By Richard J. Cebula1. Introduction. A concept .... 3 Related to this general topic, see Lowell Bassett, "The Solution of Qualitative Comparative. Static Problems: ...
Economie Stability under Conditions of a "Commodity Trap" By Richard J. Cebula1
Introduction A concept which has recently been discussed within the framework of macroeconomics is that of the commodity trap2. To understand what is meant by the commodity trap, consider an interestsensitive investment function such as i = i(0, where I is real investment and i the rate of interest. The relationship between these two variables is the usual one: i'(i) 2 . a,j < 0 for all i, akh. < 0 for some k. There exists a nonzero term in the expansion of E7.
Our fundamental interest in Section III below is with a 2 x 2 indecomposable real matrix. We then offer the following proposition. Proposition. Let E be a signed indecomposable real 2 x 2 matrix. The sufficient conditions for E to be sign stable are Condition A: Condition B:
a n , a22 < 0. E > 0 .
Proof: Given a n < 0 and a22 < 0, then &„ < 0 for all i and akk < 0 for some k. This implies Condition (3) is satisfied. Given E > 0, Condition (4) is satisfied. Since E > 0 and since this is known only on the basis of qualitative information, clearly a12a21 < 0. Thus, Condition (1) is met. Given E as a 2 x 2 matrix, Condition (2) may be ignored8.
III. We may now consider the stability of an economic system under conditions of a commodity trap9. Adopt the following notation in addition to that provided in the Introduction: Yyy = aggregate real demand YQ = aggregate real supply Nj) = demand for labor NQ = supply of labor N = level of employment Mpv = real demand for money p = price level L = speculative demand for money (in real terms) M = nominal money stock k = reciprocal of income velocity10 x = exports (real) w = money wage rate m* = real imports In our system, w is treated as exogenous, while the price level and interest rate are taken as the dependent variables in the model. Our system may be summarized by Ey = Y0  Y s =C(Y) + I(i,Y) + G + x  m * ( Y )  Y s ( N ) EM= M0  M/p = kY + L (i)  M/p,
(7)
827 where EY is the excess demand for output and EM is the excess demand for money. Equilibrium is a set of values for the dependent variables such that Er = 0
(8)
E M =0. Setting E y = 0 and EM= 0, we differentiate (7) totally. This yields (
3T ) d l + ( a?f9r )( äY
9Y"â^" 1 ) d p =
/9L ,. __, / 3Y 5 3N M , ,
( )
. ,3Yc 3N D , w dM
A tâtonnement adjustment process determines the time path of the dependent variables. Adjustment equations are given by ^=d,EY
(10)
di ~ =: d 2 E M . dt Since (10) can be approximated linearly in a sufficiently small neighborhood of equilibrium by a Taylor series expansion, it can be rewritten as J = d i a „ ( p  p f ) + diai2(ii')
(11)
j£= d2 a2i (p  p') + d2 a22 (i  i'), i' and p' being the equilibrium values of i and p, respectively11. We now let C = fajj ] for i, j = 1 , 2 and let D be a 2 x 2 diagonal matrix with di and d2 the diagonal elements. Assume the marginal propensity to spend locally to be positive but less than unity. Then, given i** as the commodity trap rate of interest, we must consider two cases: (1) when i > i**, and (2) when i < i**. In the first case, 9I/3i = 0; in the second case, dl/9i < 0. When i > i** [case (1)], the sign pattern of DC is given by DC =
 0 + 
(12)
When i < i** [case (2)], the sign pattern of DC is DC =
+

(13)
Applying our Proposition from Section II to (12) and (13), we find that our economic system — inclusive of the commodity trap — is stable.
828 Notes 1
Department of Economics, Georgia State University. See Richard J. Cebula and Paul K. Gâtons, "Aggregate Demand Under Conditions of a 'Commodity Trap'", Rivista Internazionale di Scienze Economiche e Commerciali, forthcoming; Richard J. Cebula and Paul K. Gâtons, "The 'Commodity Trap': Some Extensions and Limitations", Indian Journal of Economics, forthcoming; Richard J. Cebula and H.N. McKenzie, "A Note on Capital Depreciation, the Rate of Interest, and the IS Function", The American Economist, XIV (Fall, 1970), pp. 8186; and Richard J. Cebula and Stephen M. Renas, "A Theoretical Note on Monetary Policy", Rivista Internazionale di Scienze Economiche e Commerciali, forthcoming; Richard J. Cebula and Paul K. Gâtons, "The Commodity Trap: Some Extensions and Limitations", Indian Journal of Economics, forthcoming; Richard J. Cebula and H.N. McKenzie, "A Note on Capital Depreciation, the Rate of Interest, and the IS Function", The American Economist, XIV (Fall, 1970), pp. 8186; and Richard J. Cebula and Stephen M. Renas, "A Theoretical Note on Monetary Policy", Rivista Internazionale di Scienze Economiche e Commerciali, XVII (December, 1970), pp. 12081212. 3 Related to this general topic, see Lowell Bassett, "The Solution of Qualitative Comparative Static Problems: Comment", Quarterly Journal of Economics, LXXII (August, 1968), pp. 519523; P. Frevert, "On the Stability of Full Employment Equilibrium", Review of Economic Studies, XXXVII (April, 1970), pp. 239251; W.M. Gorman, "More Scope for Qualitative Economics", Review of Economic Studies, XXIX (January, 1964), pp. 6568; Kelvin J. Lancaster, "The Solution of Qualitative Comparative Static Problems", Quarterly Journal of Economics, LXXX (May, 1966), pp. 278295; Kelvin J. Lancaster, "The Theory of Qualitative Linear Systems", Econometrica, XXXIII (April, 1965), pp. 395408; P.J. Lloyd, "Qualitative Calculus and Comparative Statics Analysis", Economic Record, XLV (September, 1969), pp. 343353; Don Patinkin, "The Limitations of Samuelson's Correspondence Principle", Metroeconomica, IV (1952), pp. 3743; James P. Quirk, "The Correspondence Principle: A Macroeconomic Application", International Economic Review, IX (October, 1968), pp. 294306; James P. Quirk and Richard Ruppert, "Qualitative Economics and the Stability of Equilibrium", Review of Economic Studies, XXXII (October, 1965), pp. 311326; and Paul A. Samuelson, Foundations of Economic Analysis (New York: Atheneum), esp. Chapter IX. See also Robert Artoni, "The Coordination of Fiscal and Monetary Policies", Public Finance, XXV (No. 3, 1970), pp. 337360. 4 Related to result (2), see Samuelson, op. cit., p. 259, equation (4). 5 That is, lim y\ = y[. 2
t+oo 6
Related to this Section, see Ragnar Frisch, "On the Notion of Equilibrium and Disequilibrium", Review of Economic Studies, III (1936), pp. 100105; Quirk and Ruppert, op. cit., pp. 312313; or Samuelson, op. cit., pp. 258263. 7 See Quirk and Ruppert, op. cit., p. 320, for these conditions. The proposition offered below of course follows directly from these conditions. 8 Retaining Condition B and replacing Condition A by Condition A': (a n + a22 ) < 0 yields necessary but not a priori sufficient conditions for E to be sign stable. 9 See Quirk, op. cit., for the analytical foundation and point of departure for this Section. 3M, 10 Alternately, —  may be substituted for k in the analysis which follows. 3Y 11 In equations (11), ac the values dNo 31 of the coefficients are given by V 3M* 1) 3N *3p ) (3Y + 3Y 3Y dl
829 3Ys 3No + M i 3N *3p p ' 3LX
Stabilität unter dem Gesichtspunkt der "Güterfalle" Zusammenfassung In diesem Aufsatz werden die Eigenschaften der "Güterfalle" in bezug auf die Stabilität eines Modells der offenen Volkswirtschaft (oder einer Volkswirtschaft mit interregionalem Handel) untersucht Dieser Spezialfall tritt bei einer geknickten ISKurve auf, d.h. die Kurve verläuft vertikal oberhalb eines bestimmten Zinssatzes und fallend unterhalb dieses Zinssatzes. Dabei wird gezeigt, dass die "Güterfalle" keine Instabilität verursacht, sofern die einheimische marginale Konsumneigung, d.h. die gesamte marginale Konsumneigung minus die marginale Konsumneigung für Importe zwischen 0 und 1 beträgt.
Stabilité économique dans le cas d'une "trappe de biens" Résumé L'article analyse les implications d'une stabilité macroéconomique de la "trappe de biens". Dans ce cas spécial, la courbe IS fait un angle: elle est verticale en dessus d'un certain taux d'intérêt et elle tombe en dessous de ce taux d'intérêt. On montre ensuite que l'existence d'une "trappe de biens" n'est pas une source d'instabilité si la consommation marginale indigène, c'estàdire la consommation marginale totale moins la consommation marginale pour les importations, se trouve entre 0 et 1.
Economic Stability under Conditions of a "Commodity Trap" Summary This paper investigates the implications for macroeconomic stability of the commodity trap, a concept involving a commoditymarketequilibrium (IS) curve which is "kinked", i.e., vertical above a certain rate of interest and negatively sloped below that rate of interest. The investigation considers a macrosystem engaging in international (or interregional) trade. It is shown that the commodity trap is not a source of instability, so long as the marginal propensity to spend domestically, Le., the marginal propensity to spend less the marginal propensity to import, lies between zero and unity.