- Email: [email protected]
Stability and structural forms of economic models
Stability and structural forms of economic models Lambert Schoonbeek
We relate the stability of linear economic models to properties of the associated structural form matrices. An interpretation is git'en of a condition equiralent to stability when these matrices are non-negatiee. We then derive a similar condition, which is sufficient fi~r stability,for the case where the structural form matrices contain positire as well as neyative coefficients. Finally, we demonstrate that the negative coefficients have a stabili:ing impact in typical examples of macroeconomic models. Kt'.vwords: Stability: Structural forms: Positive/negative coefficients Consider the first order linear economic model in structural form: Yt =/I)"t + By,_ t + Cx,
(I)
where y, is the ((; × I ) vector of endogenous variables and x, the ( K × I ) vector of exogenous wtriables. The coellicient matrices A =(a~j), B = ( b u) and C = ( c u ) have appropriate dimensions. Let I represent the (G x G) identity matrix. Matrix (! - A) is assumed to be regular. Conscquently, the reduced form of the model is
y, = Dy, _ t + Ex,
(2)
where ( I - A ) - t B = D = ( , I # ) and ( l - A ) - l C = E = ( e u ) . It is well known that the dynamic properties of the model depend on the eigenvalues 2i (i = I . . . . . G) of the matrix D. In particular, the model is dynamically stable if and only if the spectr:d radius p(D), defined as max I,;.,I, is smaller than unity (see Gandolfo [5]). i
in that case the matrix D is called convergent. In the literature a number of conditions are known which can be used to check whether a matrix is
The author is with the Institute of Econometrics, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands. The author is greatly indebted to Professor G.F. Pikkemaat for valuable comments on an earlier version of this paper. Final manuscript received I September 1988.
182
convergent but which do not require the actual computation of the eigenvalues of the matrix. See, for instance, Gandolfo [5"1 and Murata [ 10]. We make the following observations with respect to these conditions. Necessary and sutlicient conditions for the convergence of an arbitrary matrix can be derived, taking the form of restrictions on the coetlicients of the characteristic equation of which the roots are the eigenvalues (see Gandolfo [5]). These conditions become very complicated if the dimension of the relevant matrix increases above, say, three, if the matrix is non-negative, then more useful, necessary and sufficient conditions for convergence are available, which are directly related to the matrix itself ie not to its characteristic equation [see the next section). In this study, we investigate conditions for the convergence of the reduced form matrix D of the linear model (2). We will focus attention on those conditions that relate properties of the matrices A and B to the convergence of matrix D. The reason for this is that a model is usually specified in structural form. Our approach facilitates interpretation of the stability or instability of a model in terms of the equations or coefficients of the model as it is originally given. in the next section we treat the case where A and B are non-negative. We give and discuss an intuitively appealing interpretation of a stability condition which involves the dominant diagonal concept. The interpretation extends and generalizes to the study of the structural form matrices the work of Fisher [ 4 ] on reduced form matrices. In the third section the case is considered where A and B have positive as well as negative elements. First, a sufficient ~:ondition for stability of the model is presented that is similar to the condition mentioned in the second section. The
0264-9993/89 020182-07 $03.00 ~') 1989 Butterworth & Co (Publishers) Ltd
Stability and structural forms of economic models: L. Schoonbeek
condition generalizes a result of Bear and Conlisk [ 1]. We then investigate, for some typical examples of economic models, the effects with respect to the stability conditions and the magnitude of the spectral radius if zeroes take the place of the negative elements. We argue that the latter investigations are related to the procedures used to study the dynamics of economic models, as proposed by Deleau and Malgrange [3], Schoonbeek [11] and the references given there. We summarize and conclude in the last section.
The case of non-negative structural form matrices Consider first the reduced form matrix D. Suppose that this matrix is non-negative. As known, D is convergent if and only if the matrix ( I - D ) - t exists and is non-negative, or equivalently if and only if the matrix (! - D) possesses a positive dominant diagonal (abbreviated as pdd). We recall that an arbitrary (G x G) matrix H = (hq) has a pdd if (i) there exist positive scalars m t . . . . . raa such that m~lh, I > roll hill for i = I . . . . . G, and (ii) the diagonal entries ),'i
of t l are positive (see Gandolfo [5]). As said above, we are interested in conditions for the convergence of the matrix D, which are expressed in terms of the matrices A and B. The following theorem is fundamental in this context. Theorem 1.
Let A and B be non-negative (G x G) matrices. Assume that matrix ( l - A ) has a pdd. Let D = ( I - A ) - t B . Then p(D) < 1 if and only if ( 1 - A - B) has a pdd. Proof" This follows easily from Bear and Conlisk [ 1], p 63, Theorem ! sub 1. Example I. Chow [ 2 ] considers an accelerator model of the type (1) where a o = a i ( i , j = l . . . . . G) and B = d i a g ( b t . . . . . ha). It is assumed that ai and b~ (which represent marginal propensities to spend)are G
positive (i = I . . . . . G) and ~ ai ¢= 1. Chow proves that t=l
the eigenvalues of the corresponding reduced form matrix D are real valued and in addition positive if G
In order to clarify the meaning of Theorem 1, consider again Equation ( 1). Assume that the matrices A and B satisfy the conditions of Theorem 1 and that the model is stable. Fisher [4] points out an interesting interpretation of dominant diagonals with respect to the reduced form matrix D. He demonstrates that in case of convergence, units of measurement for the endogenous variables exist such that each row sum of the corresponding matrix D is smaller than unity. We will extend his interpretation to the structural form matrices. Under the assumptions made, we know from Theorem 1 that there exist (not necessarily unique) positive scalars m t . . . . . rn6 such that rnj
I >
-j=l
(aij + hij)
i = I .....
(t-~-B)e>O
(4)
where e is a (G x 1) vector consisting of elements equal to unity. We conclude that in case of stability a matrix M exists such that the corresponding matrices/i" and /~ are small, in the sense that the row sums of(/~ + / ] ) are smaller than unity (compare with Fisher [4]). t We note that often in economic models ~,( = a , ) = 0 for all i. Hence, in those cases the elements ~o(V~.j, i # j ) and /~o(V~.j) (representing the dependency of the current endogenous variables on the other current and all lagged endogenous variables respectively), must be
i C o n s i d e r the m o d e l .V, = Ayt + BOY,- t + B l y f _ z
where A. Be and B I arc non-negative. Rcwriting this model as an equivalent lirst order model, we conclude that it is stable if and only if there exist diagonal matrices M, and M2 ( having positive diagonal entries) such that the row sums of
ial
G •
dition for the stability is ~ a~ + maxb~ < I (application i=l
i
of the wcll known Solow's condition, scc Murata [ 10]).
E C O N O M I C M O D E L L I N G April 1989
(3)
Define the matrix M = diag(m t . . . . . mr), and transform Yt into ~, = Myt(Vt). We see that the scalars m~ are interpreted as being conversion factors, used to express the endogenous variables in new units of measurement. By doing so, the matrix (I - A - B) must be transformed into the matrix (i - ~ -/~), where M A M - t = ,,l =(a0) and M B M - t =/~ = (~tj). Hence, we can rewrite (3) as
and only if ~ a~ < I. Note that the latter means that matrix ( 1 - A) has a pdd. Thus, we conclude using Theorem 1 that in that case the model is stable if and only i f ( / - A - B) has a pdd. Finally, since the latter is equivalent to p ( A + B) < 1, a simple sufficient con-
G
ml
Af,
I
0 J\ 0
0/
M,
are smaller than unity. Moreover, using a result of Harriff. Bear and Conlisk [7], Theorem 2. we can prove that M t and Mz can be chosen such that Af~ = At: = A-I. This implies that (i) the units of measurement of the endogenous variables can be chosen such that they do not change from period to period: (ii) the row sums of A3 (A + B e + B~) ~,;/-~ are smaller than unity. Other higher order models may be treated similarly.
183
Stability and structural forms of economic models: L. Schoonbeek small enough in order to guarantee that the mentioned row sums are smaller than unity. Consider now the ith structural form equation, written using the new units of measurement. For the moment we will analyse this equation on its own ie we do not take into account the impact of the other model equations. First, observe that as a result of (4), it follows that 0 < (fi',/( 1 - , ~ , ) ) < 1. This means that the equation, considered separately, is dynamically stable. Second, suppose that the 'explanatory' variables ~j,(Vj.j ~ i) and ~j,_ t(Vj) are exposed to a shock which results in a change of their values by one (new) unit. Then the ultimately induced change in the variable )7, is in absolute value smaller than or equal to G
G
+ r. ( 1 -- t~,)
(5)
which is smaller than unity because of (4). Using the chosen units of measurement we obtain an analogous result for each structural form equation. This finding reflects the contracting impact of the model within the context of the structural form. Finally, we recall that the endogenous variables are determined simultaneously by the model, it is known that the reduced form equations implicitly take account of this simultaneity. Let us proceed still using the above chosen matrix M. Because (l - A) has a pdd, (l - ,~) also possesses a pdd. it therefore follows from (4) that (l-~)-t(l-,~-B) e = ( / - / . ~ ) e > 0 where / ) = M E N - t. This demonstrates that the row sums of the matrix/) are also smaller than unity. The conclusion is that using the chosen units of measurement an analogous result holds with respect to the structural form equations as well as the reduced form equations of the model. 2
Models having positive as well as negative coefficients A condition sufficient fi~r stability In this section we consider the case where the matrices A and B contain positive as well as negative elements. As we said in the introduction, the matter is now much more complicated. In Theorem 2, which generalizes a result of Bear and Conlisk ([ 1], p 65, Theorem 2 sub ! ), we present a sufficient condition for convergence of the matrix D. In this theorem the following notation : We note that the reverse does not necessarily hold. Take for instance
hut ( l -
184
A - B ) e rkO.
is used. Let H = (h o) be a (G x G) matrix. Given H we define H+ and H_ as the matrices we obtain if we replace the negative and positive elements of H respectively by zeroes. Next, we define H [ e ] = H + + e l l - , for e 6 [ - 1, 1]. Note that H [ I ] = H and
H I - l ] >~n[O] ~>0. Theorem 2. 3 Let A and B be (G x G) matrices. Then p((l - A)- tB) <1 i f ( / - A [ - 1 ] - B [ - 1 ] ) has a pdd. Proof: See the appendix. Obviously Theorem 2 is related to Theorem 1. In the matrix (! - A [ - 1] - B [ - 1]) the negative elements of A and B are 'made non-negative' by taking the absolute values. Note that i f ( / - A [ - 1] - B [ - 1]) has a pdd, then ( I - A [ - I ] ) also has a pdd and the matrix ( t - A [ - I ] ) - I B [ - I] is convergent. In the previous section we discussed an interpretation of Theorem 1 in terms of units of measurement of the endogenous variables. It is clear that a similar interpretation holds with respect to Theorem 2. To be concrete, (I - A[ - !] B [ - 1]) has a pdd if and only if units of measurement exist such that the matrices A [ - I ] and B [ - 1 ] are transformed into , , ~ [ - ! ] and / ~ [ - I ] respectively (using similar notation to that above), such that ( , ~ [ - I] + / ] [ - I])is small in the sense that each row sum is smaller than unity. The impact of neyative structural form coefficients Suppose now that (I - A [ - l] - B [ - l ] ) does not have a pdd. We now propose to investigate the convergence of the matrix ( ! - A [ 0 ] ) - I B [ 0 ] . In this matrix the negative elements of A and B are 'made non-negative' by replacing them with zeroes. We note that in general in many economic models there are relatively few negative elements. Generally speaking, if we have to analyse the dynamic properties of a given economic model, it is interesting to evaluate the effects of deliberately chosen changes in the matrices A, B, a n d / o r D associated with the model, on its relevant eigenvalues. See. for instance, the method of the 'structural shocks' discussed by Deleau and Malgrange [3] and the procedures of Schoonbeek [ l l ] . They argue that it is worthwhile to investigate what happens with these eigenvalues, if specific elements, in particular all elements in a column or row, of A, B and/or D, are changed to zero. However, in this study, we propose to put the negative elements of the matrices A and B equal to zero. By doing this we can examine
J Bear and Conlisk [ I ] assume that sign(a,j) = sign(bo) for all i andj.
E C O N O M I C M O D E L L I N G April 1989
Stability and structural forms of economic models: L. Schoonbeek
some typical examples of macroeconomic models. We evaluate the impact of the negative elements first by comparing the convergence condition of (1 - A)-~B and (1 - A [ 0 ] ) - ~B[0] respectively. In the examples below it appears, for realistic values of the model parameters, that the convergence condition of the latter matrix is more restrictive than that of the former matrix. Furthermore, we propose to compare the magnitudes of the spectral radii p ( ( l - A [ ~ ] ) - ~B[g]) for ~ = 0 and ~ = 1 respectively. At first sight, if we alter from 0 to I. the effect on the spectral radius would be ambiguous. However. it seems, at least in the examples we will discuss below, that the appearance of the negative coefficients has a stabilizing effect in the sense that the spectral radius becomes smaller. There is another related point. Assume that the matrix I I - A [ 0 ] ) has a pdd (the examples below satisfy this requirement). Let the Frobenius eigenvalue of the matrix ( I - A [ O ] ) - t B [ O ] , denoted by At[0 ], be simple. Note that this eigenvalue equals the spectral radius of the matrix. Let the corresponding (G x I ) right and (I x G) left eigenvector, r t [ 0 ] and / t [ 0 ] respectively, be normalized such that l, [ 0 ] ' r t [ 0 ] = I. Then it can be demonstrated, using at result of Mcirovitch ( [ 9 ] , p 104), that the lirst order effect on At[() ] of the change of ~: from 0 to I is equal to
condition is more restrictive than the above mentioned condition 1 - ca > 0. Thus, as a result of the inclusion of the negative element ( - a ) in the matrix B. the convergence condition of the matrix (I - A )- ~B is less restrictive than the one ofthe matrix (I - A [ 0 ] ) - ~B[0]. Now let us compare the spectral radii of these two matrices. The spectral radius of ( I - A [ O ] ) - I B [ O ] , denoted by ,u, amounts to (1 + a ) c . The non-zero eigenvalues o f ( / - A)- IB are equal to At. 2 = ½{( 1 + a)c + x~}, where D i s c r = ( l + a ) Z c 2 - 4 a c . We will demonstrate that the absolute values of ,;-t and 2, are smaller than ii if c > ~-, a requirement satisfied by realistic values of c. Thus in normal cases the spectral radius of ( I - A ) - t B is smaller than II, or stated otherwise, the negative coefficient has a stabilizing impact. We can distinguish three cases: Discr > 0 In this case
I;.t._,l<~{(I +a)c+(l +a)c} =(I +a)c =/, Discr = 0 Then 12t..,I =
t.,(l+ a l c < l L .
Discr < 0 1,[0] { ( / - A [ 0 1 ) - 'A _{I - . - I [ 0 ] ) - ' B [ 0 ] (6)
+ (1 - : I [ 0 ] ) - ' B _ }rt[0 ]
it is easily seen that this first order effect is never positive. E x a m p l e 2. The well known multiplier-accelerator model of Samuelson reads
1, = a( C, - C, _ t )
E x a m p l e 3. Consider the model
Y, = C, + 1,
Y, = C, + 1, C, = c};-t
Now'* we have IAt.zl = ~(~:c). This square root is smaller than ~ ifand only i f g ( a ) = aZc + a(2c - I) + c > 0. A sufficient condition to ensure the latter is c > ~, because then ( 2 c - I ) ' - 4 c 2 < 0 and consequently ,q(a) > 0 for all a.
0
1
a> 0
C, = cY, 1, = a Yt - t - bK, _ 1
0
a>0, b>0
K,= K,_t +I,
where Y,, C, and I, represent national income, consumption and investment respectively. The model can be written in the form ( 1). Note that there is only one negative coefficient ie the ( - a ) in the third equation. It is easily checked that the sufficient condition of Theorem 2 is satistied if and only if I - c - 2 c a > O . This condition is rather restrictive. From the characteristic equation pertaining to the model, ie 2: - 2( I + a)c + ac = 0, it follows that the model is stable if and only if l - c a > 0 (see Gandolfo [5] for details). The matrix ( / - A [ 0 _ ] ) always has a pdd, whereas (1 - AI-0] - B [ 0 ] ) has one ifand only if I - c - c a > 0. We remark that the latter
E C O N O M I C M O D E L L I N G April 1989
where K, represents the end of period capital stock and the other symbols are already explained in example 2. The model is clearly of the type ( ! ). There is only one negative coefficient ie the ( - b) in the third equation. We note that the corresponding matrix ( i - A [ - I ] - B [ - I ] ) has no pdd, hence Theorem 2 does not help us here. The characteristic equation of the model reads
.8 Note that in this case c is smaller than 4a/( I + a) z. which may be incompatible with c > t for sufficiently high values of a.
185
Stability and structural forms of economic models: L. Schoonbeek ,;.2(1 -- c)--,;.((l - - c ) ( l - b ) + a ) + a = O Using this equation it can be demonstrated that the model is stable if and only if (1 - c ) ( b / 2 -
l)
1 -c
(See Laffargue [8]). Inspection of ( I - A [ 0 ] ) and ( l - A [ 0 ] - B [ 0 ] ) shows that only the former matrix has a pdd. From Theorem 1 it follows that ( ! - A [ 0 ] ) - t B [ 0 ] is not convergent. Thus, starting from this matrix it appears that the inclusion of the negative coefficient ( - b ) in the third model equation makes the convergence condition of ( I - A ) - t B less restrictive. Now we will compare the spectral radii o f ( / - A)- tB and (I - A [ 0 ] ) - t B [0]. The non-zero eigenvalues of the latter matrix, #t and #2 say. are equal to 1 and a/( I - c) respectively. Obviously they are both positive. The non-zero eigenvalues of (l - A)- t B are (I --c)(I - b ) + a + + _ ~ '~1.2 =
2(I - c)
where Discr -= (( 1 - c)( 1 - b) + a)' - 4a( I - c) It can be proved that ).t and 2: are in absolute value smaller than #l and #2 if b < 2 (see appendix). Note that realistic values of b are smaller than 2 (see Laffargue [8]). Thus, in normal circumstances the negative coefficient has a stabilizing effect. Example 4. The systematic part of Klein's well known model I of the US economy (model period 1921-41) reads C, = 0.017P, + 0.216P,_ t + 0.810W,, + 16.555 l, = 0. ! 50P, + 0.616P, _ t - 0. ! 58K,_ t + 20.278 W,, = 0.439E, + 0.147E,_ t + 0.130A, + 1.500 Yf = C, + 1, + G, - T, Pt= Y ~ - W I K~ = It + K t - t
w,= w,, + w2, E, = Y, + T, - W , ,
where W~, = private wage bill, Pt = profits, W, = total wage bill, E, = private product, At = time trend, G, = government expenditure, T, = taxes, W2, = government wage bill and the other symbols represent the same
186
variables as in example 3. The model is of the type of formula (1). The spectral radius pertaining to this model amounts to 0.845: thus the model is stable (see Goldberger [ 6 ] for details). It appears from our computations that the matrix (I - . A [ - 1] - B [ - 1]) does not have a pdd. The matrices A and B each contain one negative element ie ( - 1 ) and ( - 0 . 1 5 8 ) corresponding to W, in the fifth equation and K,_ t in the second equation respectively. It also appears that the matrix (I - A [0] ) has a pdd, whereas (1 - A [0] B [ 0 ] ) does not. As a result (! - A[0])-tB[0] is not convergent. As an experiment we have computed the spectral radii of the matrix ( I - A [ ~ ] ) - t B [ ~ ] for e = 0.0, 0.2 . . . . . 0.8, 1.0. They amount to 1.992, 1.674, !.213, I. 107, 0.989 and 0.845 respectively. We conclude that the negative coefficients have a stabilizing effect on the spectral radius. In order to refine this conclusion we have done the same experiment replacing the negative element of the matrix A each time with zero. The corresponding spectral radii are equal to 1.992, 1.746, 1.505, 1.264, 1.012 and 0.703 respectively. If we do the experiment by replacing the negative element of the matrix B with zero, the computed spectral radii are 1.992, 1.930, 1.864, 1.794, 1.718 and 1.631 respectively. We observe that the stabilizing effect is primarily caused by the negative element ( - I) associated with W, in the fifth equation.
Conclusion We investigated the stability of linear (macro-) economic models in terms of properties of the structural form matrices A and B of such models. A model is usually specified in structural form; our approach facilitates interpretation of the (in)stability of the model. In the second section we have seen that a model, for which A and B are non-negative, is stable if and only if by means of a suitable choice of the units of measurement of the endogenous variables, (A + B) can be transformed into a matrix of which all row sums are smaller than unity. The implications of this finding on the equations of a model, both separately and collectively, are examined. In the third section, the case was treated where A and B have positive as well as negative coefficients. A sufficient condition for stability is presented which is similar to the condition of the second section. Finally, we examined what happens with the conditions for stability and the magnitude of the spectral radius of a model if the negative coefficients are replaced by zeroes. We proved ia some typical examples of macroeconomic models that the original negative coefficients have a stabilizing impact.
E C O N O M I C M O D E L L I N G April 1989
Stability and structural forms of economic models: L. Schoonbeek
References I D.V.T. Bear and J. Conlisk, "The effect of unlagging on stability and speed of adjustment', Journal of Economic Theory, Vol 21, 1979, pp 62-72. 2 G C . Chow, ' T h e acceleration principle and the nature of business cycles'. Quarterly Journal of Economics, Vol 22, 1968, pp 403-418. 3 M. Deleau and P. Malgrange, 'M&hodes &analyse des modSles empiriques'. Annales de I'INSEE, No 20, 1975, pp 3-34. 4 F.M. Fisher, 'Choice of units, column sums. and stability in linear dynamic systems with nonnegative square matrices'. Econometrica, Vol 33, 1965, pp 445-450. 5 G. Gandolfo, Economic Dynamics. Methods and Models. North-Holland, Amsterdam, 1980. 6 A.S. Goldberger, Ectmometric Theory. John Wiley,
New York, 1964. 7 R. Harriff, D.V.T. Bear and J. Conlisk, 'Stability and speed of adjustment under retiming of lags', Econometrica, Vol 48, 1980. pp 355-370. 8 J.P. Laffargue. "lnterpr&ation 8conomique des caract&istiques quasicycliques d'un mod+le macroSconom&rique evoluant en environnement al~atoire', Annales de l"I NSEE, No 43, 1981. pp 33-65. 9 L. Meirovitch, Computational Methods in Structural Dynamics. Sijthofl'and Noordhoff, Alphen aan den Rijn, 1980. 10 Y. Murata. Mathematics fi)r Stability and Optimization of Economic Systems, Academic Press, New York. 1977. 11 L. Schoonbeek, 'Coefficient values and the dynamic properties of econometric models', Economics Letters, Vo[ 16. 1984. pp 303-308. 12 J.E. Woods, Mathematical Economics. Longman, New York. 1978.
Appendix Proofs
Theorem
('ase 1:(I - c ) ( I - h ) ~-a>O
2
Suppose that ( I - A [ - I ] - B [ - I ] ) h a s a pdd. Then ( I - .-I [ - l ] ) has a pdd. ('onscquently p( ,-I [ - 1] 1 < I, which implies that p( A ) < I (see Woods [ 12 ], p 47 ). Thus ( I - ,.! ) t
DL~cr>O In this case )-t > 0 and I , ; - , [ < 2 t . so it is sullicient to prove that 2 t ~< m a x l/it. !t2 ). Suppose lirst that 1 - c - a = O. Then \.~
exists and equals ~, ,,1'. Moreover. I)iscr
=(/-A[-I]I
t
(I-c)(l-h)+a=
0.<(It-A) 'tS)E-I].<(t-A[-I]I ' t S [ - l ]
t,(((t- A)-'thE-I]).<~,((t-.4[-
I])-'B[-1])<
I
The last inequality holds because it is assumed that (I - A [ - 1] - B [ - 1] has a pdd. The proof is completed by noting that
AI'- I ] ) 'B)[ - 1])
T h e proof of example 2
MODELLING
7.t = 1
which equals tt, ~< max(Ft, Its). Now we increase the value of b and analyse what happens with 2 t. Therefore we look again at the formula of,;.t, in particular the numerator. The term (( 1 - c)( I - b) + al decreases if h increases. The other term ie x , ~
db
=
also decreases because
~
\,,/Discr /
(a+(I-c)(l-b))(I-c)
It follows that 2 t < max(lq, It,,).
In order to prove that )-t and 2, are smaller in absolute value than 111 and it, if b < 2. we distinguish a number of cases.
ECONOMIC
h 2 - 41,
I-c+a>O
Discr=(l-c-aj-'>0and
from which it follows that
<~p(((I -
b ) + a)-" - 4 a 2 =
which is smaller than zero because b < 2. Thus, it appears that this situation is not relewmt here. Therefore suppose that 1 - c - a # 0. Let us for the moment substitute b = 0 m the above given expression of 2 t, and denote the result as 7. i. Then
Using this, we obtain
p ( ( l -- A ) - t B )
= (a( I -
April 1989
Discr=O As above, we only have to consider the case where 1 - c - a # 0. which implies that i t t # it2. Note that
187
Stability and structural forra~ of economic models: L. Schoonbeek. _,t(( 1 - - c ) ( I - b ) + a ) 0<;.t
=2:=
In both cases I,;.t I =
(l --c)
Case 2. (I - c ) ( l
= ~( 1 - b) + ~(a,,'( l - c))
I~,1
~< max(,ut, p.,).
-b)+a=O
It appears that here I i t I = 12z I = ,
If/a t > p , , then
Case 3: (l - c l ( l t(I-
b) + ,t(a/(l - c ) ) <
[(!-b}+
zt < I
Discr > 0
= Pt ~< max(Ill. It.) I;-.,I <
-b)+a
<0
N o t e that 1221 > 12 t I in this case. F u r t h e r m o r e -(1 -c)(l-b)-a
= -(l-b)---
11 - c )
O n the c o n t r a r y if lit
a 11 - c )
< m a x ( l q . ~t:)
t( l _ b ) + ½la ( l - c ) ) < [(a/l l - c ) ) + [(a/( l - c ) ) where the last inequality holds since b < 2. = (a/( I - c)) = It; ~< m a x ( l q , It.,)
Discr = 0 Discr < 0
I';-i I = I,t., 1 = ,~ ia,/( I - el) If It, ~< lti. then
w/la/(I --C') ~ I = I t I If It., > It I, then
~ f ( S i ( i --<')) < (al( I - <')) = It,,
188
Nov,
It is easily verified that now I,t~l = I';-zl =
-(I
-c)(I
-c)-a
2(I - c )
which is smaller than m a x ( / q . It,) because b < 2.
Di.v~'r < 0 I,i.i I
I lere v,c have
= I,~.21 = ,, (al(I
-<'))
See further c:.isc I wh,erc l)iscr < 0.
ECONOMIC
MODELLING
A p r i l 1989
We relate the stability of linear economic models to properties of the associated structural form matrices. An interpretation is git'en of a condition equiralent to stability when these matrices are non-negatiee. We then derive a similar condition, which is sufficient fi~r stability,for the case where the structural form matrices contain positire as well as neyative coefficients. Finally, we demonstrate that the negative coefficients have a stabili:ing impact in typical examples of macroeconomic models. Kt'.vwords: Stability: Structural forms: Positive/negative coefficients Consider the first order linear economic model in structural form: Yt =/I)"t + By,_ t + Cx,
(I)
where y, is the ((; × I ) vector of endogenous variables and x, the ( K × I ) vector of exogenous wtriables. The coellicient matrices A =(a~j), B = ( b u) and C = ( c u ) have appropriate dimensions. Let I represent the (G x G) identity matrix. Matrix (! - A) is assumed to be regular. Conscquently, the reduced form of the model is
y, = Dy, _ t + Ex,
(2)
where ( I - A ) - t B = D = ( , I # ) and ( l - A ) - l C = E = ( e u ) . It is well known that the dynamic properties of the model depend on the eigenvalues 2i (i = I . . . . . G) of the matrix D. In particular, the model is dynamically stable if and only if the spectr:d radius p(D), defined as max I,;.,I, is smaller than unity (see Gandolfo [5]). i
in that case the matrix D is called convergent. In the literature a number of conditions are known which can be used to check whether a matrix is
The author is with the Institute of Econometrics, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands. The author is greatly indebted to Professor G.F. Pikkemaat for valuable comments on an earlier version of this paper. Final manuscript received I September 1988.
182
convergent but which do not require the actual computation of the eigenvalues of the matrix. See, for instance, Gandolfo [5"1 and Murata [ 10]. We make the following observations with respect to these conditions. Necessary and sutlicient conditions for the convergence of an arbitrary matrix can be derived, taking the form of restrictions on the coetlicients of the characteristic equation of which the roots are the eigenvalues (see Gandolfo [5]). These conditions become very complicated if the dimension of the relevant matrix increases above, say, three, if the matrix is non-negative, then more useful, necessary and sufficient conditions for convergence are available, which are directly related to the matrix itself ie not to its characteristic equation [see the next section). In this study, we investigate conditions for the convergence of the reduced form matrix D of the linear model (2). We will focus attention on those conditions that relate properties of the matrices A and B to the convergence of matrix D. The reason for this is that a model is usually specified in structural form. Our approach facilitates interpretation of the stability or instability of a model in terms of the equations or coefficients of the model as it is originally given. in the next section we treat the case where A and B are non-negative. We give and discuss an intuitively appealing interpretation of a stability condition which involves the dominant diagonal concept. The interpretation extends and generalizes to the study of the structural form matrices the work of Fisher [ 4 ] on reduced form matrices. In the third section the case is considered where A and B have positive as well as negative elements. First, a sufficient ~:ondition for stability of the model is presented that is similar to the condition mentioned in the second section. The
0264-9993/89 020182-07 $03.00 ~') 1989 Butterworth & Co (Publishers) Ltd
Stability and structural forms of economic models: L. Schoonbeek
condition generalizes a result of Bear and Conlisk [ 1]. We then investigate, for some typical examples of economic models, the effects with respect to the stability conditions and the magnitude of the spectral radius if zeroes take the place of the negative elements. We argue that the latter investigations are related to the procedures used to study the dynamics of economic models, as proposed by Deleau and Malgrange [3], Schoonbeek [11] and the references given there. We summarize and conclude in the last section.
The case of non-negative structural form matrices Consider first the reduced form matrix D. Suppose that this matrix is non-negative. As known, D is convergent if and only if the matrix ( I - D ) - t exists and is non-negative, or equivalently if and only if the matrix (! - D) possesses a positive dominant diagonal (abbreviated as pdd). We recall that an arbitrary (G x G) matrix H = (hq) has a pdd if (i) there exist positive scalars m t . . . . . raa such that m~lh, I > roll hill for i = I . . . . . G, and (ii) the diagonal entries ),'i
of t l are positive (see Gandolfo [5]). As said above, we are interested in conditions for the convergence of the matrix D, which are expressed in terms of the matrices A and B. The following theorem is fundamental in this context. Theorem 1.
Let A and B be non-negative (G x G) matrices. Assume that matrix ( l - A ) has a pdd. Let D = ( I - A ) - t B . Then p(D) < 1 if and only if ( 1 - A - B) has a pdd. Proof" This follows easily from Bear and Conlisk [ 1], p 63, Theorem ! sub 1. Example I. Chow [ 2 ] considers an accelerator model of the type (1) where a o = a i ( i , j = l . . . . . G) and B = d i a g ( b t . . . . . ha). It is assumed that ai and b~ (which represent marginal propensities to spend)are G
positive (i = I . . . . . G) and ~ ai ¢= 1. Chow proves that t=l
the eigenvalues of the corresponding reduced form matrix D are real valued and in addition positive if G
In order to clarify the meaning of Theorem 1, consider again Equation ( 1). Assume that the matrices A and B satisfy the conditions of Theorem 1 and that the model is stable. Fisher [4] points out an interesting interpretation of dominant diagonals with respect to the reduced form matrix D. He demonstrates that in case of convergence, units of measurement for the endogenous variables exist such that each row sum of the corresponding matrix D is smaller than unity. We will extend his interpretation to the structural form matrices. Under the assumptions made, we know from Theorem 1 that there exist (not necessarily unique) positive scalars m t . . . . . rn6 such that rnj
I >
-j=l
(aij + hij)
i = I .....
(t-~-B)e>O
(4)
where e is a (G x 1) vector consisting of elements equal to unity. We conclude that in case of stability a matrix M exists such that the corresponding matrices/i" and /~ are small, in the sense that the row sums of(/~ + / ] ) are smaller than unity (compare with Fisher [4]). t We note that often in economic models ~,( = a , ) = 0 for all i. Hence, in those cases the elements ~o(V~.j, i # j ) and /~o(V~.j) (representing the dependency of the current endogenous variables on the other current and all lagged endogenous variables respectively), must be
i C o n s i d e r the m o d e l .V, = Ayt + BOY,- t + B l y f _ z
where A. Be and B I arc non-negative. Rcwriting this model as an equivalent lirst order model, we conclude that it is stable if and only if there exist diagonal matrices M, and M2 ( having positive diagonal entries) such that the row sums of
ial
G •
dition for the stability is ~ a~ + maxb~ < I (application i=l
i
of the wcll known Solow's condition, scc Murata [ 10]).
E C O N O M I C M O D E L L I N G April 1989
(3)
Define the matrix M = diag(m t . . . . . mr), and transform Yt into ~, = Myt(Vt). We see that the scalars m~ are interpreted as being conversion factors, used to express the endogenous variables in new units of measurement. By doing so, the matrix (I - A - B) must be transformed into the matrix (i - ~ -/~), where M A M - t = ,,l =(a0) and M B M - t =/~ = (~tj). Hence, we can rewrite (3) as
and only if ~ a~ < I. Note that the latter means that matrix ( 1 - A) has a pdd. Thus, we conclude using Theorem 1 that in that case the model is stable if and only i f ( / - A - B) has a pdd. Finally, since the latter is equivalent to p ( A + B) < 1, a simple sufficient con-
G
ml
Af,
I
0 J\ 0
0/
M,
are smaller than unity. Moreover, using a result of Harriff. Bear and Conlisk [7], Theorem 2. we can prove that M t and Mz can be chosen such that Af~ = At: = A-I. This implies that (i) the units of measurement of the endogenous variables can be chosen such that they do not change from period to period: (ii) the row sums of A3 (A + B e + B~) ~,;/-~ are smaller than unity. Other higher order models may be treated similarly.
183
Stability and structural forms of economic models: L. Schoonbeek small enough in order to guarantee that the mentioned row sums are smaller than unity. Consider now the ith structural form equation, written using the new units of measurement. For the moment we will analyse this equation on its own ie we do not take into account the impact of the other model equations. First, observe that as a result of (4), it follows that 0 < (fi',/( 1 - , ~ , ) ) < 1. This means that the equation, considered separately, is dynamically stable. Second, suppose that the 'explanatory' variables ~j,(Vj.j ~ i) and ~j,_ t(Vj) are exposed to a shock which results in a change of their values by one (new) unit. Then the ultimately induced change in the variable )7, is in absolute value smaller than or equal to G
G
+ r. ( 1 -- t~,)
(5)
which is smaller than unity because of (4). Using the chosen units of measurement we obtain an analogous result for each structural form equation. This finding reflects the contracting impact of the model within the context of the structural form. Finally, we recall that the endogenous variables are determined simultaneously by the model, it is known that the reduced form equations implicitly take account of this simultaneity. Let us proceed still using the above chosen matrix M. Because (l - A) has a pdd, (l - ,~) also possesses a pdd. it therefore follows from (4) that (l-~)-t(l-,~-B) e = ( / - / . ~ ) e > 0 where / ) = M E N - t. This demonstrates that the row sums of the matrix/) are also smaller than unity. The conclusion is that using the chosen units of measurement an analogous result holds with respect to the structural form equations as well as the reduced form equations of the model. 2
Models having positive as well as negative coefficients A condition sufficient fi~r stability In this section we consider the case where the matrices A and B contain positive as well as negative elements. As we said in the introduction, the matter is now much more complicated. In Theorem 2, which generalizes a result of Bear and Conlisk ([ 1], p 65, Theorem 2 sub ! ), we present a sufficient condition for convergence of the matrix D. In this theorem the following notation : We note that the reverse does not necessarily hold. Take for instance
hut ( l -
184
A - B ) e rkO.
is used. Let H = (h o) be a (G x G) matrix. Given H we define H+ and H_ as the matrices we obtain if we replace the negative and positive elements of H respectively by zeroes. Next, we define H [ e ] = H + + e l l - , for e 6 [ - 1, 1]. Note that H [ I ] = H and
H I - l ] >~n[O] ~>0. Theorem 2. 3 Let A and B be (G x G) matrices. Then p((l - A)- tB) <1 i f ( / - A [ - 1 ] - B [ - 1 ] ) has a pdd. Proof: See the appendix. Obviously Theorem 2 is related to Theorem 1. In the matrix (! - A [ - 1] - B [ - 1]) the negative elements of A and B are 'made non-negative' by taking the absolute values. Note that i f ( / - A [ - 1] - B [ - 1]) has a pdd, then ( I - A [ - I ] ) also has a pdd and the matrix ( t - A [ - I ] ) - I B [ - I] is convergent. In the previous section we discussed an interpretation of Theorem 1 in terms of units of measurement of the endogenous variables. It is clear that a similar interpretation holds with respect to Theorem 2. To be concrete, (I - A[ - !] B [ - 1]) has a pdd if and only if units of measurement exist such that the matrices A [ - I ] and B [ - 1 ] are transformed into , , ~ [ - ! ] and / ~ [ - I ] respectively (using similar notation to that above), such that ( , ~ [ - I] + / ] [ - I])is small in the sense that each row sum is smaller than unity. The impact of neyative structural form coefficients Suppose now that (I - A [ - l] - B [ - l ] ) does not have a pdd. We now propose to investigate the convergence of the matrix ( ! - A [ 0 ] ) - I B [ 0 ] . In this matrix the negative elements of A and B are 'made non-negative' by replacing them with zeroes. We note that in general in many economic models there are relatively few negative elements. Generally speaking, if we have to analyse the dynamic properties of a given economic model, it is interesting to evaluate the effects of deliberately chosen changes in the matrices A, B, a n d / o r D associated with the model, on its relevant eigenvalues. See. for instance, the method of the 'structural shocks' discussed by Deleau and Malgrange [3] and the procedures of Schoonbeek [ l l ] . They argue that it is worthwhile to investigate what happens with these eigenvalues, if specific elements, in particular all elements in a column or row, of A, B and/or D, are changed to zero. However, in this study, we propose to put the negative elements of the matrices A and B equal to zero. By doing this we can examine
J Bear and Conlisk [ I ] assume that sign(a,j) = sign(bo) for all i andj.
E C O N O M I C M O D E L L I N G April 1989
Stability and structural forms of economic models: L. Schoonbeek
some typical examples of macroeconomic models. We evaluate the impact of the negative elements first by comparing the convergence condition of (1 - A)-~B and (1 - A [ 0 ] ) - ~B[0] respectively. In the examples below it appears, for realistic values of the model parameters, that the convergence condition of the latter matrix is more restrictive than that of the former matrix. Furthermore, we propose to compare the magnitudes of the spectral radii p ( ( l - A [ ~ ] ) - ~B[g]) for ~ = 0 and ~ = 1 respectively. At first sight, if we alter from 0 to I. the effect on the spectral radius would be ambiguous. However. it seems, at least in the examples we will discuss below, that the appearance of the negative coefficients has a stabilizing effect in the sense that the spectral radius becomes smaller. There is another related point. Assume that the matrix I I - A [ 0 ] ) has a pdd (the examples below satisfy this requirement). Let the Frobenius eigenvalue of the matrix ( I - A [ O ] ) - t B [ O ] , denoted by At[0 ], be simple. Note that this eigenvalue equals the spectral radius of the matrix. Let the corresponding (G x I ) right and (I x G) left eigenvector, r t [ 0 ] and / t [ 0 ] respectively, be normalized such that l, [ 0 ] ' r t [ 0 ] = I. Then it can be demonstrated, using at result of Mcirovitch ( [ 9 ] , p 104), that the lirst order effect on At[() ] of the change of ~: from 0 to I is equal to
condition is more restrictive than the above mentioned condition 1 - ca > 0. Thus, as a result of the inclusion of the negative element ( - a ) in the matrix B. the convergence condition of the matrix (I - A )- ~B is less restrictive than the one ofthe matrix (I - A [ 0 ] ) - ~B[0]. Now let us compare the spectral radii of these two matrices. The spectral radius of ( I - A [ O ] ) - I B [ O ] , denoted by ,u, amounts to (1 + a ) c . The non-zero eigenvalues o f ( / - A)- IB are equal to At. 2 = ½{( 1 + a)c + x~}, where D i s c r = ( l + a ) Z c 2 - 4 a c . We will demonstrate that the absolute values of ,;-t and 2, are smaller than ii if c > ~-, a requirement satisfied by realistic values of c. Thus in normal cases the spectral radius of ( I - A ) - t B is smaller than II, or stated otherwise, the negative coefficient has a stabilizing impact. We can distinguish three cases: Discr > 0 In this case
I;.t._,l<~{(I +a)c+(l +a)c} =(I +a)c =/, Discr = 0 Then 12t..,I =
t.,(l+ a l c < l L .
Discr < 0 1,[0] { ( / - A [ 0 1 ) - 'A _{I - . - I [ 0 ] ) - ' B [ 0 ] (6)
+ (1 - : I [ 0 ] ) - ' B _ }rt[0 ]
it is easily seen that this first order effect is never positive. E x a m p l e 2. The well known multiplier-accelerator model of Samuelson reads
1, = a( C, - C, _ t )
E x a m p l e 3. Consider the model
Y, = C, + 1,
Y, = C, + 1, C, = c};-t
Now'* we have IAt.zl = ~(~:c). This square root is smaller than ~ ifand only i f g ( a ) = aZc + a(2c - I) + c > 0. A sufficient condition to ensure the latter is c > ~, because then ( 2 c - I ) ' - 4 c 2 < 0 and consequently ,q(a) > 0 for all a.
0
1
a> 0
C, = cY, 1, = a Yt - t - bK, _ 1
0
a>0, b>0
K,= K,_t +I,
where Y,, C, and I, represent national income, consumption and investment respectively. The model can be written in the form ( 1). Note that there is only one negative coefficient ie the ( - a ) in the third equation. It is easily checked that the sufficient condition of Theorem 2 is satistied if and only if I - c - 2 c a > O . This condition is rather restrictive. From the characteristic equation pertaining to the model, ie 2: - 2( I + a)c + ac = 0, it follows that the model is stable if and only if l - c a > 0 (see Gandolfo [5] for details). The matrix ( / - A [ 0 _ ] ) always has a pdd, whereas (1 - AI-0] - B [ 0 ] ) has one ifand only if I - c - c a > 0. We remark that the latter
E C O N O M I C M O D E L L I N G April 1989
where K, represents the end of period capital stock and the other symbols are already explained in example 2. The model is clearly of the type ( ! ). There is only one negative coefficient ie the ( - b) in the third equation. We note that the corresponding matrix ( i - A [ - I ] - B [ - I ] ) has no pdd, hence Theorem 2 does not help us here. The characteristic equation of the model reads
.8 Note that in this case c is smaller than 4a/( I + a) z. which may be incompatible with c > t for sufficiently high values of a.
185
Stability and structural forms of economic models: L. Schoonbeek ,;.2(1 -- c)--,;.((l - - c ) ( l - b ) + a ) + a = O Using this equation it can be demonstrated that the model is stable if and only if (1 - c ) ( b / 2 -
l)
1 -c
(See Laffargue [8]). Inspection of ( I - A [ 0 ] ) and ( l - A [ 0 ] - B [ 0 ] ) shows that only the former matrix has a pdd. From Theorem 1 it follows that ( ! - A [ 0 ] ) - t B [ 0 ] is not convergent. Thus, starting from this matrix it appears that the inclusion of the negative coefficient ( - b ) in the third model equation makes the convergence condition of ( I - A ) - t B less restrictive. Now we will compare the spectral radii o f ( / - A)- tB and (I - A [ 0 ] ) - t B [0]. The non-zero eigenvalues of the latter matrix, #t and #2 say. are equal to 1 and a/( I - c) respectively. Obviously they are both positive. The non-zero eigenvalues of (l - A)- t B are (I --c)(I - b ) + a + + _ ~ '~1.2 =
2(I - c)
where Discr -= (( 1 - c)( 1 - b) + a)' - 4a( I - c) It can be proved that ).t and 2: are in absolute value smaller than #l and #2 if b < 2 (see appendix). Note that realistic values of b are smaller than 2 (see Laffargue [8]). Thus, in normal circumstances the negative coefficient has a stabilizing effect. Example 4. The systematic part of Klein's well known model I of the US economy (model period 1921-41) reads C, = 0.017P, + 0.216P,_ t + 0.810W,, + 16.555 l, = 0. ! 50P, + 0.616P, _ t - 0. ! 58K,_ t + 20.278 W,, = 0.439E, + 0.147E,_ t + 0.130A, + 1.500 Yf = C, + 1, + G, - T, Pt= Y ~ - W I K~ = It + K t - t
w,= w,, + w2, E, = Y, + T, - W , ,
where W~, = private wage bill, Pt = profits, W, = total wage bill, E, = private product, At = time trend, G, = government expenditure, T, = taxes, W2, = government wage bill and the other symbols represent the same
186
variables as in example 3. The model is of the type of formula (1). The spectral radius pertaining to this model amounts to 0.845: thus the model is stable (see Goldberger [ 6 ] for details). It appears from our computations that the matrix (I - . A [ - 1] - B [ - 1]) does not have a pdd. The matrices A and B each contain one negative element ie ( - 1 ) and ( - 0 . 1 5 8 ) corresponding to W, in the fifth equation and K,_ t in the second equation respectively. It also appears that the matrix (I - A [0] ) has a pdd, whereas (1 - A [0] B [ 0 ] ) does not. As a result (! - A[0])-tB[0] is not convergent. As an experiment we have computed the spectral radii of the matrix ( I - A [ ~ ] ) - t B [ ~ ] for e = 0.0, 0.2 . . . . . 0.8, 1.0. They amount to 1.992, 1.674, !.213, I. 107, 0.989 and 0.845 respectively. We conclude that the negative coefficients have a stabilizing effect on the spectral radius. In order to refine this conclusion we have done the same experiment replacing the negative element of the matrix A each time with zero. The corresponding spectral radii are equal to 1.992, 1.746, 1.505, 1.264, 1.012 and 0.703 respectively. If we do the experiment by replacing the negative element of the matrix B with zero, the computed spectral radii are 1.992, 1.930, 1.864, 1.794, 1.718 and 1.631 respectively. We observe that the stabilizing effect is primarily caused by the negative element ( - I) associated with W, in the fifth equation.
Conclusion We investigated the stability of linear (macro-) economic models in terms of properties of the structural form matrices A and B of such models. A model is usually specified in structural form; our approach facilitates interpretation of the (in)stability of the model. In the second section we have seen that a model, for which A and B are non-negative, is stable if and only if by means of a suitable choice of the units of measurement of the endogenous variables, (A + B) can be transformed into a matrix of which all row sums are smaller than unity. The implications of this finding on the equations of a model, both separately and collectively, are examined. In the third section, the case was treated where A and B have positive as well as negative coefficients. A sufficient condition for stability is presented which is similar to the condition of the second section. Finally, we examined what happens with the conditions for stability and the magnitude of the spectral radius of a model if the negative coefficients are replaced by zeroes. We proved ia some typical examples of macroeconomic models that the original negative coefficients have a stabilizing impact.
E C O N O M I C M O D E L L I N G April 1989
Stability and structural forms of economic models: L. Schoonbeek
References I D.V.T. Bear and J. Conlisk, "The effect of unlagging on stability and speed of adjustment', Journal of Economic Theory, Vol 21, 1979, pp 62-72. 2 G C . Chow, ' T h e acceleration principle and the nature of business cycles'. Quarterly Journal of Economics, Vol 22, 1968, pp 403-418. 3 M. Deleau and P. Malgrange, 'M&hodes &analyse des modSles empiriques'. Annales de I'INSEE, No 20, 1975, pp 3-34. 4 F.M. Fisher, 'Choice of units, column sums. and stability in linear dynamic systems with nonnegative square matrices'. Econometrica, Vol 33, 1965, pp 445-450. 5 G. Gandolfo, Economic Dynamics. Methods and Models. North-Holland, Amsterdam, 1980. 6 A.S. Goldberger, Ectmometric Theory. John Wiley,
New York, 1964. 7 R. Harriff, D.V.T. Bear and J. Conlisk, 'Stability and speed of adjustment under retiming of lags', Econometrica, Vol 48, 1980. pp 355-370. 8 J.P. Laffargue. "lnterpr&ation 8conomique des caract&istiques quasicycliques d'un mod+le macroSconom&rique evoluant en environnement al~atoire', Annales de l"I NSEE, No 43, 1981. pp 33-65. 9 L. Meirovitch, Computational Methods in Structural Dynamics. Sijthofl'and Noordhoff, Alphen aan den Rijn, 1980. 10 Y. Murata. Mathematics fi)r Stability and Optimization of Economic Systems, Academic Press, New York. 1977. 11 L. Schoonbeek, 'Coefficient values and the dynamic properties of econometric models', Economics Letters, Vo[ 16. 1984. pp 303-308. 12 J.E. Woods, Mathematical Economics. Longman, New York. 1978.
Appendix Proofs
Theorem
('ase 1:(I - c ) ( I - h ) ~-a>O
2
Suppose that ( I - A [ - I ] - B [ - I ] ) h a s a pdd. Then ( I - .-I [ - l ] ) has a pdd. ('onscquently p( ,-I [ - 1] 1 < I, which implies that p( A ) < I (see Woods [ 12 ], p 47 ). Thus ( I - ,.! ) t
DL~cr>O In this case )-t > 0 and I , ; - , [ < 2 t . so it is sullicient to prove that 2 t ~< m a x l/it. !t2 ). Suppose lirst that 1 - c - a = O. Then \.~
exists and equals ~, ,,1'. Moreover. I)iscr
=(/-A[-I]I
t
(I-c)(l-h)+a=
0.<(It-A) 'tS)E-I].<(t-A[-I]I ' t S [ - l ]
t,(((t- A)-'thE-I]).<~,((t-.4[-
I])-'B[-1])<
I
The last inequality holds because it is assumed that (I - A [ - 1] - B [ - 1] has a pdd. The proof is completed by noting that
AI'- I ] ) 'B)[ - 1])
T h e proof of example 2
MODELLING
7.t = 1
which equals tt, ~< max(Ft, Its). Now we increase the value of b and analyse what happens with 2 t. Therefore we look again at the formula of,;.t, in particular the numerator. The term (( 1 - c)( I - b) + al decreases if h increases. The other term ie x , ~
db
=
also decreases because
~
\,,/Discr /
(a+(I-c)(l-b))(I-c)
It follows that 2 t < max(lq, It,,).
In order to prove that )-t and 2, are smaller in absolute value than 111 and it, if b < 2. we distinguish a number of cases.
ECONOMIC
h 2 - 41,
I-c+a>O
Discr=(l-c-aj-'>0and
from which it follows that
<~p(((I -
b ) + a)-" - 4 a 2 =
which is smaller than zero because b < 2. Thus, it appears that this situation is not relewmt here. Therefore suppose that 1 - c - a # 0. Let us for the moment substitute b = 0 m the above given expression of 2 t, and denote the result as 7. i. Then
Using this, we obtain
p ( ( l -- A ) - t B )
= (a( I -
April 1989
Discr=O As above, we only have to consider the case where 1 - c - a # 0. which implies that i t t # it2. Note that
187
Stability and structural forra~ of economic models: L. Schoonbeek. _,t(( 1 - - c ) ( I - b ) + a ) 0<;.t
=2:=
In both cases I,;.t I =
(l --c)
Case 2. (I - c ) ( l
= ~( 1 - b) + ~(a,,'( l - c))
I~,1
~< max(,ut, p.,).
-b)+a=O
It appears that here I i t I = 12z I = ,
If/a t > p , , then
Case 3: (l - c l ( l t(I-
b) + ,t(a/(l - c ) ) <
[(!-b}+
zt < I
Discr > 0
= Pt ~< max(Ill. It.) I;-.,I <
-b)+a
<0
N o t e that 1221 > 12 t I in this case. F u r t h e r m o r e -(1 -c)(l-b)-a
= -(l-b)---
11 - c )
O n the c o n t r a r y if lit
a 11 - c )
< m a x ( l q . ~t:)
t( l _ b ) + ½la ( l - c ) ) < [(a/l l - c ) ) + [(a/( l - c ) ) where the last inequality holds since b < 2. = (a/( I - c)) = It; ~< m a x ( l q , It.,)
Discr = 0 Discr < 0
I';-i I = I,t., 1 = ,~ ia,/( I - el) If It, ~< lti. then
w/la/(I --C') ~ I = I t I If It., > It I, then
~ f ( S i ( i --<')) < (al( I - <')) = It,,
188
Nov,
It is easily verified that now I,t~l = I';-zl =
-(I
-c)(I
-c)-a
2(I - c )
which is smaller than m a x ( / q . It,) because b < 2.
Di.v~'r < 0 I,i.i I
I lere v,c have
= I,~.21 = ,, (al(I
-<'))
See further c:.isc I wh,erc l)iscr < 0.
ECONOMIC
MODELLING
A p r i l 1989