DBJ Discussion Paper Series, No.1201

A microeconomic foundation for optimum currency areas:

The case for perfect capital mobility and immobile labor forces

Masayuki Otaki(Institute of Social Science, University of Tokyo)

May

2012

Discussion Papers are a series of preliminary materials in their draft form. No quotations,reproductions or circulations should be made without the written consent of the authors inorder to protect the tentative characters of these papers. Any opinions, findings, conclusions
or recommendations expressed in these papers are those of the authors and do not reflect
the views of the Institute.

A microeconomic foundation for optimum currency areas:
The case for perfect capital mobility and immobile labor forces
Masayuki Otaki
Institute of Social Science, University of Tokyo
ohtaki@iss.u-tokyo.ac.jp
JEL classifications: E42, E44, F33, F51
Keywords: Optimum currency area, Capital without a nationality, Non-cooperative
game between central banks, Disparity in income distribution, Inefficiency of resource
allocation
Abstract
This article provides a microeconomic foundation for Mundell’s (1961) optimum currency area
theory. We consider twin countries where labor forces are fixed to each country although the realcapital moves internationally. When the central bank in each country behaves non-cooperatively,
it will raise the domestic interest rate to attract more real capital and increase the rent of her
residences. However, the fierce competition between the central banks ultimately exacerbates
the disparity in income distribution. Moreover, when the real capital does not have a nationality,
the worsened income distribution also results in the inefficient resource allocation. Thus, such
twin countries should unify their central banks and coordinate their monetary and interest policies.
In other words, these countries constitute an optimum currency area.

1. Introduction
Income disparity is not limited to developing countries. Advanced economies also face
this growing problem. This article considers why such an undesirable economic
consequence is invoked when constructing a microeconomic foundation for the optimum
currency area theory originating from Mundell’s (1961) seminal work.
As Mundell (1961) emphasizes, the mobility of production resources plays a crucial role
when we consider which economies should constitute an optimum currency area. We
deal with the case in which both labor forces are immobile and have a nationality but
real capital with no nationality can move internationally at the owners’ discretion. Sucha setting is plausible when we observe that foreign direct investment is generally
preferable to certifying work visas.
When small twin countries with identical economic structure are in this situation,
their central banks compete to invite more real capitals to enrich their economy as longas the countries attain full employment. Nonetheless, such competition has thefollowing devastating consequence.
If one central bank pursues a high-interest policy to attract more real capital, the othercentral bank counteroffers with a higher interest rate. Such a cumulative process does
not cease until a surplus from working that is the benefits of the high interest policy
vanish entirely.
Consequently, the non-cooperative behavior of two central banks brings about a serious
income disparity between capital and labor. Furthermore, since capitals are assumed tohave no nationality, the emerging disparity also results in the large welfare losses forthese two nations.
The unification of two central banks is desirable for overcoming such a difficulty. The
small twin countries should be at least economically integrated. Then, the same amountof money is supplied to ensure full employment and the interest rates offered to
non-nationality real capitals become identical.
Accordingly, each country is supplied with an amount of real capital, and thus, theincome disparity and inefficient resource allocation within the nations are entirely
resolved. That is, the small twin countries constitute an optimum currency area
whenever the real capital mobility is complete.
The remainder of this paper is structured as follows. In Section 2, we construct a small
twin country model based on Otaki (2009). In Section 3, we compare the non-cooperativeand cooperative monetary policies, and prove the inevitability of optimum currency
areas. In Section 4, we provide brief concluding remarks.

12. The Model
2.1 Structure of the Model
We use a two-period overlapping generation model in a production economy withmoney. The world consists of twin countries A and B, whose economic structures are
identical, and the rest of the world. Each country has residents who cannot moveelsewhere and who live in two periods with the density [0, 1].
Each resident specializes in producing one differentiated goods with the help of realcapitals when he/she is young. Real capital, whose owners have no nationality, exists
with the density [0,1]  [0, 2] . Hence, each resident can potentially deploy the realcapitals with the density [0,1] . Capital income is also earned when the owner is young,and then capital itself is passed to a descendant. Once an owner determines the locationof his/her capital, he/she lives in that country even after his retirement. The minimum
rate of return from the rest of the world ( r ) is guaranteed to all capital owners.
Furthermore, for simplicity, a unit real capital combined with a resident’s businessskills produces a unit good.
The income distribution between residents and capital owners is determined by anegotiation. The negotiation process, which was developed by Otaki (2009), is assumed
to be the following two-stage game. First, a resident determines how much capital to
deploy in order to maximize his/her income from business skills. Second, given thevolume of capital deployed and goods produced, the residents and owners mutually
determine the income distribution in accordance with the asymmetric Nash bargaining
solution.
In addition, there is a central bank in each country, which pursues the social welfare ofher residents. Each central bank’s policy variables are the nominal money supply andthe real interest rate (i.e., the rate of return for capital). We assume that a central bankmanipulates the real interest rate by intervening in the negotiation process between her
residents and the non-nationality capital owners. That is, a central bank can control thebargaining power of her residents through moral suasion.
2.2 Construction of the Model
2.2.1 Individual’s Utility and Consumption Functions
We assume that all individuals (including capital owners) have the same concave andlinear homogenous lifetime function:
1
1 1U  u (c1, c2 ), ci  [  [ci ( z )]1
01 1dz ],where ci ( z ) is the consumption of good z during the i -th stage of life. We can derive2the following corresponding indirect utility function IU :

IU 

Yt
yt
Y
p

, yt  t ,   t 1 ,
 ( pt , pt 1 )  (1,  )
pt
pt

(1)

where Yt is the nominal income and pt is the price index defined by
1

1
1

pt  [  [ pt ( z )] dz ]10.

Furthermore, the consumption function of the younger generation C isC  c(  ) yt .

(2)

Finally, the demand function for good z isD( z )  [where y2.2.2

dp ( z )  d
] y ,
p

(3)

is the aggregate demand.

The Production Process by the Two-Stage GameTo develop the aforementioned production process, we consider the following two-stage
game.
I.

Each resident maximizes his/her income from business skills by deploying
non-nationality capitals.

II.

The resident and capital owners negotiate their income distribution inaccordance with the asymmetric Nash bargaining solution the threaten
point of which is [0, r ] .
*We must solve this problem by backward induction. In the second stage game, thecorresponding generalized Nash product G ( z , i ) isGP( z , i )  [ p( z )  r i ]  i [r i  r * ]1  i ,(4)where  is the genuine bargaining power of a resident and    i is the modified(actual) power of i country i ' s central bank; that is,  i denotes the power of moral
suasion of i country’s central bank. r is the domestic rate of return for a unit capital.
i

The shape of the product is derived from two properties of the model. First, theobjective function is linear on the real income as indicated by (1). Second, the production(or demand) volume is already determined by the first stage of the game.
Maximizing (4) with respect to ri

, we obtain the equilibrium domestic rate of returni*r :r i*  [1     i ] p ( z )  [   i ]r .

(5)Taking (5) into consideration, the maximization problem of the first stage can be
expressed as3 i ( z )  max p ( z ) [ p( z )  r i* ]D( z )  [   i ]max p ( z ) [ p( z )  r ]D( z ) .

(6).

The solution to (6) is

p* ( z ) 

r
, z.
1   1

(7)

Hence, the price level p is constant over time and the equilibrium inflation rate is *  1 . Substituting (3) and (7) into (6), we obtain
 i ( z) [   i ] 1 y di ,p*(8)where y denotes the real GDP of country i . From (1), it is clear that (8) corresponds todithe social welfare of residents in country i .
2.2.3

The Market Equilibrium

In both countries, money is supplied through the unexpected transfer to the older
generation; thus, taking (2) into consideration, the equilibrium condition for thedomestic aggregate goods market becomesmi,y  c(1) y  m  y 
1  c(1)
di

di

i

(9)

di

where m i denotes the real money supply within country i . The second term of (9)corresponds to the aggregate expenditure of the older generation.
Owing to the perfect mobility, the real capital market achieves equilibrium when2,k i = 1,
0,
if  i >  jif  i =  jif  i <  j(10)where k i is the amount of capital that has been invested in country i .
The model contains five types of endogenous variable ( r , p*,  , y , k ) , two types of
i*

i

di

i

exogenous variable ( , m ) , and five structural equations (5), (7), (8), (9), and (10).
ii

Thus, the model is closed.
2.3 The Non-Cooperative Game between Central Banks and the Disparity in Income
Distribution
Because of the international mobility of real capitals and the representation of social
welfare (8), each central bank is eager to attract more capital and enrich its country.
Such competition is described by the following two-stage game. In the first stage,
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central banks determine ( ,  ) . Next, they decide how much money they supply (i.e.,A

B(m A , m B ) ).
To solve the equilibrium of this game, we must begin with the second stage. Since theoutcomes of the first stage are summarized by (10), taking the social welfare (8) and theequilibrium condition for each aggregated goods market (9) into consideration, the best
response of each central bank is to maintain the full-employment equilibrium, which isdefined as the amount of real capital that is associated with its country. Hence, thefollowing dominant strategy in this game corresponds to the result of the first-stage
game. That is,if  i >  j2,im = 1,0,

if  i =  j .
if  i <  j(11)Since full-employment is assured in the second stage, central banks strive to invite asmuch real capital as possible. The following theorem holds concerning the uniqueness of
the Nash equilibrium:
Theorem 1.

The unique Nash equilibrium is characterized by( m i* ,  i* ,

 i*
p*

)  (1,  , 0).

(12)

Proof.
<Sufficiency> If (12) is satisfied, there is no active incentive to diverge the strategies
because no additional gain is obtained by lesser ( m ,  ) . Hence, (12) is a Nashi*i*equilibrium.
<Necessity> Suppose that

By selecting a 

j **

 i*
p*

is strictly positive in some Nash equilibrium. Then,

 i*

 j*

p

*


*

p

  i*   j * .

slightly larger than  i* , country j improves her social welfare as

much as[   j** ] 1  0.
Thus, there is an incentive to diverge from the equilibrium. This is a contradiction.
The economic implication of Theorem 1 is quite serious. As long as two central banks
extend the non-cooperative game to attract more capital, the income disparity deepens
5against their intentions. Owing to the competition, residents earnings from businessskills are utterly absorbed by the capital income. Since, as seen in (8), the social welfareof residents is proportional to their income, the deepening income disparity also results
in a less efficient economy.
Such a phenomenon is prominent in East Asia, for example. In this area, foreign
direct investments flow mainly from Japan and China to other countries. Although thecapital accumulation sufficiently advances and a limited number of capitalists
surprisingly become rich, the labor income of most residents including those in Japanand China stagnates. Income disparity is one of the most urgent problems in East Asia.
３． The Optimum Currency Area as the Unification of Central Banks
In the previous section, we showed that the non-cooperative actions of central banks
have quite harmful effects on both countries. In this sense, we propose the unification ofcentral banks. According to Mundell (1961), a currency area is defined as follows:``A single currency implies a single central bank (with note-issuing power) andtherefore a potentially elastic supply of interregional means of payment.” (p. 568)
We adopt this definition of a currency area.
When central banks are unified and a currency area is formed, the twin countries, A
and B, can be treated as a single country, and hence, monetary coordination becomes
possible. Because of the symmetry of the countries, the optimal coordination policy isalso symmetric. Hence the two-step game extended in 2.3 requires equal allocation ofreal capital. Thus, we obtain m A  m B  1 and  A   B  0. It is evident that the social
welfare of each country (8) becomes  . This is the maximal value that each country
1attains. In this sense, these twin countries together constitute an optimum currencyarea.
４． Concluding

Remarks

We reconsider the theory of optimum currency area from the perspective of resource
allocation and income distribution. The obtained results are as follows.
We concentrate on the case of twin countries under perfect capital mobility andimmobile labor forces. This assumption seems natural if we consider the significance ofthe existence of nation states.
When each central bank pursues its national interests, that is, the social welfare of its
immobile labor force (i.e., the residents), dire economic consequences emerge. Each6central bank is led to adopt an artificial high interest policy because more capitalinduced by a rate slightly higher than the rates of its rival central bank brings about
higher incomes for the business skills possessed by the residents of that central bank’s
nation. However, such competitive and escalating interest-raising is devastating andcumulative, and it does not end until all the residents’ income is absorbed by the realcapital without nationality. Thus, a serious income disparity and a large decrease insocial welfare occur. It is clear that the nation is not an optimum currency area.
When the two central banks are unified and the monetary coordination becomes
possible, such catastrophic competition ceases. Real capital is allocated equally byabolishing the competitive and artificial high-interest policy. Just enough externalmoney is supplied to ensure the full-employment equilibrium in each country. Thus, thesocial welfare achieves its maximum. In other words, our twin countries under perfectcapital mobility constitute an optimum currency area. It is also noteworthy that ourapproach is based on a rigorous dynamic microeconomic foundation. In this sense, we
succeed in updating and extending the theory of Mundell (1961).
Finally, we must note some limitation of our work. The first is the difficulty of central
bank unification. Bureaucrats who operate the unified central bank may be of different
nationalities. Differences due to culture, ethnicity, tradition, and etc., are not as easy to
overcome as our theory assumes. It takes more time than we expect to ensure fare policy
coordination.
The second concerns the glut of foreign direct investment. In reality, the volume andrange of mobile real capital is large and wide. It is feared that the world as a whole may
become a unique optimum currency area. Nevertheless, it is certain that such atremendous and enlarged organization would never work well. It may be more practicalto place a levy on international capital movement, like the Tobin tax.
References[1] Mundell, R. A. (1961), ``A theory of optimum currency area,’’ American Economic
Review 51, pp. 657-665.
[2] Otaki, M. (2009). ``A welfare economics foundation for the full-employment policy,’’
Economics Letters 102, pp. 1-3.

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