Money, self insurance and liquidity: Baumol and

Tobin meet Bewley

Mahdi Ben Jelloul∗†

Paris School of Economics

September 2007

Abstract

We propose a simple heterogeneous agent model with idiosyncratic risks and bor-

rowing constraints where agents hold money and bearing interest assets as government

bonds for precautionary motives. We adopt the microeconomically founded motive

for holding money provided by the Baumol-Tobin setup where agents pay a �xed

transaction cost when they trade their bonds. Even if the money is dominated by the

interest bearing asset, infrequent trading constrains the agent to hold some money

on average. We solve numerically the Bellman equations and compute the equilib-

rium distributions that show substantial dispersion in money holdings. In a partial

equilibrium, for a small economy with inelastic real interest rate, in�ation reduces

substantially the amount of money holdings and increases trades frequency.

∗benjello@univ-brest.fr†This paper is a �nal thesis of the graduate programme APE at the Paris School of Economics and was

written under the supervision Xavier Ragot, PSE-CNRS.

1

Acknowledgements

I would like to thank my friends Aurélie Ouss and Mathieu Hernu for discussions

and comments. Many thanks to Yann Algan for his kind responsiveness and to my

advisor Xavier Ragot for providing constant support and thoughtful insights.

2

Introduction

Substantial di�erential concentration in income, earnings and wealth is a major styl-

ized fact well documented in most United States surveys. Using data from the Panel

Study in Income Dynamics (PSID) and the Current Population Survey (CPS), Ro-

driguez et al. (2002) computed Gini indexes for income (standing as income before

taxes but after transfer), earnings and wealth. The most concentrated variable is

wealth, with a Gini index of 0.803, while earnings rank second with a Gini index

of 0.611, and income comes last with a value of 0.553. Krueger and Perri (2006)

note that Gini index of after-tax labor income (earnings) has increased from 0.33 in

1980 to 0.42 in 1997 while the Gini index for consumption expenditures has remained

roughly constant at 0.31 in the same period. Recent estimates from the Current Pop-

ulation Survey and the Annual Social and Economic Supplements (ASEC) conducted

by the U.S. Census Bureau (DeNavas-Walt et al., 2005, 2006) found household income

Gini index values of 0.464 in 2003, 0.466 in 2004 and 0.469 in 2005, con�rming the

persistence of an increasing trend.

Wealth is composed of di�erent assets which di�er in their liquidity. However, it

appears that liquid and illiquid asset holdings are not correlated. Still, they appears

to be very inequally distributed with Gini coe�cient around 0.8 for both of them

(SCF 2004). Roughly the Gini coe�cients can be summarized as follows:

• Consumption: 0.3�0.4

• Income (earnings after tax and transfers) : 0.4�0.5

• Earnings : 0.5�0.6

• Money: 0.8

• Financial Assets: 0.8

Another important feature is the potential large amount of liquidity-constrained

households. Although microeconomic studies are not completely conclusive, some em-

pirical results reinforce previous �ndings of Zeldes (1989) which tested a constrained

Euler equation to assess the deviation from the Friedman (1957) permanent income

hypothesis (PIH). Mulligan and Sala-i Martin (2000) notes that 50% of US house-

holds do not hold any interest bearing �nancial asset in the 1997 SCF survey. For

many households the question of adopting the technology of �nance is costly; the

relevant question is not how much interest bearing assets to hold but whether to hold

interest bearing assets at all. Gross and Souleles (2002) notes that 1/3 of households

lack bankcards, and well over half of US households with bankcards are rolling over

debt and paying high interest rates. Almost 15 percent of bankcard accounts have

3

utilization rates above 90 percent. Marginal propensity of such individuals are fairly

substantial (about .5).

Since borrowing constraints on consumption are very costly in terms of individual

welfare and their pervasiveness should trigger substantial precautionary savings for

a larger share of individuals. For example, savings increased in France form 11% of

available income in the late eighties to reach 15% in the mid nineties and are almost

entirely explained by precautionary savings against unemployment risks. It is notably

established that these risks are uninsurable because of moral hazard problems. In the

United States, the rise of non-insurable employment risks parallels the wage premium

increase.

Precautionary motives and borrowing constraints a�ect saving and thus have a

potential aggregate impact on production and welfare. In such a world, the repre-

sentative agent theory does not hold anymore and the classical welfare theorems do

not apply. As money may be used as a saving medium, money demand will react

to change in the price of money and thus on in�ation. In�ation may result in large

wealth transfer as documented by Doepke and Schneider (2006).If we assume that

money is the only available asset, credit-constrained agents do not react as money

holders and this may have real e�ects when the distribution of the seignoriage rent is

taken into accounts.

Moreover, even when �nancial assets are present, the heterogeneity in the demand

for real balances and �nancial assets between constrained and unconstrained agents

may have strong real e�ects on the long run, and thus break the super neutrality of

money present in �exible price models. Short run dynamics also should be completely

di�erent and may provide a di�erent view than the prevailing sticky price models.

The main goal of our work is to build up a simple model that includes heteroge-

neous agents that may reproduce some characteristics of the SCF. It will necessarily

include a theory of money since money holdings do play a di�erent role than other

asset holdings for economic agents. We will �st review some classical money theories

to outline their inability at explaining the �gures we mentioned above. We will then

review some literature on precautionary motives for money holdings when it is the

only available assets in presence of uninsurable idiosyncratic risk. We propose to

extend the approach to a couple of assets, money and bonds, which di�er in yield and

�xed transaction costs following Baumol (1952) and Tobin (1956).

4

1 A brief review of the literature

1.1 Classical money models

MIU and CIA models Macroeconomics used many shortcuts to introduce money

at the aggregate level in models. The simplest way is to incorporate money balances

directly in the utility functions of agents as suggested by Sidrauski (1967) to represent

somehow the convenience of holding money. It provides a solution to the problem of

generating a positive demand money without any clearly de�ned reason for individual

to hold it. One way of specifying the role of money in the utility function is brought

by shopping time models, which states that consumption is produced conjointly by

time and money. An extreme case of shopping time models are cash-in-advance mod-

els (CIA): if money can't buy goods at an endogenous price, an in�nite amount of

time is needed; if there is su�cient money, the transaction is immediate mimicking

the Clower constraint. Clower stressed the positive role of money as the only medium

of exchange �money buys goods, goods buy money, but goods do not buy goods� and

justifyed the introduction of the constraint by noting that in any given period �the

total value of goods demanded cannot [. . . ] exceed the amount of money held by the

transactor at the outset of the period�. The so-called cash in advance (CIA) model

of Patinkin (1965) can thus be seen as special case of shopping time MIU models.

Friedman's rule In deterministic versions of these models or in a complete mar-

kets setup, the optimal rate of in�ation is such that the nominal interest rate is zero

(Friedman, 1969) to avoid distortion creation. Since the steady state real return on

capital compensates the psychological discount, the optimal rate of in�ation is a rate

of de�ation that equals the real return on capital or the discount rate. For example, in

CIA models with money goods and credit goods, in�ation clearly creates a distortion.

In MIU models, the social cost of money creation is zero so the marginal utility of

money should thus be equal to zero to reach Pareto optimality. The other optimality

condition states that money private return, the nominal interest rate should also be

zero to reach optimality and Friedman's rule follows.

However, Friedman's argument essentially establishes that in an optimal equi-

librium, the marginal utility of money is zero and the optimal real interest rate is

equal to the psychological discount rate; if money holdings are positive, this requires

a vanishing nominal interest rate and therefore a de�ation at rate π = −r. But thecriticism addressed by Hahn (1965), initially targeted at Patinkin (1965) Walrasian

general equilibrium proof remains: when the �price� of money, which must equal its

5

cost, is null, there is no enough reason for a monetary general equilibrium to exist. For

example, if the money utility is a classical decreasing return function satisfying the

Inada conditions, then the optimal quantity of money is in�nite. The precise form

of this function needs thus a satiety level which is, again, not micro-economically

founded.

In�ation impact in general equilibrium The cost of in�ation in MIU and

CIA models calibrated to the US economy has been computed to be very small (Lucas,

2000; Walsh, 2003, chap. 2 and chap. 3), i.e. slightly less than a 1% reduction

in consumption for a reduction of nominal interest rates from 3% to 0 . Another

weakness of MIU and CIA models is their failure to reproduce realistic response to

monetary shocks. Output responses are wrong and adjustment clearly too quick,

even if the cash in advance constraint helps to introduce some sluggishness in the

adjustment process.

1.2 Alternatives to the classical models

Search models Instead of introducing liquidity in preferences or constraints,

search theory models rely on a microeconomic foundation of the transaction pro-

cess. The role of money as a medium of exchange is made explicit by specifying the

specialization of producers, the pattern of meetings and the informational structure.

Money arises endogenously as a medium of exchange so that the frictions and ine�-

ciencies in money transaction are theoretically identi�able at the equilibrium and its

existence conditions . The �rst formal treatment is due to Kiyotaki and Wright (1989,

1993) who showed how agents can hold a �nite amount of money in the perspective

of using it to purchase goods in a an economy with specialized consumer-producers

with low probability of double coincidence of wants. However, in these models, each

good is bought for a unique amount of money. Trejos and Wright (1993) extend the

model to allow for endogenous price formation and divisible goods but still money

holdings are indivisible. The quantity of exchanged good is determined through a

bargaining process. Trejos and Wright (1995) write down a version of the model

where agents can hold any nonnegative divisible amount of money but fail to solve

it. Indeed, the full resolution necessitates to derive the distribution of agents money

holdings explicitly, so they concentrate on a model where agents can choose a binary

quantity of money. Lagos and Wright (2005) introduced an overnight Walrasian mar-

ket and quasi-linear utility function in addition to daily decentralized meetings. This

speci�cation gives forth to a degenerate distribution of money holdings, allowing to

6

derive an analytical solution to the model. There are two sources of ine�ciencies in

his model: the psychological discounting and the seller's hold up in single coincidence

meeting. Again, the �rst ine�ciency is related to the nature of money as a claim on

future consumption which is therefore discounted when the seller choose her e�ort

level. De�ation is thus a way of providing as much utility in the next time period as

if the good was consumed instantaneously. The hold up problem arises if the seller

has some bargaining power because he can then acquire a rent. Friedman's rule is

thus an optimum only when bargaining power between seller cannot extract surplus

from the buyer.

A drastic simpli�cation arises in the Lagos and Wright (2005) model, by adding

a Walrasian market to the classical decentralized market of search theory monetary

models, and a quasi-linear utility function for individuals. These two hypotheses result

in annihilating the wealth e�ect and allow the agents to perfectly insure themselves

against the idiosyncratic risk of uncertain production and consumption in meetings.

In a model where the Walrasian market would not exist, agents hold money for

precautionary motives and their holdings being history dependent, money holdings

would not be degenerate.

Since the general model is analytically untractable, Molico (2006) solved it nu-

merically. Interestingly, Friedman's rule doesn't hold anymore. Any money injection

mechanism can be decomposed between a proportional component and redistribu-

tive component. Proportional transfers are clearly neutral but if the redistributive

component is predominant, both money holdings and prices are less dispersive. The

process has thus a real e�ect and is welfare increasing. For low in�ation a liquid-

ity e�ect dominates by decreasing money dispersion through transfers. However, for

higher levels of in�ation, the real balance e�ect dominates, but an optimum di�erent

from that of Friedman's rule still exists. Note however that the absence of any other

asset market can arti�cially overstate the role of money being held for precautionary

motives.

Neo-Keynesian models The failure of the �xed prices theory to explain the re-

sponse to in�ation shocks, was partly corrected by the neo-keynesian literature which

embedded imperfect (monopolistic) competition in model with nominal rigidities on

price or wages (menu cost, information cost, staggered wage setting). But still, many

puzzle remains (Walsh, 2003, chap. 5). The ubiquitous sluggishness of the in�ation

rate is not well predicted by sticky price model (Mankiw, 2001). Reasonable perfor-

mance are obtained for model (Christiano et al., 2005) with a many ad hoc ingredients

(MIU, variable capital utilization, habit formation, . . . ).

7

1.3 Model focusing on precautionary motives

Market incompleteness In a rational-expectations approach, a solution to the

Hahn problem can be achieved if there is a demand for �at money as a store of

value. Several motives can create such a need. Standard life-cycle theory predicts

that the consumer saves when he anticipates declining income. In the framework of

the overlapping generation model, Samuelson (1958) money can be a store of value if

agents have irregular wealth at every stage of their life and should transfer resources

across periods. The market are incomplete because only one asset is tradable and

agents face two state of nature. We now assume that endowment are not only irregular

but uncertain.

Bewley pioneering contribution (1983) A need for a store of value can also

be motivated in a framework where agents are endowed in each period with uncertain

incomes, in which case precautionary motives yield positive money holdings. Bewley

(1983) was the �rst to derive the properties of a model were money was used as a

self insurance device to cope with some idiosyncratic risk. He notably shows that

Friedman's rule cannot hold in a such a framework. If the interest rate is close to the

discount rate, the money demand explodes because of precautionary motives, as can

be seen on �gure 1.3 from Aiyagari (1994). The equilibrium thus necessitates a low

rate of return such that β(1 + r) < 1 thereby violating the Friedman rule (Huggett,

1993).

Real distributional e�ects Scheinkman and Weiss (1986) solves analytically a

simple model with heterogeneous agents to outline the real impacts of the distribution

of wealth across agents. Their model is based on the following ingredients:

• Heterogeneity of agents who can receive di�erent incomes, for example if they

are employed or unemployed.

• Some uninsurable idosyncratic risk (unemployment in this case) but no aggre-

gate risk

• Borrowing constraints that can be easily be motivated by moral hazard consid-

erations,

• In�nite horizons rational agents,

• General equilibrium framework.

The model setup is:

Agents : Two types (i = 1, 2) of agents for a total mass of 2.

8

Figure 1: (left) Asset demand as a function of interest rate; λ = β−1−1 and b the borrowinglimit. (right) Supply-demand equilibrium.

Risk structure : Agents switch between two states st(ω) with a constant probabil-

ity per unit time according to a Poisson process. Individual labor supply veri�es

li(t, ω) = 0 if st(ω) = j 6= i

Utility : Agents rationally maximize their discounted intertemporal utility

U i =∫ ∞

0e−ρt

[u(ci(t))− li(t)

]dt

Production : Goods are produced through a one factor (labor) constant return to

scale production function: Y = li(t, ω) + lj(t, ω).

This simple model is almost analytically tractable and allows the authors to com-

pare the complete market equilibrium solution with an incomplete market solution in

presence of borrowing constraints. The complete market equilibrium provides equal

consumption across agents verifying u′(c̄) = 1. On the other hand, the incomplete

market equilibrium is such that only productive agents consume as in the complete

market equilibrium, the non productive ones consuming less mainly through their

savings.

At the equilibrium, a heterogeneous distribution of money across agents is reached

where lucky agents, i.e. experiencing longer spells of employment, with higher amounts

of assets consume more than nonproductive agents, but are less numerous. The rep-

resentative consumer approach does not apply and the equilibrium is not Pareto

9

optimal as a consequence of market incompleteness. Therefore there is room for a

welfare improving policies of monetary or �scal types.

Finally, there is a real role of unanticipated money injection which are mainly

transfer from wealthy people to unwealthy ones if the seignoriage rent is redistributed

lump-sum. When output is low, which corresponds to a high level of unemployment,

money injection increases output by relaxing the credit constraint and thus fostering

demand. When output is high, money injection decreases demand since it increases

the amount precautionary savings.

Aiyagari (1994) computed numerically the equilibrium distribution in the frame-

work of a general equilibrium model calibrated to �t US data. He shows the impor-

tance of precautionary savings and recovered the strong inequality of wealth with a

Gini coe�cient surpassing the one on consumption.

In�ation impact Since in�ation depreciates money which is, for now, the only

asset that is used for precautionary savings, a higher in�ation will enhance the insur-

ance cost.

Imrohoroglu (1992) used numerical simulation to compute the welfare cost re-

lated to in�ation by using the steady state average utilities. It appears that the last

procedure might lead with results signi�cantly di�erent, higher i.e. by a factor 4,

than those obtained by Lucas (2000). Although the money demand function is quite

similar, the welfare cost can no longer be calculated using the method of the triangle

under the money demand curve. This result is another example showing that the use

of Friedman rule can be misleading in a incomplete market setup.

1.4 The coexistence of money with interest bearing assets

An inventory approach: Baumol-Tobin-Romer We now assume that agents

face some idiosyncratic risk such that there is a demand for a store of value. As pointed

by Hellwig (1993) a monetary equilibrium would not exist if another asset would pro-

vide a superior risk free return. Hellwig (1993) calls this problem the modi�ed Hahn

problem.

Friedman (1969) stresses that some of the utility of money holdings resides in

the non pecuniary value of money because it saves the agent to �pay the errand

boy that goes to the bank� when money is needed for a transaction. The �rst1

micro-economically founded model was proposed by Baumol (1952) who proposed an

1As pointed out by Baumol and Tobin (1989), Allais (1947) already exposed these principles and resultsin 1947

10

inventory model. An agent receives some initial endowment that she can consume

or buy some interest bearing assets at the bank. however those cannot be bought

without paying a �xed transaction cost. Assuming equally spaced trips to the bank,

the agent optimizes the amount of cash she should hold to consume. Baumol (1952)

then derives the famous elasticity values: an income money demand elasticity of 1/2

and in�ation money demand elasticity of −1/2 for a �xed transaction cost. Miller

and Orr (1966) extended this model to stochastic returns lower the money demand

elasticity2. Tobin (1956) completed Baumol's model by showing that the trips should

be equally spaced.

Grossman and Weiss (1983) justi�ed a simple model setup by Baumol-Tobin re-

sults. They derived a simple endowment economy with overlapping generation where

only half of the agents can adjust their interest bearing assets, and studied money

injection. Open market operations result in lower nominal interest rates, in order to

have the agents at the bank accept to hold more cash. However the process is slug-

gish and involves real e�ects through wealth transfer from those agents not making

withdrawals to the other agents.

Romer (1986) extends Baumol-Tobin in an continuous time overlapping generation

model with what he called a ��exible Clower constraint� with endogenous frequency

of trips to the bank and in general equilibrium framework. He shows that in�ation

a�ects money demand through 3 channels: the frequency of trips, the amount of

spending between trip, and the wealth e�ects. The Mundell (1963)�Tobin (1965)

e�ect is not straightforward and may be very low or even absent.

Mulligan and Sala-i Martin (2000) propose an estimation of the aggregate money

demand elasticity based on Baumol-Tobin model integrating the cross sectional dis-

tribution of assets. They show the parameter controlling the �decision to adopt� the

�nancial technology is the interest rate times the amount of assets that should be

above the transaction cost. Since the probability of holding interest bearing assets

is positively related to the total amount of assets, variation in in�ation will a�ect

only marginal participants in their decision to adopt the �nancial technology. Since

their amount of assets is relatively low, the extensive margin contribution to money

demand is modest. Adding the classical Baumol-Tobin intensive margin they found

weak in�ation elasticities of money demand for low in�ation but recover standard

values for higher levels with a minor contribution from the extensive margin.

Adopting a �nancial approach, Lo et al. (2004) study the role of a �xed transac-

tion cost on trading strategies and equilibrium prices. They show that the liquidity

2None of these models include risk aversion.

11

premium and the time between trades depends as the square root of the transaction

cost and that the no trading region depends on the cost to the power 1/4. Even small

transaction costs have therefore a strong e�ect. They also argue that the risk aver-

sion has a stronger impact on the liquidity premium than the risk premium because

it relies heavily on market clearing conditions.

1.5 Project

We have thus identi�ed the key ingredients of a model displaying a joint heterogeneity

in detention of both money and a �nancial asset together with simple micro-economic

foundation. These can be summarized as follows:

• Only money buy goods but its value depends on in�ation

• Only �nancial assets provide return on capital dominating money

• Fixed transaction costs to access �nancial markets

• Add idiosyncratic risks to obtain a non degenerate distribution of money and

interest bearing assets

• Borrowing constraints impede agent to have debt in any of these assets

The simplest model is thus a model with two contingent states and a risk free interest

bearing asset in addition to money.

2 Simple models

We �rst solve a simple model with the ingredients listed in the previous section 1.5

in simple time frames. A comparison between a two-period models and three-period

model outlines the di�erence between the two assets. However, our setting is slightly

di�erent from the classical Baumol-Tobin framework, since the agent receives at each

period wages in money (check account).

2.1 A two-period model

We consider a two-period model where the �rst period earnings w1 are deterministic

but the second period earnings are stochastic, i.e. w̃2 = w2±σ with two states of the

world with probabilities p± = 1/2. The agent can save some amount of her revenue in

money m or bonds b. The bonds are remunerated at the rate R while money evolves

with in�ation π. Agents maximize their intertemporal CRRA utility under budget

constraint. The utility is taken here as a logarithmic one to let the algebra tractable.

12

We will further assume that bonds yields always dominates money, that is RΠ > 1

where Π = 1 + π.

The model reads:

V = u(c1) + βE [u(c2)] , (1a)

u(c) =c1−γ

1− γ= log c, (1b)

c2 = [Rb− δ]χ+m

Π+ w̃2, (1c)

c1 = [−b− δ]χ−m+ w1. (1d)

where χ is a boolean variable indicating the decision to enter the bonds market at a

�xed cost δ.

Holding bonds We �rst suppose that the agent access the bond market so we set

χ = 1. The following conditions should thus hold:

w1 − b > δ, w2 +Rb > δ (2)

and the intertemporal budget constraint then reads:

c2

R+ c1 = −δ1 +R

R+w2

R+ w1 +m

(1RΠ− 1)

(3)

Since we assume here that bonds dominate money, necessarily m = 0 and the Euler

equation is thus:

u′(c1) = β RE[u′(c2)] (4)

After some algebra, we get the following consumption pro�le:

c1 =

(2 + β) (Rw1 + w2 − δ(1 +R))−√β2 (Rw1 + w2 − (1 +R) δ)2 + 4 (1 + β) σ2

2R (1 + β),

(5a)

c2 =

β (Rw1 + w2 − δ(1 +R)) +√β2 (Rw1 + w2 − (1 +R) δ)2 + 4 (1 + β) σ2

2 (1 + β), (5b)

13

b =

Rβ (w1 − δ) + (2 + β) (−w2 + δ) +√β2 (Rw1 + w2 − (1 +R) δ)2 + 4 (1 + β) σ2

2R (1 + β).

(5c)

The bond holdings b are positive when there are enough earnings in the �rst period

to be transferred to the second period. Since the amount of insurance should increase

with the risk faced, more risk means more penalty on �rst period consumption. If

the risk is too strong, no bonds will be held. A necessary condition for positive bond

holdings is computed to be:

Rw1 + w2 − (1 +R) δ > σ. (6)

A necessary condition is thus that the aggregate wealth is higher than an increasing

function (here linear) of both the transaction cost and the risk. Moreover, the bond

holdings increase with the second period revenue risk σ.

Holding money Now let us suppose that agents hold only money and do not

access the bonds market (χ = 0). Since we assume here that bond yields always

dominate money in absence of �xed transaction cost, necessarily the bond holdings

are zero (b = 0) if the agent keeps some amount of money when the �xed cost is �nite.

The Euler equation is then:

u′(c1) =β

ΠE[u′(c2)]. (7)

After some algebra, we get the following consumption pro�le:

c1 =((

Π−1 −R)m+Rw1 + w2

)−

√β2 ((Π−1 −R)m+Rw1 + w2)2 + 4RΠ (β +RΠ) σ2

2R (β +RΠ), (8a)

c2 =β((

Π−1 −R)m+Rw1 + w2

)2 (β +RΠ)

+

√β2 ((Π−1 −R)m+Rw1 + w2)2 + 4RΠ (β +RΠ) σ2

2 (β +RΠ), (8b)

14

Figure 2: Dependence of money holdings (left) and bonds (right) on in�ation π and incomevariability σ. Parameters are w1 = w2 = 3, β = 1/(1 + .04), δ = .05, r = .01. Spacebetween contours is .1 with the black color indication the 0 level and with color the level1.

m = w1 − c1 =w1 β − w2 (2 + β) Π +

√β2 (w1 + w2 Π)2 + 4 (1 + β) Π2 σ2

2 (1 + β). (8c)

A necessary condition for positive money holdings is the following condition:

σ2 > w22

(1− β

Πw1

w2

),

which is more stringent when in�ation is strong or when income is declining.

Decision Comparison of the utility attained using the alternative strategies yield

to the holdings summarized in �gure 2.1 (see also section on Fig. 4) for di�erent

values of in�ation and risk endured. An interesting feature is related to the choice of

the savings asset, as risk increases for a su�cient large in�ation/low �xed transaction

cost for which bonds are not always dominated by money. As can be seen on �gure 3,

for low risks, the agents prefer not to save: transaction cost for bonds and positive

in�ation, for money discourages savings.

As the risk increases, the need for insurance increases. For low in�ation, the

transaction cost makes bonds less appealing than money. In this case, money holdings

then increase as the risk increases. For higher in�ation at a given transaction cost,

15

Figure 3: Expected utilities if the agent saves in bonds, money or do not save.

holding money is not a valuable insurance (the agent prefers not to cover himself

at that cost) but the need for insurance is strong enough for the agent to pay the

transaction cost for a good yield. Net of transaction costs, the savings in bonds

are a limited amount but provides a valuable insurance. However, as risk increases,

the amount of income dedicated to savings increases. But as the need for insurance

increases, money can be a valuable insurance. Since using bonds instead of money

implies larger savings to a�ord the transaction cost as can be seen on Fig. 4, the

impact on the share of wealth that can be consumed is more severe; using money

becomes thus more favorable than bonds as insurance. But as the risk becomes

higher, larger amounts of money are needed, and the loss due to in�ation is stronger.

Therefore, for a su�ciently high risk, bonds are again a better insurance.

2.2 Three-period model

As we have just seen, two-period models cannot explain holdings of bonds and money

by the same agent, even with uncertainty and �xed cost. Introducing a third period

is thus mandatory. In that case, uncertainty in revenue is not essential since money

holdings are used for short term consumption and bonds as a long term saving device

(à la Baumol-Tobin-Romer). The condition such a situation where bonds are held

for longer than money are proposed in the appendix:. However, introducing some

uncertainty in revenue is mandatory to derive continuous distributions for assets and

money holdings. We thus move on to address the continuous time problem with

idiosyncratic risks on earnings.

16

Figure 4: Money and bond holdings as a function of risk σ for a �xed in�ation π = .3

3 In�nite period model with heterogeneous agents

3.1 Model description

We implement along the lines of Aiyagari (1994) and Imrohoroglu (1992) a model

with two assets that can be used as store of value. Unlike Algan and Ragot (2006) we

do not use a MIU approach but implement a Baumol-Tobin setup to provide incentive

to hold money.

We consider a continuum of agents with distinct money and bonds holdings. Every

agent occupies a state s at time t with a probability π(s′|s) of transiting to a state

s′ at t + 1. Each period t, every agent receives a wage depending on its state and

decides whether to consume, to keep some cash money or to go to the bank, paying

a �xed transaction cost, to adjust its bond holdings. Money and bonds holdings are

strictly positive so no borrowing is allowed. Agents hold mt ≥ 0 money and bt ≥ 0

bonds at the beginning of period t where they are about to consume ct to maximize

their intertemporal discounted utility:

V (mt, bt) =∞∑τ=t

βτu(cτ ), (9)

by adjusting her subsequent money mt+1 ≥ 0 and bonds bt+1 ≥ 0 holdings. We are

led to solve a coupled system of Bellman equations. The Bellman equation ruling the

17

value function V part of an agent willing to enter the �nancial market is thus:

V parts (mt, bt) =

maxmt+1,bt+1

u(ct) + β∑s′

π(s′|s) max[V parts′ (mt+1, bt+1), V exc

s′ (mt+1, bt+1)],

where

ct = ws + bt+1 − (1 + r)bt +mt+1 −mt

Π− δ,

mt+1 ≥ 0,

bt+1 ≥ 0. (10a)

If the agent is unwilling to enter the �nancial market, the Bellman equation ruling

her value function V exc is:

V excs (mt, bt) =

maxmt+1

u(ct) + β∑s′

π(s′|s) max[V parts′ (mt+1, (1 + r)bt), V exc

s′ (mt+1, (1 + r)bt)],

where

ct = ws + +mt+1 −mt

Π,

mt+1 ≥ 0,

bt+1 = (1 + r) bt. (10b)

The parameters r , Π and δ correspond to those de�ned in the previous sections. As

a �rst step, we make the small economy hypothesis hold and assume that the rate

of return capital is �xed. However we impose β(1 + r) < 1 to reach a statistical

equilibrium.

Note that we do not redistribute the seignoriage revenue of the government. We

de�nitely should include that additional ressource for households

3.2 Numerical implementation

Policy and value function The coupled Bellman equations are solved numeri-

cally using a recursive method. The modi�ed policy iteration algorithm with k steps

algorithm described in Judd (1998, p. 416) textbook is implemented. After choosing

a �rst guess of the value function, the main recursion consists in the following steps:

18

1. Compute the policy function {m′(m, b), b′(m, b)},

2. Compute the iterated value function:

V ′ =k∑l=0

βlu(cl) + βk+1V

where ct is computed according to the policy function computed a step above.

3. Stop when the value function has converged up to the desired precision.

The two value functions are discretized on a bidimensional grid and jointly solved.

The policy functions for the two problems are computed on a re�ned grid. The

optimization is performed on the re�ned grid after bilinear interpolation of the two

value functions. Iteration is carried on until the precision on the value functions allows

them to be separated by the max operator arising in (10). Two policy functions (part.

and exc.) are thus computed and the optimal one is evidently the one yielding the

highest value function.

The grid is nonlinear and re�ned for low values of m and b.

Equilibrium distributions The equilibrium distribution is computed by a Monte-

Carlo method. A direct simulation of the evolution of an agent obeying to the optimal

policy function is computed. However, since the optimal value function is available

only on a relatively coarse grid, the policy function is interpolated along the following

scheme:

1. Compute the two value functions at the state variable (mt, bt) by bilinear inter-

polation

2. Compare the two value function to determine the decision function

3. Compute (mt+1, bt+1) by interpolate the policy function corresponding to the

decision

Another algorithm was also tested:

1. If the state variable (mt, bt) is surrounded by grid points where one policy (part.

or exc.) dominates than the policy function is bilinearly interpolated on the

dominant policy function.

2. In the opposite case, the policy function is bilinearly interpolated on the domi-

nant policy function on the nearest grid point.

The algorithm was not able to yield any plausible results because of the poor resolu-

tion of the coarse grid used.

19

3.3 Parameters and calibration

Parameters are taken from Imrohoroglu (1992). The wage levels are w = (1, 0.25)

and the transition matrix is:

π =

(0.9565 0.0435

0.5 0.5

)

The psychological discount rate and the coe�cient of risk aversion are:

β = .995, γ = 1.5

The rate of return on interest bearing assets is r = 2% and the in�ation is such that

Π = 1 + 2%.

Computation are conducted on a 30 × 120 coarse grid. The re�ned grid is three

times �ner. A long period sample (106 iterations to get 3× 10−2 relative precision) is

computed and the initial part (10%) is removed. Statistical properties are computed

on the remaining part of the sample.

3.4 Some preliminary results

Policy functions The agents behavior can be inferred from the policy functions

represented in �gure 5 and Agents with high earnings accumulate money at when

they have only a little amount of bounds until they reach a su�cient level of money

such it is worth the cost of transferring it to the bank. Low earnings agents liquidate

their bonds when they do not have enough precautionary savings in money.

Typical time evolution A typical time evolution of the agent consumption,

wages, money holdings and bonds is represented on �gure 6. Looking at the bonds

curve, it is clear that the agent insure himself against shocks using by raising savings

in bonds. On the other hand the money displays the classical sawtooth pattern of

the Baumol-Tobin model although it is an accumulation rather than a progressive

dissaving process, since the agents receive a wage at each period. Money holdings are

returned to zero each time the agent goes to the bank to rebuild her precautionary

savings in bonds. Yet, note that every time a negative shock occurs, all the money is

spent and the agent restarts accumulating from zero to widen the spells between the

trip to the bank. This is due to the strong shocks experienced that always constrain

the agent to liquidate her bond holdings. Another interesting point is related to

the dependence of the amount of money held just before the trip to the bank. The

20

high earnings: money at t+1

m − Axis

b −

Axi

s

0 0.5 10

1

2

3

4

5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

high earnings: bonds at t+1

m − Axis

b −

Axi

s

0 0.5 10

1

2

3

4

5

1

2

3

4

5

6

low earnings: money at t+1

m − Axis

b −

Axi

s

0 0.5 10

1

2

3

4

5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

low earnings: bonds at t+1

m − Axis

b −

Axi

s

0 0.5 1

1

2

3

4

5

1

2

3

4

5

Figure 5: Policy function. Trading region are those where bt+1 does not depends on mt.The transaction cost is δ = .01.

21

3.98 3.985 3.99 3.995 4 4.005 4.01 4.015 4.02

x 104

−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

wage

consumption

money

bonds

participation

Figure 6: Typical time series of wage, consumption, money holdings, bonds, and partici-pation in transaction indicator (high when transacting). The transaction cost is δ = .01.

22

richer (in term of bonds) the agent is, the smaller the amount of money and the

wider the spells. When the precautionary savings reach a certain level, agents do not

hold money anymore. The process of bonds accumulation is therefore stepwise. The

joint distribution of money and bonds re�ects it by displaying aggregate around some

anchor values in the money-bonds plane (Fig. 7).

money

0.0

0.5

1.0 bonds

0.0

0.5

1.0

density

0e+00

2e+07

4e+07

6e+07

Figure 7: Equilibrium joint distribution of money and bonds. In�ation value is 2% andtransaction cost is δ = 0.1

Distribution and inequality The correlations displayed in table 1 show that

individual consumption is positively correlated with wages and bonds holdings and is

in negatively correlated with money holdings which is really di�erent from what can

be found in MIU or CIA models where money and consumption are complement. As

can be seen from the Gini's coe�cients in table 3, consumption inequality is smaller

23

wages consumption money bonds

wages 1.00 0.60 0.09 0.29consumption - 1.00 −0.34 0.81

money - - 1.00 −0.64bonds - - - 1.00

Table 1: Correlations between wages, consumption, money and bonds holdings

cost cons. wage money bonds income both only m only b none trans.

0 0.946 0.940 0.000 2.998 0.946 0.000 0.000 0.998 0.002 0.8902.5e-4 0.946 0.940 0.007 2.972 0.946 0.267 0.000 0.731 0.002 0.5995e-4 0.946 0.940 0.020 2.973 0.946 0.417 0.000 0.581 0.002 0.417

7.5e-4 0.945 0.938 0.030 2.933 0.945 0.501 0.000 0.497 0.002 0.3622e-3 0.946 0.940 0.056 2.926 0.947 0.574 0.003 0.421 0.001 0.2404e-3 0.946 0.941 0.085 2.890 0.947 0.601 0.009 0.389 0.001 0.1866e-3 0.945 0.940 0.114 2.839 0.946 0.623 0.011 0.364 0.001 0.1618e-3 0.945 0.940 0.138 2.801 0.947 0.609 0.016 0.373 0.001 0.1411e-2 0.944 0.940 0.166 2.753 0.946 0.628 0.018 0.353 0.001 0.129

Table 2: Mean values of consumption, wage, money holdings, bonds holdings, income(w + r b) and proportion of agents holding both assets, only money, only bonds, none ofthem and proportion of period where bond transaction occurs.

than wage earnings. Assets are more inequally distributed, with money holdings more

inequally distributed than bonds. We thus recover at least qualitatively the large

picture described in the introduction: bonds and money have signi�cantly di�erent

Gini coe�cients.

Sensitivity to cost We now proceed to examine the impact of a varying transac-

tion cost on the distribution of asset holdings. First note that increasing transaction

costs increases time between trips to the bank according to the Baumol-Tobin model.

Moreover, as shown in table 2 the amount of bonds held decreases with increasing

transaction cost. Since a higher transaction cost also decreases the number of trans-

actions, agents hold more money and thus also use it as an insurance against adverse

shocks. Therefore, the amount of savings in bonds is reduced accordingly. The impact

of the transaction costs is essentially to produce some inequality in bonds holdings

(Tab. 3). Money holdings are very dispersed but this dispersion doesn't evolve much

as the transaction cost increases form 2.5× 10−4 to ×10−3. The Gini coe�cients for

24

bonds increases marginally with transaction costs.

consumption wage money bonds income savings

0 0.029 0.059 0.000 0.183 0.060 0.1832.5e-4 0.030 0.059 0.811 0.180 0.060 0.1805e-4 0.030 0.059 0.685 0.183 0.060 0.180

7.5e-4 0.029 0.060 0.629 0.181 0.061 0.1782e-3 0.029 0.059 0.607 0.183 0.060 0.1754e-3 0.030 0.058 0.607 0.188 0.059 0.1736e-3 0.030 0.059 0.597 0.193 0.060 0.1728e-3 0.031 0.059 0.607 0.198 0.059 0.1711e-2 0.031 0.059 0.601 0.203 0.060 0.171

Table 3: Gini coe�cients for consumption, wage, money holdings, bonds holdings, income(w + r b) and savings ((1 + r) b+M/π) for di�erent value of transaction cost.

Sensitivity to in�ation By providing incentive to hold less money, the main

e�ect of in�ation is to produce more transaction and more agents holding only bonds.

The average level of bond holdings do not change much, but their counterpart in

money is reduced by 20%, which is a huge elasticity. However, the amount of savings

is almost stable (slightly reduced at 10−4) since the slight increase in bond holdings

o�sets the decrease of money holdings in the amount of available savings for the next

period.

in�ation/cost cons. wage money bonds income both only m only b none trans.

1%/2e-3 0.946 0.940 0.070 2.913 0.947 0.598 0.005 0.396 0.001 0.2142%/2e-3 0.946 0.940 0.056 2.926 0.947 0.574 0.003 0.421 0.001 0.2401%/4e-3 0.946 0.940 0.105 2.870 0.947 0.630 0.010 0.359 0.001 0.1722%/4e-3 0.946 0.941 0.085 2.890 0.947 0.601 0.009 0.389 0.001 0.186

Table 4: As table 2 for di�erent values of in�ation.

The resulting impact on dispersion is as expected: dispersion in money is decreased

since agents are encouraged to shift their savings toward bonds, and the latter in-

creases, since the newcomers would holds only small amounts in bonds. These results

have to be considered with caution since the seignoriage revenue are not redistributed

to households.

25

in�ation/cost consumption wage money bonds income savings

1%/2e-3 0.029 0.059 0.591 0.186 0.060 0.1752%/2e-3 0.029 0.059 0.607 0.183 0.060 0.1751%/4e-3 0.030 0.059 0.585 0.193 0.060 0.1752%/4e-3 0.030 0.058 0.607 0.188 0.059 0.173

Table 5: Gini coe�cients for consumption, wage, money holdings, bonds holdings andincome for di�erent value of in�ation.

4 Conclusion

Idiosyncratic risk provides a strong stimulus for precautionary savings. Asset deten-

tion acts as insurance against future adverse shocks. However, the role of �at money

is not clear when another dominating asset, like bonds, is present. Following the

tradition of heterogeneous agents model, we adopted a framework where agents are

subject to some uninsurable idiosyncratic risk. The technology to store value from

one period to another is nevertheless di�erent from what is provided in this literature.

We adopted the approach of Baumol (1952) and Tobin (1956) and introduced a �xed

transaction cost to save in interest bearings assets as government bonds.

We recovered the classical sawtooth pattern of money retention typical of Baumol-

Tobin model conjugated with the asset accumulation precautionary behaviour of het-

erogeneous agents models. Even with low values of wage dispersion, money holdings

show strong dispersion with Gini in the range 0.6�0.8 without altering the large, but

smaller, dispersion, in interest bearing assets.

Our tentative model undergoes a number of shortcomings that should be addressed

seriously before confrontation to data:

• The wage-earning process yields too little dispersion to be realistic. Extending

our model to a much richer earning process by adding more contingent states is

a �rst step. Heathcote (2005) used a three states model to study �scal policies

in a heterogeneous agents model that we could rely on. His calibration better

represents the distribution of poor households and yields stronger Gini coe�cient

for wealth. Yet, the time period is a year and might be too long for a Baumol-

Tobin setup like ours.

• As stated previously, we also need to redistribute the seignoriage revenue to

households. How it should be done in our setup is not completely clear. We

could assume a lump sum redistribution, but whether this redistribution should

occur in �at money or government bonds is still an open question. The precise

26

speci�cation of the model could have huge welfare implications.

• Finally, the rate of return of the interest bearing assets should be endogenously

set to its equilibrium value.

These corrections are de�nitely needed in order to obtain a better calibration

to use our framework to address its macrodynamics implications in terms of output

response to in�ation and welfare. Potential applications include:

• Evaluation of key variables, such as the in�ation elasticity of aggregate money

demand and real interest rate sensitivity.

• Estimation of the cost of in�ation, the existence of an optimal in�ation target

in a utilitarian or Rawlsian framework.

• The implication for �scal studies.

• The output in�ation trade-o� in the long run if any along the lines of Algan

and Ragot who studied it in a MIU framework. How strong is the Mundell

(1963)�Tobin (1965) e�ect in a Baumol-Tobin economy ?

• The adjustment to an in�ation shock dynamics in presence of idosyncratic risk,

borrowing constraint have been studied by Algan et al. (2006) and show a slug-

gish macrodynamic response. The question if whether the result is robust to

the introduction of an interest bearing asset.

27

A Model setup

Consider an agent initially endowed with w0, who lives 3 periods, who do not consume

in its initial period and whose utility is therefore:

V = u(c1) + β u(c2), u(c) =c1−γ

1− γ(11)

During the three periods, he faces the following evolution of consumption:

m1 + b1χ0 = w0− δχ0, (12)

b1(1− χ0) = 0, (13)

(b2 −Rb1)χ1 +m2 + c1 = m1/Π− δχ1, (14)

(b2 −Rb1)(1− χ1) = 0, (15)

c2 = Rb2χ2 +m2/Π. (16)

where χ is a boolean variable indicating the decision to enter the bonds market (at

a �xed cost δ) which value increase at the rate R. In the following we assume that

bonds always dominates money, that is RΠ > 1.

There are theoretically eight possible values of the triplets (χ0, χ1, χ2). The ones

with only one nonzero χ are clearly dominated. We are then left with 5 possible

sequences: 000, 011, 101, 110, 111.

B Using only money

b! = b2 = 0, (17a)

m1 = w0, (17b)

m2 =w0 β

1α

β1α Π + Π

1α

, (17c)

c1 =w0

Π + β1α Π2− 1

α

, (17d)

c2 =w0 β

1α

Π(β

1α Π + Π

1α

) , (17e)

V000 =w0

1−α Π−2+α(β

1α Π + Π

1α

)α1− α

(17f)

No conditions on wealth.

28

C Using always bonds

b1 = w0 − δ, (18a)

b2 =(Rβ)

1α (Rw0 − (1 +R) δ)

R+ (Rβ)1α

, (18b)

m1 = m2 = 0, (18c)

c1 =R (Rw0 − (1 +R) δ)

R+ (Rβ)1α

, (18d)

c2 =R (Rβ)

1α (Rw0 − (1 +R) δ)

R+ (Rβ)1α

, (18e)

V111 =

(1 + β

1αR

1α−1)α

(Rw0 − (1 +R) δ)1−α

1− α(18f)

Condition:w0

δ>

1 +R

R. (19)

D Using bond in the �rst two periods

Necessarily m1 = b2 = 0.

b1 = w0 − δ, (20a)

b2 = 0, (20b)

m1 = 0, (20c)

m2 =β

1α (Rw0 − (1 +R) δ)

Π(β

1α Π + Π

1α

) , (20d)

c1 =(Rw0 − (1 +R) δ) Π

1α

β1α Π + Π

1α

, (20e)

c2 =β

1α (Rw0 − (1 +R) δ)

β1α Π + Π

1α

, (20f)

V110 =(Rw0− (1 +R) δ)1−α

(β

1αΠ1− 1

α + 1)α

1− α(20g)

Condition:w0

δ>

1 +R

R. (20h)

29

E Using bond in the last two periods

Necessarily b1 = m2 = 0.

b1 = 0, (21a)

b2 =(Rβ)

1α (w0 − δΠ)(

R+ (Rβ)1α

)Π, (21b)

m1 = w0, (21c)

m2 = 0, (21d)

c1 =R (w0 − δΠ)(R+ (Rβ)

1α

)Π, (21e)

c2 =R (Rβ)

1α (w0 − δΠ)(

R+ (Rβ)1α

)Π

, (21f)

V011 =

(R+(Rβ)

1α

R

)α (−δ + w0

Π

)1−α1− α

(21g)

Condition:w0

δ> Π. (22)

30

F Using bonds in the �rst and last period

It can be proven that m2 should be zero (the �nal utility decreases with m2).

b1 =R

2α (w0− δ) (βΠ)

1α

R2 Π +R2α (βΠ)

1α

, (23a)

b2 =R1+ 2

α (w0− δ) (βΠ)1α

R2 Π +R2α (βΠ)

1α

, (23b)

m1 =R2 (w0 − δ) Π

R2 Π +R2α (βΠ)

1α

, (23c)

m2 = 0, (23d)

c1 =R2 (w0 − δ)

R2 Π +R2α (βΠ)

1α

, (23e)

c2 →w0 − δ

R−2 + Π

R2α (βΠ)

1α

, (23f)

V101 =(w0 − δ)1−α

(R2 Π +R

2α (βΠ)

1α

)αR2α Π (1− α)

(23g)

Condition:w0

δ> 1 (23h)

G Comparison

Let us �rst compare the ones with the more stringent constraint (assuming R + 1 >

RΠ), i.e. V111 and V110. One can check that if ΠR > 1 then V111 > V110.

Further comparisons yield:

w0

δ>

1 +R−

(“R2 Π+R

2α (βΠ)

1α

”α“R“R+(Rβ)

1α

””αΠ

) 11−α

R−

(“R2 Π+R

2α (βΠ)

1α

”α“R“R+(Rβ)

1α

””αΠ

) 11−α

=⇒ V111 > V101 (24)

w0

δ>

RΠRΠ− 1

=⇒ V111 > V011 (25)

31

w0δ>

1 +R

R−

(Π−2+α

(R“β

1α Π+Π

1α

”R+(Rβ)

1α

)α) 11−α

=⇒ V111 > V000 (26)

w0 > δ +δ (−1 + Π)

1−

0@ R+(Rβ)1α

R

„R2 Π+R

2α (βΠ)

1α

«1A α−1+α

R2α

} =⇒ V101 > V011, (27)

w0 >δ

1−

R2α

0@ Π

„β

1α Π+Π

1α

«R2 Π+R

2α (βΠ)

1α

1AαΠ

1

1−α=⇒ V101 > V000, (28)

w0 >δΠ

1−Π

(Π−2+α

(R“β

1α Π+Π

1α

”R+(Rβ)

1α

)α) 11−α

=⇒ V011 > V000 (29)

32

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