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TIED AND UNTIED FOREIGN AID: A THEORETICAL AND EMPIRICAL ANALYSISARVIN, B. Mak
BAUM, Christopher F.
Keio Economic Society, Keio University
1997
Keio economic studies Vol.34, No.2 (1997. ) ,p.71- 79
This paper develops a model of foreign assistance where untied aid generates goodwill for a donor
in a recipient country, thereby enhancing exports of the donor. Using an intertemporal maximizing
model, we identify the optimal adjustment paths for tied and untied aid and the parameters uponwhich they depend. Estimates of these parameters are obtained from data on aid from seventeenOECD countries over the period 1972-1990. Results suggest that donor countries maintain aconstant flow of untied aid in order to continually replenish the stock of goodwill, while adjusting theflow of tied aid over time.
Journal Article
https://koara.lib.keio.ac.jp/xoonips/modules/xoonips/detail.php?koara_id=AA00260492-19970002-0
071

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KEIO ECONOMICTIEDSTUDIES 34(2), 71-79 (1997)

AND

UNTIEDAND

FOREIGNEMPIRICAL

B. MakAID:

A THEORETICALANALYSIS

ARVIN

Department of Economics Trent University
Ontario, Canada
andChristopher

F. BAUM

Department of Economics Boston College
Massachusetts, U.S.A.
First version received April 1996; final version accepted July 1997

Abstract: This paper develops a model of foreign assistance where untied aid
generates goodwill for a donor in a recipient country, thereby enhancing exportsof the donor. Using an intertemporal maximizing model, we identify the optimal
adjustment paths for tied and untied aid and the parameters upon which they
depend. Estimates of these parameters are obtained from data on aid fromseventeen OECD countries over the period 1972-1990. Results suggest that donor
countries maintain a constant flow of untied aid in order to continually replenish
the stock of goodwill, while adjusting the flow of tied aid over time .

1.

INTRODUCTION

This paper develops a model of foreign aid and describes how different typesof aid may serve to enhance the exports of a donor country . The bulk of theliterature on foreign aid (often called ODA for official development assistance) isof an ideological nature, debating whether foreign aid is totally discredited or if
there are certain types of aid that may be generally supported (see, e.g., Gillis etal., 1992). One strand of the literature focuses on recipient needs , investigating
how aid stimulates public sector growth by supplementing low domestic savings
with foreign capital inflows, and how it accelerates capital formation by adding
to the recipients' absorptive capacity over time . The other strand focuses on donor
Acknowledgements:
The authors are grateful for helpful suggestions from Sand Choudhry, Robert M
urphy, an anonymous referee, and especially Richard Arnott . Seminar participants at Boston College
provided useful comments. Saner Tu nay provided able research assistance. The usual disclaimer applies .
71

72

B. MAK

ARVIN

andCHRISTOPHER

F. BAUM

interests, observing that aid is often tied to donor country exports and hence may
be regarded as a trade creation device facilitating formation of foreign markets
for perhaps over priced or lower-quality domestic products. It is also observedthat aid disbursements are often linked to the achievement of specific foreign
policy objectives, thereby favoring some countries for political or strategic reasons.
These different motives for giving aid are discussed in McKinlay and Little (1979),
Maizels and Nissanke (1984), and Kemp and Kojima (1985) inter alia. Hogendorn(1995).and Hook (1996) provide recent surveys of the literature.
Given the importance of tying, the literature distinguishes between foreign aidthat is tied---where recipients are formally obligated to reciprocate by buying thedonor's exports---and aid that is untied, where there is no such formal obligation.
While the donor interest theories have addressed the importance and incentive
for giving tied aid, an economic motivation for giving untied aid has not yet beenformally analyzed.
In this paper we argue that giving untied aid generates a stock of goodwill fora donor. The idea parallels the marketing literature where brand loyalty (orgoodwill) may be created through advertising and sponsorship of sporting andcultural events. A model of foreign aid is developed where tied and untied aid,
each in their own way, increase the exports of a donor country. Unlike tied
assistance, untied aid benefits the donor through future exports generated bybuilding and maintaining a stock of goodwill in the recipient country. The donor's
choice between tied and untied aid is modelled as an optimal control problem,
and the nature of the optimal adjustment path for tied and untied aid ischaracterized.
The model generates a pair of non-linear simultaneous differential equations
describing the evolution of the two types of aid, which are estimated using dataon foreign aid by seventeen OECD countries over the period 1972-1990. The
estimates provide support for the analytical model that is developed and show
equilibrium to be a saddlepoint with no change in the level of untied aid along
the stable adjustment path.
This paper is organized as follows. Section II contains a brief discussion on thehistorical origin of "goodwill". Section III presents the theoretical analysis andresults. Section IV provides a description of the data and estimation procedure,
and reports the empirical findings. Section V concludes.
2.

BACKGROUND

The literature on private income transfers offers two alternative hypotheses
concerning motivation for inter vivos (i.e., between living persons) transfers:
altruism and exchange (see, e.g., Cox 1987). Exchange contains a quid pro quo.
In the case of altruism, transfers are made based on care, principles of ethics, or

TIED

AND

UNTIEDFOREIGN

AID

73

simple charity.' We view giving of both types of foreign aid as an exchange. The
difference is that while tied aid involves an explicit quid pro quo, untied aid works
through generating goodwill and entails an implicit quid pro quo.
The concept of goodwill is far from new. It was first taken up by accounting
theorists.2 Adam Smith used the word "sympathy" which serves just as well.
Goodwill has been described as an intangible asset that capitalizes on preferential
use by customers based on certain facts of human nature (habits, propensities,
beliefs, social customs, etc.). Commons (1919) observed that as well as being
intangible, goodwill is fragile and needs continuous upkeep through continuous
repetition of some type of activity.
Nerlove and Arrow (1962) introduced the idea of goodwill in an advertisingmodel. Advertising creates a stock of goodwill which affects the demand for a
product. However, the stock of goodwill is subject to depreciation. Subsequent
work presented and/or estimated models dealing with optimal advertising andgoodwill over time, as well as a discussion of brand loyalty, purchase decisions,market shares, and competition. Examples of this large literature can be foundin work cited by Sethi and Thompson (1981), Fershtman (1984), and Thomas(1989) inter alia.
The argument used in this paper was inspired by Nerlove and Arrow's paperon advertising. An intertemporal optimizing model for donor countries isdeveloped to analyze the dynamics of tied and untied aid. By distinguishing between
these two types of aid, our setting permits a richer treatment of adjustment
dynamics of foreign aid than if aid in the aggregate were considered .
3.

THEORETICAL ANALYSISA.

The Model
Consider a world with two types of countries , those which give and those which
receive foreign aid. There are two types of foreign aid: tied (Al) and untied (A2).
We focus our attention on the role of each type of aid in generating sales/exports(S) for a donor country in a recipient country. S depends on the flow of Al andthe stock of goodwill (G) the donor has generated in the recipient country , where
G is produced from a flow of A2 over time. This is analogous to Nerlove andArrow's formulation where advertising goodwill is generated from a flow ofadvertising messages. The stock of goodwill is assumed to depreciate at a constantrate [3>0 since recipient countries over time forget the generosity of a donor . The
model that is developed is dynamic, since goodwill is fundamentally a dynamic
concept. It is set in continuous time. Since the model is partial equilibrium innature, the analysis is exclusively positive .
1 There are some reaso

ns why even charity may have motives besides altruism . The reasons are
based on genetics, signalling , or insurance (see Nelson 1984 for a discussion).
2 This paragraph draws
on Enders (1985) which contains a discussion of some early definitions of
goodwill.

74

B. MAK

ARVIN

andCHRISTOPHER

F. BAUMA donor country choosesits levelsof tied and untied aid over time to maximize
the present discounted value of its net benefits(the differencebetween generated
salesand foreignaid costs).AssumingS takes a Cobb-Douglasform, the problemismax[{At(t)}a{G(t)}b-AJO- A2(t)]e-P`disubject to6(t)=a[A2(t)]v— f3G(t)(1)

and G(0)= g (the initial levelof goodwillis given at some positive levels).This is
an infinite horizon optimal control problem with G(t) as the state variable andAI(t) and A2(t) as the control variables. a and b are respectively,tied aid andgoodwill elasticitiesof sales, p> 0 is the constant instantaneous rate of discountand a > 0, y > 0 are parameters. Equation (1) characterizes the evolution of
goodwill,which depends on the flow of untied aid, taking into account naturaldepreciation on the stock of goodwill.'
The maximization problem above is solved by forming the current-value
Hamiltonian and writing the following set of first-order conditions (time scripts
are suppressedfor convenience).
Al: aAl- lGb-l=0(2)A2: —1+ payil - t = 0(3)G:,u:+µ13+,up=µ(4)cAz-iG=G(5)with the transversality conditions R(0) free, rim,„ kt(t)e-Pt=0, where ,u is thecurrent-value co-state variable associated with (1).
Equations (2) and (3) state that, at each t, the marginal benefit of giving any
type of aid must equal its marginalcost. Equation (4) has a standard interpretation
as the adjustment equation for the marginal value of goodwillalong an efficient
path.
Using(2) and (3)to find expressionsfor G and u respectively,time differentiatingthese, and using (4) and (5) allows us to findA,---=-------—fib+ abAl [1 -a]
[l-a]

A?

[p + ]

ab

A2

[l-y]

a[l -y][a] t/bAla- t1/bAz(6)[arbArb—ti/bAz-l(7)

3 An economic justification for this is that a donor country has already established some goodwill
through a positive reputation for the quality of its products.
Throughout the paper round rackets enclose arguments of a function, and a dot over a variable
denotes the derivative of that variable with respect to time t.

TIED

AND

UNTIEDFOREIGN

75

AID

Paired together, (6) and (7) constitute a system of two simultaneous non-linear
differential equations in A, and A2 describing the optimal evolution of the control
variables. They are used to analyze the optimal dynamic paths of tied and untied
aid.

B. The Results
Equations (6) and (7) could be rewritten in the form
Al=f (Al, A2) ,A2 =g(Al, A2)

To solve the model, linearize these equations around (Ai , Al)---the point whereA1= 0 and A2= 0—to obtain[All=J(Al,A2)dA1
A2dA2
where dAi(i =1, 2) is deviation of Al from its steady state value A*, and whereJ(A i , Ai ), the Jacobian of the above system of equations evaluated at (Ai , AD, is
Of (A*
12 A*) OA,Of(At,AD
1'
0A2

ag

ag
(At,A*)
1~z
aA2

aA,(Al,Al)The partial derivatives, evaluated at the steady state values of A, and A2 are given
by
of(Ar

, AD=[a—l]zof-----(At,,bAlo,42ag A*[1—a—b]2
AD=---------------A,(1,2)bA1

Az)— yz OA
A2
cg[1—yr ----il) ---------- O
3A2(A' A—A2

where
Z =---------al)[—a]1 a] 1/bAla+b-1 ]/bAz = fib`4lat—aequilibrium , [1andz=ayb[]1/b[
a+b-lllbY_a[ l— Y]aAlA2—1—at[P+~]A2
Y

equilibrium.

Setting A, = 0 in (6) and A2=0 in (7) and combining them gives A
2= byfsAl/a[f3+p]. Using all of the above allows calculation ofTrace of

J(Al , AD= p(8)76

B. MAK

ARVIN

Determinant ofand

CHRISTOPHER

F. BAUM

J(A*, A2)_ ~[fl+P][a+by —1][1 -a][1 -y](9)Equations (8) and (9) are used to prove the following two propositions.
PROPOSITION
1. Equilibrium is a saddlepoint.
Proof: From (8) the trace is positive since p> 0. For a saddlepoint equilibrium
the determinant has to be negative which, from (9), is satisfied since a+ by < 1.5'6
PROPOSITION
2. The movement of tied and untied aid along the stable arm is in
opposite directions if there are decreasing returns to scale, and in the same direction
if there are increasing returns to scale. Furthermore, untied aid will not adjust if
there are constant returns to scale.

Proof: With a saddlepointequilibrium,the solutionfor A, and A2 along thestable arm is givenin general form as
dA, = cf2ie''l ,dA2= cQ2e'`where c is an arbitrary constant determined by the initial value of the program,
Ais the stable (negative)eigenvalueof the J(Al, Al) matrix, and Qt(i=1, 2) are
the elements of the eigenvectorassociatedwith A.Using the equations above, therelationship between A, and A2 along the stable trajectory can be expressed as
dA2= [S22/S2,]dA
t, where—cg(Al, A<)
Q2=
3A, -- =yf3[a+b-l](10)Q, ag(A*,A'2) —A,{[1—y]a}{[/3+p]—A}
3A2

Equation (10) characterizes the slope of the stable trajectory. Since A<0, it isevident from (10) that sign[S22/Q,] = sign[a + b —11. As a result, A, and A2 will
move in opposite directions along the stable adjustment path if a + b < 1; they will
move in the same direction if a + b> 1; and A2 will not adjust if a + b = 1, that is,with constant returns to scale.
4.

EMPIRICAL ANALYSIS

We test our model on a sample of annual data from 17 OECD countries for
1972-1990. The original data series include donors' nominal levels of tied anduntied aid, measured in U.S. dollars (OECD 1973-91). These are transformed
5 If a + by > 1, the second-order condition in general is not satisfied.
6 The trace is the sum of the eigenvalues associated with J(A;, Az). The determinant is the productof the eigenvalues. For the system to exhibit saddlepath (local) stability, the Jacobian must possess
one negative and one positive eigenvalue. Hence the determinant of the Jacobian must be negative.

TIED ANDUNTIEDFOREIGN

77

AID

into real values using the country-specific GDP deflator and U.S. dollar exchange
rate, both from IMF (1993). Percentage rates of change of the resulting real series
were created to provide discrete approximations to the left-hand side values ofequations (6) and (7), reducing the time series for each country to the 18 years
1973-1990. The data set is thus a panel of 17 countries over 18 years, with 211
usable observations.
Equations (6) and (7) relate, respectively, percentage changes in tied and untied
aid (DTIED, D UNTIED) to the levels of each aid category via six parameters(/3,p, y, b, a, a), five of which appear in both equations, as p only appears in (7).
The equations are highly nonlinear, with the parameters appearing in both additive and multiplicative forms. Our estimates are derived from the pooled
cross-section/time-series data set, using a nonlinear least squares systems estimator(RATS' NLSYSTEM) which takes advantage of the cross-equation correlationsof the error processes.
We find that the data contain insufficient information to support joint estimationof the entire parameter vector. Thus, we have relied on a strategy of fixing one
parameter, /3, at a plausible value and estimating the other five parametersconditional on /3. We then vary /3 through a range of values to evaluate thesensitivity of the empirical results to those arbitrary values. Values for /3 between
zero and 0.20 represent sensible values for the depreciation of goodwill, given thatsome donor-client relationships may be very long-lasting, and others may markedly
decay unless they are rejuvenated with fresh infusions of untied aid.
With this modification, the estimation is quite successful, with values for eachof the model's parameters which are qualitatively plausible and reasonably precise.
Table 1 presents our estimates for selected values of /3. As this parameter is varied
from 0.01 to 0.20, the fit of the two equations (as judged by their respective
standard errors of estimate) varies only slightly. The discount rate p can only be
distinguished from zero at levels of /3 below 0.08, but possesses a positive point
estimate throughout the range of /3values. Both y and b are very precisely estimated:both differ from zero at better than the one percent level throughout, and theirpoint estimates are quite similar for all /3 values. The b parameter (the goodwill
elasticity of sales) can never be distinguished from unity. Parameter a, the tiedaid elasticity of sales, cannot be precisely estimated, as it cannot be distinguishedfrom zero for any value of /3. The sum of estimated parameters a and b cannot
be distinguished from the threshold value of unity above the 70 per cent level ofconfidence for any value of /3. Thus, we cannot reject constant returns to scale
for the sales function. Last, estimates of parameter a are quite stable throughout
the range of /3 values, and can readily be distinguished from zero .'To summarize, since our empirical investigation revealed the value of a + b to
be near unity, there is no adjustment in the level of untied aid along the stabletrajectory. Hence to reach steady state, donor countries maintain a constant flow
It is worthwhile

noting

that

point

estimatesof a + by are

all less than

one .

78

B. MAK

TABLE

0.03

0.05
0.08

0.10

0.12

0.15
0.18

0.20

and

CHRISTOPHER

F. BAUM

1. Estimates of the Model for 17 OECD Countries, 1973-1990/3 value
0.01

ARVIN

p

y

baa

0.260
(0.11)
0.234(0.11)
0.209

0.710(0.20)
0.728

1.151(0.18)
1.131(0.19)
0.742(0.18)
0.758(0.16)
0.766(0.16)
0.774(0.15)
0.783(0.14)
0.791(0.14)
0.796(0.13)(0.17)
1.115(0.10)
0.172(0.10)
0.149(0.10)
0.125(0.10)
0.090(0.10)
0.056(0.10)
0.033(0.10)(0.16)
1.095(0.15)
1.084(0.14)
1.073(0.14)
1.059(0.13)
1.045(0.13)
1.037(0.12)

0.033(0.06)
0.039(0.06)
0.045(0.06)
0.053(0.06)
0.059(0.06)
0.064(0.06)
0.072(0.06)
0.078(0.06)
0.084(0.06)

0.104(0.09)
0.116(0.09)
0.126(0.09)
0.139(0.09)
0.147(0.09)
0.155(0.09)
0.165(0.09)
0.174(0.09)
0.180(0.09)

SER„

SER,

1.70

1.96

1.70

1.96

1.70

1.97

1.70

1.97

1.71

1.98

1.71

1.98

1.71

1.99

1.72

2.00

1.72

2.00

Notes: Estimates via nonlinear least squares applied to (6) and (7), conditional on the value of/3. Asymptotic standard errors are given in parentheses. The columns SER„ SER, contain standard
errors of estimate of the untied and tied aid equations, respectively.

of untied aid, while adjusting the flow of tied aid over time. The intuition for thisfinding is that tied aid has a more direct impact on exports than untied aid.
Nonetheless, untied aid persists through time at a constant rate in order to help
maintain a donor country's level of goodwill.
5.

CONCLUSION

Development economics has begun to recognize the importance of distinguishingthe impact of different types of foreign aid. As White's (1992) survey article notes,
this has been a neglected area of research. This paper distinguished between tiedand untied aid by hypothesizing the latter to generate goodwill. Using anintertemporal maximizing model it was demonstrated that a mix of tied and untied
aid is given in order to maximize the return to a donor. In particular, it was shownthat a donor maintains a constant flow of untied aid in order to continually
replenish its stock of goodwill. Our estimates of the model's parameters weregenerally in accordance with theoretical restrictions on their values. This sample
of major donor countries provides support for the analytical model presented
above. Our findings, it should be stressed, are only suggestive given the limitationof the data in terms of the length of the time series, as well as omission of otherfactors that impact on foreign aid. As indicated in Section I, some of these factors
include political and other economic considerations besides export promotion.

TIED

Nonetheless,thedifferentiation

oughtpromotingexports

results
to

do

AND

providebe madeis a primary

UNTIEDFOREIGNsome

betweenmotivation

the

79

AID

supportforeffectsof tiedbehindprovidingthenotionandforeignuntied

that

a

aid

if

assistance.

REFERENCES

Commons, J. R. (1919) Industrial Goodwill, New York.
Cox, D. (1987) "Motives for Private Income Transfers." Journal of Political Economy 95, 508-46 .
Enders, A. M. (1985) "Veblen and Commons on Goodwill: Case of Theoretical Divergence." History
of Political Economy 17, 637-49.
Fershtman, C. (1984) "Goodwill and Market Shares in Oligopoly." Economica 51, 271-81.
Gillis, M., D. H. Perkins, M. Roemer, and D. R. Snodgrass (1992) Economics of Development , 3rdedition, New York: W. W. Norton and Co.
Hogendorn, J. S. (1995) Economic Development, 3rd edition, New York: Harper and Row Publishers ,Inc.
Hook, S. W. (1996) Foreign Aid Toward the Millennium, Boulder, Colorado: Lynne Rienner Publishers .
IMF (1993) International Financial Statistics, International Monetary Fund , Washington, D.C.
Kemp, M. C., and S. Kojima (1985) "Tied Aid and the Paradoxes of Donor-Enrichment andRecipient-Impoverishment." International Economic Review 26, 721-9.
Maizels, A., and M. Nissanke (1984) "Motivations for Aid to Developing Countries ." World
Development 12, 878-900.
McKinlay, R. D. and R. Little (1979) "The U.S. Aid Relationship: A Test of the Recipient Need andDonor Interest Models." Political Studies 27, 236-50.
Nelson, P. (1984) "Charity and Self-Interest," Mimeographed, SUNY Binghamton , New York.
Nerlove, M. and K. J. Arrow (1962) "Optimal Advertising Policy Under Dynamic Conditions ."Economica 29, 129-42.
OECD (1973-91) Development Co-Operation: Efforts and Policies of the Members of the Development
Assistance Committee, Annual Reports, Paris: OECD.
Sethi, S. P., and G. L. Thompson (1981) Optimal Control Theory: Management Science Applications .
Boston: Martinus Nijhoff Publishing Company .
Thomas, L.G. (1989) "Advertising in Consumer Goods Industries: Durability , Economies of Scale,and Heterogeneity." Journal of Law and Economics XXXII , 163-93.
White, H. (1992) "The Macroeconomic Impact of Development Aid: A Critical Survey ." Journal of
Development Studies 28, 163-240.