Macroeconomics, Comparative Statics and the Correspondence Principle: A Critique

G. P. BROWN and C. ROGERS*(1)IN SURVEYING the development of Keynesian macroeconomic theory it is possible to distinguish at least two broad lines ofadvance, each of which in turn contains various sub-divisions. *(2) The two major lines of advance have been labelled"Fundamentalist" and "Reductionist" and in this article we will be concerned with a particular sub-division of the "Reductionist"approach: viz. traditional general equilibrium theory. *(3) The method of analysis used in this field is comparative statics and it isfair to say that the technique has reached a high degree of sophistication compared with the simple diagrammatic treatment foundin most textbooks. Large equation systems are handled with apparent ease and the method is applied to a wide range of topics.*(4) Because of the implicit practical relevance of this method of analysis we feel that it should be subjected to closer scrutiny; todetermine

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whether or not it has a sound theoretical base. Furthermore, this examination might provide some insight into the reasons why anincreasing number of economists *(5) are attempting to develop a "new" paradigm.Turning to the method of analysis proper we see that it is based on two well-known concepts: Samuelson's correspondenceprinciple and the Routh-Hurwitz theorem. *(6) These two concepts are concerned with the stability of the system under review andare essential to the analysis as it is well known that comparative static analysis is meaningless unless the system is stable. TheRouth-Hurwitz theorem, in particular, is of critical importance to the analysis as it provides the necessary and sufficient conditionsfor stability. *(7) We claim that because of practical difficulties in applying the Routh-Hurwitz theorem to large equation systems ,the policy conclusions derived from these models are of limited value. The difficulty arises because the models used are purelyqualitative in character, i. e. only general forms of the functions are specified. In this situation the Routh-Hurwitz theoremproduces unwieldly conditions that are practically impossible to interpret in economic terms. *(8) Thus we cannot establishmeaningful sufficient conditions for stability in this type of analysis. A corollary to this point is that in order to apply thecomparative statics method to large equation systems quantitative information must be introduced into the analysis. *(9)Unfortunately this information is not usually available as the variables appearing in these models have seldom been subjected toany empirical research. *(10) Thus to proceed with the analysis the practitioners are forced into making simplifying

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assumptions in an effort to reduce the complexity of the model. *(11) This, however, has the effect of emasculating the analysis byreducing it to a study of special cases; cases which seem to us to be of questionable interest to policy makers. Alternatively, policyconclusions are derived from analysis in which only necessary conditions for stability have been established. *(12) We thereforeconclude that the comparative static analysis of large equation systems is, at present, quite unsuited to the task to which it hasbeen put.To substantiate these arguments we will briefly consider the mathematics of the comparative static method and indicate theimportance of the correspondence principle in deriving comparative statics information. The argument will then be illustrated byapplying the method to the analysis of a relatively simple macroeconomic model.

1 Comparative StaticsTo illustrate the comparative static method we assume a model with n endogenous variables and m exogenous variables or shiftparameters. *(13) Let, be the unique set of equilibrium values of the endogenous variables corresponding to the initial set ofparameter values . The equilibrium position of the system is then described by the following system of equations written in implicitform. *(14)

In comparative static analysis we now wish to know how the equilibrium values of the endogenous variables change as a result ofa change in one or

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more of the shift parameters. *(15) To obtain this information we take the total derivative of each fi with respect to any parameter;that is from(2) fi [x1 (a1, a2.....am), x2 (a1, a2.....am),.....xn (a1, a2.....am); a1, a2,.....am] = 0i = 1, 2,..... n

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we obtain the following system of equations *(16)

Rearranging and rewriting equation (3) in matrix form produces,

The equations in (4) are a linear system of n equations in the n unknowns, ∂Xi/∂aj, and we may solve for the ∂Xi/∂aj by applyingCramer's Rule. This gives

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us solutions of the form, *(17)

where |J| is the determinant of the Jacobian matrix of system (4) .The object of the comparative static method is to sign one or more of the elements of ∂Xi/∂aj. If all the signs of ∂Xi/∂aj can bedetermined then complete comparative statics information is said to be available. Clearly if all the functions are completelyspecified then (4) could be solved not only for the signs but for the quantitative magnitudes as well. In economic theory, however,the, functions are not completely specified.*(18) For example we usually have information on the signs of the elements of |J| inequation (4), but that is all.Any system where only such qualitative information is available is called a purely qualitative system. And, in general, with systemsof this type it is not always possible to sign |J| or |Ji| by simply expanding the determinants. Thus to proceed any further with theanalysis we will require some method of signing |Ji| and |J|. As far as |J| is concerned it is well known that it can always be signed byapplying the correspondence principle and we will discuss this aspect in more detail below. For the moment |Ji| poses a moreinteresting problem. In this respect a number of comparative statics theorems have been developed in an effort to determine theclass of qualitative matrices to which the correspondence principle may be applied to produce complete comparative staticsinformation. *(19) Progress in this field has, however, not come up to expectations as is apparent from the comment by Lloyd: *(20)"Attempts have been made recently to find whether the hypothesis of stability will, in general, lead to useful theorems incomparative static analysis. The results have been disappointing except for simple two-equation models and other models in whichthe matrix J has a negative diagonal."The implications of this conclusion for the comparative static analysis of large equation systems is rather damaging and has ledsome authors to argue

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that the range of application of this type of analysis is severely limited. *(21) This conclusion follows simply because as thesystem of equations increases in size it becomes more and more unlikely that the Jacobian matrix will satisfy any of the comparativestatics theorems. Interestingly this possibility is seldom mentioned in the literature and the impression is given that completecomparative statics information is unattainable because certain elements in the Jacobian matrix are a priori indeterminate in sign.*(22) But this is not necessarily the case - even if all the signs of the Jacobian are known it may still be impossible to extractcomplete comparative statics information. This possibility will arise whenever the Jacobian matrix does not satisfy the relevantcomparative statics theorem. And, as these theorems are rather restrictive, we have no a priori reason far expecting a particularmacroeconomic model to satisfy the necessary conditions.Thus we can conclude, on the basis of the known comparative statics theorems, that it is unlikely that the qualitative analysis oflarge macroeconomic models will yield complete comparative statics information. This is a difficulty that arises independently of theemergence of elements in the Jacobian which are a priori indeterminate in sign. Clearly the ambiguity of these results could beremoved by introducing quantitative information and we will comment on this point again below. For the moment it is sufficient forour purposes to emphasize the rather narrow base on which the method of analysis rests.

2 The Correspondence PrincipleIt is well known that comparative static analysis has nothing to say about the path of adjustment between equilibria or even, forthat matter, whether the new equilibrium will be attained. It is for this latter reason that it is usually argued that to apply thecomparative static method we must first assume that the system is stable. *(23) That is, before we can apply the comparative staticmethod with any degree of confidence, we must first examine the dynamic properties of the system. These points are well expressedby Baumol: *(24) "Professor Samuelson has carried the argument further and has shown that where our knowledge of the relevantcurves and functions is inadequate to

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enable us to describe the equilibrium position directly, we can, on the dynamic hypothesis that the situation is stable, derivemeaningful results in comparative statics."To proceed with the analysis it is usually assumed that in addition to the static equations given in (I) we have a dynamicadjustment process of the type,

where the ki's are positive constants. *(25) More specifically, in an economic model, the ki's represent the speed of adjustment of

the ith variable to the excess demand in the ith market. The adjustment process can then be approximated by the linear terms in aTaylor series expansion *(26) and in this case (6) becomes,

where and i.e. the are the deviations from the equilibrium values .Equation (7) indicates that we have a set of simultaneous differential equations from which we can infer *(27) that the characteristicequation of the system is of the form,

Expanding the determinant in equation (8) produces the polynomial equation which has the form,

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(9) (- 1)nλn + (- 1)n-1 C1λn-1 + ... + (- 1)n-r Crλn-r + ... Cn = 0

Of particular importance in regard to the correspondence principle is the relation between the roots of the characteristic polynomial(9) and its Ci coefficients. This relation is summarized below:C1 = λ1 + λ2 + . . . . . λnC2 = λ1 λ2 + λ1 λ3 + . . . . . λ1 λn+ λ2 λ3 + . . . . . λ2 λn+ . . . . . λn-1 λnCr = sum of all possible products of λ1 . . . . .λn taken r at a time.Cn = λ1 λ2 λ3 . . . . . λnFrom the properties of characteristic equations we also establish that Cn = |KJ| = |K||J| and as |K| is unambiguously positive we are

left with the task of signing |J|. The interesting point here is that |J| produced by the dynamic analysis corresponds exa ctly with the|J| in equation (5) of the comparative static analysis. Fortunately it is now a simple matter to sign |J| by applying the properties ofcharacteristic equations summarized above. It is immediately apparent that,

and as |K| > 0 we can establish the following relation between the roots of the characteristic equation and |J|.

Now, by introducing the stability assumption, we establish that all the real parts of the roots of (9) must be negative. *(28) Thus weestablish the simplerule that |J| > 0if n is even and |J| < 0 if n is odd.

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It should be noted, however, that this simple rule which shows the relation between n and the signing of |J| holds if, and only if, thestatic system is arranged in such a form that it corresponds with the dynamic form in which all the k's are positive and K is adiagonal matrix. When the characteristic polynomial is arranged in the form as indicated by (9) then |J| is uniquely signed for each ngiven the assumption that the real parts of all the roots are negative. This signing of |J| by means of the stability assumption is theessence of the correspondence principle.The correspondence principle is, however, not without its limitations. The first attack came from Patinkin *(29) who argued that thedynamic counterpart of the static model may not involve a diagonal K matrix. This is an interesting criticism but as indicated byBenavie *(30) it does not make sense when interpreted in a comparative static framework - for the simple reason that a non-diagonalK matrix would amount to the assumption that the rate of change of Xi depended on the excess demands in all or some of theother markets. This assumption seems at odds with the conventional treatment of comparative statics where all influences on theprice of the ith good are subsumed in its excess demand function; by definition. Thus Benavie is correct to point out that Patinkinis mistaken on this point if his criticism is interpreted in terms of comparative static analysis. A careful reading of Patinkin, *(31)however, indicates that he is arguing for what we would now recognize as a non-tâtonnement type of adjustment process: he isdescribing the very inter-market pressures which cannot be captured in Walrasian general equilibrium analysis. *(32) Viewed in this

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light we feel that Patinkin's criticism is quite valid; an important limitation of the correspondence principle does result if theadjustment process is not of the Walrasian tâtonnement type. In this case there is no possibility of a correspondence between theJacobian matrix of the static system and any terms in the dynamic non-tâtonnement adjustment analysis.Nevertheless, even if we are prepared to assume that the Walrasian tâtonnement adjustment mechanism is a reasonable descriptionof the actual adjustment process, then we still cannot afford to stop at this point. For, as is quite clear from the literature, the signconditions provided by the stability

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assumption are not sufficient to ensure stability of the system. Thus we are certainly justified in continuing our enquiry into theusefulness of the correspondence principle. To appreciate the significance of this conclusion it is necessary to consider theRouth-Hurwitz theorem as it provides a test of the necessary and sufficient conditions for stability. *(33)

3 The Routh - Hurwitz TheoremThe Routh-Hurwitz theorem states that given the characteristic equation, *(34)(12) R0λn + R1λn-1 + ... Rn-1λ + ... Rn = owhere R0 > o, then, if the real parts of the roots are to be negative, the first n determinants in the following sequence must all bepositive:

To illustrate the application of these determinants we can consider a 3x3system in which case the Routh-Hurwitz conditions reduceto,(14) R1 > 0, R3 > 0, R1R2 - R0R3 > 0

or R2 > 0, R3 > 0, R1R2 - R0R3 > 0Inspection of the expressions in (14) gives some indication of where the difficulty may lie in applying the Routh-Hurwitz theorem tomacroeconomic models. For although it may be possible to determine that all of the following conditions hold - viz. R1 > 0, R2 > 0,

R3 > 0 - it is highly unlikely that we can show that R1R2 - R0R3 > 0. Thus, even in a 3x3system, it is highly unlikely that we can

establish sufficient conditions for stability in a qualitative

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analysis. This limitation is acknowledged in the literature. For example consider the following statements by Turnovsky: *(35)The system we have developed is a third order system and its stability can, in principle, be determined using the Routh-Hurwitzconditions. Unfortunately, given the complexity of the model, these turn out to be extremely complicated and consequently notvery enlightening. . . . Its stability properties can in principle be established by applying the Routh-Hurwitz conditions, although for a fifth ordersystem of this complexity, this would obviously be a hopelessly intractable task. Thus, although we are able to make somestatements below about certain necessary conditions for stability, apart from that, the question must simply be taken for granted.On the basis of these arguments we feel justified in questioning the usefulness of the correspondence principle when applied tomacroeconomic analysis. For, clearly, in large equation systems it is virtually impossible to produce sufficient conditions forstability that have any ready economic interpretation. Therefore analyses that take these sufficient conditions for granted run therisk of dealing with a system that is in fact unstable; which immediately renders the comparative static method meaningless!The important corollary that emerges from the discussion up to this point is that ambiguity, whether it arises because of an inabilityto extract complete comparative statics information or an inability to establish necessary and sufficient conditions for stability, canbe removed by introducing quantitative information. *(36) This point has been recognized by Quirk: *(37)The fundamental conclusion that arises from the preceding comments is that the scope of purely qualitative predictive models ineconomic

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theory is in fact very limited, certainly not a new or startling conclusion but one that is now arguable on rather strong grounds. Ifthis conclusion is admitted then the issue of the manner in which and the extent to which quantitative information should beintroduced into comparative statics analysis becomes quite important.Unfortunately these words of caution have not been heeded by most authors using the comparative static method.

4 A Macroeconomic ApplicationTo illustrate the points we have been discussing above we will use a model which includes wealth effects and regards assets as

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imperfect substitutes.The model is specified as follows:

Model

(1') S = S(y, w)

(2') I = I(rB, rK, y) Goods market

(3') I = S

(4') MD = L(rB, rK, y, w)

(5') MS = M0 Money market

(6') MD = MS

(7') BD = B(rB, rK, y, w)

(8') BD = pB0 Bond market

(9') BD = BS

(10') VD = V(rB, rK, y, w)

(11') VS = q.K0 Equity market

(12') VS = VD

Definitions

(13') (Non-human wealth)

(14') (Price of bonds)

(15') (Market valuation of equities)

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The notations are

S = saving VS = supply of equities

I = Investment expenditure w = non-human wealth

y = national income q = market valuation of equities

MD = demand for money (price of equities)

MS = supply of money p = market valuation of govern-

M0 = outside money ment bonds (price of bonds)

BD = demand for government bonds R0 = marginal efficiency of capital

BS = supply of government bonds K0 = capital stock

B0 = number of government bonds rB = rate of interest on bonds

outstanding rK = equity yield

VD = demand for equities

The above model incorporates the goods market as well as the financial sector of the economy which is assumed to consist of threeasset markets.*(38) The model is representative of a closed economy without government intervention and the money supply isassumed to be exclusively of the "outside" variety. The price level is assumed to be fixed so that all variables can be expressed inreal terms. Wealth is explicitly introduced into the saving function, as well as all the asset demand functions, so as to capture thewealth effects of monetary disturbances. The assets comprising the wealth definition are assumed to be gross substitutes suchthat BrB > 0, BrK < 0, VrK > 0, VrB < 0, etc. This assumption is typical of the portfolio approach. Government bonds outstanding

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are assumed to be consols yielding one "dollar" per period and are therefore not independent of their interest rate. Equation (14')describes this inverse relation between the price of bonds (π) and their return (rB). A similar assumption applies to the supply ofequities - equation (15') is well known from Tobin. *(39)The functions in the model are assumed to satisfy the following properties:

(1") 0 < Sy < 1, Sw < 0

(2") 0 < Iy < 1, IrB < 0, IrK < 0

(3") Ly > 0, 0 < Lw < 1, LrB < 0, LrK < 0

(4") By < 0, 0 < Bw < 1, BrB > 0, BrK < 0

(5") Vy < 0, 0 < Vw < 1, VrB < 0, VrK > 0

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(6") Ly + By + Vy = 0(7") Lw + Bw + Vw = 1(8") Lrj + Brj + Vrj = 0, for j = K, BProperties (1") and (2") are well known in the literature and need no further elaboration. Properties (3")-(5") state that the assets aregross substitutes and it is assumed that an increase in income causes wealth-holders to draw down their bond holdings as well astheir equity holdings in order to meet the increased demand for money. That is why we have By < 0, Vy < 0

The conditions (6")-(8") follow from the wealth constraint on the asset demand functions. Since the total of the desired holdings ofthe three assets by owners of wealth is constrained to be equal to net private non-human wealth, w, only two of the three assetmarket equilibrium conditions are independent: the remaining one can be derived from the other two and the definition of wealth.For the purpose of illustration we let equation (6'), and with it equations (4') and (5'), be eliminated by Walras's Law. Hence, the 12independent equations (equations 1'-3' and 7'-15') determine the equilibrium values of the 12 endogenous variables S, I, y, rB,

rK, w, VD, VS, Bd, BS, π and q. *(40) The variables K0, M0, B0 and R0 are assumed to be exogenously determined.The model can be solved in principle for the 12 endogenous variables but here we encounter the paradox of comparative statics:We cannot proceed without first examining the dynamics of the system. Our first step in applying the correspondence principle isto make the traditional dynamic assumptions. *(41)

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The equations in (16') can now be approximated by the linear terms in a Taylor series expansion so that (16') can be written as

where and represent deviations from the equilibrium values. Substituting for the derivatives on the L.H.S. of the equations in (17'),and rearranging, produces the characteristic equation. *(42)

Equation (18') is in the form of equation (8) above. Recall that equation (8) can also be written in matrix form as:|KJ - λI|= 0where I is an identity matrix:

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By applying the properties of characteristic equations we verify that the determinant of the Jacobian of this system must benegative for stability. To verify that this result could not be obtained by simply expanding the Jacobian determinant of the system,consider its sign pattern. *(43)

The term responsible for the question mark in the top left corner is Iy - Sy which is clearly ambiguous without additional

specification.Samuelson indicated, however, as early as 1947, that in a stable system Sy > Iy. This follows automatically since in a stable system

the sum of the first order principal minors (i.e. the trace of the Jacobian) must be negative. We are now left with the sign pattern

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and by expansion along the first row this yields,|J| = - (+-) + (--) - (++)Five of the six terms are negative and numerous side-conditions can be specified which will render |J|< 0. However, one must becareful not to specify side-conditions which are difficult to interpret in economic terms. It is always useful to make the leastrestrictive side conditions. Fortunately it can be shown *(44) that

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which has the sign pattern

is in fact positive so that |J|< 0, given the assumption that Sy > Iy.

As we have pointed out, however, the fact that we have established |J|< 0 only provides a necessary condition for stability in thismodel. For necessary and sufficient conditions we must apply the Routh-Hurwitz theorem. As our model consists of threeequations its characteristic equation can be written as,

This equation is now in the form to which we can apply the Routh-Hurwitz theorem, i.e. it is in the form

R0λ3 + R1λ2 + R2λ + R3 = 0

and we can apply the conditions given in expression (14) in section 4. In this respect we can see immediately that R1 > 0; by simple

inspection of the sign pattern given in the Jacobian above. As far as the remaining terms are concerned, however, the position isnot so clear. It can be shown that R2 > 0, R3 > 0 but the position becomes hopeless when we consider R1R2 - R0R3 > 0; where we

now need to show that R1R2 > R3! *(45) Clearly this difficulty is easily

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overcome if we have some quantitative information on the terms in the Jacobian. But without this information we must be contentwith the fact that we can only establish necessary conditions for stability.

5 ConclusionsIn this article we have pointed out a particular limitation of the correspondence principle. When it is applied to equation systemslarger than the simple 2x2 analysis found in most textbooks the correspondence principle can only provide necessary conditionsfor stability. The difficulty arises because attempts to provide necessary and sufficient conditions for stability, by applying theRouth-Hurwitz theorem, are simply not feasible in a purely qualitative analysis of large equation systems.We therefore claim that recent work with large equation systems by authors such as Benavie *(46) and Turnovsky *(47) produceconclusions of questionable value. These authors are simply taking stability for granted; a procedure that is hardly scientific andcertainly has no theoretical justification. In this respect we support the following claim by Hansen: *(48)No set of reasonably realistic, sufficient conditions for stability of the general equilibrium system has ever been presented;nonetheless some economists are content to assume that the system is stable on the basis of casual observations of the behaviourof modern economies, assuming that since they appear stable, we can safely assume that the general equilibrium system is stable.Such observations do not prove anything, of course, and there is reason to warn against this kind of reasoning.Two further corollaries follow from our analysis. Firstly it would seem that to establish necessary and sufficient conditions forstability in these large equation systems quantitative information must be introduced into the analysis. A practice which mosteconomists in this field seem to resist on the grounds that it reduces the generality of the analysis. Whatever the merits ofthis approach the fact remains that at present little use is made of the available empirical estimates in a systematic way.Secondly, and more significantly, we have pointed out that the correspondence principle constrains the adjustment process to theWalrasian tâtonnement type. Once the Walrasian tâtonnement assumption is given up

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it is highly unlikely that the dynamic counterpart of the static analysis will contain any terms corresponding to the Jacobian of thestatic analysis. This point is not new and credit is due to Patinkin for recognizing the importance of intermarket pressures fordynamic analysis, pressures which cannot be captured in a Walrasian tâtonnement adjustment mechanism.We therefore suggest that these two major shortcomings of traditional macroeconomic analysis - the inability to establish sufficientconditions for stability in large models and the restriction of the adjustment process to the Walrasian tâtonnement type - severelylimit its practical relevance. It is therefore not surprising that attempts are being made to develop an alternative approach tomacroeconomic analysis within the "Reductionist" framework. Although these efforts will not, of course, satisfy the

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"Fundamentalists".University of South AfricaPretoria

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Endnotes1Senior Lecturer and Lecturer, respectively, Department of Economics, University of South Africa, Pretoria. The authors would liketo thank Professors P. D. F. Strydom and J. A. Döckel, Mr A. V. Seeber, Mr C. S. W. Torr and a referee of this journal for helpfulcomments on an earlier draft. As is usual the authors remain responsible for any errors and omis sions.

2 The classification used here is based on Coddington, A., "Keynesian Economics: The Search for First Principles", Journal ofEconomic Literature, 14, 1976, pp. 1258-1273.

3 Two more or less distinct approaches can be distinguished within the "Reductionist" framework; the equilibrium and disequilibriumapproaches. To some extent the term disequilibrium is misleading because in many cases the analysis simply involves analternative definition of equilibrium. The traditional general equilibrium approach to macroeconomic theory forms the nucleus of thefirst approach and a good current example of this work is the book by Turnovsky, S. J., Macroeconomic Analysis andStabilization Policy, (Cambridge: Cambridge University Press, 1977).The disequilibrium approach appears more diverse but most of the current work in this field draws its inspiration from Clower, R.W., "The Keynesian Counter-Revolution: A Theoretical Appraisal", in Clower, R. (ed.), Monetary Theory, (Harmondsworth:Penguin Books, 1969), and Leijonhufvud, A., On Keynesian Economics and the Economics of Keynes: A Study inMonetary Theory, (New York: Oxford University Press, 1968). Current examples are the work by Malinvaud, E., The Theory ofUnemployment Reconsidered, (Oxford: Basil Blackwell, 1977). Barro, R. J. and Grossman, H. I., Money, Employment andInflation, (Cambridge: Cambridge University Press, 1976).

4 For example see Benavie, A., "Monetary and Fiscal Policy in a Two-Sector Keynesian Model", Journal of Money Credit andBanking, 8, 1976, pp. 63-83, and Park, Y. C., "The Transmission Process and the Relative Effectiveness of Monetary and FiscalPolicy in a Two-sector Neoclassical Model", Journal of Money Credit and Banking, 5, 1973, pp. 595-622.

5 gSee Harcourt, G. C. (ed.), The Microeconomic Foundations of Macroeconomics, (London: MacMillan, 1977).

6 See Samuelson, P. A., Foundations of Economic Analysis, (Cambridge: Harvard University Press, 1947).

7 See for example Baumol, W. J., Economic Dynamics (3rd ed.), (London: MacMillan, 1970), p. 365 and Dernburg, T. F. andDernburg, J. D., Macroeconomic Analysis: An Introduction to Comparative Statics and Dynamics, (Addison-Wesley PublishingCompany, 1969), p. 253. The Liapunov Method can also be applied to establish necessary and sufficient conditions for stability.See Benavie, A., Mathematical Techniques for Economic Analysis, (Englewood Cliffs, N.J.: Prentice Hall, 1972), pp. 231-236.Note that in this article differential equations are used to describe the dynamic behaviour of the systems studied.

8 This point is recognized in the literature; see for example Quirk, J. and Saposnik, R., Introduction to General EquilibriumTheory and Welfare Economics, (New York: McGraw-Hill, 1968), p. 215 and Kingston, G. H. and Turnovsky, S. J., "A SmallEconomy in an Inflationary World: Monetary and Fiscal Policies under Fixed Exchange Rates", The Economic Journal, 88, 1978,p. 28.

9 Strictly speaking it is not the size of the system that is important. Even in the simple IS-LM model quantitative information isintroduced in the form of the restriction; MPC + MPI<I. See Samuelson, P. A., op. cit., p. 279.

10 See Brown, G. P., The Transmission Mechanism in Monetary Theory. Unpublished M.Comm. thesis, the University of SouthAfrica, 1978.

11 A classic example is found in Kingston, G. H. and Turnovsky, S. J., op. cit., p. 42, where, by assuming fixed expectations,numerous principal minors and the Jacobian of the system are reduced to zero. Even so, the stability conditions that remain defyinterpretation!

12

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See in particular Benavie, A., (1976), op. cit.

13 Explanations of the comparative static method can be found in most texts on mathematical economics. See for example Gandolfo,G., Mathematical Methods and Models in Economic Dynamics, (London: North Holland, 1972), pp. 342-347.

14 Furthermore if the conditions of the implicit function theorem are satisfied then we can express the xi's as differentiable functions ofthe a's in the neighbourhood of the point

as functions which have the form, xi; = xi (a1, a2 . . . . . am) i = 1, 2, . . . . . n

15 Usually in economic analysis we alter the parameters one at a time. The method can be extended to take account of thesimultaneous variation of all the parameters. See for example Lloyd, P. J., "Qualitative Calculus and Comparative Static Analysis",The Economic Record, 45, 1969, p. 344.

16 By applying the chain rule to the differentiation of composite functions. The total derivative is evaluated at the equilibrium point

17 Note that one of the conditions of the implicit function theorem is that |J| ≠ o.

18 See for example Lloyd, P. J., op cit., p. 344 and Quirk, J., "The Correspondence Principle: A Macroeconomic Application",International Economic Review, 9, 1968, p. 295.

19 Most progress has been made with matrices that have a negative diagonal. See for example Comparative Statics Theorem 2' andTheorem 3 in Quirk, J. and Saposnik, R., op. cit., chapter 6. This chapter contains a comprehensive discussion of comparativestatics theorems and clearly indicates their restrictive nature.

20 Lloyd, P. J., op, cit., p. 352.

21 See for exa mple Quirk, J. and Saposnik, R., op. cit., p. 205.

22 For example Turnovsky, S. J., (1977), op. cit., makes no specific mention of the comparative statics theorems in his discussion ofthe methodology of comparative statics. For an example of where the source of the ambiguity is inadequately investigated seeBenavie, A., (1976), op. cit., p. 73. The same article contains a good example of how |Ji| can be signed in some cases by makingsimplifying assumptions on the vector of exogenous variables.

23 See Gandolfo, G., op. cit., p. 346.

24 Baumol, W. J., op. cit., p. 122.

25 That is, a Walrasian tâtonnement type adjustment mechanism is assumed to operate. See for example Quirk, J., op. cit., p. 296.

26 If we consider a sufficiently small neighbourhood of the equilibrium point.

27 See Dernburg, T. F. and Dernburg, J. D., op. cit., chapter 14. In matrix form we could write (8) as |KJ - λI| = 0 where K is a diagonalmatrix of the ki's, J is the Jacobian of the system, and I is an identity matrix.

28 Solutions to the system of equations in (7) above are of the form, i = 1, 2, . . . n in the case of real roots and

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i = r, 2, . . . n in the case of complex roots.If the system is assumed to be stable then all the real parts of the roots λ1 λ2 . . . . . λn must be negative. This conclusion followsfrom inspection of the conditions given above where if and only if λj < o. j = 1, 2, . . . n.

29 See for example Patinkin, D., "Limitations of Samuelson's 'correspondence principle' ", Metro-economica, 4, 1952, pp- 37-43

30 Benavie, A., (1972), op. cit., pp. 62-63.

31 Patinkin, D., (1952), op. cit. p. 41.

32 See Torr, C. S. W., Aspects of Macroeconomic Disequilibrium Analysis. Unpublished M.A. thesis, the University of South Africa,1977, chapters 1 and 2.

33 See Gondolfo, G., op. cit., p. 268. For examples of other necessary and sufficient conditions that are unlikely to hold inmacroeconomic analysis see Dernburg, T. F. and Dernburg, J. D., op. cit., chapter 15 and Gandolfo, G., op. cit., pp. 266-268.

34 Equation (12) is easily produced from equation (9) by multiplying through by (-1)n for the cases when n is odd. We see then that

R1 is always equal to -C1 and Rn = (-1)nCn = (-1)n|K||J|.As |K|> o this leaves us with another simple rule for signing Rn. If n is odd

we know |J|<o and so Rn must be positive. If n is even |J|> o therefore Rn is again positive, i.e. Rn is always positive in a stablesystem.

35 Turnovsky, S. J., (1977), op. cit., p. 175 and Kingston, G. H. and Turnovsky, S. J., (1978), op. cit., p. 28. Another interesting pointconcerning the Routh-Hurwitz theorem has been raised by Takayama. A. He argues that the Routh-Hurwitz theorem cannot ingeneral be used to obtain comparative statics results from the linear approximation system. He supports this point on the basis thatthe original (non-linear) system may in fact be stable, as a result of the favourable effect of higher order terms, when the linearapproximation system appears unstable. The Routh-Hurwitz theorem cannot then be used as a necessary condition for stability.See Takayama, A., Mathematical Economics, (Hinsdale, Illinois: The Dryden-Press, 1974).

36 For an example of how quantitative information is used to argue for the stability of the system see Blinder, A. S. and Solow, R.,"Does Fiscal Policy Matter?", Journal of Public Economics, 2, 1973, pp. 319-337. For an interesting counter argument, namely thatintroducing quantitative information into comparative static analysis renders the correspondence principle redundant see Finger, J.M., "The Correspondence Principle: A Superfluous Tool of Economic Analysis", Journal of Economic Issues , 1971, pp. 47-56.

37 Quirk, J., (1968), op. cit., p. 298.

38 For the implications and arguments in favour of equations (13')-(15') see Brown, G. P., (1978), op. cit., pp. 19-27, 75-84, 97-99 andTobin, J., "A General Equilibrium Approach to Monetary Theory", Journal of Money Credit and Banking, I, 1969, pp. 15-29.

39 Tobin, J., (1969), op. cit.

40 Throughout we use the convention that subscripts denote partial derivatives, i.e. Ly = ∂L/∂y.

41 Notice that in the Bond and Equity markets we have dr/dt as increasing functions of excess supply in those markets as there is aninverse relationship between the price and the rate of return on these assets. Although apparently trivial this point has importantimplications for the signing of the Jacobian since all the k's must be positive if the general rule, n odd, |J| < 0, n even |J| < 0, is toapply for a stable system.

42 For example, in the case of the solution to equation (16') is of the form

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= Aeλt ∴ d/dt = λAeλt = λ.

43 For an example of where this is in fact possible in a 3x3 system consider the following case:

44 This requires substitution of the adding up requirements given in (7") and (8") into the expressions in the above determinant.

45 This expression contains numerous terms and its economic interpretation is an intractable task.

46 Benavie, A., (1976), op. cit.

47 Turnovsky, S. J., (1977), op. cit.

48 Hansen, B., A Survey of General Equilibrium Systems, (New York: McGraw Hill, 1970), p. 35.

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