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

and for the cost function to be convex this matrix must be positive definite, i.e. we
must have:

H1 > 0      H2 > 0

With reference to the first condition we have:

i.e. it is satisfied if ξ>1, while with reference to the second condition we have:

-P_tη (ξ - 1) (η+θ)(1- P_t) - P_t² (ξ- 1) (η+θ)(θ - 1) +

^-ξn2 ⁺ 2^ξnPt (n ^{+ θ}) ^{+ ξ} (n2 ⁺ 2nθ ^{+ θ}^ Pt²] > 0

The fraction outside the square bracket is positive, while considering the expression
inside the square bracket and letting n tend to 0 (both in the case of n positive and
inthecaseofn negative) we get:

-P_t² (ξ - 1) θ (θ - 1) + ξθ²P_t² > 0,

which holds when ξ>1 and θ<1, and hence the cost function is convex under the
conditions of the proposition. ■

At this point the profit function writes:

∏t = At • £(1 - ρt) • ht • Tt^d • L]^1-a n? - nξ ^ (1 ^-_d^pt) ^ ^pη - wt • ht • T^d • Lt

where d_t > 0 and where T_t^d denotes now the working time demanded by the firm.
In the decentralized economy the firm’s optimization program is then given by:

max    At £(1 - ρt) h_tTt^dL_t] 1-^α n? - ^П (1 -^{Pt) P} - WthT^dLt

nt ,Tt^d,ρt                                                               dt

12

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