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More on the welfare economics of foreign aid
JOURNAL
OF THE
JAPANESE
AND
INTERNATIONAL
ECONOMIES
More on the Welfare Economics
1,
97-109 (1987)
of Foreign Aid*
MURRAY C. KEMP School
of Economics, New
University of New South Wales, South Wales 2033. Australia
Kensington,
AND SHOICHI Japanese
KOJIMA
Delegation
to the OECD
Received July 27, 1986; revised September 10, 1986
Kemp, Murray C., and Kojima, Shoichi-More Foreign Aid
on the Welfare Economics of
It is widely believed (i) that it is always possible to enrich low-income countries by means of incremental international transfers, and (ii) that low-income countries are best served by any given increment of aid if the latter is given directly to those countries. It is shown that if the transfer is lump-sum then neither (i) nor (ii) is generally true but that if aid is suitably tied then both (i) and (ii) are necessarily true. J. Japan. Int. &on., March 1987, l(l), pp. 97-109. School of Economics, University of New South Wales, Kensington, New South Wales 2033, Australia; and Japanese Delegation to the OECD. 6 1987 Academic PESS, IIIC.
I.
INTRODUCTION
In the recently available Report of the Task Force on Concessional Flows’ it is proposed that a major purpose of future official development assistance (ODA) should be the raising of living standards in low-income * The present paper is a sequel to Kemp and Kojima (1985a,b). We have benefited from the comments of Daniel Leonard, Ngo Van Long, Edward Tower, Albert Schweinberger, and Shigemi Yabuuchi; from correspondence with Richard Brecher and Tatsuo Hatta; and, above all, from the constructive suggestions of the Editor, Michihiro Ohyama. I The reference is to the Task Force on Concessional Flows, Joint Ministerial Committee 97
0889-1583/87 $3.00 Copyright 0 1937 by Academic Press. Inc. All rights of reproduction in any form reserved.
98
KEMP
AND
KOJIMA
countries; to that end it is recommended that all increases in ODA during the current decade be directed to those countries. Implicit in the recommendation are two propositions: (I) It is always possible to enrich low-income countries by means of incremental international transfers. (II) Low-income countries are best served by any given increment of aid if the latter is given directly to those countries. This being so, one must ask whether the propositions are reliable. It is well known that if there are just two countries and if there are no impediments to trade then both propositions are valid. Suppose then that there are three countries, (Y, & and y. Country (Y (the donor) wishes to assist country /? by means of an international transfer. The transfer can be directed to /3, so that p is the recipient and y the bystander; or it can be directed to y. In the first case aid is direct; in the second case it is indirect. It will be shown that each of the following outcomes is possible; necessary and sufficient conditions for each outcome will be provided. (i) Any transfer from (Y, whether to /3 or to y or to both, impoverishes p. In this case, clearly, it is impossible for (Yto aid /3. (Negative transfers by (Y are ruled out.) (ii) A transfer from (Yto /? impoverishes /3 but a transfer from Q! to y enriches /3. In this case Q! can aid p only indirectly, by making a grant to y. (iii) A transfer from (Y to /? enriches p but a transfer from (Y to y impoverishes p. In this case (Y can aid p only directly, by making a grant to p. (iv) Any transfer from CX,whether to /3 or to y, enriches p. In some circumstances a transfer of given size wll do most good to p if it is made directly [outcome (iva)]; in other circumstances the transfer will be most efficacious if made indirectly [outcome (ivb)]. Thus neither proposition (I) nor proposition (II) is generally valid. On the other hand, they may be valid under interesting special conditions. Thus it will be shown that both propositions are correct if the donor and recipient are required to be small or if either consumption inferiority or offer-curve inelasticity is ruled out. There is now an extensive body of writing on the so-called “transfer paradoxes”; for a fairly complete list of references, see Kemp and Kojima (1985b). In that tradition one studies the implications of a transfer from (Yto /3 on the welfare of (Y,p, and y. In contrast, our concern is with of the Boards of Governors of the World Bank and the International Monetary Fund on the Transfer of Real Resources to Developing Countries. The Task Force presented its final report at the Committee’s meeting in Seoul, October 1985.
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1
the effects on p’s welfare of a transfer from (Ywhich may be divided in any way between /3 and y; that is, our task is to assess and compare the response of a single country’s welfare to each of a whole set of alternative transfers by another country. The scope of our analysis can be clarified by means of a simple diagram. Figure 1 displays the world utility possibility frontier ABC. If an initial, pretransfer equilibrium is represented by point P then any other point on the frontier can be attained by imposing some vector of international lump-sum transfers. In our case, the transfers are constrained: they flow from one country only. Given that constraint, in what directions is it possible to move from P? In particular, is it always possible to move into the interior of the surface A'BC'? Our answer will be in the negative. Section II contains a discursive but easy-to-comprehend statement and proof of our primary results. In Section III the same material is treated from a slightly different point of view and in a much more compact manner. In Section IV it is shown that the “impossibility” results of Section II can be circumvented by suitably tying the aid. In Section V parts of our earlier analysis are extended to accommodate joint donors and joint recipients. And, finally, Section VI indicates some of the directions, possibly game theoretical, which might be taken by further research.
II.
ANALYSIS
The three countries (II, /I, and y produce and trade two consumption goods, 1 and 2. Country (Y exports the first commodity and countries p and y jointly (but not necessarily separately) export the second. Country
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KOJIMA
o (the donor) disturbs an initial balanced free-trade equilibrium by making a small transfer to /3 and y (the joint recipients). The following additional notation will be employed. T the amount of the transfer, in terms of the second commodity (T Z 0) the proportion of T directed to country /I (0 s E 5 1) & the price of the first commodity in terms of the second P l.d the utility of country j (j = CY,p, -y) the expenditure of country j, in terms of the second comeqp, uj) modity (j = (Y,p, 7) the revenue of country j, in terms of the second commodr’(p) ity (.i = a, P, Y) d’(p) the supply of the ith commodity by the jth country @(p, d) the compensated demand for the ith commodity by the jth country (i = 1, 2; j = (Y,p, y) zqp, uj) the excess demand of the jth country for the ith comE &(p, uj) modity (i = 1, 2; j = a, p, y); .zal < 0 - d’(p)
The transfer is financed in (Yand distributed in /3 and/or y by means of lump-sum taxes and subsidies. The budget constraints of (Y,p, and y are therefore ea(p, u”) = r”(p)
- T
(1)
efi(p, u@) = rfi(p)
+ ET
(2)
e”(p,
+
and uy) = U(p)
(1 - E)T,
(3)
respectively. The description of world equilibrium is completed by the market-clearing condition z”‘(p,
u”) + zqp,
up) + zqp,
UY) = 0.
(4)
Equations (l)-(4) contain the four variables P, UP, UY,and p, as well as the parameters T and E. It is assumed that, for any sufficiently small T and for any E E [O, 11,the system possessesisolated solutions with positive price ratio. Suppose that an initial equilibrium is disturbed by a small additional transfer from (Yto p. Differentiating (l)-(4) with respect to T and setting E= l,wefindthat
WELFARE
i
ECONOMICS
a e,
0
0
0
eu6
0
0
0
al
Z?
e2: Y' ZU
ZU
OF
FOREIGN
101
AID
dT>
(5)
where subscripts indicate differentiation ($ = &Flap, rp” = drmldp, etc.). Recalling the envelope result that e; - ri = zj’ (j = (Y,0, y), choosing units of utility so that ej = 1, and writing Xj z$ = zh, (5) reduces to
Solving,* du”
AZ=
-z:, + zqz;’
dup = z:, + zqzy AdT
- z!‘) - 22;‘)
du” A dT = zy’(z,8’ - z;‘) dP ATT=
zual - ZY,
(7d)
where
and is negative as a necessary condition of local Walrasian stability and where pz’,‘lei = pz’,’ is the marginal propensity to buy the first good in the jth country. Equation (7b) tells us the effect on p’s welfare of a direct transfer from 2 The equations of change (7) are standard; see, for example, Bhagwati et al., 1983. For alternative interpretations of Eqs. (7), see Yano, 1983; Bhagwati et al., 1983.
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KEMP AND KOJIMA
a; permuting superscripts /3 and y, we obtain from (7~) the effect on p’s welfare of an indirect transfer. Thus
A do
E=O= -z’Yz;’
- z:‘).
These are our basic equations. Inspection of (9) reveals that, even with A constrained to be negative, dufildTI,=l and du@ldTI,=o can both be negative, both positive, or (either) one positive, the other negative; that is, none of the outcomes (i)-(iv) distinguished in Section I can be ruled out. We can look beyond the binary possibilities @-(iv) and show that if (Y cannot help /3 [outcome (i)] then y can do so, at least indirectly. In other words, p always has at least one potential friend. (Of course, the potential friend may be even poorer than p.) The plausibility of this proposition can be established by observing that indirect aid from y to /3 reverses the direction of indirect aid from cyto p; in effect, the former removes the natural sign restriction from Ta3 Whether y can also help j3 directly is another matter. It can be verified that if (Yand y export the samecommodities then at least one of them can help /3 directly.4 Finally, we note that CY may be able to help neither /I nor y, so that all feasible movement is into the interior of the surface AB’C” of Fig. 1.5 While each country has a friend, it is not always the case that each country can serve as a friend. It remains to consider the optimal mode of assistance in case (iv). To this end, let us return to system (l)-(4), with T positive and E E (0, 1). We 3 To construct a formal demonstration of this plausible proposition, permute the superscripts OLand y in (9b), thereby obtaining an expression for the change in UP when y makes a transfer to (Y.The new expression and (9b) are of opposite sign. 4 Suppose that a! cannot help p directly. Then, from (7b),
2”(Z:’ - z:‘) > 0.
(*I
Permuting the superscripts (Yand y in (7a), we see that a transfer from y to p helps p if
Z”‘(Z$- z:‘) > 0.
(**)
But if a! and y export the same commodity, (*) implies (**). J For CYto be a friend to neither p nor y it is necessary that /3 and y export different commodities and that, for each commodity, (I’S marginal propensity to consume be intermediate to those of p and y. By way of proof, assume that (7b) and (7~) are positive with and without the superscripts p and y interchanged and then note that the four inequalities imply the conditions for the proposition.
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seek the effect of a small change in E on UP, with T held constant. Diierentiating (l)-(4), and simplifying,
whence duP
= T[z:, - z4’(zt1 -
G-
z:‘N.
(11)
Thus if and only if I ZP
-
za’(z;’
-
2;‘)
>
0,
(12)
du@/d& < 0 and the most efficient way for (Yto help /3is to give everything to Y.~ Since z/r < 0, a necessary condition for (12) is al ZU - zz’ > 0,
(13)
which is compatible with case (iv) if zP1> 0. Alternatively, making use of the Slutzky decomposition ip”’ = zp*’- Pzf, where $ is the price slope of the uncompensated excess demand by country cxfor the first commodity, we can rewrite (12) as (zf’ + $1) + 2;’ + pz$ > 0.
(12’)
Since zal < 0, (12’) can be satisfied only if ip”’ > 0 and/or z:’ < 0, that is, only if (Y’Soffer curve is inelastic and/or the first good is inferior in y. Generalizing, if it is optimal for the donor (Yto aid /3 indirectly then the donor’s offer curve is inelastic and/or the donor’s export good is inferior in the bystander country y.7 Thus outcome (ivb) requires elements of 6 From (9) we see that (12) is a necessary condition for duUdTI,=, > duYdT\.,, Thus (12) is implied not only by case (ivb) but also by case (ii). ’ To justify this generalization, we must show that it covers the case zal > 0. By Walras’ law, p?‘(p) + i&(p) + T = 0, so that, differentiating with respect to p, Z”’
+ pfp”’
+ ip”*
= 0.
(***)
On the other hand, the marginal propensities to consume sum to one in each country. In particular,
KEMP AND KOJIMA
104
consumption inferiority and/or offer-curve inelasticity. In this respect it resembles the paradoxical outcomes (i) and (ii). For those outcomes, however, it is necessary that the bystander’s offer curve be inelastic and/ or that the bystander’s export good be inferior in the donor country.* III.
A COMPACT STATEMENT OF RESULTS
The findings of Section II can be summarized and proved in the following more streamlined way. PROPOSITION.
(1) (Y can aid p if and only if
(a) zbi(zE’ - z:‘) > 0, or (b) z@(z:’ - z:‘) I 0 and z:, + zy’(z:’ - zl;‘) < 0. (2) In Case (a) direct or indirect aid is optimal according
as z; -
za’(zf - z$‘) is negative or positive, respectively. (3) In Case (b) direct aid is optimal. Proof.
Differentiating (l)-(4) with respect to T and solving, dup A dT = -zB 1(ZUa’ - z;‘) + E[Z:,- zqz;’ - z:‘)l.
If zqz:i - zl’) > 0 then dup/dT > 0 when E = 0. Whether E = 0 is optimal depends on the sign of z; - zQ1(zE1 - zl’); if it is positive then E = 0 is optimal and if it is negative then E = 1 is optimal. This establishes (la) and (2). If, on the other hand, zpi(z,a’ - z:‘) Z 0 then du WdT > 0 if and only if
E[Z:, - zqz:’ - z;‘)] < zqz:’ - ZI’).
(14)
Since 0 5 E s 1, (14) implies that I -
zal(&
ZP
-
ZI;‘)
<
zqz:’
-
zf);
that is, pz:‘le:
+ z:2/e: = pz:’ + 2:’ = 1.
(****)
Hencesubstitutingfrom (***) and(****) into (12’)andrearranging terms, (z,B’ + z;‘, - ($/p)
- (Z~‘ZWP) > 0.
Evidently (123is satisfiedonly if zg < 0 and/orzi2 < 0. 8SeeBhagwatiet al., 1983.
(12”)
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AID
z; + zY’(zf- ZI;‘)-=I0. In this case, E = 1 is optimal.
IV.
This proves (lb) and (3).
CIRCUMVENTION
BY AID
n
TYING
We have seen that it may be impossible for a! to aid /3, either directly or indirectly. However, that conclusion was derived from the assumption that the aid is “clean” or untied. Now it is usually taken for granted that aid is most beneficial to the recipient when it is completely untied. In fact this is not the case. In particular, even when it is impossible for (Yto help p by means of an untied transfer it yet may be possible to help with a tied transfer. Thus, while tying can create paradoxes in a two-country setting (Kemp and Kojima, 1985b) it can remove them when there are three countries. To see this, let us suppose that (Yinsists that the government of /I spend a proportion rnp of the aid on the first commodity (0 5 rnfl5 1). Then the conditions of international equilibrium take the revised form
P(p,
u”) + z@‘(p, up) + z”(p,
ea(p, u”) = r”(p) - T
(1’)
e%,
up) = rP(p)
(2’)
eQ,
uy> = rY(p)
(3’)
uy) + mPT/p = 0.
(4’)
It must be noted that (2’) is the private budget constraint of p and up is that part of /3’s welfare generated by that budget. The total welfare of p, say w@, is greater than U@by the contribution of (Y’S aid. Let us set T = 0 initially. Then, differentiating (l’)-(4’) with respect to T and solving for du”ldT and duYldT, we find that A(du”ldT)
= -zL + zY*z:’ + zfi’z&’ + za’m@lp
A(duYldT)
= (zY’/p)(m@ - pz:‘).
Since T = 0 the initial equilibrium is Pareto-optimal; hence an infinitesimal transfer has the same effect on the recipient’s welfare however it is spent and
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KEMP AND KOJIMA
dwP -=dT
!g+!g
)
= ; [z:, + p’(z;’
- z;‘) + (z~‘lp>(rd - pzC’)l.
(16)
It is easy to verify that (16) may be positive even when dufi/dTI,,, and duPIdTl,=o are negative. For that comforting outcome it is necessary that z@i(mp - pzt’) be negative, that is, that the government of p be required to spend more on /3’s exported commodity than would p-individuals. This is a thoroughly plausible condition. If the aid had been tied both in the donor and in the recipient, the condition would have been even weaker. Thus by tying its aid a donor can circumvent the impossibility results of Sections II and III. Note, however, that if cr and /3 export different commodities then circumvention can be achieved only by (unconventionally) tying aid to the good imported by the donor.
V.
JOINT DONORS AND JOINT RECIPIENTS
We have concentrated on a few basic questions raised by the desire of a single country to aid another. However, the poorer countries typically receive aid from several quarters, and most wealthy countries spread their aid widely. These well-known facts suggest additional questions. If both (Yand y help j3 and wish to extend their aid, cooperatively sharing the utility burden, either equally or in any other fashion, how should the additional aid be shared among the donors? If (Y helps both /3 and y and wishes to extend the aid, sharing the additional utility benefits either equally or in some other fashion, how should the additional aid be shared among the recipients? Each question is grist for the mill constructed in Section II. Thus suppose that in an initial equilibrium p receives Ta from (Yand TY from y, both Ta and TY positive. The equilibrium is described by the equations
zqp,
u”) + zqp,
ea(p, u”) = r”(p) - T”
(17)
ep(p, up) = rP(p) + Ta + Ty
W-9
eY(p, uy) = F(p) - TV
(19)
UP) + zqp,
uy) = 0.
cm
Treating up as a parameter and T*, TY as variables, we can differentiate the system with respect to UP, add the equality-of-sharing condition du” =
WELFARE
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C&Y,and solve for the additional contributions of the donors (Y and y. Omitting the detailed calculations, we have A’(&““/&@) = z”*(& -
22:'
+ z;‘) - d
= -A + (z”’ - zY’)(z;’ A’(dTY/&a)
=
zy’(Z;’
=
-A
+
(~7’
22;' -
+
z:‘)
za’)(zf
W$
Z!‘>
@lb)
-
z:,
(224
-
Zt’>,
(22b)
where A’ = -22; > 0. If zul = zY*or if z$ = z!’ = zz’ then dT*lduP = dT?l du@ > 0. Otherwise, the vector (dT*, dTP) of incremental aid can have any pattern of signs. In particular, it is possible that the recipient p can be made better off with less aid from each donor. However, for that outcome it is necessary that the recipient’s offer curve be inelastic and/or that the commodity exported by the recipient be inferior in at least one of the donor countries;9 and, as (21b) and (22b) reveal, it is necessary also that the recipient’s marginal propensity to buy any good fall between the donors’ marginal propensities to buy the same good. Of course, whatever the manner in which aid changes, both donors find themselves worse off: du” = duy = -dufi/2.
Suppose alternatively that in an initial equilibrium the single donor a! gives Tp to p and Ty to y, both To and Ty positive. The equilibrium is described by the equations ea(p, u”) = r*(p) - Tp - Ty
(23)
eP(p, UP) = rP(p) + Tp
(24)
ey(p, uy) = C(p) + Ty
(25)
zqp, LP”)+ zqp, UP) + zqp, UY)= 0.
(26)
9 If both dTa/d& and dTyldu@ are negative then, adding (21) and (22) and making use of (201, -z@‘(zf- 2~:’ + z:‘) - 2~: < 0. Substituting from the Slutzky equation zj? = it’ + z@‘z:‘, -zyzy
+ z2;‘) - 237’ - 2(z;’ + z;‘) < 0.
(t)
If zP1 < 0, the proposition follows immediately. Suppose then that zB’ > 0. By Wahas’ law, p,fBr(p) + i@(p) - Ta - TV = 0, so that, differentiating with respect to p, z@ + pz,-8’ + ipa’ = 0.
(tt)
Substituting from (tt) and recalling that, for each country, the two marginal Propensities to consume add to one, (t) becomes (z@‘lp)(z~* + zL*) + (2/p)i,8* - 2(t,“’ + z;‘) < 0. The proposition then follows from the twin facts that z@’ > 0 and zp”’ + z;’ < 0.
108
KEMPANDKOJIMA
Treating ua as a parameter and T,, Ty as variables, both positive, we can differentiate (23)-(26) with respect to P, add the equality-of-benefit condition dufl = duy, and solve for the additional aid received by the recipients /3 and y. Thus -A’(dTBldu”> = zfil(-Zz;’ + zf’ + z:‘) - z; -A’(dT,l&)
(27a)
= -A + (zfil ;- zy’)(z;’ - z:‘)
Wb)
= zy’(-2~:’ + zt’ + z;‘) - z;
GW
= -A + (z@i- zy’)(z;’ - zf’).
(2W
lfzB*
= zY* or if& = zf’ = z!’ then dT,Jdu” = dT,ldu” < 0. Otherwise, the vector (dTB , dT,) of incremental aid can have any pattern of signs. In particular, it is possible that both recipients can be made better off with a smaller total outlay by the donor. However, for that outcome it is necessary that the donor’s offer curve be inelastic and/or that the commodity exported by the donor be inferior in at least one of the recipient countries;lO and, as (27b) and (28b) reveal, it is necessary also that the donor’s marginal propensity to buy any good fall between the recipients’ marginal propensities to buy the same good. Of course, whatever the manner in which aid changes, the donor is left worse OR du” = -duS/2 = -duY/2. In the traditional stable two-country setting additional aid implies an additional transfer. What we have established in this section is that, when there are more than two countries, there is no comparable result, even when donors or recipients share the burden or benefit equally.
VI.
RELATED QUESTIONS
Important questions have been neglected, at least for the time being. Some of these questions can be posed only when additional countries 6, E, . . . , o are recognized. Thus, given that LYwants to help p and that direct aid is inefficient, to which of the remaining countries should the transfer be directed? Other questions are game theoretical in nature. They arise, for example, when two or more countries offer aid to a third. They also can arise when a single donor seeks to secure some quid for its quo (a tariff reduction, perhaps). Finally, one can drop the requirement that aid be financed and disposed of in lump-sum fashion and reconsider the questions of Section I under alternative fiscal assumptions. We have given our model a purely static interpretation. However, the two commodities might differ only in the dates at which they become lo The proof follows the general lines of footnote 9.
WELFARE
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109
available. Thus our analysis and conclusions apply equally in the dynamic context of “now” and “then.” However, in such an application it is implied that there is a perfect world capital market. It remains to consider the dynamics of aid under alternative assumptions about the capital market. REFERENCES J. N., BRECHER, R. A., AND HATTA, T. (1983). The generalised theory of transfers and welfare: Bilateral transfers in a multilateral world, Amer. &on. Rev. 83, 606-618. KEMP, M. C., AND KOJIMA, S. (1985a). The welfare economics of foreign aid, in “Issues in Contemporary Microeconomics and Welfare” (G. R. Feiwel, Ed.), pp. 470-483, Macmillan & Co., London. KEMP, M. C., AND KOJIMA, S. (1985b). Tied aid and the paradoxes of donor-enrichment and recipient-impoverishment, Znt. Econ. Rev. 26, 721-729. YANO, M. (1983). Welfare aspects of the transfer problem, J. Znt. Econ. 15, 277-289.
BHAGWATI,
OF THE
JAPANESE
AND
INTERNATIONAL
ECONOMIES
More on the Welfare Economics
1,
97-109 (1987)
of Foreign Aid*
MURRAY C. KEMP School
of Economics, New
University of New South Wales, South Wales 2033. Australia
Kensington,
AND SHOICHI Japanese
KOJIMA
Delegation
to the OECD
Received July 27, 1986; revised September 10, 1986
Kemp, Murray C., and Kojima, Shoichi-More Foreign Aid
on the Welfare Economics of
It is widely believed (i) that it is always possible to enrich low-income countries by means of incremental international transfers, and (ii) that low-income countries are best served by any given increment of aid if the latter is given directly to those countries. It is shown that if the transfer is lump-sum then neither (i) nor (ii) is generally true but that if aid is suitably tied then both (i) and (ii) are necessarily true. J. Japan. Int. &on., March 1987, l(l), pp. 97-109. School of Economics, University of New South Wales, Kensington, New South Wales 2033, Australia; and Japanese Delegation to the OECD. 6 1987 Academic PESS, IIIC.
I.
INTRODUCTION
In the recently available Report of the Task Force on Concessional Flows’ it is proposed that a major purpose of future official development assistance (ODA) should be the raising of living standards in low-income * The present paper is a sequel to Kemp and Kojima (1985a,b). We have benefited from the comments of Daniel Leonard, Ngo Van Long, Edward Tower, Albert Schweinberger, and Shigemi Yabuuchi; from correspondence with Richard Brecher and Tatsuo Hatta; and, above all, from the constructive suggestions of the Editor, Michihiro Ohyama. I The reference is to the Task Force on Concessional Flows, Joint Ministerial Committee 97
0889-1583/87 $3.00 Copyright 0 1937 by Academic Press. Inc. All rights of reproduction in any form reserved.
98
KEMP
AND
KOJIMA
countries; to that end it is recommended that all increases in ODA during the current decade be directed to those countries. Implicit in the recommendation are two propositions: (I) It is always possible to enrich low-income countries by means of incremental international transfers. (II) Low-income countries are best served by any given increment of aid if the latter is given directly to those countries. This being so, one must ask whether the propositions are reliable. It is well known that if there are just two countries and if there are no impediments to trade then both propositions are valid. Suppose then that there are three countries, (Y, & and y. Country (Y (the donor) wishes to assist country /? by means of an international transfer. The transfer can be directed to /3, so that p is the recipient and y the bystander; or it can be directed to y. In the first case aid is direct; in the second case it is indirect. It will be shown that each of the following outcomes is possible; necessary and sufficient conditions for each outcome will be provided. (i) Any transfer from (Y, whether to /3 or to y or to both, impoverishes p. In this case, clearly, it is impossible for (Yto aid /3. (Negative transfers by (Y are ruled out.) (ii) A transfer from (Yto /? impoverishes /3 but a transfer from Q! to y enriches /3. In this case Q! can aid p only indirectly, by making a grant to y. (iii) A transfer from (Y to /? enriches p but a transfer from (Y to y impoverishes p. In this case (Y can aid p only directly, by making a grant to p. (iv) Any transfer from CX,whether to /3 or to y, enriches p. In some circumstances a transfer of given size wll do most good to p if it is made directly [outcome (iva)]; in other circumstances the transfer will be most efficacious if made indirectly [outcome (ivb)]. Thus neither proposition (I) nor proposition (II) is generally valid. On the other hand, they may be valid under interesting special conditions. Thus it will be shown that both propositions are correct if the donor and recipient are required to be small or if either consumption inferiority or offer-curve inelasticity is ruled out. There is now an extensive body of writing on the so-called “transfer paradoxes”; for a fairly complete list of references, see Kemp and Kojima (1985b). In that tradition one studies the implications of a transfer from (Yto /3 on the welfare of (Y,p, and y. In contrast, our concern is with of the Boards of Governors of the World Bank and the International Monetary Fund on the Transfer of Real Resources to Developing Countries. The Task Force presented its final report at the Committee’s meeting in Seoul, October 1985.
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1
the effects on p’s welfare of a transfer from (Ywhich may be divided in any way between /3 and y; that is, our task is to assess and compare the response of a single country’s welfare to each of a whole set of alternative transfers by another country. The scope of our analysis can be clarified by means of a simple diagram. Figure 1 displays the world utility possibility frontier ABC. If an initial, pretransfer equilibrium is represented by point P then any other point on the frontier can be attained by imposing some vector of international lump-sum transfers. In our case, the transfers are constrained: they flow from one country only. Given that constraint, in what directions is it possible to move from P? In particular, is it always possible to move into the interior of the surface A'BC'? Our answer will be in the negative. Section II contains a discursive but easy-to-comprehend statement and proof of our primary results. In Section III the same material is treated from a slightly different point of view and in a much more compact manner. In Section IV it is shown that the “impossibility” results of Section II can be circumvented by suitably tying the aid. In Section V parts of our earlier analysis are extended to accommodate joint donors and joint recipients. And, finally, Section VI indicates some of the directions, possibly game theoretical, which might be taken by further research.
II.
ANALYSIS
The three countries (II, /I, and y produce and trade two consumption goods, 1 and 2. Country (Y exports the first commodity and countries p and y jointly (but not necessarily separately) export the second. Country
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o (the donor) disturbs an initial balanced free-trade equilibrium by making a small transfer to /3 and y (the joint recipients). The following additional notation will be employed. T the amount of the transfer, in terms of the second commodity (T Z 0) the proportion of T directed to country /I (0 s E 5 1) & the price of the first commodity in terms of the second P l.d the utility of country j (j = CY,p, -y) the expenditure of country j, in terms of the second comeqp, uj) modity (j = (Y,p, 7) the revenue of country j, in terms of the second commodr’(p) ity (.i = a, P, Y) d’(p) the supply of the ith commodity by the jth country @(p, d) the compensated demand for the ith commodity by the jth country (i = 1, 2; j = (Y,p, y) zqp, uj) the excess demand of the jth country for the ith comE &(p, uj) modity (i = 1, 2; j = a, p, y); .zal < 0 - d’(p)
The transfer is financed in (Yand distributed in /3 and/or y by means of lump-sum taxes and subsidies. The budget constraints of (Y,p, and y are therefore ea(p, u”) = r”(p)
- T
(1)
efi(p, u@) = rfi(p)
+ ET
(2)
e”(p,
+
and uy) = U(p)
(1 - E)T,
(3)
respectively. The description of world equilibrium is completed by the market-clearing condition z”‘(p,
u”) + zqp,
up) + zqp,
UY) = 0.
(4)
Equations (l)-(4) contain the four variables P, UP, UY,and p, as well as the parameters T and E. It is assumed that, for any sufficiently small T and for any E E [O, 11,the system possessesisolated solutions with positive price ratio. Suppose that an initial equilibrium is disturbed by a small additional transfer from (Yto p. Differentiating (l)-(4) with respect to T and setting E= l,wefindthat
WELFARE
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a e,
0
0
0
eu6
0
0
0
al
Z?
e2: Y' ZU
ZU
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AID
dT>
(5)
where subscripts indicate differentiation ($ = &Flap, rp” = drmldp, etc.). Recalling the envelope result that e; - ri = zj’ (j = (Y,0, y), choosing units of utility so that ej = 1, and writing Xj z$ = zh, (5) reduces to
Solving,* du”
AZ=
-z:, + zqz;’
dup = z:, + zqzy AdT
- z!‘) - 22;‘)
du” A dT = zy’(z,8’ - z;‘) dP ATT=
zual - ZY,
(7d)
where
and is negative as a necessary condition of local Walrasian stability and where pz’,‘lei = pz’,’ is the marginal propensity to buy the first good in the jth country. Equation (7b) tells us the effect on p’s welfare of a direct transfer from 2 The equations of change (7) are standard; see, for example, Bhagwati et al., 1983. For alternative interpretations of Eqs. (7), see Yano, 1983; Bhagwati et al., 1983.
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a; permuting superscripts /3 and y, we obtain from (7~) the effect on p’s welfare of an indirect transfer. Thus
A do
E=O= -z’Yz;’
- z:‘).
These are our basic equations. Inspection of (9) reveals that, even with A constrained to be negative, dufildTI,=l and du@ldTI,=o can both be negative, both positive, or (either) one positive, the other negative; that is, none of the outcomes (i)-(iv) distinguished in Section I can be ruled out. We can look beyond the binary possibilities @-(iv) and show that if (Y cannot help /3 [outcome (i)] then y can do so, at least indirectly. In other words, p always has at least one potential friend. (Of course, the potential friend may be even poorer than p.) The plausibility of this proposition can be established by observing that indirect aid from y to /3 reverses the direction of indirect aid from cyto p; in effect, the former removes the natural sign restriction from Ta3 Whether y can also help j3 directly is another matter. It can be verified that if (Yand y export the samecommodities then at least one of them can help /3 directly.4 Finally, we note that CY may be able to help neither /I nor y, so that all feasible movement is into the interior of the surface AB’C” of Fig. 1.5 While each country has a friend, it is not always the case that each country can serve as a friend. It remains to consider the optimal mode of assistance in case (iv). To this end, let us return to system (l)-(4), with T positive and E E (0, 1). We 3 To construct a formal demonstration of this plausible proposition, permute the superscripts OLand y in (9b), thereby obtaining an expression for the change in UP when y makes a transfer to (Y.The new expression and (9b) are of opposite sign. 4 Suppose that a! cannot help p directly. Then, from (7b),
2”(Z:’ - z:‘) > 0.
(*I
Permuting the superscripts (Yand y in (7a), we see that a transfer from y to p helps p if
Z”‘(Z$- z:‘) > 0.
(**)
But if a! and y export the same commodity, (*) implies (**). J For CYto be a friend to neither p nor y it is necessary that /3 and y export different commodities and that, for each commodity, (I’S marginal propensity to consume be intermediate to those of p and y. By way of proof, assume that (7b) and (7~) are positive with and without the superscripts p and y interchanged and then note that the four inequalities imply the conditions for the proposition.
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seek the effect of a small change in E on UP, with T held constant. Diierentiating (l)-(4), and simplifying,
whence duP
= T[z:, - z4’(zt1 -
G-
z:‘N.
(11)
Thus if and only if I ZP
-
za’(z;’
-
2;‘)
>
0,
(12)
du@/d& < 0 and the most efficient way for (Yto help /3is to give everything to Y.~ Since z/r < 0, a necessary condition for (12) is al ZU - zz’ > 0,
(13)
which is compatible with case (iv) if zP1> 0. Alternatively, making use of the Slutzky decomposition ip”’ = zp*’- Pzf, where $ is the price slope of the uncompensated excess demand by country cxfor the first commodity, we can rewrite (12) as (zf’ + $1) + 2;’ + pz$ > 0.
(12’)
Since zal < 0, (12’) can be satisfied only if ip”’ > 0 and/or z:’ < 0, that is, only if (Y’Soffer curve is inelastic and/or the first good is inferior in y. Generalizing, if it is optimal for the donor (Yto aid /3 indirectly then the donor’s offer curve is inelastic and/or the donor’s export good is inferior in the bystander country y.7 Thus outcome (ivb) requires elements of 6 From (9) we see that (12) is a necessary condition for duUdTI,=, > duYdT\.,, Thus (12) is implied not only by case (ivb) but also by case (ii). ’ To justify this generalization, we must show that it covers the case zal > 0. By Walras’ law, p?‘(p) + i&(p) + T = 0, so that, differentiating with respect to p, Z”’
+ pfp”’
+ ip”*
= 0.
(***)
On the other hand, the marginal propensities to consume sum to one in each country. In particular,
KEMP AND KOJIMA
104
consumption inferiority and/or offer-curve inelasticity. In this respect it resembles the paradoxical outcomes (i) and (ii). For those outcomes, however, it is necessary that the bystander’s offer curve be inelastic and/ or that the bystander’s export good be inferior in the donor country.* III.
A COMPACT STATEMENT OF RESULTS
The findings of Section II can be summarized and proved in the following more streamlined way. PROPOSITION.
(1) (Y can aid p if and only if
(a) zbi(zE’ - z:‘) > 0, or (b) z@(z:’ - z:‘) I 0 and z:, + zy’(z:’ - zl;‘) < 0. (2) In Case (a) direct or indirect aid is optimal according
as z; -
za’(zf - z$‘) is negative or positive, respectively. (3) In Case (b) direct aid is optimal. Proof.
Differentiating (l)-(4) with respect to T and solving, dup A dT = -zB 1(ZUa’ - z;‘) + E[Z:,- zqz;’ - z:‘)l.
If zqz:i - zl’) > 0 then dup/dT > 0 when E = 0. Whether E = 0 is optimal depends on the sign of z; - zQ1(zE1 - zl’); if it is positive then E = 0 is optimal and if it is negative then E = 1 is optimal. This establishes (la) and (2). If, on the other hand, zpi(z,a’ - z:‘) Z 0 then du WdT > 0 if and only if
E[Z:, - zqz:’ - z;‘)] < zqz:’ - ZI’).
(14)
Since 0 5 E s 1, (14) implies that I -
zal(&
ZP
-
ZI;‘)
<
zqz:’
-
zf);
that is, pz:‘le:
+ z:2/e: = pz:’ + 2:’ = 1.
(****)
Hencesubstitutingfrom (***) and(****) into (12’)andrearranging terms, (z,B’ + z;‘, - ($/p)
- (Z~‘ZWP) > 0.
Evidently (123is satisfiedonly if zg < 0 and/orzi2 < 0. 8SeeBhagwatiet al., 1983.
(12”)
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z; + zY’(zf- ZI;‘)-=I0. In this case, E = 1 is optimal.
IV.
This proves (lb) and (3).
CIRCUMVENTION
BY AID
n
TYING
We have seen that it may be impossible for a! to aid /3, either directly or indirectly. However, that conclusion was derived from the assumption that the aid is “clean” or untied. Now it is usually taken for granted that aid is most beneficial to the recipient when it is completely untied. In fact this is not the case. In particular, even when it is impossible for (Yto help p by means of an untied transfer it yet may be possible to help with a tied transfer. Thus, while tying can create paradoxes in a two-country setting (Kemp and Kojima, 1985b) it can remove them when there are three countries. To see this, let us suppose that (Yinsists that the government of /I spend a proportion rnp of the aid on the first commodity (0 5 rnfl5 1). Then the conditions of international equilibrium take the revised form
P(p,
u”) + z@‘(p, up) + z”(p,
ea(p, u”) = r”(p) - T
(1’)
e%,
up) = rP(p)
(2’)
eQ,
uy> = rY(p)
(3’)
uy) + mPT/p = 0.
(4’)
It must be noted that (2’) is the private budget constraint of p and up is that part of /3’s welfare generated by that budget. The total welfare of p, say w@, is greater than U@by the contribution of (Y’S aid. Let us set T = 0 initially. Then, differentiating (l’)-(4’) with respect to T and solving for du”ldT and duYldT, we find that A(du”ldT)
= -zL + zY*z:’ + zfi’z&’ + za’m@lp
A(duYldT)
= (zY’/p)(m@ - pz:‘).
Since T = 0 the initial equilibrium is Pareto-optimal; hence an infinitesimal transfer has the same effect on the recipient’s welfare however it is spent and
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dwP -=dT
!g+!g
)
= ; [z:, + p’(z;’
- z;‘) + (z~‘lp>(rd - pzC’)l.
(16)
It is easy to verify that (16) may be positive even when dufi/dTI,,, and duPIdTl,=o are negative. For that comforting outcome it is necessary that z@i(mp - pzt’) be negative, that is, that the government of p be required to spend more on /3’s exported commodity than would p-individuals. This is a thoroughly plausible condition. If the aid had been tied both in the donor and in the recipient, the condition would have been even weaker. Thus by tying its aid a donor can circumvent the impossibility results of Sections II and III. Note, however, that if cr and /3 export different commodities then circumvention can be achieved only by (unconventionally) tying aid to the good imported by the donor.
V.
JOINT DONORS AND JOINT RECIPIENTS
We have concentrated on a few basic questions raised by the desire of a single country to aid another. However, the poorer countries typically receive aid from several quarters, and most wealthy countries spread their aid widely. These well-known facts suggest additional questions. If both (Yand y help j3 and wish to extend their aid, cooperatively sharing the utility burden, either equally or in any other fashion, how should the additional aid be shared among the donors? If (Y helps both /3 and y and wishes to extend the aid, sharing the additional utility benefits either equally or in some other fashion, how should the additional aid be shared among the recipients? Each question is grist for the mill constructed in Section II. Thus suppose that in an initial equilibrium p receives Ta from (Yand TY from y, both Ta and TY positive. The equilibrium is described by the equations
zqp,
u”) + zqp,
ea(p, u”) = r”(p) - T”
(17)
ep(p, up) = rP(p) + Ta + Ty
W-9
eY(p, uy) = F(p) - TV
(19)
UP) + zqp,
uy) = 0.
cm
Treating up as a parameter and T*, TY as variables, we can differentiate the system with respect to UP, add the equality-of-sharing condition du” =
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C&Y,and solve for the additional contributions of the donors (Y and y. Omitting the detailed calculations, we have A’(&““/&@) = z”*(& -
22:'
+ z;‘) - d
= -A + (z”’ - zY’)(z;’ A’(dTY/&a)
=
zy’(Z;’
=
-A
+
(~7’
22;' -
+
z:‘)
za’)(zf
W$
Z!‘>
@lb)
-
z:,
(224
-
Zt’>,
(22b)
where A’ = -22; > 0. If zul = zY*or if z$ = z!’ = zz’ then dT*lduP = dT?l du@ > 0. Otherwise, the vector (dT*, dTP) of incremental aid can have any pattern of signs. In particular, it is possible that the recipient p can be made better off with less aid from each donor. However, for that outcome it is necessary that the recipient’s offer curve be inelastic and/or that the commodity exported by the recipient be inferior in at least one of the donor countries;9 and, as (21b) and (22b) reveal, it is necessary also that the recipient’s marginal propensity to buy any good fall between the donors’ marginal propensities to buy the same good. Of course, whatever the manner in which aid changes, both donors find themselves worse off: du” = duy = -dufi/2.
Suppose alternatively that in an initial equilibrium the single donor a! gives Tp to p and Ty to y, both To and Ty positive. The equilibrium is described by the equations ea(p, u”) = r*(p) - Tp - Ty
(23)
eP(p, UP) = rP(p) + Tp
(24)
ey(p, uy) = C(p) + Ty
(25)
zqp, LP”)+ zqp, UP) + zqp, UY)= 0.
(26)
9 If both dTa/d& and dTyldu@ are negative then, adding (21) and (22) and making use of (201, -z@‘(zf- 2~:’ + z:‘) - 2~: < 0. Substituting from the Slutzky equation zj? = it’ + z@‘z:‘, -zyzy
+ z2;‘) - 237’ - 2(z;’ + z;‘) < 0.
(t)
If zP1 < 0, the proposition follows immediately. Suppose then that zB’ > 0. By Wahas’ law, p,fBr(p) + i@(p) - Ta - TV = 0, so that, differentiating with respect to p, z@ + pz,-8’ + ipa’ = 0.
(tt)
Substituting from (tt) and recalling that, for each country, the two marginal Propensities to consume add to one, (t) becomes (z@‘lp)(z~* + zL*) + (2/p)i,8* - 2(t,“’ + z;‘) < 0. The proposition then follows from the twin facts that z@’ > 0 and zp”’ + z;’ < 0.
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Treating ua as a parameter and T,, Ty as variables, both positive, we can differentiate (23)-(26) with respect to P, add the equality-of-benefit condition dufl = duy, and solve for the additional aid received by the recipients /3 and y. Thus -A’(dTBldu”> = zfil(-Zz;’ + zf’ + z:‘) - z; -A’(dT,l&)
(27a)
= -A + (zfil ;- zy’)(z;’ - z:‘)
Wb)
= zy’(-2~:’ + zt’ + z;‘) - z;
GW
= -A + (z@i- zy’)(z;’ - zf’).
(2W
lfzB*
= zY* or if& = zf’ = z!’ then dT,Jdu” = dT,ldu” < 0. Otherwise, the vector (dTB , dT,) of incremental aid can have any pattern of signs. In particular, it is possible that both recipients can be made better off with a smaller total outlay by the donor. However, for that outcome it is necessary that the donor’s offer curve be inelastic and/or that the commodity exported by the donor be inferior in at least one of the recipient countries;lO and, as (27b) and (28b) reveal, it is necessary also that the donor’s marginal propensity to buy any good fall between the recipients’ marginal propensities to buy the same good. Of course, whatever the manner in which aid changes, the donor is left worse OR du” = -duS/2 = -duY/2. In the traditional stable two-country setting additional aid implies an additional transfer. What we have established in this section is that, when there are more than two countries, there is no comparable result, even when donors or recipients share the burden or benefit equally.
VI.
RELATED QUESTIONS
Important questions have been neglected, at least for the time being. Some of these questions can be posed only when additional countries 6, E, . . . , o are recognized. Thus, given that LYwants to help p and that direct aid is inefficient, to which of the remaining countries should the transfer be directed? Other questions are game theoretical in nature. They arise, for example, when two or more countries offer aid to a third. They also can arise when a single donor seeks to secure some quid for its quo (a tariff reduction, perhaps). Finally, one can drop the requirement that aid be financed and disposed of in lump-sum fashion and reconsider the questions of Section I under alternative fiscal assumptions. We have given our model a purely static interpretation. However, the two commodities might differ only in the dates at which they become lo The proof follows the general lines of footnote 9.
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available. Thus our analysis and conclusions apply equally in the dynamic context of “now” and “then.” However, in such an application it is implied that there is a perfect world capital market. It remains to consider the dynamics of aid under alternative assumptions about the capital market. REFERENCES J. N., BRECHER, R. A., AND HATTA, T. (1983). The generalised theory of transfers and welfare: Bilateral transfers in a multilateral world, Amer. &on. Rev. 83, 606-618. KEMP, M. C., AND KOJIMA, S. (1985a). The welfare economics of foreign aid, in “Issues in Contemporary Microeconomics and Welfare” (G. R. Feiwel, Ed.), pp. 470-483, Macmillan & Co., London. KEMP, M. C., AND KOJIMA, S. (1985b). Tied aid and the paradoxes of donor-enrichment and recipient-impoverishment, Znt. Econ. Rev. 26, 721-729. YANO, M. (1983). Welfare aspects of the transfer problem, J. Znt. Econ. 15, 277-289.
BHAGWATI,