On the Dynamic Specification of Disequilibrium Econometrics

Copyright © IFAC Dynamic Modelling and Control of National Economies. Budapesl. H lIngar)' 1986

ON THE DYNAMIC SPECIFICATION OF DISEQUILIBRIUM ECONOMETRICS R. Orsi Dl'j){[rl'IlI' 1I1 or E(()I/IIII/ics. L'lIil'l'I"sily or ,\1olil'llll. ,\JOlil'llll. IIII/\"

Abstract. The specification of a dynamic disequilibrium model is proposed where dynamics is introduced both by quantity adjustments and by a price adjustment mechanism allowing for different upward and downward adjustment speeds. Past unsatisfied demand and supply, which represent the effects of present rationing passed on from the past, are considered as explanatory variables giving rise to effective demand and supply functions. The solution of the model and the specification in terms of observable variables, suitable for estimation purposes, is presented; it is shown that the problems to solve are of the same order as those appearing in a two markets disequilibrium model. ~eywords.

Econometrics; disequilibrium; dynamic specification; partia l adjustment models; rationing models.

1.

INTRODUCTION

will be a kind of spill-over effect from past disequilibrium into the current demand and supply . These aspects, if they are of undoubted importance for analyzing interactions within a multimarket economy, are also of som interest in a single market disequilibrium model where spill-over effects are ignored (see Quandt's survey, 1982).

A large part of recent econometric research on disequilibrium theory makes use of the concept of fixprice equilibrium, and on this basis the short run consistency of the model is achieved by quantity rationing. This underlines the fact that quantity adjustments are much faster than price adjustments. In fact, under the two assumptions that prices and wages are fixed in the short run and that the outcome is an equilibrium with rationing, it follows that it will be reached through quantity adjust ments.

The purpose of this paper is to extend dynamic specification to a single market disequilibrium model by putting together the two types of adjustment. Using a simple disequilibrium framework a la Fair and Jaffee we shall consider the main features of the dynamic behaviour of the market by assuming that: - prices react to disequilibrium and partially adjust towards equilibrium, allowing for possible differences in upward and downward adjustment speeds; - past rationing on quantities influences current effective demand and supply.

On the other hand, some theoretical works on dynamics of disequilibrium in macromodels analyze the short-run dynamics under the assumption that prices and wages respond to discrepancies between effective demand and supply . This points out that, even if prices and wages do not react completely in order to permit the achievement of an equilibrium, they are not rigid to such a point that they can be considered completely exogenous.

The dynamics of the model is thus introduced via price and quantity adjustments, and, by taking advantage of this specification, several aspects can be analyzed which can be briefly summarized as follows:

In our opinion disequilibrium theory must also be concerned with the transmission of quantity constraints. Ideally the econometric specification of a disequilibrium model should describe the nature of the transmission process along with the condi tions which must prevail if the transmission process is incomplete. Without entering into details regarding the nature of the transmission process, it seems evident that the fixed price model gene rates a rationing which, in case it persists, will be passed on to the next period. As a consequence, decisions taken in one period cannot be regarded as independent of those taken in another period. If rationed agents try to restore the desired equilibrium , their behaviour will affect their effective demand and supply in the next period, and there DM ("N E - K

(i)

estimation of the impact of past rationing, considered as sort of overall spill - over effect across the whole economy; (ii) comparison of price and quantity adjustments; (iii) test of the equilibrium hypothesis; (iv) comparison of upward and downward adjustment speeds, and (v) comparison of disequilibrium models of different markets or for different co untri es . In the next section we shall describe the main features of the basic disequilibrium model which is proposed. In particular sub-se ction 2.1 deals with

305

R.Orsi

306

the role played in disequilibrium econometrics by the price adjustment equation, while 2.2 treats the problem of model solvability, namely the derivation of a form where unobserved variables disappear. Some concluding remarks can be founded in section 3 of the paper. 2.

THE MODEL

Let us consider a standard disequilibrium model where dynamic behaviour is explicitly introduced via quantity adjustments, i.e. past unsatisfied demand or supply enters as an explanatory variable for the actual demand and supply equations:

In the present disequilibrium econometric literature the use of adjustment models is restricted to price behaviour and the basic model (1) to (3) is often implemented with a price equation, getting a result that is twofold ; introduce dynamics in the model vi a changes in prices and, at the same time, have access to an observable disequilibrium regime indicator. A first model is obtained by integrating equations (1), (2) and (3) with a partial adjustment model on prices analogous to that proposed by Bowden (1978 a,b), allowing fOl different upward and downward adjustments . when

(4) D t

a'xD+a ,p + a (D _ - Qt"l) + u 1t 1 t 2 t 3 t 1

(1)

S

S 13 'X + 13 p + 13/S _ - Qt-l) + u 2t t 1 t t 2 t

(2)

min (Dt,St)

(3)

t

Qt where

Ult

ror term iances

ot

and

U2t

are serially independent erzero mean and varrespectively.

= 1, 2, ... , T, with and

a~

This disequilibrium model specification constitutes a departure from the customary framework, but the underlying economic behaviour is quite standard. The only new aspect consists in taking into account past rationing as a constraint conditioning the utility and profit maximization processes. This leads to demand and supply functions containing spill-over effects. In the context of a single market disequilibrium model this may be interpreted as the fact that past history of rationing affects the actual effective demand and supply. In other words a3 and 13 3 may be considered as overall spill-over coefficients. Such effects have as yet not been fully analyzed in t 'he one market disequilibrium models that appear in the econometric literature. The particular specification (1) and (2) assumes, moreover, that spillover is linear in unsatisfied demand or supply; more precisely, it is proportional to the value of dissatisfaction. We will also assume that the multiple effect of the spill-over is less than unity, permitting the convergence of the spill-over process in the market, according to the definition of the coherence condition given by Gourieroux, Laffont and Monfort (1980), that in this case becomes: (a3i33) < 1. 2.1.

when with

o~

D(p~)

=

llPt< 0

1, i = 1, 2, and P~ defined as i.e. the price at which the market would clear without rationing. l1 i

~

S(p~),

Another model, which is frequently encountered in applications, consists in the Fair and Jaffee quantitative model where the price change, adapted for upward and downward sluggishness coefficients, is proportional to excess demand:

p = t with

J

Pt-l

+ A (D -S ) 1 t t

when

D > S t t

LPt-l

+ A (D -S ) 2 t t

when

D < S t t

(4' )

A. > 0, i = 1, 2. 1-

Following Mouchart and Orsi (1986) the additional condition 13 2 > a 2, besides those on l1i and Ai, i = 1, 2, are sufficient for glv1ng rise to a oneto-one mapping between models (4) and (4') above, even if their economic interpretation is different. In fact while the Fair and Jaffee model assumes a competitive market, where prices react to the presence of excess demand because prices do not fully adjust to their equilibrium value, and the system acts as minimizing the costs of adjustment. Moreover the model (1), (2), (3) and (4) implies a recursive causal structure between endogenous variables; when replacing (4) with (4') one gets a simultaneous determination of Dt , St and Pt. Apart from this, it is evident that inference in the structural model may be conducted indifferently in terms of (4) or (4') and the estimation techniques which lead to the identification of 111' 112, a2 and 13 2 will also lead to the identification of Al and A2, the relationship between the adjustment coefficients being expressed by:

The Price Adjustment Equation i = 1,2

Partial adjustment models have been proposed for a long time as disequilibrium models, before the most recent formulation of disequilibrium econometric models in terms of "quantity rationing models", as introduced by Fair and Jaffee (1972). These models were proposed for specifying disequilibrium, essentially as an incomplete adjustment to the equilibrium values. In a previous work (Orsi, 1982), we try to implement this approach by extending it in a simultaneous context which permits to take into account cross-variable adjustments other than direct adjustments.

(5 )

Coming to the point of the regime indicators, since the price adjustment equation proposed is a nonstochastic version, through models (4) and (4') the observable variable llPt may be considered as disequilibrium regime indicator for both models. When Dt>St (or P~>Pt) one will observe llPt>O and through short-side rule (3) one can write:

307

Disequilibriulll ECOIlOIllClrics

D -(D -S ) t t t

S

t

(6)

when

I'lp t > 0

and similarly, in the case of e~cess supply Dt < St (or P~ < Pt) the observed quantlty may be expressed as: St + (D t - St) - St+ ( 13 2-a2)(p~ -Pt)

In the case that one is only interested in the estimation of the price adjustment speed and/or in testing the equilibrium hypothesis, assuming for simplicity ill = l-J 2 = il , it is easy to solve the system of equations (11) and (12) for the endogenous varible Pt. One first has to equate (11) and (12) in order to get the expression for the equilibrium price:

(7)

St+ f( ",pt) when

I'lPt < 0

Then the price equation (4) or (4'), by proving signals about excess demand and excess supply, permits to sp lit the sample in demand and supply periods; through the clearing rule (3) one knows which quantity is observed at each period. From an inferential viewpoint, testing the equilibrium hypothesis for models like (1) to (4) or (1) to (4') is quite straightforth sin ce it becomes a case of nested hypotheses. The equilibrium model corresponds to a particular subset of the parameter set of the disequi lib rium model (see Ito and Ueda, 19 81).

Let us now try to ge t a structural form for model (1) to (4) s uitabl e for estimation purposes, i.e. where unobserved variables are e liminated from equation (1) and (2). Remembering (6) and (7), ex tending them to the period t-l and using (5) one

il l ( S2-a 2)

S when

Qt-l

+ -

The error terms Ult and U2t are assumed t o be uncorrelated, one can consistently estimate (14) by a non linear technique; the equilibrium hypothesis can be tested simply by using a standard significance test for the adjustment parameter il . In order to derive demand and supply equations expressed in terms of transacted quantities it is convenient to ref er to expr e ssions (6) and (7) that along with (5) permit to write:

can write:

D t--l

in (4)

then, substituting this expression for one obtains :

- - - I'l P t_l 1 - il l (8)

t - 1 = Qt-l

Qt

D t

Qt

S

Qt

D t

Qt

S

l-J (13 - ex ) 2 l 2 I'lPt 1 - il

I'l P t -l >0,

and similar ly :

when

I'lPt> 0

(15 )

"'Pt < 0

( 16)

t

and D t-l

Qt-l

S

l-J ( 13 - ex ) 2 2 2 Qt-l 1 - l-J 2

t-l

(9)

I'l P t_l

when I'l Pt-l < 0 where the lagged price change appears e ither in the demand function or in the supply function (but not both). Defining the switching variables: Max (0, I'lPt . 1) Min (0, I'lPt_l)

( 10)

the demand and supply equations (1) and (2) can be rewritten as: D D = ex' X + ex p + ex t It Lt 3

(11 )

t

+

il (13 - ( ) 2 2 2 I'lPt 1 - l-J

when

Since also in this case the contemporaneous price variation appears in only one function at a time, one can define the same kind of swi thin g variables as above: + Max (0, I'lPt) I'l Pt (17)

and then the dynamic demand-supply model incorporating the price adjustment process (4) or (4'), can be rewritten as:

R. Orsi

308

(18)

The system (18) and (19) now incorporates the price equation (4) or (4') and represents an alternative structural form for the equation system described by (1) to (4) written in term of observable variables. The FIML (or NL3SLS) estimation method applied directly to system (18), (19) yields the efficient structural parameter estimates. These results permit to test several family of hypotheses; among them it is worthwhile to mention: a) test equilibrium versus disequilibrium comparing the value of II I and 112 with the null hypothesis II I = 0 (no excess demand) and/or 112 = 0 (no excess supply); b) test whether upward ( Il l) and downward (1l 2) adjustment speeds are the same; c) test the impact of past rationing on actual disequilibrium. 3.

SOME CONCLUDING REMARKS

In this paper we have tried to implement the specification of a dynamic disequilibrium model by putting forward quantity adjustments and a price adjustment mechanism allowing for different upward and downward adjustment speeds. The specification of a price equation like (4) or (4') permits to "discover" the sample separation and consequently, after some manipulations, to solve the model and hypotheses testing in this context become quite standard. Such an equation, among other things, avoid the unboundedness of the likelihood function in the parameter space (see Quandt, 1982). Other economic variables like inventories, capacity utilization, unemployment , demand pressure indicators or other variables can provide signals about excess demand reg ime c la ss if ica t ion. The problem con cerning the most appropriate variable for providing useful information about regime classification is strongly related to the problem of a correct specification of an economic model. The choice of a variable apt to solve the indeterminacy regime classification must come from economic theory. REFERENCES Bowden, R.J. (1978a). Specification, estimation and inference for models of markers in disequilibrium. Inter national Economic Review , 19 , 711-726. Bowden, R. J . (1978b). The Econometr ics of Diseq uilibr i um. North-Holland. Fair, R.C., and D.M. Jaffee (1972). Methods of estimation for markets in disequilibrium. Econome trica, 40 . 497··514. Gourieroux, C., J . J . Laffont, and A. Monfort (1980) Disequilibrium econometric in simultaneous equati on systems . Econometr ica , 48 , 75-96.

Ito, T., and K. Ueda (1981). Tests of the equilibrium hypothesis in disequilibrium econometries: An international comparison of credit rationing. International Economic Review, 22, 691-708. Mouchart, M., and R. Orsi (1986) . A note on price adjustment models in disequilibrium econometrics. Journa l of Econometr i cs, 31 , 209-217. Orsi, R. (1982). A simultaneous disequilibrium model for Italian export goods. Empi rical Economics , 7, 139-154. Quandt, R.E. (1982) . Econometric disequilibrium models. Econometr ic Reviews, 1 , 1-96.