Foreign aid and transboundary pollution in a
dynamic factor endowment model
Kenji Fujiwara∗
School of Economics
Kwansei Gakuin University

Norimichi Matsueda
School of Economics
Kwansei Gakuin University

December, 2006

Abstract
Constructing a dynamic model of international trade with transboundary pollution, this paper examines the welfare effects of untied foreign aid. By allowing forthe dynamic effect of foreign aid on capital and pollution accumulation, foreign has
various possibilities. Two outstanding results are: depending on the conditions, (i)both the donor and recipient lose from foreign aid, i.e., aid is Pareto inferior, and(ii) both gain from it, i.e., aid is Pareto improving. These results shed light on the
dynamic effect neglected in the existing literature on foreign aid.
Keywords: transboundary pollution, capital accumulation, Pareto-improvement,
Pareto-impoverishment.
JEL classification: F10.

∗

Corresponding author.
School of Economics, Kwansei Gakuin University.
1-1-155 Uegahara, Nishinomiya, Hyogo, 662-8501, Japan. Tel: 81-798-54-7066. Fax: 81-798-51-0944. Email:kenjifujiwara@kwansei.ac.jp.

11Introduction

It has been argued that international trade and foreign aid have been two significant
forces for economic growth and development of the world as a whole. In a neoclassicalframework, it is normally concluded that the opening of free trade benefits all the trading
countries and the recipient gains from foreign aid, while the donor loses from it.1 Looking
at the actual cases, however, this theoretical prediction has not always been observed.
Then, what causes the prediction to fail?
The question raised above has brought voluminous literature which shows a possibility of losses from trade and the transfer paradox. Focusing on the transfer paradox, theviolation of Warlasian stability of the world commodity market, the presence of tariffs
and other trade impediments, untied aid, and the presence of the third party can potentially induce the welfare improvement for the donor and the welfare deterioration for
the recipient. From another perspective, by constructing a two-country dynamic model
without market distortions, Shimomura (1999) demonstrates that indeterminacy of thesteady state can become a source for the paradox.
In addition to such development of the literature, there are recent works which investigate under what conditions foreign aid becomes Pareto-improving for the world. Naito(2003) shows this possibility in a static two-country model with transboundary pollution.
With the aid of a dynamic two-country model, Shimomura (2006) proves that foreign aid
benefits both the recipient and donor when indeterminacy is present.
This paper also explores the welfare consequences of foreign aid in a dynamic modelwith transboundary pollution. However, our approach has a clear deviation from Naito(2003) and Shimomura (2006) in the sense that we assume two small open countries sothat the terms of trade effect is a priori abstracted. It is well-known that the termsof trade effect plays a key role for the transfer paradox and Pareto-improving foreign
aid. Nevertheless, we show that the terms of trade effect is not the only source forthese outcomes. Instead, we pay close attention to the dynamic effect of foreign aid on
capital accumulation as well as pollution accumulation. To grasp our argument concisely,
1

We sometimes call this recipient-enrichment and donor-impoverishment result ”Samuelson’s theorem”, following Brakman and Marrewijk (1998).

2

suppose a Heckscher-Ohlin world with transboundary pollution. In this framework, weshow: foreign aid increases the steady state capital stock in the donor country, which
accelerates the transboundary pollution when the dirty good is capital-intensive throughthe Rybzcynski theorem. This pollution expansion effect has the following possibility
neglected in the existing literature. Foreign aid is Pareto inferior, namely, not only thedonor but also the recipient loses from it.
The plan of the paper is as follows. Section 2 lays out the basic model. Section
3 addresses the saddle point stability of the steady state. Based on these preliminary
results, Section 4 considers the welfare effects of permanent foreign aid and Section 5
provides a concluding comment. Furthermore, the Appendix contains a detailed proof ofthe saddle point stability of the steady state.

2

The model

The model constructed here is comprised of two countries (Home and Foreign), two goods(Goods 1 and 2), and two primary factors (capital and labor). Good 1 is a consumption
good and Good 2 is an investment good. Throughout this paper, we assume that bothcountries are so small that they take the world price as exogenously given.2 Good 2
serves as a numeraire so that its price is normalized to unity and p denotes the world
price of Good 1. The two goods are produced subject to the standard neoclassical production technologies. Then, the production side can be completely described by the GDP
function:3
G(p, K, L) ≡max {pF1 (K1 , L1 ) + F2 (K2 , L2 ) | K1 + K2 ≤ K, L1 + L2 ≤ L},

Ki ,Li ,i=1,2

where K and L denote the Home endowment of capital and labor, Ki and Li , i = 1, 2, theinputs of capital and labor in each sector, and Fi (·) each sector’s production function,
which satisfies all the neoclassical properties including the linear homogeneity. Then,
2This assumption has been adopted in the existing literature, e.g., Hatzipanayotou et al. (2002, 2005).
On the other hand, Chao and Yu (1999, 2005) consider a similar model which explicitly incorporates theterms of trade effect.
3
Any interested reader in the detailed properties of the GDP function is referred to Dixit and Norman(1980), Woodland (1982), and Wong (1995).

3the equilibrium output of Good 1 equals the partial derivative of G(·) with respect to p,
which is denoted by Gp (p, K, L).
Concerning the GDP function, it is instructive to make a brief comment here since,
while we have given a general definition of the GDP function above, we shall specify it in
the subsequent sections to sharpen the analysis. Concretely, we employ two types of the
GDP function in this paper. The first is a standard Heckscher-Ohlin type. As long as thetwo goods are produced from both factors and the diversification condition is satisfied,
the factor prices becomes a function of only the commodity price, namely, r = r(p) andw = w(p), where r and w respectively give the capital rental rate and wage rate. Then,the GDP function takes the form ofG(p, K, L) ≡ r(p)K + w(p)L,(1)

and the equilibrium output of Good 1 is equal to r0 (p)K + w0 (p)L. The sign of r0 (·)
and w0 (·) depends on whether Good 1 is capital-intensive from the Stolper-Samuelson
Theorem.
The second type of the GDP function to be considered is obtained from the assumptionthat only Good 1 is produced from two factors and one unit of labor produces one unitof Good 2. Under this circumstance, the GDP function becomesG(p, K, L) ≡ r(p)K + L,

r0 (·) > 0,

r00 (·) > 0.

(2)

Hence, the output of Good 1 is r0 (p)K. These convenient properties of the specified GDP
functions are made use of as the argument proceeds.
The production of Good 1 causes pollution and its effect is transboundary.4 Without
loss of generality, we assume that one unit of Good 1 produced in Home generates thesame unit of domestic pollution and one unit of the Foreign production contributes to
α > 0 units of pollution in Home. Thus, the total amount of pollution in Home at each
instant is
Gp (p, K, L) + αG∗p (p, K ∗ , L∗ ).
We are now ready to set up our dynamic general equilibrium model. Following Shimomura (2006), we assume that only the representative consumer in Home faces an
4

Naito (2003) also shares the same presumption.

4

intertemporal utility maximization problem, while the Foreign consumer behaves as astatic utility maximizer. Then, the Home consumer’s problem is formulated asmax
subject toZ ∞0e−ρt [ln C − v(Z)] dt,

v 0 (·) > 0,v 00 (·) > 0

K̇ = G(p, K, L) − pC − δK − T
Ż = Gp (p, K, L) + αG∗p (p, K ∗ , L∗ ) − γZ,

δ, γ ∈ (0, 1),

where C and Z are respectively the consumption of Good 1 and the pollution stock,T > 0 the international transfer from Home to Foreign, and δ and γ the depreciation rateof capital and the pollution stock.
In order to solve this problem, let us set up the current-value Hamiltonian:H ≡ ln C − v(Z) + λ[G(p, K, L) − pC − δK − T ] + µ[Gp (p, K, L) + αG∗p (p, K ∗ , L∗ ) − γZ].
where λ and µ are respectively the co-state variables associated with Home’s capital stock,K, and its pollution stock, Z. Then, the optimality conditions are derived as follows:
1
C

= λp

=⇒

C=

1
λp

1
− δK − TλŻ = Gp (p, K, L) + αG∗p (p, K ∗ , L∗ ) − γZ(4)λ̇ = λ[ρ + δ − r(p)] − µr0 (p)(5)µ̇ = µ(ρ + γ) + v 0 (Z)(6)

0 = lim e−ρt λK = lim e−ρt µZ.

(7)K̇ = G(p, K, L) −

t→∞

t→∞

(3)

Note that (5) is obtained by the use of GK (p, K, L) = r(p) from the foregoing restriction
on the GDP function. Four equations consisting of (3)-(6) determine four variables, e.g.,
K, Z, λ and µ.
This section is closed by describing the Foreign consumer’s behavior. The assumptionthat the Foreign consumer solves a static problem of utility maximization makes its
indirect utility function be given byu∗ (C ∗ ) − v ∗ (Z ∗ )
!
∗
∗
∗


∗ G (p, K , L ) + T= u− v ∗ α∗ Gp (p, K, L) + G∗p (p, K ∗ , L∗ ) ,
p
5

(8)

where u∗ (·) and v ∗ (·) stand for the utility/disutility function of Foreign. The equality
above follows from the ‘static’ budget constraint:pC ∗ = G∗ (p, K ∗ , L∗ ) + T,
and the definition of the ambient pollution level in Foreign country:
Z ∗ = α∗ Gp (p, K, L) + G∗p (p, K ∗ , L∗ ).
Thus, we assume that Foreign regards this pollution issue simply as flow pollutionin which the scale of environmental damages is determined by the amount of current
emission amounts alone. We can interpret such a case as the one where the environmentis in such a healthy state there that its “assimilative capacity” can prevent the pollutant
from accumulating in its environmental body. Alternatively, we can also think of thissetup as a case where this nation behaves rather myopically in the sense that it completely
ignores the possible future consequences of the current emissions.
This completes the description of the world economy in consideration. From the next
section, we shall make use of this model to investigate the welfare effects of foreign aid
in a dynamic context.

3

Stability of the steady state

Before conducting the main task of this paper, this section examines the stability propertyof the steady state in our model. Since only Home faces a dynamic optimization problem,the present dynamic general equilibrium model is comprised of four equations (3)-(6). The
characteristic equation associated with the system can be obtained as
1
0
r−δ−x
0
λ2
r0
−γ − x00
00
ρ+δ−r−x
−r0
00
0
v
0
ρ+γ−x
i
1 h
2
= − 2 λ2 (γ + x)(ρ + γ − x)(r − δ − x)(ρ + δ − r − x) − (r0 ) v 00
λ1
≡ − 2 Γ(x) = 0,
λ

6

(9)

where x denotes the eigenvalue of the Jacobian matrix evaluated at the steady state suchthat K̇ = Ż = λ̇ = µ̇ = 0.
Based on the characteristic equation of (9), we can prove the saddle point stability ofthe steady state. This statement is formally summarized in:
Proposition 1. The steady state of the present dynamic general equilibrium model issaddle point stable if ρ + δ − r < 0.
Proof. See Appendix. Q. E. D.

4

Welfare effects of foreign aid

Having confirmed the saddle point stability of the steady state, we are now in a position
to do a comparative statics analysis in the steady state and evaluate the effects of apermanent increasing in the amount of foreign aid.
Totally differentiating (3)-(6) with respect to the endogenous variables and T yields










1
0
dK
1
r−δ 0
λ2




 r0
−γ
0
0   dZ   0 



=
 dT.


0
0 ρ + δ − r −r0   dλ   0 
0
v 00
0
ρ+γ
dµ
0

(10)

Letting the determinant of the coefficient matrix be denoted by ∆, it is given by
1
0
r−δ 0
λ2
r0
−γ
0
0
∆ ≡
0
0 ρ + δ − r −r0
0
v 00
0
ρ+γ
i
1 h
2
= − 2 λ2 γ(ρ + γ)(r − δ)(ρ + δ − r) − (r0 ) v 00 > 0,λunder the saddle point stability. By resorting to the standard procedure, the effects of asmall change in T on K, Z and λ are respectively derived as follows:
dK
dT

=

−γ(ρ + γ)(ρ + δ − r)
>0
∆

(11)

dZ
dT

=

−r0 (ρ + δ − r)(ρ + γ)
∆

(12)

7

dλ
dT

=

(r0 )2 v 00
>0
∆

(13)

r0 v 00 (ρ + δ − r)
dµ
=
.
dT
∆
These comparative statics results have clear interpretations. From (11), foreign aid
enhances capital accumulation in the donor country. This is because the permanent
provision of foreign aid reduces Home’s disposable income in each point of time, which
decreases the consumption as well. Thus, saving and capital accumulation are enhanced,
from which the capital stock at the steady state increases. This, in turn, favorably affects
its welfare from (13) since λ represents the marginal utility of K, namely, the shadow
price of K and the steady state capital increases as a result of aid. A similar argument
applies to (12). The effect of aid on Z and its shadow price µ depends on the factor
intensity ranking between the goods.
In the first benchmark case of the Heckscher-Ohlin technology, dZ/dT > 0 if and only
if Good 1 is capital-intensive, i.e., r0 > 0. On the other hand, in the second example of(2), dZ/dT > 0 necessarily follows from r0 > 0. We can intuitively understand this byrecalling the Rybzcynski type relationship in a factor endowment model. That is, (11)convinces us that foreign aid increases the steady state capital stock. Then, the output
of Good 1 also increases under r0 > 0.
The comparative statics results above are now utilized in order to assess the welfare
effects of foreign aid in the steady state. For this purpose, let us define each country’s
welfare as a function of T . The felicity of Home in the steady state isln C − v(Z)!
1
− v(Z)= ln
λp= − ln λ − v(Z) + constant≡ U (λ(T ), Z(T )).
The Foreign counterpart is given by (8):u∗ (C ∗ ) − v ∗ (Z ∗ )8(14)G∗ (p, K ∗ , L∗ ) + T= up∗≡ U (T, K(T )).
∗!

− v ∗ (α∗ Gp (p, K, L) + G∗ (p, K ∗ , L∗ ))(15)

Accordingly, a permanent increase in foreign aid affects each country’s welfare as
follows:
dU
dT

= Uλ

dλ
dZ
+ UZ
dT
dT

=

−1 dλ
dZ
− v0
λ dT
dT

=

−1 (r0 )2 v 00
r0 (ρ + δ − r)(ρ + γ)
+ v0
,
λ
∆
∆

(16)

and
dK
dU ∗
≡ UT∗ + UK∗
dT
dT
0

= u∗

1
γ(ρ + γ)(ρ + δ − r)0
+ v ∗ α∗ r 0
.
p
∆

(17)

First of all, (16) implies that we can decompose the impact of foreign aid on Home’s
welfare into two terms. The first term in (16), which captures the direct income effect, isalways negative, whereas the second term can be either positive or negative, dependingon the factor intensity ranking between the two goods. Especially, it is negative (resp.
positive) if and only if r0 > 0 (resp. r0 < 0). The intuition behind this result is quite
simple. As mentioned above, foreign aid increases the steady state pollution under r0 > 0
due to the Rybczynski effect. This enhances the negative externality due to the pollution
and makes the second term come out negative. The parallel applies to the case withr0 < 0.5
In a similar way, (17) indicates that foreign aid has two distinguishable effects on
Foreign’s welfare. The first term in (17) is a direct income effect, which is necessarily
positive. On the other hand, the second term is negative (resp. positive) if and only if
5Note that the second term never appears in a static model since Z = Gp (p, K, L) + αG∗p (p, K ∗ , L∗ )is not affected by the change in T .

9

r0 > 0 (resp. r0 < 0). The intuition goes as follows. An increase in T induces the steadystate value of K to go up because of (11). This increase in K in turn contributes tothe rise in Z if Good 1 is capital-intensive and hence its effect on the Foreign welfare is
detrimental.
Based on the results derived above, we can prove two propositions as regards thewelfare implications of foreign aid. The first is:
Proposition 2. Foreign aid is Pareto-improving under r0 < 0 and v 00 ≈ 0.
Proof. Under r0 < 0, Foreign gains from the transfer regardless of the factor intensity
ranking. Moreover, an increase in T decreases Z under r0 < 0, which favorably affects
the Home welfare. Hence, this positive effect dominates and Home also gains from giving
foreign aid under v 00 ≈ 0. Q. E. D.
While Proposition 2 makes an affirmative assessment on foreign aid in the presenceof transboundary pollution and dynamics, completely the opposite conclusion is also
possible. That is:
Proposition 3. Foreign aid is Pareto-inferior under r0 > 0 and if α∗ is sufficiently large.
Proof. It is trivial that Home loses from the transfer since both the direct effect and theeffect through the increased pollution have a negative impact on U . Furthermore, if α∗
is sufficiently large, the negative effect dominates in the overall impact on the Foreign
welfare. Hence, both countries lose from the international transfer in this case. Q. E. D.
Remark 1. While this paper has focused on the theoretical possibility of financial aid’s
being Pareto-inferior as well as Pareto-improving, Samuelson’s Theorem and the transfer
paradox are also possible in our model.6 For example, suppose r0 < 0, i.e., Good 1 is
6

Samuelson’s Theorem is referred to the result that aid has a negative effect on the donor’s welfare
and positive effect on the recipient’s welfare. This terminology is borrowed from Brakman and Marrewijk(1998).

10

labor-intensive. Then, Foreign necessarily gains from aid but Home possibly loses from it.
In particular, this is likely to occur if v 00 is sufficiently large. In an analogous fashion, thetransfer paradox, namely, the donor-enrichment and recipient-impoverishment transfer
tends to be possible if r0 > 0 and α is sufficiently large.
Remark 2. We should now differentiate our result from those of the predecessors. As
we have repeatedly mentioned above, our result is not obtainable in a static model. If wereduce our model into a static one, Samuelson’s Theorem always holds. This is because
we have assumed way the terms-of-trade effect of foreign aid. Hence, the presence oftransboundary pollution plays no important role in such a model.7
On the other hand, Shimomura (2006) proves that Pareto-improving foreign aid istheoretically possible in a dynamic model with the terms-of-trade effects. In his argument,
indeterminacy of the steady state plays an decisive role in obtaining the outcome of
the Pareto-improving transfer. However, we have shown that a Pareto-improvement ispossible even if the steady state is saddle point stable.

5

Concluding remarks

Highlighting the environmental interrelationship of the modern world economy, this paper
has identified the potential welfare effects of foreign aid in a dynamic model. We haveshown that foreign aid can have drastically different implications via the channels ofcapital and pollutant accumulations. Under certain conditions, foreign aid immiserizes
both the donor and recipient by worsening the transboundary pollution issue. In contrast,
it can be Pareto-improving in the sense that both countries gain under other sets ofconditions. We have seen in this article that the factor intensity ranking and the degreeof negative externalities from the pollution play important roles in determining the welfare
implications of foreign aid.
Our analysis complements the recent contribution made by Hu, Nishimura and Shimomura (2006) which explores the dynamic effect of capital accumulation on the comparative
7

Note that the same does not apply to another model. Naito (2003) shows the Pareto-improving
transfer in the context of transboundary pollution.

11

statics results addressed by Jones (1971). In this sense, it is inappropriate to conjecturethat the static results carry over to dynamic models even if the analysis is confined tothe steady state.
In this article, we deliberately ignored the terms of trade effect of foreign aid so
as to facilitate and sharpen the analysis, following Hatzipanayotou et al. (2002, 2005).
However, this simplifying assumption could be unacceptable depending on the situations.
It is left to our future study to incorporate the terms of trade effect although the model
becomes immensely complicated.

Appendix: proof of Proposition 1
What we shall show below is that Γ(x) in the characteristic equation (9) has two negative
and two positive eigenvalues. To this end, let us expand Γ(x) as follows:2

Γ(x) ≡ λ2 (γ + x)(ρ + γ − x)(r − δ − x)(ρ + δ − r − x) − (r0 ) v 002

= −λ2 [x2 − ρx − γ(ρ + γ)][x2 − ρx + (r − δ)(ρ + δ − r)] − (r0 ) v 00
≡ −λ2 X 2 − λ2 [(r − δ)(ρ + δ − r) − γ(ρ + γ)]X + λ2 γ(ρ + γ)(r − δ)(ρ + δ − r)2

− (r0 ) v 00
= 0,
where X ≡ x2 − ρx = x(x − ρ). Under ρ + δ − r < 0, the above quadratic equation of X
has two positive solutions. From the definition of X, this implies that there can be threepossibilities such that (i) all values of x are negative, (ii) all values of x are larger thanρ, and (iii) two values of x are negative and the other two are larger than ρ. The rest of
our tack is to show that only (iii) survives.
To see this, take another careful look at the characteristic equation:
Γ(x) = −λ2 x2 (x − ρ)2 − λ2 [(r − δ)(ρ + δ − r) − γ(ρ + γ)]x(x − ρ)2+λ2 γ(ρ + γ)(r − δ)(ρ + δ − r) − (r0 ) v 00
= 0.
It is easily seen that Γ(ρ) = Γ(0) = λ2 γ(ρ + γ)(r − δ)(ρ + δ − r) − (r0 )2 v 00 < 0. Relating
this fact to the foregoing arguments, the locus of Γ(x) = 0 is drawn as in Figure 1. The
12

figure convinces us that there are two positive and two negative roots associated with
Γ(x) = 0. This means that the steady state exhibits saddle point stability because wehave two state variables, K and Z, and two co-state variables, λ and µ. This establishes
Proposition 1.

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[12] Shimomura, K. (2006), ‘A Pareto-improving foreign aid in a dynamic North-Southmodel’, in Lahiri, S. (ed.), Theory and practice of foreign aid, 2, Amsterdam: Elsevier
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14

Γ(x)

ρ
O

Figure 1: The characteristic equation: Γ(x) = 0

15x