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Immiserizing growth in a many-commodity setting
Journal
of International
Economics
IMMISERIZING
17 ( lY84)
GROWTH
Received
May
335-345.
North-Holland
IN A MANY-COMMODITY
1983. revised
version
received
September
SETTING
I983
M;lthematical economists have rrccntly shed new light on the phenomenon of immiserizing growth. independently of the trade-theoretic literature on this subject. The present paper aims IO build ~1 beidge between the two literatures. In particular. Mantel’s set of sullicient conditions for non-imnliserizill~ growth is extended to a trade-theoretic model with a smooth production frontier. and Bhagwati’s trade-theoretic set of sullicient conditions to an n-commodity hamework. It is shown that the condition of normality in consumption in the rormer substitutes for the stability condition in the latter. The concept of the semi-compensated substitutability condition plays an essential role in our analysis.
1. Introduction
In a celebrated paper on immiserizing growth, Bhagwati (1958) proved that in a two-commodiiy open economy with a smoothly substitutable production frontier and smooth preferences, immiserizing growth is impossible if (a) growth does not reduce the domestic production of importables at constant relative commodity prices (i.e. normality holds for production); (b) the offer curve of the rest of the world is elastic; and (c) the equilibrium is Walras-stable. Recently, Aumann and Peleg (1975) have rediscovered the phenomenon of immiserizing growth in the context of a closed exchange economy, and constructed a numerical example of this phenomenon. Subsequently, MasColell (1976) and Mantel (1984) have extended the Aumann-Peleg analysis and established new sufficient conditions to preclude immiserizing growth. Translated into the international trade framework, Mas-Colell has shown that, in a two-commodity exchange economy, an increase in initial endowments of the home country will not reduce its welfare if (a) the foreign excess demand functions satisfy gross substitutability and (b) all goods are normal in home consumption. Mantel has proved that this proposition of Mas-Colell’s will *I would like to thank Jagdish Bhagwati and Avinash Dixit ror bringing an earlier version of (1984) and Disit’s unpublished comments on that version IO my attention. Comments on earlier drafis by Bela Balassa. Jngdish Bhagwati. Richard Brecher, Avinash Disit. Ali Khan. Peler Newman. and two anonymous referees are greatly appeciated. Correspondence with Rob Fccnstrn. Masnyoshi Hirota. and Murray Kemp was also helpful.
Mantel
0022-lYY6/84/$3.00
8’
1984.
Elsevier
Science
Publishers
B.V. (North-Holland)
hold in the ri-commodity framework only if a third condition is added. that (c) all commodity pairs are net-substitutes in home consumption.’ The aim of the present paper is to build a bridge between the trade-theory literature and the mathematical economics literature on the phenomenon of immiserizing growth. First, we will generalize Mantel’s theorem to an ncommodity trade world where the home country has a smoothly substitutable production frontier (instead of initial endowments of goods in an exchange economy as in Mantel’s analysis). With this production frontier an assumption on normality in home production is shown to become essential, as in Bhagwati’s Theorem [namely his condition (a)]. Second, we will generalize a slightly weakened version of Bhagwati’s Theorem to an IIcommodity economy. It will then be shown that the only difference between the Generalized Bhagwati Theorem and the Generalised Mas-Colell-Mantel1 Theorem thus obtained is that the former assumes Walras-stability in place of the normality of home consumption in the latter.2 In this paper, we employ a duality framework in constructing the model and proving the theorems.3 The concept of the semi-compensated substitutability condition, to be defined on the text, will play an important role in the present paper. This condition is implied either by the notion of the ‘elastic foreign offer curve’ in the trade-theory literature or by the various substitutability concepts used in the mathematical economics literature. We will define this condition by singing the cross-substitution terms of the semi-compensated demandfunction, also to be defined in the text. The use of this function in the statement of the model is essential for our analysis. The next section presents the model. Two principal theorems are given in section 3. The theorems of Bhagwati, Mas-Colell, and Mantel, as stated by them, are presented in section 4 and related there to the two principal theorems derived in section 3. Appendices A and B prove two lemmas stated in section 3. 2. The model
Assume that there are n goods, l,..., n; and that there are two countries, the home country and the foreign country. Define the functions xi, yi, and 5*i by the following: x’(q,u)= the compensated demand function of the home country for good i, ‘Mantel established, by constructing a concrete numerical example, that a perverse welfare effect may result if(c) is not satisfied. ‘Bhagwati, Brecher and Hatta (1982) briefly compares the theorems of Bhagwati and MasColell in the trade-theoretic framework with two commodities. 3This proor method was introduced into the welfare analysis of international trade theory by Hatta (1973. 1977). Takayama (1974). Chipman (1979) and, most comprehensively, by Dixit and Norman (1980) and Woodland (1982). It was first applied to the study of immiserizing growth by Dixit and Norman.
7: HARA,
Immiserizing
growth
in A many-commodity
331
setting
y’(q, r) = the supply function of the home country for good i, and f*‘(q) = the uncompensated import demand function of the foreign country,
where q =(ql,..., q”)‘= the price vector of the n goods, t =a scalar representing the degree of technological advancement or factor accumulation of the home country, and u =a scalar representing the utility level of the home country.
Also define the function z’ by Z’(q, u, t) E xyq, u) - y’(q, t) + L?*‘(q).
(1)
We will call Z’ the semi-compensated excess demand function of the world for good i. This is the summation of the compensated import demand function of the home country and the uncompensated import demand function foreign country. The following set of equalities must be satisfied at an equilibrium Z’(q, u, t) = 0,
of the
i=l 7..-, n.
(2)
Let the first good be the numeraire, and
Then the mode1 of (2) has n equations and n variables: q2,. . . , q”, and x4 We will analyze effects of a change in t upon q2,. . . , q” and u through this model. 3. The main results
In the present section we prove two theorems: one generalizing the MasColell-Mantel Theorem and the other the Bhagwati Theorem. 3.1. Assumptions
Various combinations present section. (S)
of the following
All commodity
assumptions will be made in the
pairs are substitutes in the functions Z’, i.e.
Zj > 0, if i #j. ‘The budget equation for all the equilibrium conditions satisfied, i.e. cqi(xi - .v’) = 0.
the foreign country, i.e. 1~1 ir*i _ =0, are’satislied, the budget equation
is always satislied. Thus, when for the home country is also
338
‘I: Hatta,
(I,,)
Immiserizing growth in a many-commodity setting
In the home country, there is no inferior good in production,
i.e. yf 20, (I,)
for all i.
In the home country, there is no inferior good in consumption,
i.e. XI 20,
for all i.
We will call (S) the semi-compensated sunstitutability condition. As will be shown in section 4, this condition is implied by various substitutability concepts assumed in the literature. When (4)
an increase in t represents economic growth, since it raises the value of the national output measured at constant prices. In the theorems of this paper, we will effectively assume this inequality by means of (I,). 3.2. A generalized Mas-Cole&Mantel
Theorem
Lemma 1 gives an expression for the welfare effect of an increase in t in the home country. Theorems in this paper will be obtained by signing the determinants INI and IDI defined in this lemma. Lemma 1. In the model of (3), the effect of an increase in t of the home country is represented by
d” INI -=-, dt IDI
(5)
where5
r-z: 2: ... q % the present note, unless otherwise stated, a subscript given to a functional notation indicates the variable or the parameter with respect to which the function is partially differentiated, e.g.
T Hatta,
and
Immiserizing
growth
in a many-commodity
setting
339
z,lz: ... 2-j r D=“:“I ; ... ““, ..
..
Lz: z; ... z;_1 ProoJ Totally differentiating (2) and applying Cramer’s rule, we immediately obtain the lemma. Q.E.D. Lemma 2, whose proof is given in appendix A, signs the denominator
of
(5). Lemma 2.
If the model of (2) satisfies assumptions (S) and (I,), then
(-l)“-‘pI>o.
(6)
Lemma 3 signs the numerator of (5). The proof is practically identical to that of lemma 2, assumption (I,) now playing the role of (I,) in lemma 2. Lemma 3.
If the model of (2) satisfies assumptions (S) and (I,), then
(-l)“-rlN/>o. Lemmas 1, 2 and 3 immediately yield theorem 1. Theorem I (Generalized Mas-Colell-Mantel). If the model of (2) satisfies (S), (I,), and (I,), then immiserizing growth is impossible. 3.3. A generalized Bhagwati Theorem
We now generalize a modified version of the Bhagwati Theorem to the ncommodity setting. For this purpose, we have to define the Walrasian adjustment mechanism and stability condition in our model. Let 2’ be the ordinary excess demand function for good i in the world as a whole,6 and let E’ be the corresponding excess demand. E’ = 2’(q, t),
i= 1,. . . , n.
(7)
Define the Walrasian price adjustment mechanism by $=g’(E’), 6Eq. demand
i=2,...,n,
(B.6) in appendix B gives the exact relationship and supply functions delined earlier.
(8) between
the function
2’
and
the varidus
where gi is a differentiable g’(O) = 0,
sign-preserving
function of E’ satisfying
i=7 -, . . . , Il.
(9)
and dg’ dE’ > 0,
i=3 -, . . . ) Il.
(10)
This adjustment mechanism gives rise to the following equations with 4’ = 1: cj’ =fi(q, t),
system of differential
i=2,...,n,
(11)
where .J”(q, f) -gi(Zi(q,
t)).
(12)
Note that an equilibrium of (2) has to be an equilibrium E2=...=E”=0 implies j2=...=q=0 from (7), (9), and (12).
of (1 l), since
Definition.’ We say an equilibrium of (2) is Walrus-stable if it is a locally stable equilibrium of (11) and if 14 #O, where
In this section we make the following assumption: (W)
The equilibrium
of (2) is Walras-stable.
Lemma 4, whose proof is given in appendix for assumptions (S) and (I,) in lemma 2. Lemma 4.
B, replaces assumption
If‘ the model of (2) satisjies (W), t/Tell
(-l)“-‘IDI>o. Inequality condition.”
(W)
(13) is the n-commodity
(13) generalization
of the Marshall-Lerner
‘In international trade theory, the equiiibrium where the Jacobian of the excess demand function vanishes is excluded from the definition of a W&as-stable equilibrium. Thus, the Marshall-Lerner condition is stated in the form of a strict inequality, even though an equality may be satisfied at a locally stable equilibrium in the mathematical sense. In the present paper, we extend this trade-theoretic usage of the phrase ‘Walras-stability’ to the n-commodity case. “Inequality (13) implies zf
Lemmas 1, 3 and 4 immediately Theorer?~ 2
yield theorem 2.
!f th mxfel qf(2)
(Generalized Bhagwati).
(IV), tlverv irvvrvviserizing growth
satkfies
(S), (I,,), rind
is ivvvpossiblr.
Note that the only difference between theorem 1 and theorem 2 is that assumption (W) in the latter replaces assumption (I,) in the former. A comparison of lemma 2 and lemma 4 raises a natural question as to what the relationship is between (S) plus (I,) on the one hand and (W) on the other. When n=2, it can be readily shown that (I,) and (S) together imply (W).9 Whether or not the same relationship holds among these assumptions, even when n>2, is an open question. Also, from the theorem of Hahn (1958) and Negishi (1958), we know that (W) follows from the assumption of gross substitutability of the world excess demand. The latter assumption, however, does not follow from the combination of(S) and (I,). 4. Notes on the literature Let us now examine the relationship between the results of the present paper and those of the existing literature. The mathematical economics literature on this subject has not explicitly made assumption (S), unlike our theorems 1 and 2. Instead, it made one or more of the following assumptions: (S,) In the home country, all commodity tion, i.e.
xj > 0,
if i#j.
(S,,) In the home country, no commodity tion i.e. - -J1~~. 20, (G*) ‘When
(BS)
2: = c.z;
pairs are complements
in produc-
if i#j
In the foreign ,I=?,
pairs are substitutes in consump-
country,
in Appendix + z;(l)’
- 2).
the import
B reduces
demand
functions
satisfy gross
to (*I
Without loss of generality, we can assume thal the home country imports the second good, i.e. x2 -JJ >O. Thus, (I,) implies the negativity of the second term on the right-hand side of (*). On the other hand, since the function Z’ is homogenous of degree zero with respect to q’ and q’ , we have q’Z:+q’Zi =O. Hence, (S) implies the negativity of the lirsl term on the right-hand side of (*). Therefore. (I,) and (S) together imply the negativity of p:, and hence (IV). See also Bhagwali. Brecher and Hatta (1984) on this.
substitutability, :*i. -
> ,=
i.e. 0 1
if i #.i.
Partially differentiating
(2) with respect to yj, we obtain:
+.+yj+q;. Hence, assumption (S) is implied by the combination (S,), and (G*). Thus, Theorem 1 implies:
of assumptions
(S,),
Mantel’s Theorem. ” Suppose that the home country in the model of (3) has a fised endowment rather thtm (I smoothly substitutable production frontier. If this economy satisfies (S,), (G*) and (I,), then nn increase in some of the initial endowments oj’ the home country never immiserizes this country.
Assumption (S,,) is automatically satisfied under the fixed endowment assumption, while (I,,) is implicitly assumed by the way the endowment bundle is changed in Mantel’s Theorem. Mas-Colell’s Theorem, which is the twocommodity version of this theorem, does not explicitly assume (S,), since this assumption is always satisfied in a two-commodity world due to the Hicksian demand rule. Theorem 2, on the other hand, is a generalization of the following proposition.” Corollary to Bhagwati’s Theorem. two-commodity version oj the model
Immiserizing growth is impossible in the of(Z) if it sati.sfies ( W), (I,), cmd (G*).
It is readily seen that this is contained in Bhagwati’s original theorem:12 Bhogwati’s Theorem. Immiserizing growth is impossible in the two-commodity version of the model of(Z) f it satijes (W) und the jbllowing conditions: (a) The home importable is a normal good in production. (b) The.foreign qfjer curve is elastic. ‘“As explained in the Introduction, both Mantel and Mas-Colell are concerned with the welrare or a consumer in an exchange economy. rather than that ol a country in a trading world. Our statements of Mantel’s and Mas-Colell’s Theorems are restated versions of their originals in the framework of a trading world. Also note that Mantel assumes, rather than our (S,), only weak substitutability in home consumption together with a minor additional assumption on the demand functions or the two countries. “Since (S) can be replaced by the combination of (S,), (S,). and (G*), and since the twocommodity economy always satislies (S,.) and (S,), we fmd that (S) can be replaced by (G*) in the two commodity economy. “(I,.) obviously implies (a) in Bhagwati’s Theorem. On the olher hand, (G*) implies (b). Without loss or generality. let the second good be the home importable. Then (G*) implies 2: 50. which is the statement of(b) in our no(ation.
343
Appendix A: Proof of lemma 2 Define the matrix Z by
Since Z’ is homogeneous
of degree zero with respect to q, (S) implies:
ji,qjZj=-Z:-cO,
i=2 ,..., n.
These inequalities and the fact that all the off-diagonal positive mean:”
elements of Z are
(-1)“-qq>o,
(A.11
and Z-’
is a negative matrix.
64.2)
Noting the nonsingularity of Z, we can apply to IDI a well-known on the determinant of a partitioned matrix to obtain:‘”
theorem
Zi IDI= z;-(z;,...,z;)z-’ ( t 1)M.
(A.3)
Z::
Partially differentiating Zl = .~I 2 0,
(1) with respect to u and noting (I,), we obtain: for all i,
with at least one strict inequality (S), (A.3) implies lemma 2.
(A.4) holding. In view of (A.l), (A.2) (A.4), and
Appendis B: Proof of lemma 4 The Walras-stability of an equilibrium istic roots of F have negative real parts,” ‘-‘Let negative Nikaido ‘*See “The nonpositive
of (I 1) requires that all characterand hence
A be a square matrix with positive OK-diagonal elements. If Ax<0 for some nonvector x, then A is negatively invertible and I- AI>O. See theorems 6.1 and 7.4 in (1968). proposition 30 in Dhrymes (1978). for example. local stability of an equilibrium of (11) requires that all characteristic roots or F have real roots. See theorem I9 on p. 208 and the remarks on p. 213 in Pontryagin
7: Hatto,
344
Immiserizing growfh in o many-commodity serrirtg
(-1)“-‘pp-o.
(B.1)
From (12) we have
where
&[:
;:I
;]
and
Cf.;].
Since JGI> 0 from (lo), we obtain: (-I)“-llq-O. From (B.l) and (B.2). Therefore, lemma 4 is an implication equality:
of the following
IDI=c$$
(B.3)
where c, is the inverse of the marginal utility of money for the home county. Let us now prove (B.3). When
is added to the first row of the matrix D, the value of its determinant remains the same. The first row of the thus modified matrix is:r6 ~4’21,~q’Z’,,...,~q’Zf 1 1
=&,x2-y2
)...) x”-y”).
(B.4)
1
Hence, applying the theorem on the determinant
of a partitioned
matrix to
(1962), for example. This, together with the condition that IFI does not vanish, implies that all roots of F have negative real roots. The following alternative definition of Walras-stability also implies that all roots of F have negative real parts. ‘An equilibrium of (2) is Wlros-stable if the linear approximation of (11) at that point is asymptotically stable to this point.’ ‘eFrom (2) and then (I), we have have:
Cq'zj=aCq'zi/aqJ=aCq'x'iab-aCq'yi/a~+aCqii*i/aqi. The last expression is equal to xJ -y’ +0 from Shephard’s Lemma and Cq’i*‘(q)=O. get Cq’Z;=x’--y’.
Thus, we
345
the modified matrix, we get:”
IDI=c; z-c;’
z; ;
(x2-y2 )..., x”-y”)
.
(B.5)
(>z:: Let Z’(q,t) denote the ordinary home country. Then, since Zi(q, t)=?(q,
demand function
for commodity
i of the
t) -y’(q, t) +2*‘(q),
applying the Slutsky decomposition .?(q, r) - y’(q, t), we have:
to the home import
Thus, (B.5) may be restated as (B.3).
Q.E.D.
demand function,
“See footnote 14.
References Aumann, R.J. and B. Peleg. 1975, A note on Gale’s example. Journal of Mathematical Economics 2. 209-211. Bhagwati, J.N., 1958, Immiserizing growth: A geometrical note. Review of Economic Studies 25, 201-205. Bhagwati, J.N., R.A. Brecher and T. Hatta, 1984, The paradoxes of immiserizing growth and donor-enriching (recipient-immiserizing) transfers: A tale of two literatures, Weltwirtschaftliches Archiv 120, 228-243. Chipman, J.S., 1979, The theory and application of trade utility functions, in: J. Green and J.A. Scheinkman, eds., General equilibrium growth, and trade (Academtc Press, New York) 277296. Dhrymes, P., 1978, Mathematics for econometrics (Springer-Verlag). Dixit, A.K. and V. Norman, 1980, Theory of international trade (Cambridge University Press). Hahn, F.H., 1958, Gross substitutes and the dynamic stability of general equilibrium, Econometrica 26, 1699170. Hata, T., 1973, A theory of piecemeal policy recommendations, Ph.D. Thesis, Johns Hopkins University. Hatta, T., 1977, A recommendation for a better tariff structure, Econometrica 4.5, 1859-1869. Mantel, R., 1984, Substitutability and the welfare effects of endowment increases. Journal of International Economics 17. 325-334, this issue. Mas-Colell, A., 1976, En torno a una propiedad poco atractiva del equilibrio competitive, Moneda y Credit0 (Madrid) 136, 1 l-27. Negishi, T., 1958, A note on the stability of an economy where all goods are gross substitutes, Econometrica 26,445-447. Nikaido, F., 1968, Convex structures and economic theory (Academic Press). Pontryagin, L.S., 1962, Ordinary differential equations (Addison Wesley). Takayama, A., 1974, On the analytical framework of traiffs and trade policy, in: G. Horwich and P.A. Samuelson, eds., Trade stability and macroeconomics, essays in honor of Lloyd A. Metzler (Academic Press) 153-178. Woodland, A.D., 1982, International trade and resource allocation (North-Holland).
of International
Economics
IMMISERIZING
17 ( lY84)
GROWTH
Received
May
335-345.
North-Holland
IN A MANY-COMMODITY
1983. revised
version
received
September
SETTING
I983
M;lthematical economists have rrccntly shed new light on the phenomenon of immiserizing growth. independently of the trade-theoretic literature on this subject. The present paper aims IO build ~1 beidge between the two literatures. In particular. Mantel’s set of sullicient conditions for non-imnliserizill~ growth is extended to a trade-theoretic model with a smooth production frontier. and Bhagwati’s trade-theoretic set of sullicient conditions to an n-commodity hamework. It is shown that the condition of normality in consumption in the rormer substitutes for the stability condition in the latter. The concept of the semi-compensated substitutability condition plays an essential role in our analysis.
1. Introduction
In a celebrated paper on immiserizing growth, Bhagwati (1958) proved that in a two-commodiiy open economy with a smoothly substitutable production frontier and smooth preferences, immiserizing growth is impossible if (a) growth does not reduce the domestic production of importables at constant relative commodity prices (i.e. normality holds for production); (b) the offer curve of the rest of the world is elastic; and (c) the equilibrium is Walras-stable. Recently, Aumann and Peleg (1975) have rediscovered the phenomenon of immiserizing growth in the context of a closed exchange economy, and constructed a numerical example of this phenomenon. Subsequently, MasColell (1976) and Mantel (1984) have extended the Aumann-Peleg analysis and established new sufficient conditions to preclude immiserizing growth. Translated into the international trade framework, Mas-Colell has shown that, in a two-commodity exchange economy, an increase in initial endowments of the home country will not reduce its welfare if (a) the foreign excess demand functions satisfy gross substitutability and (b) all goods are normal in home consumption. Mantel has proved that this proposition of Mas-Colell’s will *I would like to thank Jagdish Bhagwati and Avinash Dixit ror bringing an earlier version of (1984) and Disit’s unpublished comments on that version IO my attention. Comments on earlier drafis by Bela Balassa. Jngdish Bhagwati. Richard Brecher, Avinash Disit. Ali Khan. Peler Newman. and two anonymous referees are greatly appeciated. Correspondence with Rob Fccnstrn. Masnyoshi Hirota. and Murray Kemp was also helpful.
Mantel
0022-lYY6/84/$3.00
8’
1984.
Elsevier
Science
Publishers
B.V. (North-Holland)
hold in the ri-commodity framework only if a third condition is added. that (c) all commodity pairs are net-substitutes in home consumption.’ The aim of the present paper is to build a bridge between the trade-theory literature and the mathematical economics literature on the phenomenon of immiserizing growth. First, we will generalize Mantel’s theorem to an ncommodity trade world where the home country has a smoothly substitutable production frontier (instead of initial endowments of goods in an exchange economy as in Mantel’s analysis). With this production frontier an assumption on normality in home production is shown to become essential, as in Bhagwati’s Theorem [namely his condition (a)]. Second, we will generalize a slightly weakened version of Bhagwati’s Theorem to an IIcommodity economy. It will then be shown that the only difference between the Generalized Bhagwati Theorem and the Generalised Mas-Colell-Mantel1 Theorem thus obtained is that the former assumes Walras-stability in place of the normality of home consumption in the latter.2 In this paper, we employ a duality framework in constructing the model and proving the theorems.3 The concept of the semi-compensated substitutability condition, to be defined on the text, will play an important role in the present paper. This condition is implied either by the notion of the ‘elastic foreign offer curve’ in the trade-theory literature or by the various substitutability concepts used in the mathematical economics literature. We will define this condition by singing the cross-substitution terms of the semi-compensated demandfunction, also to be defined in the text. The use of this function in the statement of the model is essential for our analysis. The next section presents the model. Two principal theorems are given in section 3. The theorems of Bhagwati, Mas-Colell, and Mantel, as stated by them, are presented in section 4 and related there to the two principal theorems derived in section 3. Appendices A and B prove two lemmas stated in section 3. 2. The model
Assume that there are n goods, l,..., n; and that there are two countries, the home country and the foreign country. Define the functions xi, yi, and 5*i by the following: x’(q,u)= the compensated demand function of the home country for good i, ‘Mantel established, by constructing a concrete numerical example, that a perverse welfare effect may result if(c) is not satisfied. ‘Bhagwati, Brecher and Hatta (1982) briefly compares the theorems of Bhagwati and MasColell in the trade-theoretic framework with two commodities. 3This proor method was introduced into the welfare analysis of international trade theory by Hatta (1973. 1977). Takayama (1974). Chipman (1979) and, most comprehensively, by Dixit and Norman (1980) and Woodland (1982). It was first applied to the study of immiserizing growth by Dixit and Norman.
7: HARA,
Immiserizing
growth
in A many-commodity
331
setting
y’(q, r) = the supply function of the home country for good i, and f*‘(q) = the uncompensated import demand function of the foreign country,
where q =(ql,..., q”)‘= the price vector of the n goods, t =a scalar representing the degree of technological advancement or factor accumulation of the home country, and u =a scalar representing the utility level of the home country.
Also define the function z’ by Z’(q, u, t) E xyq, u) - y’(q, t) + L?*‘(q).
(1)
We will call Z’ the semi-compensated excess demand function of the world for good i. This is the summation of the compensated import demand function of the home country and the uncompensated import demand function foreign country. The following set of equalities must be satisfied at an equilibrium Z’(q, u, t) = 0,
of the
i=l 7..-, n.
(2)
Let the first good be the numeraire, and
Then the mode1 of (2) has n equations and n variables: q2,. . . , q”, and x4 We will analyze effects of a change in t upon q2,. . . , q” and u through this model. 3. The main results
In the present section we prove two theorems: one generalizing the MasColell-Mantel Theorem and the other the Bhagwati Theorem. 3.1. Assumptions
Various combinations present section. (S)
of the following
All commodity
assumptions will be made in the
pairs are substitutes in the functions Z’, i.e.
Zj > 0, if i #j. ‘The budget equation for all the equilibrium conditions satisfied, i.e. cqi(xi - .v’) = 0.
the foreign country, i.e. 1~1 ir*i _ =0, are’satislied, the budget equation
is always satislied. Thus, when for the home country is also
338
‘I: Hatta,
(I,,)
Immiserizing growth in a many-commodity setting
In the home country, there is no inferior good in production,
i.e. yf 20, (I,)
for all i.
In the home country, there is no inferior good in consumption,
i.e. XI 20,
for all i.
We will call (S) the semi-compensated sunstitutability condition. As will be shown in section 4, this condition is implied by various substitutability concepts assumed in the literature. When (4)
an increase in t represents economic growth, since it raises the value of the national output measured at constant prices. In the theorems of this paper, we will effectively assume this inequality by means of (I,). 3.2. A generalized Mas-Cole&Mantel
Theorem
Lemma 1 gives an expression for the welfare effect of an increase in t in the home country. Theorems in this paper will be obtained by signing the determinants INI and IDI defined in this lemma. Lemma 1. In the model of (3), the effect of an increase in t of the home country is represented by
d” INI -=-, dt IDI
(5)
where5
r-z: 2: ... q % the present note, unless otherwise stated, a subscript given to a functional notation indicates the variable or the parameter with respect to which the function is partially differentiated, e.g.
T Hatta,
and
Immiserizing
growth
in a many-commodity
setting
339
z,lz: ... 2-j r D=“:“I ; ... ““, ..
..
Lz: z; ... z;_1 ProoJ Totally differentiating (2) and applying Cramer’s rule, we immediately obtain the lemma. Q.E.D. Lemma 2, whose proof is given in appendix A, signs the denominator
of
(5). Lemma 2.
If the model of (2) satisfies assumptions (S) and (I,), then
(-l)“-‘pI>o.
(6)
Lemma 3 signs the numerator of (5). The proof is practically identical to that of lemma 2, assumption (I,) now playing the role of (I,) in lemma 2. Lemma 3.
If the model of (2) satisfies assumptions (S) and (I,), then
(-l)“-rlN/>o. Lemmas 1, 2 and 3 immediately yield theorem 1. Theorem I (Generalized Mas-Colell-Mantel). If the model of (2) satisfies (S), (I,), and (I,), then immiserizing growth is impossible. 3.3. A generalized Bhagwati Theorem
We now generalize a modified version of the Bhagwati Theorem to the ncommodity setting. For this purpose, we have to define the Walrasian adjustment mechanism and stability condition in our model. Let 2’ be the ordinary excess demand function for good i in the world as a whole,6 and let E’ be the corresponding excess demand. E’ = 2’(q, t),
i= 1,. . . , n.
(7)
Define the Walrasian price adjustment mechanism by $=g’(E’), 6Eq. demand
i=2,...,n,
(B.6) in appendix B gives the exact relationship and supply functions delined earlier.
(8) between
the function
2’
and
the varidus
where gi is a differentiable g’(O) = 0,
sign-preserving
function of E’ satisfying
i=7 -, . . . , Il.
(9)
and dg’ dE’ > 0,
i=3 -, . . . ) Il.
(10)
This adjustment mechanism gives rise to the following equations with 4’ = 1: cj’ =fi(q, t),
system of differential
i=2,...,n,
(11)
where .J”(q, f) -gi(Zi(q,
t)).
(12)
Note that an equilibrium of (2) has to be an equilibrium E2=...=E”=0 implies j2=...=q=0 from (7), (9), and (12).
of (1 l), since
Definition.’ We say an equilibrium of (2) is Walrus-stable if it is a locally stable equilibrium of (11) and if 14 #O, where
In this section we make the following assumption: (W)
The equilibrium
of (2) is Walras-stable.
Lemma 4, whose proof is given in appendix for assumptions (S) and (I,) in lemma 2. Lemma 4.
B, replaces assumption
If‘ the model of (2) satisjies (W), t/Tell
(-l)“-‘IDI>o. Inequality condition.”
(W)
(13) is the n-commodity
(13) generalization
of the Marshall-Lerner
‘In international trade theory, the equiiibrium where the Jacobian of the excess demand function vanishes is excluded from the definition of a W&as-stable equilibrium. Thus, the Marshall-Lerner condition is stated in the form of a strict inequality, even though an equality may be satisfied at a locally stable equilibrium in the mathematical sense. In the present paper, we extend this trade-theoretic usage of the phrase ‘Walras-stability’ to the n-commodity case. “Inequality (13) implies zf
Lemmas 1, 3 and 4 immediately Theorer?~ 2
yield theorem 2.
!f th mxfel qf(2)
(Generalized Bhagwati).
(IV), tlverv irvvrvviserizing growth
satkfies
(S), (I,,), rind
is ivvvpossiblr.
Note that the only difference between theorem 1 and theorem 2 is that assumption (W) in the latter replaces assumption (I,) in the former. A comparison of lemma 2 and lemma 4 raises a natural question as to what the relationship is between (S) plus (I,) on the one hand and (W) on the other. When n=2, it can be readily shown that (I,) and (S) together imply (W).9 Whether or not the same relationship holds among these assumptions, even when n>2, is an open question. Also, from the theorem of Hahn (1958) and Negishi (1958), we know that (W) follows from the assumption of gross substitutability of the world excess demand. The latter assumption, however, does not follow from the combination of(S) and (I,). 4. Notes on the literature Let us now examine the relationship between the results of the present paper and those of the existing literature. The mathematical economics literature on this subject has not explicitly made assumption (S), unlike our theorems 1 and 2. Instead, it made one or more of the following assumptions: (S,) In the home country, all commodity tion, i.e.
xj > 0,
if i#j.
(S,,) In the home country, no commodity tion i.e. - -J1~~. 20, (G*) ‘When
(BS)
2: = c.z;
pairs are complements
in produc-
if i#j
In the foreign ,I=?,
pairs are substitutes in consump-
country,
in Appendix + z;(l)’
- 2).
the import
B reduces
demand
functions
satisfy gross
to (*I
Without loss of generality, we can assume thal the home country imports the second good, i.e. x2 -JJ >O. Thus, (I,) implies the negativity of the second term on the right-hand side of (*). On the other hand, since the function Z’ is homogenous of degree zero with respect to q’ and q’ , we have q’Z:+q’Zi =O. Hence, (S) implies the negativity of the lirsl term on the right-hand side of (*). Therefore. (I,) and (S) together imply the negativity of p:, and hence (IV). See also Bhagwali. Brecher and Hatta (1984) on this.
substitutability, :*i. -
> ,=
i.e. 0 1
if i #.i.
Partially differentiating
(2) with respect to yj, we obtain:
+.+yj+q;. Hence, assumption (S) is implied by the combination (S,), and (G*). Thus, Theorem 1 implies:
of assumptions
(S,),
Mantel’s Theorem. ” Suppose that the home country in the model of (3) has a fised endowment rather thtm (I smoothly substitutable production frontier. If this economy satisfies (S,), (G*) and (I,), then nn increase in some of the initial endowments oj’ the home country never immiserizes this country.
Assumption (S,,) is automatically satisfied under the fixed endowment assumption, while (I,,) is implicitly assumed by the way the endowment bundle is changed in Mantel’s Theorem. Mas-Colell’s Theorem, which is the twocommodity version of this theorem, does not explicitly assume (S,), since this assumption is always satisfied in a two-commodity world due to the Hicksian demand rule. Theorem 2, on the other hand, is a generalization of the following proposition.” Corollary to Bhagwati’s Theorem. two-commodity version oj the model
Immiserizing growth is impossible in the of(Z) if it sati.sfies ( W), (I,), cmd (G*).
It is readily seen that this is contained in Bhagwati’s original theorem:12 Bhogwati’s Theorem. Immiserizing growth is impossible in the two-commodity version of the model of(Z) f it satijes (W) und the jbllowing conditions: (a) The home importable is a normal good in production. (b) The.foreign qfjer curve is elastic. ‘“As explained in the Introduction, both Mantel and Mas-Colell are concerned with the welrare or a consumer in an exchange economy. rather than that ol a country in a trading world. Our statements of Mantel’s and Mas-Colell’s Theorems are restated versions of their originals in the framework of a trading world. Also note that Mantel assumes, rather than our (S,), only weak substitutability in home consumption together with a minor additional assumption on the demand functions or the two countries. “Since (S) can be replaced by the combination of (S,), (S,). and (G*), and since the twocommodity economy always satislies (S,.) and (S,), we fmd that (S) can be replaced by (G*) in the two commodity economy. “(I,.) obviously implies (a) in Bhagwati’s Theorem. On the olher hand, (G*) implies (b). Without loss or generality. let the second good be the home importable. Then (G*) implies 2: 50. which is the statement of(b) in our no(ation.
343
Appendix A: Proof of lemma 2 Define the matrix Z by
Since Z’ is homogeneous
of degree zero with respect to q, (S) implies:
ji,qjZj=-Z:-cO,
i=2 ,..., n.
These inequalities and the fact that all the off-diagonal positive mean:”
elements of Z are
(-1)“-qq>o,
(A.11
and Z-’
is a negative matrix.
64.2)
Noting the nonsingularity of Z, we can apply to IDI a well-known on the determinant of a partitioned matrix to obtain:‘”
theorem
Zi IDI= z;-(z;,...,z;)z-’ ( t 1)M.
(A.3)
Z::
Partially differentiating Zl = .~I 2 0,
(1) with respect to u and noting (I,), we obtain: for all i,
with at least one strict inequality (S), (A.3) implies lemma 2.
(A.4) holding. In view of (A.l), (A.2) (A.4), and
Appendis B: Proof of lemma 4 The Walras-stability of an equilibrium istic roots of F have negative real parts,” ‘-‘Let negative Nikaido ‘*See “The nonpositive
of (I 1) requires that all characterand hence
A be a square matrix with positive OK-diagonal elements. If Ax<0 for some nonvector x, then A is negatively invertible and I- AI>O. See theorems 6.1 and 7.4 in (1968). proposition 30 in Dhrymes (1978). for example. local stability of an equilibrium of (11) requires that all characteristic roots or F have real roots. See theorem I9 on p. 208 and the remarks on p. 213 in Pontryagin
7: Hatto,
344
Immiserizing growfh in o many-commodity serrirtg
(-1)“-‘pp-o.
(B.1)
From (12) we have
where
&[:
;:I
;]
and
Cf.;].
Since JGI> 0 from (lo), we obtain: (-I)“-llq-O. From (B.l) and (B.2). Therefore, lemma 4 is an implication equality:
of the following
IDI=c$$
(B.3)
where c, is the inverse of the marginal utility of money for the home county. Let us now prove (B.3). When
is added to the first row of the matrix D, the value of its determinant remains the same. The first row of the thus modified matrix is:r6 ~4’21,~q’Z’,,...,~q’Zf 1 1
=&,x2-y2
)...) x”-y”).
(B.4)
1
Hence, applying the theorem on the determinant
of a partitioned
matrix to
(1962), for example. This, together with the condition that IFI does not vanish, implies that all roots of F have negative real roots. The following alternative definition of Walras-stability also implies that all roots of F have negative real parts. ‘An equilibrium of (2) is Wlros-stable if the linear approximation of (11) at that point is asymptotically stable to this point.’ ‘eFrom (2) and then (I), we have have:
Cq'zj=aCq'zi/aqJ=aCq'x'iab-aCq'yi/a~+aCqii*i/aqi. The last expression is equal to xJ -y’ +0 from Shephard’s Lemma and Cq’i*‘(q)=O. get Cq’Z;=x’--y’.
Thus, we
345
the modified matrix, we get:”
IDI=c; z-c;’
z; ;
(x2-y2 )..., x”-y”)
.
(B.5)
(>z:: Let Z’(q,t) denote the ordinary home country. Then, since Zi(q, t)=?(q,
demand function
for commodity
i of the
t) -y’(q, t) +2*‘(q),
applying the Slutsky decomposition .?(q, r) - y’(q, t), we have:
to the home import
Thus, (B.5) may be restated as (B.3).
Q.E.D.
demand function,
“See footnote 14.
References Aumann, R.J. and B. Peleg. 1975, A note on Gale’s example. Journal of Mathematical Economics 2. 209-211. Bhagwati, J.N., 1958, Immiserizing growth: A geometrical note. Review of Economic Studies 25, 201-205. Bhagwati, J.N., R.A. Brecher and T. Hatta, 1984, The paradoxes of immiserizing growth and donor-enriching (recipient-immiserizing) transfers: A tale of two literatures, Weltwirtschaftliches Archiv 120, 228-243. Chipman, J.S., 1979, The theory and application of trade utility functions, in: J. Green and J.A. Scheinkman, eds., General equilibrium growth, and trade (Academtc Press, New York) 277296. Dhrymes, P., 1978, Mathematics for econometrics (Springer-Verlag). Dixit, A.K. and V. Norman, 1980, Theory of international trade (Cambridge University Press). Hahn, F.H., 1958, Gross substitutes and the dynamic stability of general equilibrium, Econometrica 26, 1699170. Hata, T., 1973, A theory of piecemeal policy recommendations, Ph.D. Thesis, Johns Hopkins University. Hatta, T., 1977, A recommendation for a better tariff structure, Econometrica 4.5, 1859-1869. Mantel, R., 1984, Substitutability and the welfare effects of endowment increases. Journal of International Economics 17. 325-334, this issue. Mas-Colell, A., 1976, En torno a una propiedad poco atractiva del equilibrio competitive, Moneda y Credit0 (Madrid) 136, 1 l-27. Negishi, T., 1958, A note on the stability of an economy where all goods are gross substitutes, Econometrica 26,445-447. Nikaido, F., 1968, Convex structures and economic theory (Academic Press). Pontryagin, L.S., 1962, Ordinary differential equations (Addison Wesley). Takayama, A., 1974, On the analytical framework of traiffs and trade policy, in: G. Horwich and P.A. Samuelson, eds., Trade stability and macroeconomics, essays in honor of Lloyd A. Metzler (Academic Press) 153-178. Woodland, A.D., 1982, International trade and resource allocation (North-Holland).