Technological progress, organizational change and the size of the Human Resources Department

α                                                                          ξ(1 —α)

= ^ξ-a /adʌ ^ξ-α a⅛E_π-δ⅛⅛L'    P (1 - ^δ> ⁽¹ - . ʌ     • (21)

ξ ∖ξ J                        ∖ (1 + β - 2βδ)2 J

α(ξ + θ) —ξ         αη                                α(ξ+η + θ) —ξ

• (α (ξ + θ) - ξ) ^α-ξ (an)^α-ξ (α (ξ + П + θ) - ξ)   ^ξ-α

β ξζ — a _{τ 1} ʌ aad∖ ^ξ-α _λ-L- ,. ^ξ(1^-α) _τ- α(1-ξ) ββ (1 — δ) (1 — βδ)δ ^ξ-“

=          L--L - 1 + all         A ξ-α E (1—^δ)(ξ-α) L ξ-α                 (22-    (22)

¹ — β V ξ         /U/                    к (1 + β — 2βδ)2 )

ξ-α(ξ + θ)         αη                              α(ξ+η + θ) —ξ

• (a (ξ + θ) - ξ) ^ξα   (an)^a—ξ (a (ξ + n + θ) - ξ) ^ξα

The following result can then be stated:

Proposition 2 Provided the following restrictions on the parameters hold:

1 ^{1 + β} .
δ <  —   and

2β

ξ

<—< < a < ξ if n > 0
ξ + θ

ξ

^a < ξ+θ < ^ξ if n < ⁰

there exists a unique steady state of the model with 0 <T< 1 and 0 <ρ< 1 where
the values of the different variables are given by the expressions (15)-(22).

Proof. The steady-state values of the variables are obtained in Appendix 7.2.

Concerning the restrictions on the parameters, given the expression obtained for T :

T 1 - βδ

1 + β - 2βδ

the fact that 0 < T < 1 implies 0 < 1+β-2βδ < 1, and since the numerator is positive
(because 0 <β< 1 and 0 <δ < 1) we must have (for the first inequality to hold):

1+β

1 + β - 2βδ> 0 ⇒ δ< +~p
2β

while the second inequality is always verified (being δ<1). Given the expression
obtained for ρ:

an

^P a (ξ + n + θ) - ξ

then, the fact that 0 < ρ < 1 implies 0 < _α(ξ+^α+θ)_-ξ < 1. At this point it is necessary
to distinguish the case n> 0 and the case n< 0.Ifn> 0 the restriction 0 <ρ< 1
requires:

ξ

a (ξ + n + θ) - ξ> 0 ⇒ a> -------

ξ+ n + θ

and also:

ξ

an<a (ξ + n + θ) - ξ ⇒ a> ξ+θ

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