The Long Run of Macroeconometric Models

CoplTig ht © I F.-\ C DI II,,,nic \I od eltill g alld Co ntro l o f !'\a tio ll a l Erono llli es. Edill b urg h . C K . 1 9~( 1

THE LONG RUN OF MACROECONOMETRIC MODELS M. Deleau*, C. Le Van** and P. Malgrange** "' l:' umj )('al/ I m 'I'S / IIII' II/ B al/k. LII.wlII/nll g

'''''C.\ RS

(/1/1/

CEPRE,\IAP. Paris.

Fi ((I/(('

~ This paper which constitutes a substantially abridged version of Deleau ~ Al. (~988) snxiies the existence and properties of Steady State Growth Paths (SSGP) ~ted. W1th a given macroeconometric model. The analysis of these SSGP y1e1ds a powerful .I1'lSIght mto the specification and theoretical structure of the model. In parti~, it ~s that the un?erly~g long run structure of typical "neokeynesian" macroeconometr1C models 15 not :WairasWl ~ 15 often asserted, but that it should rather be interpreted as a monopolistic competition equilibrium.

Key Words : Economics; (macroeconometric) models; steady state growth path; structural analysis of models; long run equilibrium. IN1RODUCfION In spite of their weaknesses, macroeconometric dynamical models constitute a useful device without any dominant alternative, and still widely used for shedding light on the preparation of quantitative economic policy. These models aim at providing a simplified numericalrepresentatlon of t~ global dynamical mechanisms at work m an ~' . Thell" structure is generally based on the so-called "disequilibriUm growth theory". an attempt at synthesis between short run keynesian theory and neo-c/assical growth theory. These models have been criticized on several grounds -see e.g. Malinvaud (1981)-. the most dramatic one being the numerous defects in their theoretical foundations. Indeed, the theory -or theories- behind a typical large scale macromodel is largely obscured by the numerous and complex dynamical interrelations. There is then a need of methcx.ls especially designed to reveal the theory hidden in the "black box".

The aim of the present paper is to deliver a convenient way to reduce a given macromodel to its theory based core. The basic idea is to exhibit the long structure of the model taking opportunity of the fact that. most of the time. the derivation of the typical behavioural relations start from a long runlequilibrium theory, on which dynamical mechanisms are plugged We derme the concept of long run for macroeconometric models in the usual way of dynarruc equilibrium of all adjustment mecanisms.-Indeed. when the model is dynamically stable. this corresponds to 1ts asymptotical behaviour-. Deriving long run mechanisms associated with dynamical relationships is obviously not a new quest. Most of the time however. it has concerned single equations and seldom an entire macromodel, a smal1 number of studies excepted (see e.g. Bergstrom-Wymer (1976). Gandolfo (1981), Currie (1982) and more recently Wa!lis-

Whitley (198])). A word now about what is meant by equilibrium. It is well known since Harrod. Hicks and Solow that the best concept of equilibrium adapted to models embedding capital accumulation and technical progress is the concept of a growth path at constant rate or steady slate grorvtIJ path called in the following SSGP. Indeed, only such trajectories preserve the invariance of the structure. More especially. they keep stock-flow ratios oonstant This imposed definition implies first that the economic prq>erties of the long nul

equilibrium cannot be set a priori but must result from the analysis of the whole system. second that we restrict our analysis to the behaviour of its "deterministic part" only. Our study is divided into two sections. The first gives the fonnal setting of the problem. It amounts to a static simultaneous equations system which sets the initial conditions for a SSGP. We show that the existence of a SSGP rests on numerous specification prerequisites. The second section deal! with the economic interpretation of the long run system of a typical macroeronometric model. The common assertion: "the short run is Keynesian, the long run is Wa!rasian" is shown to be false for usual macroeconometric models. Indeed, their long run is closer to some kind of monopolistic oompetition equilibrium.

I - METHODOLOGY 1.1 -Formalization Let us consider a macroeconometric model written as follows: (1)

yCt) ,. f( yCt). yet-I). x(t); t).

with yet) (x(t)), vector nxI (mxI) of the endogenous (exogenous) variables. and t index of -discrete- time. The search for a Steady State Growth Path (SSGP) amounts to exhibiting particular solutions of (1) of the type: • • t xi(t) ,. xiCO)ai

. . .

y.(t) ,. y.(O) b· J J J

i-I, ... , m

j . 1. .... n

In other ~ In' loak for a set of n initial caxiilioos y'(O" and n grorvtIJ rates (bJ, ..., b"J for the endOf!t!111OUS variables, knowing the ~ eKOf!t!111OUS elements x'(O) and

raJ, ..., a",), so thal (2) verilJes system (J) al each time period.

For a solution of (I) of type (2) to exist it is necessary that

each equation of this system fulfills some homogeneity

266

\1. lkic'l ll.

c:.

Le \ 'a ll alld 1'. I\ Ldgr
oonditions. They are imposed by an elementary mathematical property of exponentials: except for pathological cases, a sum of teI'rm evolving through time at constant rates follows an exponential pattern only if all rates are identical. An example is provided by the usual consumption function:

with real disposable income R(t) and inflation rate net). A solution of C(t) at a non zero rate b exists only if P2 = 133 ..0, sinre in the long run n must be constant. Indeed specification (3) implicitely assumes that the elasticity of consumption with respect to inflation, P2(n(tYC(t», decreases when living standards increase. The same reasoning applies to the constant term P3, supposed to represent all unexplained factors. In presence of that type of heterogeneity it is of interest to ask wether the specification is not in contradiction with the Wlderlying theory. FlD1hermore, it is necessary to substitute for the original equation another one as similar as possible which allows for a SSGP. One can, for instance, substitute for the formula (3) the consumption function (3') with time varying coefficients coinciding with (3) at time t - 01. (3')

C(t) - Pl R(t) + P2 (1 +b' n(t) + Ps (1 +b)t.

1.2- Determination of Growth Rates. It comes out that the structure of usual macroeconometric models is such that the long run growth rates (bl,... ,b,,) for the n endogenous variables can be determined independently from the initial conditions (the reverse being not true -see below). Moreover the m rates (aU of the exogenous variables cannot be freely choosen. Actually, all the rates (aU and (1);) derive in a straightforward way from three basic exogenous growth rates: n

: exogenous growth rate of population,

y

: exogenous growth rate of technical progress (necessarily Harrod neutral),

m

: exogenous growth rate of money supply : exogenous growth rate of foreign prices.

orp

The foll
1+g-(1+nX1+y). For one sector models, this uniqueness results from supplydemand equilibrium, except for very special cases (abovementioned property of sums of exponentials). The same applies to multisectoral models where fixed ratios between quantities of goods are involved, for production (infU-
Exceptions to these general rules imply very peculiar specifications, such as an open economy with flexible exchange rates where no nominal aggregate is exogenously fixed, or a smooth homogeneous multisector economy (e.g. of the Cobb-Douglas type) where distortions between quantities can be exactly oompensated by opposite distortions between prices.

U· Detennjoation of Initial Conditions The gr
y" (0) -

.

re y"(0), (Yj(W bj), x"(O), 0)

or, omitting the time index: (5)

y'-f(y",x',n,y,p)

lhe system (5) CY' *long run system" 35S0Ci31Bd with model (J), lXK1SisIs in a set of simultaneous equadons, whose evr!!TJ' solutioo y~, giW:!rJ the exogenous ~ x':. def'mes a set of

imli3J cmditioos fCY' a SSGP. One can notice that long run (exogenous) growth rates intervene in this long-run system. The static multipliers associated with (5) correspond to the asymptotic effect of a sustained multiplicative shock on the exogenous variables, which could be obtained through long run simulations of the initial model (1), provided it is dynamically stable. Let us make a few preliminary remarks about this long run system (S}. G) An equation of model (1) between growth rates yields a long run relationship between long run growth rates only, without any implication for initial levels. (ii) In a relation involving both rates and levels, only the last ones subsist in the long run system. For instance an equation of the foll
y.. f( z),

ZE

R.

with ... meaning growth rate,

implies an autonomous determination of z': z· - f-l(

y") - g( n, y, p)

A typical case is the Phillips curve. It links the growth of wages and prices with the unemployment rate, and thus leads to a long run determination of the unemployment rate independently from the rest of the system.

CiiD In a dynamic progressive adjustment of a variable y to its target Y, y -1jI( y), a stationary target is generally used for the reference ie. 1jI(l) - 1. For the long run system, this leads to: y' -ljI(lIb) Y'

{a

y'.

This discrepancy between actual and target long run values is a well known phenomenon, intensively studied in recent years Wlder the headings of FIror Correction Mechanisms and Even if the long run Cointegration (see e.g. Hendry system (5) admits a mathematical solution y', the associated SSGP is not necessarily economically meaningful, unless this phenomenon has been taken care of at the very specification stage of (1).

(986»,

267

Th e LOll g RUIl o r l\ lanoecOIl Olllelric l\ lodcl s

II.- THE ECONOMICS OF THE LONG RUN SYSTEM 2.1 - A Reference Core Model

We turn now to the ecooomic properties of a typic:al long run system We use a representative core model derived from the ana1ysis of the long run of several maaoeconometric models, in particular the french METRIC and DMS models, as well as of the CEPREMAP "maquette" (see Deleau etaL (1984), Kuh et a1 (1984), MaIgrange (1988)). This core model describes the long run structure of a keynesian open economy with ftxed exchange rates. All variables below refer to long run equilibrium values in the sense of paragraph 1.3. The asterix is omitted for the sake of simplicity, as well as the explicit presence of long run growth rates. Finns, equipped with constant return to scale technology, derive their demand for labour Net and for investment I from cost minimisation at given demand for output Q and given capital labour price rati02. They set the supply price P for their output through mark-up on Wlit costs. Workers supply a

P4 the predetermined unemployment rate (deriving from the phillips rurve as noticed in section 1.3). Substituting in relations (M!) and (0) the labour demand Nci by its expression in terms of pY'W and Pc, and eliminating Pc and PI. give the following system with the three variables Q, piP, and W/p:

(Ml') Q - Q d( W/p, piPe) - Q m( piPe) + ~ x( piP,) + G, (M2')

(plP,JII1· h( W/p),

(M3') (W/p) - (piPe) -u 1 t (QI N),

with d'1, d'2 > 0, h' > 0 and t' > O. In the first equation, the propensity to spend depends positively on the real wages and on the relative prices. It is worth noting that the autonomous expenditure multiplier at giVEn fX1Ces is Cl-
ftxed amount of labour N and consume a fraction of their disposable income -wage plus non wage income. Foreign trade -exports X, imports M- depends on domestic demand

The model includes four exogenous variables, Government

Q, foreign demand ~, and competitiveness piPe, with p, foreign general price level 01 domestic atr'Tl!T1CY The prices of the different demands are given by the usual fonnulre:

Foreign Prices Pe.

PilE p1-e

P:, where a

corresponds to the import share in

the relevant aggregate demand.Wages deflated by pu can then be written

independently of the other variables Q and piP•.

Expenditures

G,

Foreign Demand ~, Labour Force

2.2 - The Working of the

N and

Model.

We first observe that a variation of Pe implies a proportionnal shift in prices and wages, without any effect on real variables.

N, G

W/pu'" W/p [plP.~·

In the same way, simultaneous proportional shocks on

Lastly, the interest rate is assumed exogenously fixed by international competition.

and ~, affect Q only and leave prices and wages unchanged. Two important cases can be distinguished:

According to our experience, this representation describes rather well the long term structure associated with most usual macroeconometric models (see the afore-mentioned references). It can be synthetized by a three equations system:

(M})

Q = (~1 W Net + ~2P QV Pc + Q i(WI pi) + ~ x (piP,)

(Demand)

(C)

(I)

(X)

(j) Cl,= 0 -no import content in investment-.In this case, real wages result from (M2') independently from the exogenous variables. The relation (M3') then gets the production Q which

does

not

p

= ~3 W NcljQ

(ll) General ~ Cl, > 0

(M3) N - ~4Q nCpYW) - ~4 Net (Equilibrium on the labour market) PI'"' p1-e 1 pU1 , e

N,

where output results obviously from the pre-determination of N,.can be put in this category.

(Supply price)

with

N.

In this Classical Regime, variations of G and/or ~, affect prices only, the greater the effect, the lower the price elasticities of foreign trade and the greater the import content of consumption (al) The case of a Leontief technology,

Q-n (M2)

depend on demand but only on the labour force

The system is now interdependent. If we eliminate wages through relation (M2'), we get a traditional Aggregate Demand-Aggregate Supply system:

Pc· p1-u2 P~,

(+)

,

and iD, nO [optima) investment and labour demand per unit of production. It is supposed that ~1 + ~ S; 1, i' > 0 and i', n', x', m' > O.

The ftrst equation describes equilibrium of supply and demand on the market for goods in a quite traditional manner except that the various elements of the aggregate demand are here supposed to woespond to a SSGP. The two other relations correspond respectively to the long run outcome of the dynamics of prices and wages, with ~3 the mark-up rate and

(Ml") Q

(+)

(-)

= Q ac pip,) - Q m( piP.) + ~ x( piP.) + G (+) 11

-

(M?) (piP,) u 1.. t(QI N)

It can be observed that the Aggregate Demand relation (Ml") is not necessarily downward sloping wr.t. prices, because of

a

their positive influence on domestic demand This positive relative price effect, direct, and indirect through the real wages, called "Laursen-Metzler effect", expresses that an adverse inflation gap w r .t. foreign prices mechanically implies a smaller increase in relative user price because of the import share, and act then as a lever on real income. The aggregate

268

M . De leau. C . Le Va n an d P. '\ Ia lgra n gc

demand function is less decreasing when the economy gets more open (a1 and a2largerlThis phenomenon is reinforced if the constant prire multiplier is small The secood equation (0) colJespouds to the traditional aggregate 5Upply function of textbooks. It incorporates three basic elements: description of the labour market, labour demand function and supply prire relation. The only, rut important, difference lies in the replacement of the Phillips curve by a fixed unemployment rate which is its long run counterpart

The main comparative summarized:

static results can

be

briefly

1/- A positive shock on Gor on Q. induces a translation of the demand curve, Le. a move of the equilibrium along the supply curve, and then an increase in both prices and quantities, the greater the price increase the less open the economy. '}/- An exogenous increa3e in labour forre N shifts the supply function upward and generates a rise of production and a

drop of prices.

3/-

A movement in foreign prices

increase of public expenditure leads to a permanent deterioration of the government budget, and an unbounded increase of the public debl The so called long run public spending multiplier has thus a dubious meaning. This phenomenon is well known sinre Olrist (1968) This same policy exercise implies a permanent wcnening of the balanre of payments, with an inf"mite foreign debt in the long run. Taking into account these two constraints would enhance the economic relevanre of the long run core model. It would also lead to a substantial extra complexity: money and public bonds, interest rate, exchange rate and capital movements being then endogenous variables.

Pe

generates an equal variation of domestic prices, without any effect on the quantities. 23 - The Lon~ Run Core Model and Maqoecooomic 1beor:y In this section, the specification of our core model is confronted with the suggestions that would emerge from macroeconomic theory. (j) The Phillips curve implies, in the long run, an involontary unemployment equilibrium rate independent from the state of the economy. This rate may be greater than its mean historical value if the progression of wages has been actually higher in the sample period than the productivity growth (see Deleau et Ai (1988»). Furthermore, on the goods market, the

hypothesis of monopolistic competition between fl/ll1S leads to a mark-up pricing corresponding to an excess supply of output at 81'1!a7 p:ices. Hence the following remarks: 1/- It is clear that the long run of maaoeconometric models is not Walrasian: it would be advantageous for firms to produre more and for CXlI'lSUI'Ilel" to work more at equili/;yjum pices; this is blocked by the monopolistic behaviour of the agents.

'}/- The black-box nature of the Phillips curve already discernible for short run analysis, becomes impeding in a long run setting. The representative core model would much gain in explanative power from a real dynamical theory of wages and employmenl In the present state of maaotheory and maaomodelling, the law IVl1 of maorX;!CXK10metric models hetlviIy rests aJ a lII1m1pioymtnt rale determined cnIy by a b/a&-bax reiah{xJ, IIeJ' often of rather JXX:Y econometric quality

3/ - In the core model, the elasticity of the demand perceived by firms is directly linked with the value of i>3, while the realised demand (unction is given by (M1) or (M'1) . 11Je 1.ITXkr¥ng I1XX1OfXJIistic a:wpetih"oo scheme rwu/d g;Un Jrrm beiJ?g slated 0Ja'e exp/icitely (in Another important problem consists in the limits to he

drawn Ix!trn!en the exogmous and endogmous danains In a short run perspective, it is usual to leave aside some phenomena which look largely pre- or post-determined with respect to the system under consideration This practire may affect the long run properties and consistency of the model The intertemporal constraints of the internal and external debt are typical examples. In the previous core model, a sustained

To summarize, macroecooometric models have long run economic properties which parallel some macrcKheory models however in an often less sophisticated and consistent way.

CONQ.USION

The previous analysis leads to a certain number of conclusions, negative as well as positive, about the long run structure of mac oeco! lOI1Ietric models. The negative conclusions consist in the numerous prerequisites refering to the specification as well as the estimation level, which a given macroecooometric model has to satisfy in order that an acceptable long run can be associated with it. Furthermore, the long run analysis of maaoeconometric models is a very acid test for applied maaoeconomics. The limitations of black-box approaches or approximations, which can be considered tolerable for short run simulation, appear strongly binding when FM in a long term per.!peCtive. On the positive side, our study offers a methodology to incorporate long run considerations in the elaboration of macroeconometric models. In this respect, it points out at potential improvements. Some are linked with research in

progress. If, clearly, existing maaoeconometric models should be considered with extreme caution for interpreting long run phenomena, maaoeconometric modelling should in the future benefit from fruitful lines of research.

BIBLIOGRAPHY

BERGSTROM, kR. and CR WYMER (1976). "A Model of Disequilibrium Neoclassical Growth and an Application to the United Kingdom", in Bergstrom A.R Eel., Statistical Infermce in Continuous Time Econanic M0de4 North Holland, Amsterdam. CHRIST, C.F. (968), "A Simple Macroeconometric Model with a Government Budget Restraint", Jouma/ ofIbIitical Ecax.my 76, pp.5~7. CURRIE, D.A. (1972). "The Long-run Properties of the Bank of England's Small Monetary Model of the UK Ecooomy", Afp!ied ~ 14, pp. 6}-72

DELEAU, M., C. LE VAN and P. MALGRANGE (1988). "The Long Run of Macroeconometric Models', Watirw hper CEPHEMAP, October 1988, to appear in a volume in honour of E. Malinvaud, MIT Press. DELEAU, M, P. MALGRANGE and PA MUET (1984), "A Study of Short Run and Long Run Properties of Macroecouometric Models by Means of an Aggregative Core Model", in P. Malgrange and PA Muet Eds., Onn:npAllry MIIcroetxnamic J./odeIlirw, Basil BIackwell, Oxford.

Tilt'

LOIl~

RUI! of l\Iacroecol!ol11etric I\foclels

GANDOLFO G. (1981). Quajilati~ Analysis and Econometric Estimation of ContinlJOl.lS Time Dynamic Models, North

Holland, Amsterdam. HENDRY, D.F. Ed. (1986), Econometric Modelling with CoIntegrated Variables, Oxfad BuUetin of Ea:X1omics and SlJitistjcs, sped3J issue, 48. KUH, E., C. LE VAN and P. MALGRANGE (1984). "Une etude de la dynamique structurelle des IJIO({eles

maaoeoooometriques",

WaXing Paper

CEPHEMAP,

July 1984. P. (1988). "The Structure of Dynamic Macroeoooometric Models", CORE WaXing Paper, to appear in a volume in honour of CORE's XX 1h Anniversary.

MALGRANGE,

MALINVAUD, E. (1981). "Econometrics Faced with the Needs of Macroeconomic Policy", Eax1on1etrica, 49, 6, pp.

1363-1375. WAlLIS, K.F. and J.o. WHITLEY (1987). "I.oog-Run Properties of Large-Scale Macroeconometric Models", AnnaJes d'Economie et de Slatistique, 6-7, pp.207-224.

269