Labour Market Flexibility and Regional Unemployment Rate Dynamics: Spain (1980-1995)

The unemployment rate (not in logs) is¹³

ut = lt - nt.                                          (4)

Let us rewrite the labour demand and real wage equations (1)-(2) as

(1 - α1L) nt = -γwt + ε_tⁿ,                           (5)

(1 - α₂L) w_t = -δu_t + ε_t^w ,                            (6)

where L is the lag operator. Substitution of (6) into (5) gives

(1 - α1L) (1 - α2L) n_t = γδu_t + (1 - α2L) ε_tⁿ - γε_t^w.               (7)

Next, rewrite the labour supply (3) as

(1 - α₁L) (1 - α₂L) l_t = (1 - α₁L) (1 - α₂L) ε^l_t.                    (8)

Finally, subtract from (8) the labour demand eq. (7) to get the reduced form unem-
ployment rate equation:¹⁴

[γδ + (1 - α1L) (1 - α2L)] u_t = - (1 - α2L) ε_tⁿ + γε_t^w + (1 - α1L) (1 - α2L) ε^l_t.    (9)

Note that the above equation is dynamically stable since (i) products of polynomials
in L which satisfy the stability conditions are stable, and (ii) linear combinations of
dynamically stable polynomials in L are also stable.

Alternatively, the reduced form unemployment rate equation (9) can be written as

u_t = φ₁u_t-1 - φ₂u_t-2 - β₁ε_tⁿ + β₂ε_t^w + β₁ε^l_t + θ₁ε_tⁿ_-1 - φ₁ε^l_t-1 + φ₂ε^l_t-2      (10)

where Φ1 = ^ααi+δ^{2 ,}Φ2 = α+αδ^, β 1 = ⅛ ^,β2 = ⅛^, and θι = ⅛^.

The above reparameterisation of the reduced form unemployment rate equation helps
to explain the characteristic features of the chain reaction theory. First, the autoregres-
sive parameters φ₁ and φ₂ embody the interactions of the employment and wage setting
adjustment processes (α₁ and α₂, respectively).

Second, the coefficients β ₁ and β₂ are the short-run elasticities and are a function of
the feedback mechanisms that give rise to the spillover effects. When γ and δ are non
zero, all labour market shocks generate spillover effects. If δ = 0, i.e. unemployment does
not influence wages, then labour demand and supply shocks do not spillover to wages. If
γ = 0, i.e. labour demand is completely inelastic with respect to wages, then shocks to

¹³ Since labour force and employment are in logs, we can approximate the unemployment rate by their
difference.

¹⁴ The term "reduced form" means that the parameters of the equation are not estimated directly -
they are simply some nonlinear function of the parameters of the underlying labour market system.

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