0.143542408

1985

8.01

0.150059711

1986

6.39

0.165774178

1987

6.85

0.161484123

1988

7.68

0.157358047

1989

8.8

0.147793378

1990

7.95

0.14521859

1991

5.85

0.152707684

1992

3.8

0.164985089

1993

3.3

0.173235347

1994

4.925

0.166032211

1995

5.925

0.155740433

1996

5.425

0.141290153

1997

5.625

0.132086389

1998

5.05

0.128062879

1999

5.925

0.123845971

2000

26

Interest rate

Money demand

6.325

0.113277783

FEDERAL RESERVE BANK OF CLEVELAND

Appendix 2: Parameter Values

REGRESSIONS

A

η

R2

Log–Log: m( r ) = A−η r

0.097835 0.29953

0.6238

Semilog: m( r ) = A−η r

0.43056

0.6750

11.027

Nonparametric

0.6795

ESTIMATION OF THE SEARCH MODEL

(ii) Nash bargaining

(i) Proportional bargaining

θ

A

η

θ

η

A

1.0

1.8248

0.14421

1.0

1.8249

0.14423

0.8

1.9096

0.17601

0.8

1.8167

0.17676

0.5

2.1876

0.26441

0.5

1.7722

0.26453

0.3

2.8112

0.40346

0.3

1.6200

0.37878

(iv) Endogenous participation and
proportional bargaining

(iii) Constant markup

μ

A

η

θ

η

A

0

1.8249

0.14422

0.9

1.8166

0.1

1.0367

0.14423

0.8

1.5574

0.33898

0.2

0.61861

0.14423

0.5

0.85057

0.2855

0.3

0.44921

0.35956

(v) Endogenous participation and
Nash bargaining

θ

A

η

0.50674

(vi) Distributional effects

p

A

η

0.9

1.5699

0.53586

1.0

2.1876

0.26441

0.8

1.6757

0.47177

0.8

1.7461

0.26128

0.5

0.88848

0.27391

0.5

1.0875

0.25658

0.3

0.48668

0.29534

0.3

0.65116

0.25358

27

POLICY DISCUSSION PAPERS

NUMBER 12, JANUARY 2006

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Friedman, Milton.“The Optimum Quantity of Money,” in The Optimum Quantity of Money and Other Essays, Chicago:Aldine Publishing Company, 1969.
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Rocheteau, Guillaume, and Randall Wright.“Inflation and Welfare in Models with Trading Frictions,” in
Monetary Policy in Low-Inflation Economies, edited by David E.Altig and Ed Nosal, Cambridge: Cambridge University Press, forthcoming.
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(1999), pp. 81–103.

30

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