wage-setting do not spillover to unemployment. In this case the influence of the shocks
(εtn, εtw , and εlt ) on unemployment can be measured through individual analysis of their
respective equations. In other words, the main feedback mechanism in this toy model is
provided by the wage elasticity of labour demand.
Third, the emergence of "moving average" terms (lags of the shocks) in the reduced
form unemployment rate equation emphasizes the interplay of the lagged adjustment
processes and the spillover effects.
In a dynamic labour market model shocks are not absorbed instantly - their effects are
felt through time. The concept of unemployment persistence captures the after effects of
the labour market shocks. The impulse response function of unemployment describes the
responses of unemployment through time to a specific shock (impulse). For a temporary
shock occuring at period t, we define unemployment persistence (σ) as the sum of its
responses for all periods t + j in the aftermath of the shock (j ≥ 1):15
∞
σ ≡ Rt+j, (11)
j=1
where the series Rt+j , j ≥ 0 is the impulse response function (IRF) of unemployment.16 If
the unemployment model (i) is static, then the shock is absorbed instantly and so σ = 0,
(ii) is dynamically stable, then the effects of the shock gradually die out and persistence is
a finite quantity, and (iii) displays hysteresis, then the temporary shock has a permanent
effect and thus σ = ∞.
For illustrative purposes, we derive the measure of unemployment persistence in the
special case of no spillover effects from wages to labour demand (γ = 0). Consider a one-
off unit labour demand shock at period t, i.e. εtn = 1, εtn+j = 0 for j ≥ 1. The infinite
moving avarage representation (IMA) of employment is
nt = εtn + α1εtn-1 + α1εtn-2 + α1εtn-3 + ...
Since unemployment is defined as the difference between labour supply and demand
in eq. (4), and there are no spillover effects, the response of unemployment t periods after
the occurence of the shock is
Rt+j = -αj1, j ≥0. (12)
Thus the immediate impact of this shock on unemployment is -1 and persistence equals
-α1/ (1 - α1) .
It is important to note that if we interpret the shock at period t as the change in a
specific explanatory variable, say x, over that period then (i) the immediate response (Rt)
is simply the short-run elasticity of the unemployment rate with respect to x and (ii) the
15 See Karanassou and Snower (1996, 1998) for definitions of temporal as well as quantitative measures
of persistence and their application. See also Pivetta and Reis (2004) for a discussion of various persistence
measures.
16 In other words, Rt+j , j ≥ 0, denotes the coefficients of the infinite moving average representation of
unemployment with respect to the shock.