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Contents lists available at ScienceDirect

Journal of Monetary Economics

Journal of Monetary Economics 55 (2008) 1038–1053

0304-39

doi:10.1

$ The

the Fed

comme� Cor

E-m

journal homepage: www.elsevier.com/locate/jme

Monetary policy and distribution$

Stephen D. Williamson a,b,c,�

a Department of Economics, Washington University in St. Louis, St. Louis, MO 63130, USAb Federal Reserve Bank of Richmond, USAc Federal Reserve Bank of St. Louis, USA

a r t i c l e i n f o

Article history:

Received 2 April 2007

Received in revised form

1 July 2008

Accepted 7 July 2008Available online 23 July 2008

JEL classification:

E4

E5

Keywords:

Monetary policy

Segmented markets

32/$ - see front matter & 2008 Elsevier B.V. A

016/j.jmoneco.2008.07.001

author thanks anonymous referees and confe

eral Reserve Bank of Cleveland, the University

nts and suggestions. Some proofs and details

responding author at: Department of Econom

ail address: swilliam@artsci.wustl.edu

a b s t r a c t

A segmented markets model of monetary policy is constructed, in which a novel feature

is goods market segmentation, and its relationship to conventional asset market

segmentation. The implications of the model for the response of prices, interest rates,

consumption, labor supply, and output to monetary policy are determined. As well,

optimal monetary policy is studied, as are the costs of inflation. The model features

persistent nonneutralities of money, relative price effects of increases in the money

supply, persistent liquidity effects, and a negative Fisher effect from a money supply

increase. A Friedman rule is in general suboptimal.

& 2008 Elsevier B.V. All rights reserved.

1. Introduction

In this paper, a model of the monetary transmission mechanism is constructed, based on market segmentation. Thisbuilds on ideas in the literature on financial market segmentation and limited participation, but includes an important newelement—goods market segmentation. Goods market segmentation, and its relationship to financial market segmentation,is critical in this model in determining the effects of monetary policy actions on prices, interest rates, consumption, laborsupply, and output.

Why do we need another model of the monetary transmission mechanism? Some might argue that New Keynesiansticky price models, as represented for example in Woodford (2007), provide an adequate account of the key short-runnonneutralities of money and perform well in guiding monetary policy. There are good reasons to doubt these viewshowever. First, Bils and Klenow (2004) find evidence on price-setting behavior that seems inconsistent with NewKeynesian models. Second, Golosov and Lucas (2007) show, in an explicitly formulated and calibrated menu-cost model,that the real effects of monetary policy are quantitatively unimportant. Third, it seems important in analyzing themonetary transmission mechanism and monetary policy to capture the key frictions in exchange that make money matter.New Keynesian models do not model these frictions and are therefore at odds with modern monetary theory (Wallace,

ll rights reserved.

rence and seminar participants at the University of Toronto, the Federal Reserve Bank of Richmond,

of Western Ontario, the SED meetings in Budapest, 2005, and the University of Minnesota for helpful

of the analysis are omitted and supplied as a technical appendix in Williamson (2008) for brevity.

ics, Washington University in St. Louis, St. Louis, MO 63130, USA. Tel.: +1314 935 9283.

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S.D. Williamson / Journal of Monetary Economics 55 (2008) 1038–1053 1039

1998; Lagos and Wright, 2005). Thus it seems important to explore nonneutralities of money that arise for reasons otherthan price stickiness, in models with explicit frictions that matter for monetary exchange.

The key ideas at work in the model are the following. Some economic agents are connected to financial markets, in thatthey frequently trade financial assets, and are on the receiving end of the first-round effects of changes in monetary policy.In practice, these connected agents are banks and other financial intermediaries and the consumers and firms that tradefrequently with these financial intermediaries. Unconnected economic agents trade infrequently in financial markets, andare affected by monetary policy only indirectly. In practice, of course, there is a varying degree of connectedness acrosseconomic agents in the economy, but in our model we assume only two types of agents, who are at the two extremes.Connected economic agents are assumed to trade in each period in financial markets, while unconnected economic agentsnever do.

In contrast to a Friedman helicopter drop, which distributes money uniformly across economic agents, outside moneyinjected into the economy by the central bank is initially received just by connected economic agents. How does this moneyeventually become dispersed through the economy? The new money will find its way to unconnected economic agents byway of transactions, and since unconnected agents are not trading in financial markets, such transactions must involve theexchange of goods. That is, the rate at which the new money finds its way from connected to unconnected agents isdetermined by the frequency with which the two groups trade in goods markets. An important element of our theory isthat connected agents are more likely to trade with connected agents, and similarly for unconnected agents. The moreeconomic agents tend to trade with their own types (connected or unconnected) the slower will be the process by whichthe new money is ultimately distributed across the population.

In the short run, a central bank money injection results in a redistribution of wealth towards connected economic agentsfrom the unconnected ones. An important feature of this model is that, once an increase in the money supply occurs,whether it was anticipated or not is irrelevant for the effects on real and nominal variables. The fact that goods markets aresegmented implies that relative prices change in the short run. That is, in the markets in which connected economic agentstrade more frequently there will be increases in prices that are initially larger than those observed in unconnected markets.Then, over time, as the size of the money stock decreases in connected markets and increases in unconnected markets, theconnected market prices fall and the unconnected market prices rise. Connected market prices initially overshoot theirlong-run values, while unconnected market prices adjust gradually to the increase in the aggregate money stock. Thechanges in the relative prices of goods that occur in the short run as a result of a money injection bear some similarity towhat occurs in menu-cost models (e.g. Golosov and Lucas, 2007). However, the friction that permits these relative pricechanges is quite different. Prices are perfectly flexible in our model, but goods markets are segmented.

A central bank money injection increases the dispersion in consumption across the population. As the behavior ofconsumption of connected economic agents and the goods prices faced by these agents determines asset prices, we willobserve a liquidity effect—a decrease in the nominal interest rate. This liquidity effect is obtained for two reasons here.First, when the money injection occurs and consumption increases for connected agents, these agents expect theirconsumption to fall over time, so the real interest rate falls, just as in many other models of segmented asset markets.Second, there is a negative Fisher effect in our framework, which is novel in the literature. That is, because connectedmarket prices overshoot, the average market price of goods faced by a connected agent falls over time after the moneyinjection occurs, so that a connected agent expects deflation, which contributes to the drop in the nominal interest rate.

What about the real effects of a central bank money injection on labor supply and output? When a money injectionoccurs, a connected (unconnected) agent faces an effectively higher (lower) expected real wage. Labor supply responses tothese changes in expected real wages roughly net out across the population and give a negligible effect on aggregate laborsupply and output. However, a central bank money injection also leads to a short run increase in uninsurable real wage riskfor all economic agents. Conditions are established under which this yields an increase in aggregate labor supply andoutput, so that agents self-insure in the face of increased real wage risk by working harder.

In addition to the dynamic responses of prices, interest rates, consumption, labor supply, and output, to a one-timeincrease in the money stock, the consequences of long-run money growth are also studied in this model. Given the relativeprice distortions that result when the aggregate money supply is not constant over time, a Friedman rule for monetarypolicy is not optimal. The welfare losses from inflation are potentially large at low inflation rates. A version of the model isalso examined with stochastic money growth, and equilibrium solutions are computed. As this illustrates, the virtues of themodel include analytical and computational tractability.

The main purpose of this paper is to explore the theoretical properties of this model, but it is important to providesupport for the theory in terms of its plausibility and consistency with basic empirical evidence. Three key elements of thetheory are financial market segmentation, goods market segmentation, and the link between the two. First, it seemsobvious that financial market participation, in the United States for example, is limited. Evidence from the Federal ReserveSystem’s Survey of Consumer Finance indicates that 12.7% of families did not hold a checking account in 2001, only 21.3%held publicly tradeable stocks, and only 17.7% held mutual funds (see Aizcorbe et al., 2003). Clearly, a large fraction of theU.S. population sees no initial effect on their portfolio of assets when the Fed intervenes in financial markets. Second, goodsmarkets are clearly segmented, due to spatial frictions and differences in consumption across income and wealth classes.Third, financially connected sellers tend to sell to financially connected buyers, for example financial intermediaries morefrequently sell services to financially connected consumers. Thus, reality seems consistent with the key frictions at work inthis model.

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This work in part builds on research on models of asset market segmentation. This literature is quite large, with recentcontributions including Alvarez and Atkeson (1997) and Alvarez et al. (2002), and Khan and Thomas (2007). The keyinnovation in our paper relative to this literature is the role played by goods market segmentation, which in part acts togive persistence in nonneutralities of money, without the assumption of implausibly infrequent asset market transactions.

Recent research in monetary theory is aimed at developing models of monetary economies that capture heterogeneityand the distribution of wealth in a manner that is tractable for analytical and quantitative work. One approach is to use aquasi-linear preferences as in Lagos and Wright (2005), while another approach is to use a representative household withmany agents, as in Shi (1997) (also see Lucas, 1990). Work by Williamson (2006) and Shi (2004) uses the quasi-linear-utilityand representative-household approaches, respectively, to study some implications of limited participation for optimalmonetary policy, interest rates, and output. Other related work is Head and Shi (2003), and Head and Lapham (2005). Fortractability, the model studied in this paper makes use of Lucas/Shi households with many agents, but there is sufficientheterogeneity that the distribution of money balances across the population matters. In the equilibria we study, thisdistribution and its evolution are very easy to characterize.

In Section 2 the model is constructed, while the effects of level changes in the money supply and changes in moneysupply growth in the absence of aggregate risk are studied in Section 3. Then, in Section 4 a version of the model withstochastic money growth is examined. Section 5 is a conclusion.

2. The model

There is a continuum of households with unit mass, with households indexed by i 2 ½0;1�. Each household is infinite-lived and consists of a producer and a continuum of consumers with unit mass. A consumer in household i is indexed byði; jÞ, where j is uniformly distributed on ½0;1�. The preferences of household i are given by

E0

X1t¼0

btZ 1

0uðcij

t Þdj� vðnitÞ

" #, (1)

where t indexes time, 0obo1, cijt is the consumption of consumer ði; jÞ, and ni

t is the labor supply of the producer inhousehold i. Assume that uð�Þ is twice continuously differentiable and strictly concave, with u0ð0Þ ¼ 1. Also suppose thatvð�Þ is twice continuously differentiable and strictly convex, with v0ð0Þ ¼ 0 and v0ð1Þ ¼ 1. Each household produces adifferent perishable good and resides at a different location.1 The producer in a household can supply an unlimited quantityof labor, and each unit of labor supplied yields one unit of the household’s own good.

There is a fraction a of connected households, where 0oao1. At the beginning of the period, each household has somequantity of divisible fiat money. All of the connected households then receive an identical money transfer tt from thecentral bank. Unconnected households never receive transfers. After receiving transfers, each consumer in the householdreceives a preference shock, and these preference shocks are i.i.d. across consumers in a given household. After receivinghis or her preference shock, a particular consumer then knows that he or she will receive utility uðcÞ from consuming c unitsof the good produced by one other household, and consuming any quantity of goods produced by a household other thanthat one gives uð0Þ units of utility. For a consumer from a connected household, the probability is 1� ð1� aÞp that theconsumer likes the good produced by another connected household, and ð1� aÞp that he or she likes an unconnectedhousehold’s good. Similarly, a consumer in an unconnected household will like a good produced by a connected householdwith probability ap and the good produced by another unconnected household with probability 1� ap. Thus, p capturesthe degree of preference of consumers for goods produced by their own type (either connected or unconnected) or thedegree of local preference. If p ¼ 1 then a consumer’s preferred good is a random draw from the population of goods, but ifpo1 then consumers will tend to prefer the goods produced by their own type. In general we will assume 0opp1. Theparameter p is key, as it determines the degree of interaction between connected and unconnected households, and willtherefore govern the rate at which a money transfer to connected households finds its way to unconnected households.

We will assume sufficient lack of communication that credit transactions are impossible between consumers and otherhouseholds. That is, there is a double-coincidence problem that can be mitigated through monetary exchange. Think of theproducer in the household playing the role of parent, while the consumers are children. The job of the producer is to workto produce output and to act to maximize the expected utility of the household, as represented by (1). The producer hascontrol of the household’s stock of money at the beginning of the period, and must decide how to allocate this moneyamong the consumers in the household. However, the consumers’ preference shocks are private information, and thehousehold has no memory, so that reports and actions of household members cannot be recorded to be recalled in thefuture. As well, the producer cannot observe the consumption of consumers from the household (consumption must occurwhen and where a good is acquired from its producer), but can confiscate any money holdings that consumers carry back tothe household at the end of the period. Finally, each consumer in the household will act to maximize his or her own utility.Given our assumptions, this implies that a consumer will take the money he or she receives at the beginning of the period

1 Ultimately, we will assume competitive pricing at each location. If readers find it more assuring to have a large number of agents in a competitive

market, it is straightforward to write down an essentially identical model with a continuum of households at each location. As the model is specified, the

household at each location is just the standard representative stand-in.

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from the producer, go to the household that produces the good he or she likes, and spend all of his or her money tomaximize consumption of that good. We assume that a given consumer can only visit one other household each period.

While consumers shop for goods, producers remain at home and sell goods to consumers arriving from otherhouseholds. The household’s consumers cannot share risk by returning to their home location and pooling theirconsumption goods, even if they wished to do so.

3. No aggregate uncertainty

In this economy, the only potential source of intrinsic aggregate uncertainty is the behavior of the money stock, and itsdistribution across the population. We will first consider the case where the actions of the monetary authority for each t areknown from the first date.

Consumers from a given household will in general face different prices for consumption goods, depending on whathousehold they buy from. In the equilibria we study, the prices of goods sold by all connected households will be identical,as will the prices charged by all unconnected households. Then, let p1

t ðp2t Þ denote the price of goods sold by connected

(unconnected) households in terms of money, and assume that prices are determined competitively. If the producer in thehousehold knew consumers’ preference shocks, then he or she would in general like to allocate different quantities ofmoney to different consumers before they go shopping. However, our assumptions guarantee that consumers cannot beinduced to reveal their preference shocks truthfully if their money allocations differ according to their preference shocks.Further, it cannot be optimal for money balances to be allocated randomly among consumers in the household, as thiswould reduce household expected utility. It is therefore optimal for the producer to allocate the same quantity of money toeach consumer.

A connected household has m1t units of money balances at the beginning of period t, and receives a nominal transfer tt.

We will also suppose that connected households can trade among themselves at the beginning of the period on a bondmarket. Each bond sells for qt units of money in period t and is a claim to one unit of money in period t þ 1. Unconnectedhouseholds do not have access to a communications technology that allows them to trade bonds. Further, bonds cannot betraded for goods as it is costless to produce counterfeit bonds that are indistinguishable from genuine bonds to the agentsselling goods.

Let m̂1t denote the quantity of money allocated by a connected household to each consumer, and let bt

denote the nominal bonds acquired by the household that mature in period t. Then, a connected household faces theconstraint

qtbtþ1 þ m̂1t pm1

t þ tt þ bt , (2)

which is effectively a cash-in-advance constraint, derived here from the fundamental frictions present in the environment.When preference shocks are realized for a connected household, 1� ð1� aÞp consumers in the household will buy fromconnected households, with each consuming c11

t goods which are purchased at the price p1t . As well, ð1� aÞp consumers

buy from unconnected households and consume c12t goods purchased at the price p2

t . As each consumer wishes to consumeas much as possible, we have

p1t c11

t ¼ p2t c12

t ¼ m̂1t . (3)

The producer remains at the home location, supplying n1t units of labor to produce n1

t consumption goods, which are thensold at the price p1

t . The household enters period t þ 1 with m1tþ1 units of money. The household’s budget constraint is then

qtbtþ1 þ m̂1t þm1

tþ1 ¼ p1t n1

t þm1t þ tt þ bt . (4)

Similarly, an unconnected household begins period t with m2t units of money, and allocates m̂2

t units of money to eachconsumer in the household. Given that an unconnected household does not have access to the bond market and receives notransfer from the government, its cash-in-advance constraint is

m̂2t pm2

t . (5)

After receiving preference shocks, ap consumers from an unconnected household buy from connected households andconsume c21

t consumption goods each, while 1� ap consumers buy from unconnected households, with each consumingc22

t . Each consumer spends his or her entire money allocation from the household, so that

p1t c21

t ¼ p2t c22

t ¼ m̂2t . (6)

For an unconnected household, the budget constraint is

m̂2t þm2

tþ1 ¼ p2t n2

t þm2t , (7)

or money balances allocated to the household’s consumers plus end-of-period money balances equals total receipts fromsales of goods plus beginning-of-period money balances.

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From (1) and (2)–(7), the first-order conditions for an optimum for connected and unconnected households,respectively, give

v0ðn1t Þ ¼ b

p1t ½1� ð1� aÞp�u0ðc11

tþ1Þ

p1tþ1

þp1

t ð1� aÞpu0ðc12tþ1Þ

p2tþ1

( ), (8)

v0ðn2t Þ ¼ b

p2t apu0ðc21

tþ1Þ

p1tþ1

þp2

t ð1� apÞu0ðc22tþ1Þ

p2tþ1

( ). (9)

Each household supplies labor each period to produce consumption goods, which it sells for money. The money is thenspent in the following period for consumption goods at connected and unconnected locations. Thus, Eqs. (8) and (9) statethat, at the optimum, each household equates the current marginal disutility of labor with the discounted expected payofffor the household from spending current money income next period.

The bond price qt is determined, from (1) and (2)–(7) and the first-order conditions for an optimum by

qt

½1� ð1� aÞp�u0ðc11t Þ

p1t

þð1� aÞpu0ðc12

t Þ

p2t

� �¼ b

½1� ð1� aÞp�u0ðc11tþ1Þ

p1tþ1

þð1� aÞpu0ðc12

tþ1Þ

p2tþ1

8>>>><>>>>:

9>>>>=>>>>;

. (10)

In Eq. (10), the price of a nominal bond is determined by the choices of a connected household. The left-hand side ofEq. (10) is the value of what a connected household gives up in the current period when it purchases one nominal bond,that is the average of the product of the quantity of consumption goods foregone and the marginal utility of consumptionacross consumers in the household. The right-hand side is the discounted expectation of the value of the payoff on thenominal bond in the forthcoming period. The real payoff on the bond is 1=p1

tþ1 if the money return on the bond is used topurchase goods from connected households, and 1=p2

tþ1 if used to purchase goods from unconnected households. This realpayoff is valued according to the marginal utility of consumption of the consumer who is purchasing the goods. Inequilibrium, the nominal bond must be priced so that the value of what a connected household gives up in purchasing thebond in the current period is equal to the expected discounted value of the payoff on the bond for the connected householdin the forthcoming period.

In all of the equilibria we examine, the cash-in-advance constraints (2) and (5) hold with equality.2 As well, in thesymmetric equilibria that we study, we will have bt ¼ 0 for all t. That is, in a symmetric equilibrium the fact that connectedhouseholds can trade bonds among themselves is irrelevant for the equilibrium allocation. Trading of nominal bonds ispermitted here so that we can determine a nominal interest rate, and it seems natural that the households that areconnected to the central bank are also the ones who trade interest-bearing assets.

With binding cash-in-advance constraints, each household always spends all of its money on consumption goods, so thepath for the money stock at each location is exogenous. In period t, let M1

t denote the supply of money per connectedhousehold after the transfer from the central bank, and let M2

t denote the supply of money per unconnected household. Fora connected household, during period t there will be a total of 1� ð1� aÞp consumers who will arrive from connectedhouseholds, and each of these consumers will spend M1

t units of money in exchange for goods, while ð1� aÞp consumerswill arrive from unconnected households and will spend M2

t units of money each. Similarly, for an unconnected household,1� ap consumers will arrive from other unconnected households, with each of these consumers spending M2

t units ofmoney, and M1

t units of money will be spent by each of the ap agents arriving from connected households. Therefore, thestocks of money per connected and unconnected household, respectively, evolve according to

M1tþ1 ¼ c1

t þ ttþ1, (11)

M2tþ1 ¼ c2

t , (12)

where c1t and c2

t are the quantities of money received in exchange for goods by, respectively, each connected andunconnected household, with

c1t ¼ ½1� ð1� aÞp�M

1t þ ð1� aÞpM2

t , (13)

c2t ¼ apM1

t þ ð1� apÞM2t . (14)

Note p will govern how quickly a money injection by the monetary authority to connected households finds its way tounconnected households. If p ¼ 1, in which case a consumer’s preferred good is a random draw from the population, thenfrom (11)–(14) each household will receive the same quantity of money in exchange for goods in each period, so a central

2 We will not call attention to this explicitly in what follows, but for all of the analysis and the examples we study, we have checked that the cash-in-

advance constraints hold with equality.

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bank money injection alters the distribution of money balances between connected and unconnected households for onlyone period. At the other extreme, if p ¼ 0 so that connected and unconnected households never trade with each other, thenM2t ¼ M20 for all t and money injections affect the distribution of money balances across the population permanently.

The remaining equilibrium conditions are that all goods markets clear in each period, or that in each segmented goodsmarket the nominal quantity of goods produced pi

tnit is sold in exchange for the quantity of money ci

t for each i ¼ 1;2, andeach t ¼ 0;1;2; . . . : This then determines prices in terms of labor supplies and monetary quantities, or

pit ¼

cit

nit

for i ¼ 1;2. (15)

Consumption quantities are then determined by

cijt ¼ nj

t

Mit

cjt

for i; j ¼ 1;2. (16)

In (16), consumption is determined by the quantity of output produced in the market in which the consumer buys, and theshare of that output the consumer can purchase. This share is determined by how much money the consumer has withhim/her relative to the average quantity of money held by buyers in the market.

Then, using (15) and (16), we can substitute in (8) and (9) for prices and consumption quantities and derive twodifference equations that solve for the equilibrium sequence of labor supplies fn1

t ;n2t g1t¼0 given fM1

t ;M2t ;c

1t ;c

2t g1t¼0

determined by (11)–(14), and given M10, M2

0, and fttg1t¼0. Similarly Eq. (10) can be used to solve for nominal bond prices.

Details are in the appendix, available in Williamson (2008).

3.1. Permanent level increase in the money supply

Money will not be neutral in this economy, because of the distributional effects of a one-time increase in the moneysupply. It is important to note that whether the increase was anticipated or not is of no consequence for the paths followedby real and nominal variables after the money supply increase. Further, because the distributional effects of monetarypolicy persist, so do the nonneutralities of money. As we will show, there will be effects on aggregate labor supply, output,consumption, and the nominal interest rate. As well, a level increase in the money supply will have persistent effects on thedistribution of employment across sectors and the distribution of consumption across economic agents. In this subsectionwe will characterize these effects.

To study the effects of a level increase in the money supply, compare two alternative economies with different initialquantities of money for connected and unconnected households. In the baseline economy, all households initially haveM units of money, while in the second economy, connected households each initially have M þD=a units of money whileunconnected households have M. Thus, in the second economy the aggregate money stock is higher by D units, and themoney supply increase is received initially only by the connected households, the households who are on the receiving endof money transfers from the government. In our experiment, there will be no transfers at dates t ¼ 1;2;3; . . . ; so that theaggregate money stock remains constant for all t in each economy, but is higher by D units in the second economy.

Now, we can determine the distribution of money balances across the population at each date in the second economy.Since there are no transfers in periods t ¼ 1;2;3; . . . ; from (11)–(14), we have Mi

tþ1 ¼ cit for i ¼ 1;2, and t ¼ 0;1;2; . . . ; and

M1tþ1 ¼ ½1� ð1� aÞp�M

1t þ ð1� aÞpM2

t , (17)

M2tþ1 ¼ apM1

t þ ð1� apÞM2t . (18)

Then, from (17) and (18), we have

Da þM ¼ M1

04M114M1

24 � � �4DþM4 � � �4M224M2

14M20 ¼ M, (19)

and

limt!1

M1t ¼ lim

t!1M2

t ¼ DþM.

Thus, note that the money stock per household falls over time for connected households, and rises for unconnectedhouseholds. That is, since po1, connected households interact relatively more with connected households thanunconnected households, so that the money transfer received by connected households at the first date persists. However,as t!1 all households will hold the same quantity of money.

In the baseline economy where all households hold the same quantity of money for all t, in equilibrium cijt ¼ ni

t ¼ n� forall t and i; j ¼ 1;2, where n� is the solution to

v0ðn�Þ ¼ bu0ðn�Þ. (20)

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Note here that n� is not optimal labor supply, it is labor supply given a fixed money stock. In the second economy with ahigher money supply, we will also have cij

t ¼ nit ¼ n� in the limit when money is neutral. Our aim is to characterize the

nonneutralities of money that we will observe on the path to the steady state.

3.1.1. Effects on labor supply and output

In period 0, the money transfer received generates a positive wealth effect for connected households and a negativewealth effect for unconnected households. However, this initial wealth effect shows up only in higher consumption forconnected households vs. unconnected households in period 0. Any effects of the money supply increase on labor supplyand output in periods 0;1;2; . . . ; will occur through the response of labor supply to prices. From (19), since the money stockinitially increases in connected markets, and then falls, while the money stock in unconnected markets rises over time, theprice of goods will tend to rise in period 0 in connected markets, and then fall over time, while the price of goods inunconnected markets rises over time.

Connected (unconnected) households sell their goods in connected (unconnected) markets for money, then exchangemoney for goods in connected and unconnected markets in the following period. Thus, as a result of the money supplyincrease, connected households will tend to face a higher rate of return on money (i.e. a higher effective real wage) whileunconnected households face a lower rate of return. Further, as a result of the money supply increase, both types ofhouseholds face increased uncertainty about the rate of return on money. Thus, we wish to analyze the implications of ahigher (lower) mean rate of return on money for the labor supply decisions of connected (unconnected) households,combined with increased rate-of-return uncertainty for both households.

To study the above labor supply effects, consider the effects on labor supply in period 0, n10 and n2

0, for connected andunconnected households, respectively. To characterize these effects, we need to resort to approximation. It will prove usefulin our analysis to ignore the effects of future labor supply decisions on prices, letting ni

t ¼ n� for t ¼ 1;2;3; . . . ; and i ¼ 1;2,and then to use Taylor series approximations.

Proposition 1. The solutions to n10 and n2

0 can be approximated by

Vðni0Þ � bUðn�Þ þ bU0ðn�Þn�ri þ bU00ðn�Þ

ðn�Þ2

2si, (21)

where for i ¼ 1;2, where r140, r2o0, and si40 for i ¼ 1;2; UðcÞ � u0ðcÞc, and VðnÞ � v0ðnÞn.

Proof. See the appendix, available in Williamson (2008). &

The second term on the right-hand side of (21) captures the effect of the money supply increase on mean rates of returnon money faced by connected and unconnected households, respectively. The fact that r140 and r2o0 reflects thatconnected households face higher mean rates of return and unconnected households lower mean rates of return than in thebaseline case. In turn, the effect of mean rates of return on labor supply depends on income and substitution effects. IfU0ðn�Þo0, then the substitution effect dominates and the effects of altered mean rates of return is for labor supply toincrease (decrease) for connected (unconnected) households, as V 0ðnÞ40. If U0ðn�Þ40, then the income effect dominatesand these effects are reversed. In any case, this first effect of the money supply injection acts to increase the dispersion inlabor supply across sectors of the economy.

In (21), the third term on the right-hand side captures the effect of the money supply increase on the dispersion in ratesof return. Here, since si40 for i ¼ 1;2, the effect works in the same direction for connected and unconnected households. IfU00ðn�Þ40 ðU00ðn�Þo0Þ then labor supply increases (decreases) for all households as a result of the increase in dispersion inrates of return.

Our analysis thus far shows that monetary policy is important for the dispersion in labor supply across sectors.However, we also care about the effects on aggregate labor supply, and conventional wisdom suggests that an increase inthe money supply will lead to a short-run increase in labor supply. What can we say about the effects of the money supplyincrease on aggregate labor supply? From (21) we obtain

aVðn10Þ þ ð1� aÞVðn

20Þ � bUðn�Þ þ bU00ðn�Þ

ðn�Þ2

2ðas1 þ ð1� aÞs2Þ. (22)

Note that the first-order terms do not appear in (22). This implies that, for example, if VðnÞ is linear in n (utility is linear inlabor supply), then the effects of changes in mean rates of return net out across households, and the only effect of themoney supply increase on aggregate labor supply will be due to the effect of increased dispersion in rates of return. Thefollowing proposition provides sufficient conditions for the money supply increase to generate an increase in aggregatelabor supply.

Proposition 2. If V 00ðn�Þp0 and U00ðn�Þ40, then aggregate labor supply increases as a result of the money supply increase.

Proof. If V 00ðn�Þo0 and U00ðn�Þ40, then from (22) we have

an10 þ ð1� aÞn

20XaVðn1

0Þ þ ð1� aÞVðn20Þ4bUðn�Þ: &

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Note that the sufficient conditions in the proposition are satisfied in the particular case vðnÞ ¼ on, and uðcÞ ¼

ðc1�g � 1Þ=ð1� gÞ with o40 and g41.

Here, it is possible for a level increase in the money supply to result in an increase in aggregate output, and the key effectworks through the response of labor supply to an increase in the dispersion in prices across markets. A money injectioninduces changes in relative prices in the short run, which creates increased risk for all households, as individual consumersare uncertain concerning the goods markets they will be participating in, and this uncertainty is not insurable. Providedthat the third derivative of the utility function is sufficiently large, a household self-insures by increasing labor supply. Notethat the segmentation of goods markets is critical to the monetary transmission mechanism at work here.

3.1.2. Effects on the nominal interest rate

From (19) the money supply increase at t ¼ 0 implies that the cash balances of connected households rise at the firstdate, and then fall over time. At each date, the consumers from connected households will be buying goods in marketswhere their nominal spending is greater than average nominal spending in that market, but average consumption forconsumers in a connected household will be falling over time. Thus, there will be a liquidity effect—a negative effect on thereal interest rate. This is similar to the liquidity effect that holds in models with limited asset market participation. Amoney injection from the central bank increases the consumption of market participants, and the real interest rate falls, asconsumption for market participants is then expected to fall over time.

A novel effect that obtains in this model, due to goods market segmentation, is a negative Fisher effect. That is, becausethe money stock falls on average in the goods markets in which connected households trade, after the money supplyincrease, a connected household on average expects deflation after a money injection occurs. Therefore, in contrast to whatoccurs in the absence of goods market segmentation, where the liquidity effect and the Fisher effect tend to work inopposite directions, in this model both effects act to reduce the nominal interest rate.

To characterize these nominal interest rate effects it proves useful, without doing much harm, to simplify the model byeliminating the endogeneity in labor supply. That is, suppose that each household receives an exogenous endowment y ofperishable consumption goods each period. Then, we have the following proposition.

Proposition 3. In the fixed-endowment economy, the bond price in period 0 can be approximated by

q0 � bþ LðDÞ þ FðDÞ, (23)

where

LðDÞ ¼bð1� aÞpDgðyÞ½1� ð1� pÞ2�

aM, (24)

FðDÞ ¼bð1� aÞpDð1� pÞ2

aM, (25)

and gðyÞ ¼ �yu00ðyÞ=u0ðyÞ is the coefficient of relative risk aversion.

Proof. See the appendix, available in Williamson (2008). &

In (23), LðDÞ denotes the liquidity effect on the nominal interest rate, while FðDÞ denotes the Fisher effect. We areinterested in how these nominal interest rate effects vary with parameters. First, note from (24) and (25) that the size of theliquidity and Fisher effects increases as a decreases, i.e. as the degree of financial market participation (the fraction ofconnected households in the population) decreases. Next, from (24), the size of the liquidity effect increases with gðyÞ, sincethe intertemporal elasticity of substitution decreases as gðyÞ increases. As well, from (24) the size of the liquidity effectincreases with p. This is because p determines the persistence in the effects of the money supply increase, so that as pincreases consumption will fall for connected households at a lower rate. Higher p then implies a higher impact liquidityeffect of the money supply increase on the nominal interest rate, but the effect dies out faster. Next, from (25), the Fishereffect is zero both for p ¼ 0 (connected and unconnected households do not trade, so there is a one-time increase in theprice level at t ¼ 0) and p ¼ 1 (connected and unconnected households have the same cash balances at t ¼ 1 and thereafter,so the price level adjusts fully after one period). The size of the Fisher effect is increasing and then decreasing in p, with themaximum Fisher effect achieved when p ¼ 1

3.It seems that the presence of an additional effect from goods market segmentation on nominal interest rates—a

negative Fisher effect—would increase the negative effect on the nominal interest rate of a money injection. However, thisneed not be the case. If we compare (23) to what we obtain when p ¼ 1 and there is no goods market segmentation, then itis straightforward to show that, if gðyÞog�, then the net effect on the nominal interest rate is larger with goods marketsegmentation, while if gðyÞ4g� the net effect is smaller, where

g� ¼ pð1� pÞ1þ pð1� pÞ

o1

5.

Thus the net effect on the nominal interest rate will be larger only if the intertemporal elasticity of substitution is verylarge.

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3.2. Money growth

In the previous section, the nonneutralities of money—some of which are novel—were studied, in response to a levelincrease in the money supply in our model. In this section, the implications of changes in the money growth rate areexplored. Again, goods market segmentation adds some interesting features to the monetary transmission mechanism. Ifthe money supply is changing over time as a result of money injections in connected markets, then there will be permanenteffects on relative prices. That is, prices will tend to be higher (lower) in connected markets than in unconnected marketswhen the money supply is growing (shrinking) over time. Thus, goods market segmentation adds a distortion to the typicalmonetary distortion which is corrected with a Friedman rule. That is, the typical distortion is due to the fact that, with toolow a rate of return on money, economic agents hold too little money, and the allocation of resources is inefficient.Reducing the nominal interest rate to zero (typically through deflation) corrects this standard distortion. In our model,correcting the standard monetary distortion involves reducing the money supply over time, but this introduces a relativeprice distortion. An optimal money growth rule involves trading off the welfare losses due to the typical distortion againstthe welfare losses from the relative price distortion.

If the aggregate money stock grows at the gross rate mt in period t, then the money transfer received by each connectedhousehold in period t is

tt ¼ ðmt � 1Þ M1t�1 þ

1� aa

� �M2

t�1

� �. (26)

The distribution of money balances across the population can be summarized by

dt �M1

t

M2t

.

Then, from (11)–(14), the distribution of money balances evolves according to

dtþ1 ¼a½mtþ1 � ð1� aÞp�dt þ ð1� aÞðmtþ1 � 1þ apÞ

a2pdt þ að1� apÞ. (27)

Letting mt ¼ m for all t, it is straightforward to show that dt converges to

d ¼m� 1þ ap

ap. (28)

Thus, given an arbitrary distribution of money balances between connected and unconnected households, if the aggregatemoney stock grows at a constant rate, then there will be convergence to a unique long run distribution of money balances.In this steady state, the distribution of consumption across consumers, and the distribution of labor supply across sectorsare constant.

Now, one approach to analyzing the effects of money growth would be to consider equilibria with a constant moneygrowth rate, and an arbitrary initial distribution of money balances. However, the implications that the initial distributionhave for the equilibrium allocation on the path to the steady state have essentially been studied in the previous section. Forour purposes here, therefore, it will be convenient to study the effects of a particular monetary policy experiment wherethe money supply growth rate is constant, but where the equilibrium consumption allocation is stationary. That is, for agiven gross money growth rate m, assume that the government makes a transfer to connected households at t ¼ 0 thatdelivers the steady state distribution of money balances given by (28) at the first date. Then dt ¼ d for all t. Note that (28)implies that if the money supply growth rate is positive ðm41Þ then there will be a higher quantity of money per capita ateach date in connected markets than in unconnected markets, and vice versa if mo1.

Given this constant money growth policy, there exists an equilibrium where labor supply, output, and consumption foreach type of consumer are constant for all time for each household (though individual consumers will have stochasticconsumption in equilibrium). Letting n1 and n2 denote labor supply by a connected and unconnected household,respectively, then it is straightforward to derive a set of equations from (8)–(10), (11)–(14), (15), (16), and (28) that solve forn1, n2, and the constant bond price q, in terms of m, the money growth factor. The equilibrium bond price is q ¼ b=m. Detailsare in the appendix, available in Williamson (2008).

If the money supply is fixed for all t ðm ¼ 1Þ, which implies that d ¼ 1 from (28), so that the distribution of moneybalances across the population is uniform, then n1 ¼ n2 ¼ n�, where n� is determined by (20), and consumption is n� for allagents in each period. However if ma1 then consumption will differ according to a consumer’s home market and themarket in which he or she purchases goods. A higher money growth rate implies that consumers from connectedhouseholds consume larger shares of output, and consumers from unconnected households consume smaller shares. Asm!1, connected households consume all output. Higher money growth transfers consumption from connectedhouseholds to unconnected households in an inefficient manner. In particular, consumption is transferred in a way thatincreases consumption risk for individual consumers.

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Proposition 4. Letting dni=dm for i ¼ 1;2 denote the derivatives of labor supply with respect to the money growth factor,evaluated at m ¼ 1, the effect on aggregate labor supply is given by

adn1

dmþ ð1� aÞdn2

dm¼�bu0

v00 � bu00o0, (29)

and the effect on dispersion in employment across sectors from an increase in m, for m ¼ 1, is proportional to

að1� aÞ dn1

dm �dn2

dm

� �2

¼1� aa

� �bn�½ð2� pÞn�u00 þ u0�

v00n� þ bpu0 � bð1� pÞu00n�

� �2

. (30)

Proof. See the appendix, available in Williamson (2008). &

In (29), note that the effect that we get on aggregate labor supply is the same as what would occur with no marketsegmentation, i.e. if a ¼ 1: That is, when m ¼ 1 the first-order effect on aggregate labor supply of an increase in m is just thenegative substitution effect from a decrease in the rate of return on money. In general, there will be income effects from achange in the rate of return on money and effects due to increased dispersion in the rate of return on money on aggregatelabor supply, when m changes, but these effects are second order when m ¼ 1:

In (30) the dispersion measure is decreasing in a and decreasing in p. As the connected sector of the economy becomessmaller (a decreases) and as connected households and unconnected households interact less in goods markets(p decreases), the effect on dispersion in employment of an increase in the money growth rate increases. Decreases in a andp both make market segmentation more important. A decrease in a concentrates money injections among a smallerfraction of the population, and a decrease in p slows the rate at which a money injection spreads from the connected sectorof the economy to the unconnected sector.

3.2.1. Optimal money growth

In this model, resources are misallocated in two ways in equilibrium. First, labor supply is in general suboptimal becauseof a standard type of monetary distortion. With a fixed money supply and discounting, the rate of return on money is toolow, and households supply too little labor relative to what is optimal. This standard distortion is reflected in a positivenominal interest rate, and it can be corrected in the usual way, by implementing a Friedman rule. However, the secondsource of misallocation arises from goods market segmentation, in that consumption goods are not allocated efficientlyacross consumers. The inefficiency arises (relative to first-best) because households would like to allocate differentquantities of money to different consumers (depending on which market they purchase in), but cannot do this forinformational reasons. Consumption goods are allocated efficiently across consumers (given labor supply) when the moneysupply is fixed ðm ¼ 1Þ, but if ma1 then wealth is reallocated inefficiently, either from connected to unconnectedhouseholds if mo1 or vice versa if m41. This second inefficiency is reflected in a relative price distortion, in that the pricelevel will be permanently higher in unconnected markets if mo1 and in connected markets if m41.

The first distortion creates welfare losses from inflation above the Friedman rule ðm ¼ bÞ, while the second distortioncreates welfare losses from a nonconstant money supply. As we will show in what follows, this implies that a small amountof deflation is optimal, but deflation at the rate of time preference ðm ¼ bÞ is either infeasible or suboptimal.

Suppose that we look for an optimal monetary growth rule in the class of constant money growth polices with constantd that we have been studying. If we weight expected utilities of households equally, then welfare as a function of the moneygrowth factor is

WðmÞ ¼a½1� ð1� aÞp�uðc11

t Þ þ að1� aÞpuðc12t Þ

þð1� aÞð1� apÞuðc22t Þ þ ð1� aÞapuðc21

t Þ

�avðn1Þ � ð1� aÞvðn2Þ

8><>:

9>=>;, (31)

where the cijt are determined from (16) and (28). The optimal money growth factor m maximizes WðmÞ, and given (28), the

choice of m is constrained by

mXmaxð1� ap;bÞ.

Here, note that a Friedman rule need not be feasible. That is, if

bp1� ap, (32)

then an equilibrium does not exist when m ¼ b. Thus, if (32) holds, then in the class of equilibria we examine, the only onesthat exist have strictly positive nominal interest rates. The possibility of an infeasible Friedman rule arises because, if ap issufficiently small, then the taxes required to support a Friedman rule deflation would be greater than the money balancesthat households in connected markets have available at the beginning of the period.

Proposition 5. W 0ð1Þo0.

Proof. Available in the appendix, Williamson (2008). &

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From Proposition 5, we can say that a small amount of negative money growth is optimal. The flavor of these results onoptimal monetary policy is similar to what can be derived in Keynesian sticky price models, for example Khan et al. (2003),though the forces at work here are quite different. In sticky price models, deviation from a constant money supply will ingeneral produce price dispersion, as some goods prices cannot be changed in any given period. Thus, in sticky price modelsthere is a standard Friedman-rule motivation for deflation, but a positive nominal interest rate is generally optimal, due to atradeoff between the relative price distortions arising from sticky prices and the intertemporal distortion from inflation. Inour model, we obtain a similar tradeoff, but all prices are perfectly flexible, and relative price distortions arise because ofgoods market segmentation and frictions that impede the flow of money across markets.

3.2.2. Numerical experiments

The solution in the appendix, available in Williamson (2008), is used to solve for n1 and n2 given m, and then forconsumptions and welfare. For now, we wish to learn something about the quantitative properties of the model, withoutdoing a serious calibration exercise.

We let uðcÞ ¼ ðc1�g � 1Þ=ð1� gÞ, with g40 and vðnÞ ¼ n. To begin, let b ¼ 0:99, and a ¼ p ¼ 0:5. Figs. 1 and 2 show resultsfor different levels of risk aversion, since curvature in the utility function will be critical in determining the implications ofmoney growth, which will in general affect the riskiness of consumption. In Fig. 1, we graph welfare relative to optimalmoney growth, measured in units of consumption relative to what is achieved with an optimal money growth rate, fordifferent levels of the coefficient of relative risk aversion. Note that the optimal money growth rate increases with thecoefficient of relative risk aversion, and that for a fairly moderate level of risk aversion ðCRRA ¼ 2Þ a fixed money supply isvery close to optimal. That is, it does not take a high degree of risk aversion for consumption risk to become the dominantforce in determining optimal monetary policy. Higher risk aversion of course also increases the welfare costs of deviationsfrom the optimal money growth rate. In Fig. 2 we show the same picture as in Fig. 1, but we include higher levels for themoney growth rate. With CRRA ¼ 0:5, the welfare loss from a 10% per period inflation is somewhat more than 1% ofconsumption, but this number increases to more than 7% of consumption for CRRA ¼ 3, a cost which is very large relative towhat is typically obtained in the literature. Cooley and Hansen (1989), using a cash-in-advance model similar to thebaseline model here ða ¼ 1Þ with log utility, obtain a welfare cost of inflation of at most 0.4% of GDP from a 10% inflation,while Lucas (2000) estimates that reducing the inflation rate from 10% to 0% per annum would yield a welfare benefit ofsomewhat less than 1% of income.

The welfare cost of inflation also depends critically on a and p: The larger is a, the lower is the cost of inflation. Notein particular that a ¼ 1 gives us a standard cash-in-advance model. Since p determines the rate at which a money injectionfinds its way to unconnected households, higher p will imply a lower cost of inflation. Fig. 3 shows how money growthaffects consumption risk for the same parameterization as in Figs. 1 and 2, and with g ¼ 1:5. Note, for example, that for a1% money growth rate, there is a very large difference between the consumption of agents from connected householdswho buy from unconnected households ðc12Þ, and those from unconnected households who buy from connectedhouseholds ðc21Þ.

In Figs. 1 and 2, the welfare measure we look at is average expected utility of households, converted into consumptionunits per period. It is useful to see how the expected utility of each type of household varies with the money growth rate, as

0.99 0.995 1 1.005 1.01 1.0150.9993

0.9994

0.9995

0.9996

0.9997

0.9998

0.9999

1

Money growth factor

fract

ion

of c

onsu

mpt

ion

CRRA=.5

CRRA=1

CRRA=2

Fig. 1. Welfare.

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0.98 1 1.02 1.04 1.06 1.08 1.1 1.120.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

money growth factor

fract

ion

of c

onsu

mpt

ion

CRRA=.5

CRRA=1

CRRA=2

CRRA=3

Fig. 2. Welfare.

0.99 0.995 1 1.005 1.01 1.0150.94

0.96

0.98

1

1.02

1.04

1.06

money growth factor

cons

umpt

ion

c11

c12

c21

c22

Fig. 3. Consumption.

S.D. Williamson / Journal of Monetary Economics 55 (2008) 1038–1053 1049

depicted in Fig. 4. To construct this figure, we use the same parameterization as in Figs. 1–3, but with g ¼ 1:5. Expectedutilities of a connected household and an unconnected household were calculated for m 2 ½b;2�, and then the curve in Fig. 4is an expected utility frontier, with the end point of the curve in the upper left-hand corner of the figure representing theconfiguration of expected utilities for m ¼ b, and the other end point representing the configuration with m ¼ 2. Note that,for low money growth rates, the expected utility of connected households rises while the expected utility of unconnectedhouseholds falls. In this range a redistribution effect dominates, as connected households receive money transfers andunconnected households do not, with the size of the money transfer increasing with m. However, for sufficiently highmoney growth rates, an increase in m reduces the expected utility of both types. This occurs because the negative effect ofinflation on expected utility on both types of agents dominates the redistribution effect when m is high.

In the experiments studied here, the optimal money growth factor is always greater than the discount factor, that ism4b. Since q ¼ m=b, therefore qo1 and the nominal interest rate is strictly positive at the optimum. In other words, aFriedman rule is in general suboptimal.

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-3.05 -3 -2.95 -2.9 -2.85 -2.8 -2.75 -2.7 -2.65 -5

-4.5

-4

-3.5

-3

-2.5

Expected Utility, Connected Household

Exp

ecte

d U

tility

, Unc

onne

cted

Hou

seho

ld

Fig. 4. Welfare tradeoff.

S.D. Williamson / Journal of Monetary Economics 55 (2008) 1038–10531050

4. Stochastic money supply

In the previous sections, we studied the effects of a permanent level increase in the money supply and of a permanentincrease in the money supply growth rate in our model. One of the attractions of this model as a framework for studyingmonetary policy is that it is very straightforward to permit stochastic monetary policy.

For notational convenience, use primes to denote variables dated t þ 1, with d ðd0Þ denoting the ratio of money holdingsof connected and unconnected households in period t ðt þ 1Þ. As well m ðm0Þ denotes the gross growth rate in the aggregatemoney stock in period t ðt þ 1Þ. Then, under the maintained assumption that cash-in-advance constraints bind in all statesof the world, from (27) d0 depends only on m0, and d. Now, if m ¼ mðd; �Þ, where � is a first-order Markov process, then thestate vector is ð�; dÞ and we can look for a recursive competitive equilibrium where quantities and nominal interest ratesdepend only on the state.3 The set of equations determining equilibrium quantities and prices is just a minor modificationof what we obtained in the previous section with constant money growth. Details are available in Williamson (2008).

4.1. Numerical experiments

In this model, the effects of a monetary injection by the central bank from the date the injection occurs are notdetermined in any important way by whether the money injection is anticipated or unanticipated. Thus, the qualitativeresponses of consumption, output, labor supply, prices, and the nominal interest rate, to a money growth rate shock will bemuch like the effects of an increase in the money supply in the deterministic version of the model that we studiedpreviously. For this numerical example, we will plot impulse responses which can also be understood as capturing theforces at work when there is a level increase in the money supply in a deterministic setup.

Here, as with the constant-money-growth experiments, we use uðcÞ ¼ ðc1�g � 1Þ=ð1� gÞ and vðnÞ ¼ n. We set b ¼ 0:99,g ¼ 1:5; a ¼ 0:1, and p ¼ 0:3. The small value of a concentrates a money injection among few connected households and alow value of p makes the nonneutralities of money more persistent than they would be for higher values of p. Thus, this is aparameterization which will make market segmentation potentially quantitatively important. In the example, moneygrowth shocks are i.i.d. It is particularly instructive to consider this case, as output, consumption, labor supply, and thenominal interest rate will be constant under i.i.d. money growth shocks when there is no market segmentation ða ¼ 1Þ. Themoney growth factor m is assumed to have mean 1.04 and to be uniformly distributed over the interval ½1:03;1:05�. Thus,the mean money growth rate is 4% per period in the experiment.

For the numerical experiment we first compute equilibrium labor supply, consumption, and the nominal interest rate,across states. Then, we are interested in plotting the impulse responses to a money growth rate shock. To obtain these, weassume that the money growth rate has been constant at 4% for a long period of time, implying that the ratio of moneybalances per household in connected vs. unconnected markets will have converged to a value given by (28). The impulse

3 We do not explore issues of existence and uniqueness of equilibrium, due to space limitations, but in the numerical experiments the model

appeared to be well behaved.

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responses then result from an increase in the money growth rate to 5% for one period, with the money supply growth ratethen returning to 4% forever.

The results are shown in Figs. 5–7. Recall from Section 3 that we have assumed a parameterization that implies that alevel increase in the money supply will increase aggregate output, due to the response of labor supply to increaseduncertainty. In Fig. 5, labor supply and output rise for both connected and unconnected households, and the effects persist.Note however, that the aggregate output effect is small, in that a 1% temporary increase in the money growth rate increasesaggregate output on impact only by approximately 0.12%.

In Fig. 6 we see the increase in the dispersion in consumption across the population as the result of the money growthshock. Consumption increases for connected households and decreases for unconnected households. The largest changes inconsumption occur for consumers from one type of household (connected or unconnected) who purchase goods in theother type of market. This is because a consumer from a connected (unconnected) household has a relatively large (small)

0 5 10 151

1.0005

1.001

1.0015

period (money shock occurs inperiod 1)

Labo

r Sup

ply

and

Out

put

connected

unconnected

total

Fig. 5. Labor supply and output, money growth shock.

0 5 10 150.95

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

period (money shock occurs in period 1)

cons

umpt

ion c11

c12

c21

c22

Fig. 6. Consumption, money growth shock.

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0 5 10 15-1.5

-1

-0.5

0

period (money shock occurs in period 1)

nom

inal

inte

rest

rate

in %

Fig. 7. Nominal interest rate, money growth shock.

S.D. Williamson / Journal of Monetary Economics 55 (2008) 1038–10531052

quantity of money, and buys at a relatively low (high) price. The changes in individual consumption are large, withincreases and decreases in individual consumption of up to 4%. Therefore, while the effects of money growth shocks onlabor supply and output (at least for this parameterization) are small, the effects on the distribution of consumption acrossthe population are large.

Finally, the effect on the nominal interest rate is shown in Fig. 7. There is a large reduction in the nominal interest rate,which declines on impact by almost 150 basis points in response to the 1% money growth rate shock. Further, as the figureshows, the effect persists, with a decline in the nominal interest rate of about 50 basis points remaining after 5 periods.

5. Conclusion

Here, a framework for studying short-run nonneutralities of money and the role of monetary policy was constructed andanalyzed. The model incorporates ideas from the asset market segmentation literature, and a key novel element in themodel is goods market segmentation.

The purpose of this paper was to explore the theoretical properties of the model, leaving empirical research for thefuture. It was shown that there are persistent nonneutralities from a level increase in the money supply, with increases inthe dispersion in consumption and labor supply. Further, conditions were established under which aggregate output willdecrease, and it was shown how a negative Fisher effect contributes to a persistent decrease in the nominal interest rate. Animportant feature in the monetary transmission mechanism is the increase in relative price dispersion following a changein the money supply, which arises because of goods market segmentation.

In the model, there is a typical monetary distortion. When the nominal interest rate is greater than zero, householdstend to supply too small a quantity of labor, and hold real cash balances that are too low relative to the social optimum. Thisdistortion can be corrected if the monetary authority implements a Friedman rule. However, in this model, there is a seconddistortion, in that if the money supply is not constant, then goods market segmentation implies that prices differ acrossmarkets. If the money supply grows or shrinks over time, consumption goods are inefficiently allocated across consumers.It was shown that a Friedman rule is in general suboptimal, as is positive money growth. Potentially, the welfare lossesfrom inflation are much larger than what is obtained with alternative models.

A virtue of this model is its tractability and we illustrated, with some numerical exercises, the model’s ability to handleaggregate risk, in this case associated with the money growth rule. In another paper, Williamson (2007), it is shown howthe model can be modified to incorporate credit transactions, clearing and settlement, and more sophisticated instrumentsof monetary policy. In that paper the role of monetary policy in the context of aggregate payments technology shocks isalso investigated.

Appendix A. Supplementary data

Supplementary data associated with this article can be found in the online version at 10.1016/j.jmoneco.2008.07.001.

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