Discussion Papers In Economics
And Business

A Planner of Global Income Transfers:
International Public Goods and Productivity Differentials

Tatsuyoshi Miyakoshi

Discussion Paper 08-38

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANA Planner of Global Income Transfers:
International Public Goods and Productivity Differentials

Tatsuyoshi Miyakoshi

Discussion Paper 08-38

December 2008

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANA Planner of Global Income Transfers:
International Public Goods and Productivity Differentials
byTatsuyoshi Miyakoshi
Osaka University
December 4, 2008
Abstract
The purpose of this paper, by introducing the adjustment expense of global income
transfers across N countries, is to produce an explicit rule for the planner country
regarding income transfers, and to investigate the effects of income transfers on eachcountry’s welfare. The findings are: (i) when country i has a productive advantage inproducing public goods, country i becomes an income receiver; (ii) specifying theparticular level of the adjustment expense for global income transfers, the planner can
decide the values of income transfers for all countries; (iii) even though any country can
become a planner of income transfers, all countries get the same utility level, while thelow adjustment expense under a particular planner country leads to a Pareto-improving
outcome; (iv) all conclusions are derived based on well-known information regarding thecost of producing public goods and income levels for all countries.
Keywords: international public goods; productivity differentials; planner; global income
transfer; adjustment expense; welfare
JEL classification: H41, F13

Earlier versions of this paper were presented at a seminar in the Helsinki School of
Economics, 2008; the East Asian Economic Association Conference in Manila, 2008;
conference on Dynamics, Economic Growth, and International Trade (DEGIT) in Manila,
2008; and a seminar at Osaka University’s Institute of Social Economic Research, 2008. I
am grateful to Juuso Välimäki, Pekka Ilmakunnas, Pertti Haaparanta, Matti Pohjola,Prince Christian Cruz, Bjarne Sloth Jensen, Shinsuke Ikeda, Charles Yuji Horioka andparticipants at the conferences for their helpful comments. The research of the author was
supported by a grant-in-aid from the 2007 Zengin Foundation for Studies on Economics
and Finance, and Grant-in-Aid 20530271 from the Ministry of Education, Culture, Sport,
Science and Technology of Japan.
Tatsuyoshi Miyakoshi, Prof. of Monetary Economics
Osaka School of International Public Policy, Osaka University
1-31, Machikaneyama-machi, Toyonaka, Osaka, 560-0043, Japan.
tel:+81-6-6850-5638; fax:+81-6-6850-5656
E-mail: miyakoshi@osipp.osaka-u.ac.jp1

1.

Introduction
Since Warr’s (1983) seminal work, a large body of literature has evolved on theneutrality result that the real equilibrium is unaffected by income transfers when public
goods are privately provided. This framework is generalized and reinforced by giving theproof of existence and uniqueness of the equilibrium by Bergstrom et al. (1986, 1992).
However, income transfers are not completely neutral in some cases.1 In one of thesecases, Ihori (1996) investigates the welfare effects of changes in productivity differentials
associated with the production of international public goods across countries.2 It is shownthat an income transfer from a country with low productivity (i.e., a high cost of
producing public goods) to a country with high productivity (i.e., a low cost) produces aPareto improvement. His analysis on income transfers used a two-country model.
After Ihori (1996), Boadway and Hayashi (1999), Ihori (1999), Caplan et al.
(2000), Itaya et al. (2002) and Kim and Shim (2006) investigate the effect of income
transfer under productivity or population differentials, among others. They mainly
investigate the effects of local income transfers between two countries, i.e., the effects
between two countries in an N-country model. However, when they investigate theeffects of global income transfers across N countries, the previous analysis between thetwo countries cannot answer the following questions. First, how are the different
productivities among the N countries compared, and which countries among the N
countries are income senders and income receivers? Second, when some countries areincome receivers, how much do they receive? Third, which country manages the income
transfer? To my knowledge, no papers investigate the effects of global income transfers
across N countries with an explicit planner rule.
The purpose of this paper is to develop an explicit rule for the planner country of
global income transfers among N countries, and to investigate the effects of income
transfers on each country’s welfare, in order to answer the three questions above. We
introduce the concept of adjustment expense for income transfers across countries. The
finding for the first question is, when country i has a productive advantage in producing
public goods, country i becomes an income receiver. The finding for the second questionis, specifying the particular level of the adjustment expense, the planner can decide thesize of income transfer for all countries. The finding for the third question is, even though
any country can become the planner of income transfers, each country gets the same
utility level, while the low adjustment expense under a particular planner country (theUSA or the United Nations) leads to a Pareto improvement. All conclusions are derived
based on well-known information on the cost of producing public goods and income
levels for all countries.
The paper is organized as follows. Section 2 outlines the model with thenoncooperative Cournot–Nash equilibrium of voluntary contributions to international
public goods when global income transfers are implemented. Section 3 investigates theeffects of global income transfers on national welfare under different productivities of the
N countries, comparing the effects of local transfer between two countries. Section 4
concludes the paper.
1

A useful survey in this field is Ihori (1996).

2

Murdoch and Sandler(1986,p.84) consider the weapons with the North Atlantic Treaty
Organization’s (NATO) allies as the international public good and investigate its demand..
2

2.

The Model
Consider a model where there is one public good, one private good, and N
countries (i = 1,2,…,N). Country i consumes an amount xi of the private good andprovides an amount gi to the supply of the international public good. The total supply ofinternational public good, G, is just the sum of gi provided by each country. Country i’sutility is given by Ui  ( xi , G) , where Ui is strictly increasing and quasi-concave, and

G and xi are normal goods for each country. Country i’s budget constraint is given byxi  pi gi  yi , where yi  0 is the exogenously given national income of country i andpi  0 is the relative price (cost of production) of public goods in terms of private
consumption in country i. As Ihori (1996) interpreted, low (high) pi means a high (low)productivity in producing the public good. We also make the Cournot–Nash assumptionthat each country believes that the contributions of others are independent of its own.
Then, we let G  g i   g j denote the total public good, where  g j is the sum of
j i

j i

g j provided by countries j other than i.
Definition 1. A Cournot–Nash equilibrium in this model is a vector of { gi* : i = l,...,N},
such that for each i, (xi* , gi* ) solves the following:

max U i U i ( xi , g i   g *j )xi , g is.t . x i  p i g i  y i ,j ii  1,2,.., N .

xi , g i  0 :(1)Each country consumes the same level of g *   j  i g  at the equilibrium, which is aijcharacteristic of the international public good. The solution to problem (1) yields thereaction function, which we can denote by gi ( pi , yi , j i g *j ) . The solution gi (i =1,2,…,N) satisfies gi*  gi ( pi , yi , j i g *j ) , i  1, 2, ..., N at the equilibrium. Then, thesolution gi in (1) can be written as gi ( p1 , p2 ,....pN , y1 , y2 ,...y N ) by substituting the othersolutions into it. Furthermore, by inserting gi* into the budget constraint, we get thesolution of xi* . The existence and uniqueness of this solution is assumed here.3 Because
the reaction function gi ( pi , yi , j i g *j ) derived from an unspecified utility function has ageneral form, the solution gi* is more general, and it is then difficult to investigate the3

Miyakoshi (2008) tries to prove the existence and the uniqueness of this equilibrium.

3effect of income transfers on country i’s utility. We use an alternative method, following
Bergstrom et al. (1986, 1992).
Let us consider the maximization problem of a given country at a Cournot–Nashequilibrium in (1). We can reformulate (1) as follows, by adding  j  i g j on both sidesof the budget equation:
Definition 2. A Cournot–Nash equilibrium in this model is such that for each i, (xi* , G* )i = l,2,...,N, solves:

maxU i  U i ( xi , G )xi ,Gs.t. xi  piG  yi  pi  g *j ,j ixi  0 , G   g *j  0, i  1,2,.., N.

(2)

j i

In (2), each country implicitly chooses not only its supply gi* of international public
goods, but also the equilibrium level of G* itself. Each country consumes the same levelof G* . The first order condition here is pi Ui / xi  Ui / G (i = 1,2,..,N). Utilizing theimplicit theorem function, these N equations can be solved to give xi* (i = 1,2,…,N).4
Based on Definition 2, we follow Ihori (1996), who uses the expenditure function
to analyze the effect of the income transfers. We define the expenditure function at theCournot–Nash equilibrium as follows:

min Ei xi  piG
xi ,Gs.t. Ui ( xi ,G ) Ui,(3)

where Ui*  Ui (xi* , G* ) is the utility level at the equilibrium. Then, the expenditure
function can be written as Ei  Ei (Ui* , pi ) . At the equilibrium, this expenditure is equal
to the income in (2):Ei (Ui* , pi )  yi  pi  g *j ,

(4)

j i

G* is not decided. Warr (1983) sums the budget constraints of the N countries and then obtains
G* under pi = 1 for all i. However, pi 1 in our model and then the equilibrium G*is derived in (6)and (7).
4

4

where the income on the right-hand side contains actual income yi and the externalities
from the other country’s provision of the public good. By using Shephard’s Lemma on
the expenditure function Ei (U i* , pi ) at the equilibrium, we can derive the following:Ei (Ui* , pi )
 Gi (Ui* , pi ) , Gi (Ui* , pi )  G* ,
pi

(5)

where Gi (U i* , pi ) is the compensated demand function of country i for the international
public good, which must equal the same level of G* for each country supported by thisLemma. From (4) and (5), the equilibrium can be summarized by the following
equations:
N

N

i 1

i 1

i Ei (Ui* , pi )  i yi  (N 1)G* ,(6)

Gi (Ui* , pi )  G* , (i  1,2,...,N) ,

(7)

N

N

i 1

j i

where    Pi ,  i   Pj and multiplying (4) by  i and summing yields (6).

These N+1 equations decide the utilities U i* of the N countries and the amount of theinternational public good G* at the equilibrium. This formulation is an extension of
Ihori’s (1996) two-country model to N countries.

3.

Planner of Income Transfers: Local vs Global Transfers

We take the total derivatives of (6) and (7) consisting of N+1 equations:
N

N

i 1

i 1

i EiU dUi*  (N 1)dG*  i dyi ,(8)Giu dU i*  dG *  0 , (i = 1,2,..,N),(9)where  E i (U i * , p i ) /  U i *  E iU and Gi (U i* , pi ) / U i*  GiU . By using
G1u dU 1*  dG * in (9), we can delete dG* from (8) and (9) and decrease the number of

equations from N+1 to N. Moreover, by using the relationship of the expenditure function(3) in country 1,

E1U  x1U  p1G1U ,(10)

we can rewrite (8) and (9) as follows:

5

N

N

1 (x1U  p1G1U )dU  i EiU dU  (N 1)G1u dU  i dyi ,(11)

Giu dU i*  G ju dU j*  0 , (i,j = 1,…,N),

(12)

*
1

*
i

i 2

*
1

i 1

We arrange equation (11) and (12) in vector and matrix format,N.....,N ENU  dU1*  i dyi 1(x1U  p1G1U (2  N)), 2E2U ,....................
  *   i1


G
G
,
,
0
,
0
..........
.....,
0
1
U
2
U dU2  0 ..  
0
, G2U ,G3U ,0..........
....., 0     .
 , (13)..........
..........
..........
..........
..........
......
 ..  .
..........
 ..  
..........
..........
..........
..........
..........
..........
......
   .


, 0 , 0 ,0...G(N1)U ,GNU  dUN* 
0

0
where the determinant  of the matrix on the left hand side of (13) is:   1 ( x1U  p1G1U (2  N )) 11  G1U  21
N 1

  1 ( x1U  p1G1U (2  N )) (G2U  GNU )  G1U   i EiU  ( j i G2U  G jU  GNU ),

(14)

i 2

where  11 ,  12 are sub-determinants of  . Because of (10),  i EiU   i xiU   i pi GiU .
Inserting this relation into (14) yields:   1 ( x1U  p1G1U (2  N )) (G2U  GNU )  (G1U G2U  GNU ) NN 1

 G1U   i xiU ( j i G2U  G jU  GNU )

(15)

i 2

N 1

  1 ( x1U  2 p1G1U ) (G2U  GNU )  G1U   i xiU ( j i G2U  G jU  GNU ).
i 2

The partial derivative of the compensated demand function, xiU and GiU , is positive interms of utility because they are normal goods. Then, the determinant  is positive.

Local income transfers
The planner is interested in the local income transfer from country j to i underp j  pi , that is, in (13):6Ndy i  dy j  0, dy s  0 for all s  i, j. , then,   i dy i  ( i    j )dy i (16)i 1

The welfare effects of this transfer are derived using Cramer’s rule on (13):2i
NdU j*  ( p j  pi )(1) 2 j  SN j G SU
dU i*  ( p j  pi )(1)  S i G SU 0, 0 (17)dy i
p j  pi dy i
p j  pi 
We have derived the same effect of local income transfers as Ihori (1996) and others: atransfer from country j with low productivity (high price pj) of providing the public goodto country i with high productivity (low price pi) improves welfare in both countries.
Here, we consider global income transfers among many countries by introducing
the adjustment expense for income transfers, which to my knowledge has not been
proposed yet.
Global income transfers
We first design the planner of global income transfers, which is only one country
among N countries. The planner country h takes a first linear approximation around thepresent utility level, and approximates the utility increase dU h* from the present level byusing the income transfers and the partial derivatives of utility in terms of each country’s
income (solved from (13)) as follows:NU ( y1  y1 , y2  y2 ,..., yN  y N )  U ( y1 , y2 ,..., yN )  (Uh* / yi )yi
*
h

*
h

i 1

N

i.e., dU  U ( y1  y1 , y2  y2 ,..., yN  yN )  U ( y1 , y2 ,..., yN )  (U / yi )dyi .
*
h

*
h

*
h

i 1

(18)

*
h

Next, the planner country h maximizes the utility U h* (or dU h* ), which is alreadymaximized under a given income of each country, by transferring income to each country.
Here, prices are always fixed. However, the income transfers among the N countries
involve an expense for making the agreement between the income senders and income
receivers, or an administration expense. Then, the maximization problem of plannercountry h is formulated as follows:
N
~
Max
U

 a hi ~yi ,h~{ yi }i 1

N

s.t .  ~yi  0 ,
i 1

where

~
U h  d U h* ,N ~yi2   2 ,

(19)

j 1

~y  dy ,iia hi   U h* /  y i ( const .),  0 ( const . fixed ).

7

(20)

The first constraint in (19) means that the income transfers are implemented with fixed
total income across the N countries. The second constraint means all income transfers are
yi2 of the income
implemented with a given total expense  2 >0. The unit expense for ~y  0 (income receiver) and thetransfer for any country i and for the positive transfer ~iyi  0 (income sender) equals one for the sake of simplicity. Finally, a hi
negative transfer ~is derived from (13) as follows:a hi 

U h* 1  i  SN h G SU  0
y i(i = 1,2,.h,.,N).

(21)

Then, the Lagrange equation L and Kuhn-Tucker condition for (19) are as
follows:
NN~
L  Uh  (0   ~yi )   ( 2   ~
yi2 )
i 1

i 1

(22)

N

ahi    2 ~yi  0, (i  1,2,.., N ), 0   ~
yi ,
i 1

N

N

i 1

i 1

 2   ~yi2 ,  ( 2   ~yi2 )  0,   0 .

(23)

The multiplier  is nonnegative and is depending on the exogenously given priceparameters in (19). Note that  is zero when all pi are the same, while it is positive when
all pi are not the same.5 Then,  can be solved as follows, inserting the first conditions
into the second condition in (22) when  > 0, that is, all prices are not the same. We
consider the case of  > 0, that is, all prices are not the same. Then, the second conditionN~y.
in (23) can be rewritten as  2 
y 2 . By using the resolved  , we get ~


i 1

i

i

When  > 0,  can be solved in (24) where all prices are not the same. If all prices are the same,ahi are the same for all i because of (21) and then  in (24) is zero. Conversely, when all prices

5

are not the same (at least one is different) and then at least one ahi for i is different in (21),yi  0 in the first condition in (22). Similarly, we assume
a hi    0 for i and then  > 0 and ~that all prices are the same and then ahi for all i is the same as in (21). Aggregating the first
condition and using the second condition in (22), we can get
N

N

N

i 1

i 1

i 1

 a hi  N   2   ~y i   a hi  N   0 . Then, because ahi is the same for all i,ahi   for all i , which means  = 0 (in which case ~yi are not determined for all i) or ~yi = 0 for
all i in (22).

8a hi  
 0,

2
i 1
N

1 N
1
1 N
~
   a hi , y i  (a hi   a hi ).
N i 12
N i 1(24)

yi in (24) into the second condition in (23),  can be
On the other hand, by inserting ~solved as follows:N (ahi 
i 1



1 N
a hi ) 2
N i 1

2

(25)

.

Finally we get ~y1 , ~
y 2 ,..., ~y N as follows, by inserting (25) into (24).
  a hj 


~
yj 

1 N

a hi 
N i 1 

N

1 N
(
a

a hi ) 2hi
N i 1i 1(j = 1,2,.h,.,N).

,

(26)

Then, by inserting (21) into (26), we can get the following solution { ~y j }:~yj 

 SN h G SU 
1 N   j    i 
N i 1N  j N1
(a hi   a hi ) 2

N i 1
i 1



1 N

 i 
N i 1


N

1 N
(


 i ) 2i
N i 1i 1, (j = 1,2,.h,.,N).(27)We can easily confirm that the solution satisfies the conditions in (22) and (23).
If the planner can get more information about x1U ( for ) and {G1U , G2U ,..., G NU } ,the optimal utility level of country h is solved by inserting (27) into (19):



N



1
 j SNhGSU
j 1 

dU  *h   j 


1 N

i 

N i1


1 N




i 



j
N i1
j 1 

N

2

.

(28)

We approximate the utility level of the other country T under the planner country h byusing (18). Then, inserting (27) into (18) we have:

9



N



1
 j SNT GSU
j 1 

dU  *T   j 


1 N

i 

N i1


1 N




i 



j
N i1
j 1 

N

2

.

(29)

Here, when the planner is changed from country h to country T, the optimal utility levelof country T is the one replacing  SN hGSU with  SNT G SU in (28), which is the same asthat in (29). That is, even though any country can become the planner of income transfers,
each country gets the same maximum utility level, dU i . It is because the optimal income
transfers do not depend on the planner as shown in (27). Moreover, when the unit
adjustment expense for income transfers is decreased from one by a planner country withstrong political power, that is, the  in (19) increases, the utility levels for all countries
increase, showing a Pareto improvement. In this sense, the United Nations or the US may
be appropriate for a planner. If it is the United Nations, it decides the unit expense (then) and the optimal income transfers in (27).
Following the planner country h, which manages the global income transfers
across all countries, we obtain the following policy conclusions.
Because of (27), when the cross product of costs except for the cost of(i)
country j (   j ) is larger than the average cross product of costs except for1 N  i ), (this positive difference is called the productive
N i 1
advantage of country j), country j is an income receiver ( ~y j >0).

one country (

(ii)
(iii)

(iv)

Because of (27), specifying the particular level of the adjustment expense ζ,
the planner can decide the size of income transfers for all countries.
Because of the rule of the planner country h in (19) and the solution in (27),even though any country can become the planner, these conclusions hold,moreover because of (29), each country gets the same utility level. It does
not matter which country becomes the planner. However, when the unit
adjustment expense for income transfers is decreased from one by a plannercountry with strong political power, the utility levels for all countries
increase, creating a Pareto improvement.
All conclusions are derived based on well-known information on the cost pi
of producing public goods, and the income level yi for all countries.

An illustration of global income transfers
We provide an illustration of the planner’s rule using the example of N = 2.
Because of (19), the maximization problem of a planner country 1 is as follows:

~
~
~
Max
U
1  a11 y1  a12 y 2 ,
~ ~

(30)

s.t. ~y1  ~y2  0 ,2~~y  y 2   2.

(31)
(32)

{ y1 , y 2 }

1

2

10

Moreover, because of (21):

a11 

1
 1G2U   0 , a12  1  2 G2U   0 , and  1  p 2 ,  2  p1 . (33)Then, the objective function in (30) can be rewritten, by using (33), as follows:11~
U 1  G 2U  p 2 ~y1  p1 ~y 2   A p 2 ~y1  p1 ~y 2  : A  G 2U  0 . (34)The maximization program (34), (31) and (32) is illustrated using Figure 1.
Insert Figure 1

The objective function (34) and constraints (31) and (32) are depicted in Figure 1, where~
the term U 1 / P1 A shows the utility level, constraint (31) is the solid line and constraint(32) is the region within the circle. In the case of P1  P2 , the objective function (34)
coincides with the constraint (31), and then the optimum transfers are on the constraint(31) and within (32),but it is not uniquely decided. The optimum utility level is zero.
These findings are already shown in footnote 4.
In the case of P1  P2 , the objective function (34) with P1  P2 obtains the
maximum utility for planner country 1 at point C. The optimum income transfer exists atpoint C* with ~y1  0 and ~y 2  0 and the utility level is A p 2 ~y1  p1 ~y 2 . If there is noconstraint (32), the optimal income transfer ~y1  0 endlessly decreases to obtain the
maximum utility level, by moving upwards on the line (31). This formulation of
maximization without (32) is used in previous studies. On the other hand, the non-plannercountry 2 has obtained their utility level by receiving income from planner country 1:
B p2 ~y1  p1 ~y 2 where B  G1U /  . If country 2 becomes the planner, its objectivey p~y where B  G /  . When P  P , the optimal income
function is B p ~









2

1

1

2



1U

1



2





y1  p1 ~y 2 .
transfer is given at point C* by using Figure 1 and its utility level is B p 2 ~The utility level for the planner country 2 is the same as that of the non-planner country 2.
The reason is as follows. The optimal transfers are decided only by the constraints (31)and (32). Moreover, the increase in the utility is approximated by a linear function of theincome transfers of all countries and then, whichever country become the planner, thiscountry necessarily obtains the intersection of two constraints (31) and (32). Finally,
when the unit adjustment expense of income transfers increases, constraint (32) becomeslarger and the utility for each country increases.

4.

Concluding Remarks
We have pointed out three problems regarding local income transfers. First, how
are the different productivities among N countries compared, and which countries among
the N countries are income senders and income receivers? Second, when some countries

11

are income receivers, how much do they receive? Third, which country manages theincome transfers? To resolve these issues, this paper proposes a planner of global income
transfers and finds the optimal global income transfers for all countries. Only one planner,country h among the N countries, maximizes its utility, which is already maximized given
the income levels of all countries, by transferring income to each country for a fixedadjustment expense. The income transfers among countries involve a fixed adjustment
expense ζ for forming agreements among income-sender and income-receiver countries.
Using a planner country h, which manages global income transfers across all
countries, we obtain the following policy conclusions.
(a) The finding for the first problem is, when country i has a productive advantage inproducing the international public good, country i becomes an income receiver.
(b) The finding for the second problem is, specifying the particular level of theadjustment expense ζ, the planner country can decide the size of income transfers
for all countries.
(c) The finding for the third problem is, even though any country can become aplanner of income transfers, these conclusions hold. Whichever country becomesa planner, each country gets the same utility level. However, when the unit
adjustment expense for income transfers is decreased from one by a plannercountry (the United Nations or the US) with strong political power, the utility
levels for all countries increase, creating a Pareto improvement.
(d) All conclusions are derived based on well-known information on the cost of
producing public goods and income levels for all countries.

12References

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13~y1  ~y2  0~y2~
y12  ~y 22   2

C*( ~y1 , ~y2 )~y1~
C  U 1 / P1 A

45Figure 1. N = 2

14

P1  P2~~y U /PA ~y P /P
2

1

1

1

2

1