max
wu
ρ -Bρ])γ(Y -wuL -wsH)1-γ
(8)
The bargaining solution is obtained by maximising this expression with respect to wu , taking into account
the fact that, for a given skilled wage, changing the unskilled wage affects both skilled and unskilled
employment. The resulting first-order conditions can be expressed as (see Appendix I),
(1 -γ θ
ρ(1 -τ)ρ=∣-γ (1 -β) -θ- + ε L
V y 1 - θ
(
I (1 -τ)ρ-
V
B Y∣
wu J
u / j
(9)
where εL is the elasticity of the demand for unskilled labour. Since εL , wu , and θ are functions of H
and L , equation (9) determines wu as a function of skilled and unskilled employment.
Equilibrium and comparative statics
The equilibrium of the model is then given by equations (2b), (2c), (6), and (9), that is, by
1~~γ (1 -β) A + ε L
γ 1-θ
(9)
wu
= (1 - β)(1 - α)(α + (1 - α)xσ) ( +σ) σ xσ K
L
(10)
= β(1 -α)(α + (1 -α)xσ) ( ) xσ H
(11)
ws
(12)
where φ( B, e, p ) is implicitly defined by (6). Together these four equations determine the equilibrium
levels of skilled and unskilled employment, H and L , and the two wages as a function of model
parameters: the unemployment benefit, B , the bargaining power of the union, γ , the capital stock, K , as
well as the preference parameters, ρ and e , and the technological parameters, α,β, σ , and p .
Let u ≡ 1 - (L + H )/(L + H ) be the unemployment rate, where H is the skilled labour force.
Once H and L are determined, we can obtain our three main variables of interest, the labour share, the
relative wage, and the unemployment rate, which we can express as functions of the stock of capital and
labour market institutions (as well as of the preference and technology parameters):
θ = θ(K,B,γ),
ω = ω(K,B,γ)
(13)
(14)
u = u (K , B, γ).
(15)
All comparative statics are derived in Appendix I. Consider first the effect of union power. It is possible
to show that