A Local Currency in a Dollarized Economy

doi:10.4236/me.2012.35086

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Modern Economy, 2012, 3, 671-674

http://dx.doi.org/10.4236/me.2012.35086 Published Online September 2012 (http://www.SciRP.org/journal/me)

A Local Currency in a Dollarized Economy

Sokchea Lim1,2

1Department of Economics, Southern Illinois University Carbondale, Carbondale, USA

2Cambodian Institute for Cooperation and Peace, Phnom Penh, Cambodia

Email: [email protected]

Received July 3, 2012; revised August 2, 2012; accepted August 10, 2012

ABSTRACT

The paper uses a dual-currency search theoretic approach to demonstrate that it is possible to induce the acceptance of a

local currency in a dollarized economy. In the model, we assume that a foreign currency is in full circulation and the

government policy tool is the convertibility of the local currency to the foreign currency. We show that the economy

can achieve equilibria where two monies are in circulation if the government can raise a sufficiently high probability of

exchange between the two currencies.

Keywords: Dual-Currency Search Theoretic Approach; Dollarized Economy

1. Introduction

Curtis and Waller [1] explore the coexistence of two cur-

rencies, domestic as a legal tender and foreign as an ille-

gal one. Assume that the legal currency is always accepted,

the government uses several enforcement policies to drive

the illegal currency out of circulation. Dutu [2] constructs

a one-country search-theoretic model of two monies with

different rates of return and bargaining powers. The results

show that the weakening of the domestic currency’s bar-

gaining power can invite strong foreign currency while

its lower rate of return drives out the foreign one. Lotz

and Recheteau [3] use the search-theoretic approach to

determine how the government can replace an old currency

with a new one. Different from the above papers, we in-

troduce the convertibility between the two monies in the

model. We want to show that the government can induce

the acceptance of a local currency in an economy in which

a foreign currency is fully accepted.

The purpose of the paper is to determine under which

conditions a local currency can be accepted in a dollar-

ized economy where a foreign currency is in full circula-

tion. The policy tool of the government is to link the lo-

cal currency to the fully established foreign currency by

easing the convertibility between the two. Using search-

theoretical approach of money, we show that the econ-

omy can achieve equilibria where two monies are in cir-

culation if the government can raise a sufficiently high

probability of exchange between the two currencies.

2. The Simple Model

The model is related to the dual-currency search-theoretic

model of Kiyotaki and Wright [4]. In this model, the dou-

ble coincidence of wants between a given pair of agents

is ruled out; hence, there exist two fiat monies which play

an important role as media of exchange. The economy is

populated by a continuum of infinitely-lived agents with

total population normalized to unity. Time is continuous.

There are also a large number of consumption goods which

are indivisible and non-storable, and they come in units

of size one. Each agent is specialized in both consump-

tion and production. All agents are assumed to be homoge-

nous in their preferences. An agent gets zero utility (U =

0) from consuming her own production good or not con-

suming any goods and she gets a positive utility (U > 0)

from consuming a good that she likes. A type j consumer

consumes good j and produces good j + 1 (modulo N), for

j = 1, 2, ···, N and N 3. So, the probability that, for a

given agent, a would-be trading partner can produce her

desired good is x = 1/N (a single coincidence of wants).

The assumption is essential to rule out the double coin-

cidence of wants. For simplicity, it is assumed that there

is no labor cost of production of goods. Once consuming,

an agent can enter the production process and produce an-

other unit of consumption goods to trade in the next period.

There are two currencies in the model, local and for-

eign currencies which are denoted by subscripts of L and

F, respectively. Both monies are indivisible, costlessly stor-

able and cannot be produced by any private agents. An

agent stores only one unit of one type of money at a time.

At the beginning of period 0, a fraction of M agents is

chosen at random and endowed with one unit of money

each. Since there are two currencies in this economy, mL

denotes a fraction of private agents who are endowed with

S. LIM

672

a local currency and mF denotes a fraction of private agents

who are endowed with a foreign currency. So it follows

that M = mL + mF. mG is the fraction of agents who are

goods producers. Hence, at any point in time, the total po-

pulation can be written as:

Lt Ft

mmm



max

max π

GL LG

Gt (1)

A key feature of the model is that the government has

a policy tool in the exchange market. There is a fraction

of money changers which are set up by the government

in the market so that private agents can exchange their

currency from one to another. We can think of this as the

government establishes exchange booths throughout the

economy or the government licenses exchange businesses

which will operate in a perfectly competitive market. We

assume that the exchange spread between the two curren-

cies is zero1. That is, the two currencies are exchanged

one for one with no extra cost incurred by the private agents.



F is the fraction of money changers who hold foreign

currency and exchange for local currency.



L is the faction

of local currency changers. Once exchange occurs, a for-

eign currency changer becomes a local one and, vice versa.

We also can think of these fractions as the probabilities

that a private agent can get her currency converted so 0 



i  1 where i = L, F. Assume that money changers do not

enter the goods market, so they do not trade with the goods

holders.

History of each agent is private information. Informa-

tion that a good holder values a local or foreign currency

is private to the money holders; therefore, currency ex-

changes do not take place before the matching process.

On the other hand, because of the availability of instant

exchange with no cost in the market, it is deemed unnec-

essary for a money holder to exchange his currency be-

fore the matching process. The matching process follows

Poisson with constant arrival rate of α.

The acceptability of currency is determined endogenou-

sly. Π where 0  Π  1 is probability that a random goods

holder accept the foreign currency in trade and π repre-

sents the best response of individual goods holder to an

offer of the foreign currency. Φ where 0  Φ  1 is prob-

ability that a random goods holder accepts the local cur-

rency in trade and



represent the best response of indi-

vidual goods holder to an offer of the local currency. Given

symmetry, the Nash equilibrium is π = Π and



= Φ.

The analysis focused here is restricted to steady-state

equilibria where strategies and all aggregate variables are

constant over time. In such equilibria, all agents are hold-

ing either one unit of money or nothing (i.e. being able to

produce one unit of consumption goods), at the end of each

period. Let Vi where i = G, L, F be the value functions for

a goods holder, a local currency holder and a foreign cur-

rency holder, respectively, and they measure the expected

present discounted value of utility from trading in the fu-

ture, given the current trading position. Thus, the Bellman’s

equations specifying the steady-state returns to searching

are:

rVxmV V

mVV











(2)







LG GL

GF GL

rVxmU VV

xm ΠUV V









(3)







FG GF

GLG F

rVxm ΠUV V

mUVV









(4)

where r is the real interest rate.

Equation (2), the expected flow return to a goods

holder is equal to the sum of the surpluses (payoffs) from

trade with a local currency holder or with a foreign cur-

rency holder. αxmi, i = L, F, is the probability that the

goods holder is matched with a money holder holding

local or foreign currency and is able to provide her with

the desired goods. The former chooses



and π optimally

to maximize the payoff (VL – VG) or (VF – VG) gained

from trade with the latter.

Equations (3) and (4) have a similar interpretation. The

right hand sides denote the sum of surpluses from trade

with a goods holder who values either local or foreign cur-

rency. Note that the money holder has no information about

which currency is valued by the good holder before the

meeting. The first term is the payoff from trading with a

goods holder who values the currency she holds and swit-

ching from a money holder to a goods holder. The sec-

ond term is the payoff from trading with a goods holder

who values a different currency than the one held by a mon-

ey holder; hence, the currency needs to be exchanged with

probability



i in the same period and then traded. The

payoff (U + VG – Vi where i = L, F) is the sum of the util-

ity gained from consuming the desired goods and surplus

from switching her position from a money holder to a goods

holder.

According to the set of Equations from (2) to (4), the

acceptability of both currencies will depend on the com-

parison of value functions of monies (VL and VF) with the

value function of goods (VG). If a representative goods

holder expects a positive payoff from becoming a local

currency holder in the next period, VL > VG, her best re-

sponse is to choose to completely accept the local cur-

rency,



= 1. If the negative payoff is expected, VL < VG,

then



= 0, i.e. the representative goods holder chooses

not to accept the local currency. And, if the representa-

tive goods holder is indifferent between holding the local

1On Nov. 19, 2010, the buy-sell spread between Cambodian riel and US

dollar is 7 basis points which equivalent to 0.0017 dollar. Also, men-

tioned in Craig and Waller [5], the spread between Ukrainian hryvna

and US dollar was 0.5 cent per dollar exchanged.

S. LIM

673

Ψα

currency and remaining as a goods holder in the next pe-

riod, VL = VG, she will choose



=  where 0 <  < 1, i.e.

the currency is partially accepted. The explanation is the

same for foreign currency acceptability.

Now, there are potentially nine types of steady-state

equilibria. Each type is characterized by a pair of the best

responses of an individual goods holder to both local and

foreign currencies. Confine to the purpose of this paper,

we only look at the equilibria in which the foreign cur-

rency is completely accepted by the goods holder (π =



= 1) which requires that VF > VG while the local currency

acceptability may fall in either of three cases. Solving the

system of Equations (2) to (4) gives the reduced-form

payoffs:

mU (5)

xm αxm ΓU

rΨα

rV rαxm



 (6)

xm αxm ΞU

ΦF

ΓμΠ

ΞΠ

rΨα

rV rαxm



 (7)

where

(8)



Πμ Ψ

(9)

Using Equations (5) and (7) and setting VF  VG, the

sufficient condition for having the foreign currency ac-

cepted in the economy is to have2:

(11)

When the goods holder meets with a foreign currency

holder, she must decide whether to accept the currency or

wait until she meets with a local currency holder and trade.

From Equation (11), the left hand side is the acceptability

of the foreign currency plus the probability that it can be

converted into the acceptable local currency. The right hand

side is the probability of waiting to meet with a local cur-

rency holder and trade; then, acquiring the desired goods,

or the probability of hoping to consume a desired good in

the next period and not accepting the foreign currency

now. The intuition here is that the goods holder accepts

the foreign currency if she expects that it will help her

acquire her desired goods faster. And, if everyone com-

pletely accepts the foreign currency, exchanging it for the

local one may not be needed at all (



L = 0).Then, it fol-

lows that  < 1.

Using (5) and (6), we can derive the sufficient condi-

tions for the acceptability of the local currency (see Ap-

pendix for derivation). The three equilibria to be discussed

in this paper are given in Table 1. In column (1), we pro-

vide the necessary conditions for the acceptability of a local

currency and column (2) presents the symmetric Nash equi-

libria and the sufficient conditions are detailed in column

(3). Please note that the explanation of column (1) and (2)

is provided above and we will discuss the last column and

government policy tool to move from an equilibrium where

the local currency is not accepted to the one where it is

partially or completely accepted in the economy. Under

the sufficient conditions, the right side is the same term

as interpreted in the condition for foreign currency ac-

ceptability and determined to be less than one ( < 1).

The left hand side is the probability that a goods holder

can obtain her desired goods after trading with a local cur-

rency holder. The three conditions state that whether or

not the local currency can be accepted depends on the ag-

gregate acceptability and its convertibility into a widely

accepted currency, the foreign currency.

From Table 1, it is clear from the sufficient conditions

that the government can raise the probability of exchange

of a local currency into a widely accepted foreign currency

to a sufficient level such that the local currency will be

accepted in the economy. From the mixed strategy equi-

librium (partial acceptability of local currency), the equi-

librium value of the aggregate acceptability is associated

with the level of convertibility of the local currency. The

association between the two variables is important in de-

termining the optimal level of convertibility that could put

the local currency in circulation. If the association is very

elastic, then we may see a low optimal value of



L; oth-

erwise, a high value. The intuition here is that if people

have little trust in the value of the local currency, the gov-

ernment has to pour in big amount of a trusted foreign

currency to ensure that every unit of the local currency is

backed by every unit of a trusted value, hence raising the

liquidity. Finally, the government can also move to the

best optimal equilibrium where the local is fully accepted

in the economy if



L gets so large that a representative

agent can get her desired good by using either currency.

Table 1. Characterization of symmetric nash equilibria.

Necessary

Conditions

(1)

Symmetric Nash

Equilibria

(2)

Sufficient

Conditions

(3)

VL > VG ϕ = Φ = 1  +





> 

VL = VG 0 < ϕ = Φ < 1  +





= 

VL < VG ϕ = Φ = 0  +





< 









 

LGF G

GGFLG

αxm Γrαxm Ξαxm ΠΞ rαxm Γ

rαxm Γrαxm Ξαxm Παxmr αxm

 



(10)

2The derivation is provided in the appendix.

S. LIM

674

Therefore, in this economy, the convertibility from a local

currency to a foreign currency (



L) is an important policy

tool to move from a dollarized economy where foreign

currencies are fully accepted to an economy with both local

and foreign currencies.

3. Conclusions

The paper examines a fully-dollarized economy where the

government tries to introduce a local currency. The model

is a dual-currency theoretic approach in which one cur-

rency (foreign currency) is in full circulation. The policy

instrument of the government is the probability of exchange

between the local and foreign currencies. The result is that

the economy can achieve equilibria where two monies are

in circulation if the probability of exchange between the

two currencies is sufficiently high.

The paper does not examine the case where the gov-

ernment is in short supply of foreign currencies. This would

constrain the ability of the government to raise a suffi-

cient high value of



L. Hence, the government may not

be able to achieve the equilibria with two monies in cir-

culation. However, the game theory approach such as cred-

ible announcement by the government to back the local

currency can be introduced and the interaction between the

government and the representative agents can be analyzed.

This would go beyond the extent of this paper. Moreover,

the welfare analysis is not done in this paper.

REFERENCES

[1] E. S. Curtis and C. Waller, “A Search-Theoretic Model of

Legal and Illegal Currency,” Journal of Monetary Eco-

nomics, Vol. 45, 2000, pp. 155-184.

doi:10.1016/S0304-3932(99)00042-2

[2] R. Dutu, “Strong and Weak Currencies in a Search-

Theoretic Model of Money,” Research Paper, 2001.

[3] S. Lotz and G. Rocheteau, “On the Launching of a New

Currency,” Journal of Money, Credit and Banking, Vol.

34, No. 3, 2002, pp. 563-588.

doi:10.1353/mcb.2002.0003

[4] N. Kiyotaki and R. Wright, “A Search-Theoretic Ap-

proach to Monetary Economics,” American Economic

Review, Vol. 83, No. 1, 1993, pp. 63-77.

[5] B. Craig and C. Waller, “Dual Currency Economies as

Multiple Payment Systems,” Economic Review, Federal

Reserve Bank of Cleveland, Cleveland, 2000.

Appendix

Subtracting Equation (5) from (7) yields:



FGG L

rΨαxm

rVrVxm ΠμU

rxmΠμ

Ψαxm U





 







(A1)

Simplifying the equation further gives:



ΦΨ

FG L

αxm U

VV Πμ

rxmΠμ



 



ΦΨ

Πμ

(A2)

VF  VG requires that:







(A3)

Similarly, from Equations (5) and (6) we get,



ΦΨ

LG F

αxm U

VV μΠ

rxmμΠ







ΦΨ

μΠ

(A4)

VL  VG requires that:

(A5)



