THE ECONOMETRICS OF MACROECONOMIC MODELLING

Stability and steady state

We want to investigate the dynamics of the wage-price system consisting of equations (6.3), (6.5), and the definitional equation

Pt = Фяг + (1 - Ф)рч, 0 <ф< 1. (6.7)

Following Kolsrud and Nymoen (1998), the model can be rewritten in terms of the product real wage wq, t, and the real exchange rate

piq, t = pit - qt. (6.8)

In order to close the model, we make two additional assumptions:

1. ut follows a separate ARMA process with mean uss.

2. pit and at are random walks with drift.

The NAIRU thesis states that the rate of unemployment has to be at an appro­priate equilibrium level if the rate of inflation is to be stable. Assumption 1 is made to investigate whether that thesis holds true for the present model: with no feedback from wq, t and/or piq, t on the rate of unemployment there is no way that ut can serve as an equilibrating mechanism. If a steady state exists in spite of this, the NAIRU thesis is rejected, even though the model incorporates both the ICM and the Phillips curve as special cases.

Obviously, in a more comprehensive model of inflation we will relax assump­tion 1 and treat ut as an endogenous variable, in the same manner as in the Phillips curve case of Chapter 4. In the context of the imperfect competition model, that step is postponed until Section 6.9. Assumption 2 eliminates stabil­ising adjustments that might take place via the nominal exchange rate and/or in productivity. In empirical work this amounts to the question of whether it is valid to condition upon import prices (in domestic currency) and/or productiv­ity. Section 5.5 gives an empirical example of such valid conditioning, since we found weak exogeneity of productivity with respect to the identified long-run wage curve.

The reduced form equation for the product real wage wq, t is

Wq, t = St + £Apit + KWq, t-1 + piq, t-1 - ПЩ-1 + €Wq, t,

0 < £ < 1, 0 < к < 1, 0 < X, (6.9)

with disturbance term eWq t, a linear combination of ew, t and eq, t, and coefficients which amalgamate the parameters of the structural equations:

St [(cw + 0w mb + 0w 1at-1)(1 ^qw )

(Cq - Oqmf - 0q%-і)(1 - фwq - ФwpФ)}/X,

£ [,0wp(1 фqw)(1 Ф) фqг(1 фwq 'фwpФ')/X,

X = Ow w(1 - фqw )(1 - Ф)/х, (6.10)

к 1 [0w(1 фqw) + 0q(1 фwq фwpф')/х,

П = [hw (1 - Фqw ) - Pq (1 - фwq - ФwpФ)}/X■

The denominator of the expressions in (6.10) is given by

X = (1 - Фqw(Фwq + ФwpФ))■ (6.11)

The corresponding reduced form equation for the real exchange rate piq, t can be written as

piq, t = - dt + e Apit - kWq, t-1 + lpiq, t-1 + ПЩ-1 + Cpiq, t, 0 < e < 1, l < 1, 0 < n,

where the parameters are given by

dt — [(cq Qq mf Qq at-1) + (cw + Qw mb + Qw 1а—1)ф qw/xi

Є — 1 - [^qw^wp(1 - Ф) - Фqi]/X,

l = 1 - [^qw^wш(1 - Ф)/Х] , (6.13)

k (Qq ^qwQw)/X>

n (gw фqw + gq )/x•

Equations (6.9) and (6.12) constitute a system of first-order difference equations that determines the real wage wq, t and the real exchange rate piq, t at each point in time. As usual in dynamic economics we consider the deterministic system, corresponding to a hypothetical situation in which all shocks ew, t and eq, t (and thus ewq, t and epiq, t) are set equal to zero. Once we have obtained the solutions for wq, t and piq, t, the time paths for Awt, Apt, and Aqt can be found by backward substitution.

The roots of the characteristic equation of the system are given by

hence the system has a unit root whenever kA — 0 and either к — 1 or l — 1. Using (6.10) and (6.13), we conclude that the wage-price system has both its roots inside the unit circle unless one or more of the following conditions hold:

Based on (6.15)-(6.17), we can formulate a set of sufficient conditions for stable roots, namely

9w > 0 and 9q > 0 and ш > 0 and фqw < 1. (6.18)

The first two conditions represent equilibrium correction of wages and prices with respect to deviations from the wage curve and the long-run price-setting schedule. The third condition states that there is a long-run wedge effect in wage-setting. Finally, a particular form of dynamic response is precluded by the fourth condition: for stability, a one point increase in the rate of wage growth must lead to less than one point increase in the rate of price growth. Note that фqw — 1 is different from (and more restrictive than) dynamic homogeneity in general, which would entail фqw + фqi — 1 and ri^wp + фwq — 1. Dynamic homogeneity, in this usual sense, is consistent with a stable steady state.