In this paper, we show a similar concept of stability of the adjustment process concerning the difference between the weighted average and the actual value. Because the adjustment process includes only one eigenvalue of zero, and thus one dimension of freedom, it is difficult to apply the Routh-Hurwitz theorem to the process, at least directly. To solve the one-dimensional freedom, we fully apply Goodwin's analysis of the weighted average. One-dimensional freedom is suitable for indeterminacy issues, or the characterization of price and the like, which has only relative value and no absolute value. From this analysis, we know that it is reasonable to be concerned about the difference between the average and the actual value.Linear approximation system, Weighted average, Hicksian matrix, Linear approximation stable except choosing the absolute value

Takeshi Ogawa

English

Research Papers in Economics

     
 
 
  
  
Volume 30, Issue 4 
  
Stability of the adjustment process with the difference between the weighted 
average and the actual value 
  
 
 
Takeshi Ogawa  
Nagoya University 
Abstract 
In this paper, we show a similar concept of stability of the adjustment process concerning the difference between the 
weighted average and the actual value. Because the adjustment process includes only one eigenvalue of zero, and thus 
one dimension of freedom, it is difficult to apply the Routh-Hurwitz theorem to the process, at least directly. To solve 
the one-dimensional freedom, we fully apply Goodwin's analysis of the weighted average. One-dimensional freedom is 
suitable for indeterminacy issues, or the characterization of price and the like, which has only relative value and no 
absolute value. From this analysis, we know that it is reasonable to be concerned about the difference between the 
average and the actual value.
This paper is a spining off version from a mission for the author by his advisor. The author would like to thank his advisor. The author would 
also like to thank the members of NMW (Nagoya Macroeconomic Workshop) and so on, who provided useful feedback, as well as the 
referee and the associate editor. The author is solely responsible for any errors. 
Citation: Takeshi Ogawa, (2010) ''Stability of the adjustment process with the difference between the weighted average and the actual value'', 
Economics Bulletin, Vol. 30 no.4 pp. 3010-3017. 
Submitted: Mar 17 2010.   Published: November 11, 2010. 
 
       1 
1. Introduction 
This paper examines a similar concept to stability of the adjustment process with the 
difference between the weighted average and the actual value. To analyze the stability, it is 
usual to apply the Routh-Hurwitz theorem or the Hicksian method of stability with which we 
can derive the condition that all the roots of the  nth degree equation with real coefficients 
have negative real parts. However, at least directly, we cannot use these methods for this 
process when the process includes only one eigenvalue of zero. To analyze the stability of the 
process, we must consider the eigenvalue of zero, which means one-dimensional freedom. 
The case is similar to the price adjustment process because price has essentially relative value 
and no absolute value. We analyze the stability of the process, modifying the concept of the 
Hicksian method of stability according to the process. We show that the process is linear 
approximation stable except choosing the absolute level, which we define for analyzing the 
stability of the price adjustment system.   
In the stable case, Goodwin (1949) answered the similar case including the weighted 
average.
1
Recently, more and more areas take in some affection from other economic agents. For 
example, the areas of social status and status preference take in the preference affecting the 
average wealth. The concept of strategic complement is widely applied in various areas. The 
findings that we present in this paper may thus have wide application to a variety of research 
endeavors like these areas. Since we analyze a case including an eigenvalue of zero, our 
findings may have implications for situations characterized by indeterminacy and have 
applications to the mathematical basification of endogenous growth models. 
  Essentially, we apply Goodwin (1949) fully to the adjustment system, which is a 
form of the linear approximation system. 
The paper is organized as follows: The following section explains and analyzes  the 
model. The third section discusses several implications of our findings and the final section 
presents our conclusions. 
 
2. Model and Analysis 
One of the most important prices is the wage. Hereafter, we use the model with the form of 
the wage adjustment process. Suppose that the economy has  n  jobs, and the wage moves 
only the wage adjustment process. Let  ) (t wk   be a wage of the  k th job for any  n k , , 1 = . 
Let  k w   be a positive initial wage for any  n k , , 1 = . Let  kj b   be the weight parameter of 
                                                 
1  Recently, Goodwin’s findings have been applied to various fields of research; for example, 
they have been used in the studies of Punzo (1995), Brida et al. (2003), and Raghavendra 
(2006). In addition, his findings have been reconsidered in the studies of Velupillai (1998) 
and Punzo (2006).   2 
the  j th wage affecting the  k th wage, satisfying  1 0 ≤ ≤ kj b   and  1
1
= ∑
=
n
j
kj b . Specifically, in 
the arithmetic mean,   
n
bkj
1
=   for any  n j k , , 1 ,  =  
holds. In the economy, the wage adjustment process with the difference between the weighted 
average and the actual value is as follows: For any  n k , , 1 = ,   
                      k k k
n
j
j kj k w w w w b w = − =∑
=
) 0 ( ,
1
 ,                    (1) 
where 
dt
dw
w
k
k =  . For simplicity, we use the following assumption: 
 
Assumption 1  1 < kk b   is satisfied for any  n k , , 1 = .   
 
If  1 = kk b , then,  k w   becomes constant, which we avoid.   
(1) can be rewritten as follows: 
    












−
−
−
=












=
























=












1
1
1
,
) 0 (
) 0 (
) 0 (
,
2 1
2 22 21
1 12 11
dfn
2
1
2
1
2
1
2
1
nn n n
n
n
n n n n b b b
b b b
b b b
B
w
w
w
w
w
w
w
w
w
B
w
w
w
dt
d

   


   
.     (2) 
We consider the stability of (2). From the definition of  kj b ,  0 1= B   holds, where  1  is the 
transpose of [ ] 1 , , 1   and  0  is the zero vector. Thus,  0 det = B , which implies that the 
coefficient matrix B   is not linear approximation stable because B   has at least an 
eigenvalue of zero. Therefore, we define a similar concept of stability. For this purpose, the 
fact, which we show without proof is important to define a concept similar to stability. 
 
Fact 1 Consider one variable  x  following the linear differential equation  x x γ =  . Even if 
x  starts at any initial value, the differential equation converges not only in the case that the 
real part of  γ   is negative but also in the case  0 = γ . In the case  0 = γ , the steady state 
x
t +∞ → lim   depends on the initial value. This is different from the case that the real part of  γ   is 
negative, where the steady state is 0 uniquely, which does not depend on the initial value. 
 
With this fact, we define a similar concept to stability as follows. 
 
Definition 1 A system of ordinal linear differential equations is linear approximation stable   3 
except choosing the absolute value if and only if the following conditions are satisfied: 
1.  Any eigenvalue of the coefficient matrix except one has a negative real part. 
2.  The coefficient matrix has an eigenvalue of zero. 
3.  The eigenvalue of zero has only a one-dimensional set of eigenvectors.   
4.  1  is an eigenvector of the coefficient matrix about an eigenvalue of zero.   
 
Therefore, if a system of ordinal linear differential equations is linear approximation stable 
except choosing the absolute level, the system converges except choosing the absolute level. 
This is a better characterization because this stability has a freedom in choosing the 
numeraire. We offer the following proposition: 
 
Proposition 1 Under Assumption 1, the wage adjustment process system with the difference 
between weighted average and the actual value, (2), is linear approximation stable except 
choosing the absolute level.   
 
Proof  For the sake of simplicity, we only provide the proof of this proposition when 
0 > kj b .  0 1= B   implies that the coefficient matrix has an eigenvalue of zero and  1  is an 
eigenvector of the coefficient matrix about the eigenvalue of zero. First, we show that the 
eigenvalue of zero has only a one-dimensional set of eigenvectors. The  1 − n   th successive 
principle minors of B   have the same sign as 
1 ) 1 (
− −
n , which is non-zero. Thus, 
1 rank − = n B   holds because  0 det = B . Therefore, the eigenvalue of zero has only a 
one-dimensional set of eigenvectors. 
  Second, we show that any eigenvalue of the coefficient matrix has a non-positive real 
part. For any  0 > ε , define a new matrix  ) (
~
ε B   as follows: 
.
0
0 0
) (
~ dfn










− =
ε
ε

  

B B  
From the definition of  kj b , the  lth successive principle minors of  ) (
~
ε B   have the same sign 
as 
l ) 1 (−   for any  n l , , 1 = . Applying Theorem 4.D.3 of Takayama (1985, p.393) to this 
condition, the real parts of all eigenvalues of  ) (
~
ε B   are negative. Let  ) ( Re ε λ   be the largest 
real part of eigenvalue of  ) (
~
ε B .  0 ) ( Re < ε λ   for any  0 > ε , which means that 
0 ) ( Re lim
0 ≤
+ ↓ ε λ
ε . Therefore, the real parts of all eigenvalues of  B   are non-positive.     4 
Third, we show that any eigenvalue of the coefficient matrix except one has a negative 
real part. Suppose not.  B   has eigenvalues of  ai ± , where  i  satisfies  1
2 − = i , and  a  is a 
non-zero real number. Then,  ( ) 0 det = − n aiI B   is satisfied, where  n I   is the identity matrix. 
However, all elements of  B   are real numbers. Therefore, for any  n k , , 1 = ,   
( ) ∑
≠
= − > + − = − −
k j
kj kk kk kk b b a b ai b 1 1 1
2 2 , 
which implies that  n aiI B −   is a dominant diagonal matrix. Applying Theorem 4.C.1 of 
Takayama (1985, p.381) to this condition,  n aiI B −   is nonsingular, that is,  ( ) 0 det ≠ − n aiI B  
holds, which is in contradict to the equation that  ( ) 0 det = − n aiI B . Therefore, any 
eigenvalue of the coefficient matrix except one has a negative real part.   
We can conclude that the wage adjustment process system with the difference between 
the weighted average and the actual value, (2), is linear approximation stable except choosing 
the absolute level.                                                  (Q.E.D.) 
 
Proposition 1 shows that the adjustment process is essentially stable except the absolute 
level. The adjustment process satisfies the following: If the actual value is higher/lower than 
the weighted average, the value decreases/increases. Therefore, seeing the long term, the 
actual value converges to the weighted average. Moreover, if the actual value starts from the 
proposal of higher values, the steady state to which the system converges is also higher than 
the original convergent point.   
At least in the standard stability theory, the focus point is such that any real part of the 
eigenvalue is negative, that is, the convergent point is unique, which prohibits freedom in the 
choice of the absolute level. However, the convergent cases are such that not only is the real 
part of the eigenvalue negative but also the eigenvalue is zero, without any imaginary part. 
The case that the eigenvalue is zero is such that the differential equation stops, at least in 
some sense. This quality brings to the adjustment process of the freedom of one dimension. 
The freedom is particularly important for the price adjustment process when the price 
shows only relative value, not absolute value. Therefore, the adjustment process with the 
difference between the weighted average and the actual value is a suitable process because it 
satisfies stability essentially and includes the freedom to choose the numeraire. 
The most important case is that where the weight is equivalent to all variables, that is, 
the case of the orthodox arithmetic mean. In this case, Assumption 1 is satisfied. We can state 
the following corollary from Proposition 1 straightforwardly. 
 
Corollary 1 The wage adjustment process system with the difference between arithmetic 
mean and the actual value is linear approximation stable except choosing the absolute level. 
 
Therefore, even if we use the adjustment process to minimize the difference between the   5 
arithmetic mean and the actual value, the system becomes linear approximation stable except 
choosing the absolute level, that is, essentially stable except choosing the absolute level. 
 
3. Discussion 
In this section, we discuss some potential for applications and some backgrounds. 
Firstly, we mention the sketch of the adjustment system in the case of wage or price. In 
the case of wage, if a company shows a lower wage than an average one, then many workers 
will avoid to choose the job. Maybe, the company will raise the wage. Conversely, consider 
the case that a company shows a higher wage than an average one in the similar (but a little 
different) qualities of workers. Then, another company can collect the similar qualities of 
workers with lower wage than that of the company. The company has a probability to think as 
follows: Even if we show a little lower wage than the original one, we can collect workers. 
Thus, the company may decrease the wage. The similar mechanism can be shown in the case 
of price adjustment system. 
Second, as previously discussed, Goodwin’s findings (1949) have been widely applied 
to various areas. However, many studies have solely focused on Goodwin’s analysis of 
discrete time lags and have neglected considering his analysis of the weighted average. In the 
case of simple discrete time, if all variables choose the arithmetic mean, the system will stop 
immediately, which is non-sense.  However, in the continuous time case, the adjustment 
system is affected by an indeterminacy issue essentially, which implies that we must argue 
the convergence without point of closures. Indeterminacy issues are now widely spread to 
various dynamic areas. Especially, endogenous growth models have essentially the 
indeterminacy issues to grow their own engine. This study focuses on the potential of the 
overlooked point of Goodwin (1949) to apply to these areas. 
Thirdly, we should stress that the system can be applied not only price or wage 
adjustment system but also various systems considering an average. Recently, the area of 
social status has developed.
2  In this area, people care about some kind of status like average 
wealth. Some models are directly introduced the difference between actual wealth.
3
 
  This 
paper has a potential to the analysis of people who worrying the around.  Moreover, the 
system is suitable for strategic complementarities, which are well-known as the cases if all 
players’ variables mutually reinforce one another in game theory. Therefore, the finding of 
this paper has the potential of applying to cases satisfying strategic complementarities. 
4. Conclusion 
In this paper, we examined the stability of the adjustment process with the difference between 
the weighted average and the actual value. The orthodox stability is not satisfied because the 
                                                 
2  See for example, Easterlin (1995) and Clark and Osward (1996). 
3  See for example, Corneo and Jeanne (2001) and Fisher and Hof (2005).   6 
adjustment process matrix includes the eigenvalue of zero. However, to consider the concept 
of linear approximation stability except choosing the absolute level, we showed that the 
adjustment process is essentially stable without choosing the absolute level. The difference 
between this stability and orthodox stability is useful for a price adjustment system because 
price shows only relative value, not absolute value. The freedom of one dimension enables us 
to choose the numeraire.   
Goodwin’s analysis includes the one-dimensional freedom of the weighted average, at 
least in the static case. Therefore, our research essentially adds a new economic application of 
Goodwin’s analysis to the various  adjustment processes  with the difference between the 
weighted average and the actual value. 
Although we considered the weighted average value to be constant, our model remains 
applicable even when the weighted average value adjusts over time. 
 
References 
Brida, J.G., Anyul, M.P. and Punzo, L.F. (2003) “Coding Economic Dynamics to Represent 
Regime Dynamics. A Teach-Yourself Exercise” Structural Change and Economic Dynamics 
14, 133-157. 
 
Clark, A.E.  and Osward, A.J. (1996) “Satisfaction and Comparison Income” Journal of 
Public Economics 61, 359-381. 
 
Corneo, G.  and Jeanne, O. (2001) “On Relative-Wealth Effects and Long-Run Growth” 
Research in Economics 55, 349-358. 
 
Easterlin, R.A. (1995) “Will Raising the Incomes of All Increase the Happiness of All?” 
Journal of Economic Behavior & Organization 27, 35-47. 
 
Fisher, W.H. and Hof, F.X. (2005) “Status Seeking in the Small Open Economy” Journal of 
Macroeconomics 27, 209-232. 
 
Goodwin, R.M. (1949) “The Multiplier as Matrix” The Economic Journal 59, 537–555. 
 
Punzo, L.F. (1995) “Some Complex Dynamics for a Multisectoral Model of the Economy” 
Revue économique 46, 1541-1559. 
 
Punzo, L.F. (2006) “Toward a Disequilibrium Theory of Structural Dynamics Goodwin’s 
Contribution” Structural Change and Economic Dynamics 17, 382–399. 
   7 
Raghavendra, S. (2006) “Decomposition Methods for Analyzing Intra-regional and 
Inter-regional Income distribution” Working Papers from National University of Ireland 
Galway, Department of Economics 108. 
 
Velupillai, K.V. (1998) “The Vintage Economist” Journal of Economic Behavior & 
Organization 37, 1–31. 
 
Takayama, A. (1985) Mathematical Economics, Second Edition, Cambridge University Press: 
the United Kingdom. 

Coding Economic Dynamics to Represent Regime Dynamics.

Decomposition Methods for Analyzing Intra-regional and Inter-regional Income distribution” Working Papers from

Some Complex Dynamics for a Multisectoral Model of the

Status Seeking in the Small Open Economy”

The Multiplier as

The Vintage Economist”

Toward a Disequilibrium Theory of Structural Dynamics

Will Raising the Incomes of All Increase the Happiness of

Stability of the adjustment process with the difference between the weighted average and the actual value

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