The first-order conditions of this program are:
∂∏t = αA £(1 - ρ,) ' '. nα-1 - ξnξ-1 (1d-ρt)θ ρη =0
∂Td = (1 - α) a, £(1 - ρ,) h,TfLt]--α (1 - ρt) ⅛⅛? — WhtLt = 0
∂π nξ
∂ = - (1 - α) At [(1 - ρt) htTt Lt] htTt Ltnt - dj- η (1 - ρt) ρt - θ (1 - ρt) ρt =0
∂ρ, d,
from which we obtain:
αAt (1 - ft)1-α (htTJ*Lt)1-α n,α = ξnξ (1 - ρt) ^1 (4)
dt
(1 - α) At (1 - ρt)1-α (htTtdLt)-α ntα = wt (5)
(1 - α) At (1 - ρt)-α (htTdlLt)1~α nα = nξ h« (1 - ρ1)θ-1 ρt' - η (1 - ρt)θ ρ7^1l (6)
dt
The second-order conditions of the problem of the firm, that guarantee the presence
of a maximum, are checked in Appendix 7.1.
Equation (4) gives the optimality condition for the number of tasks by equalizing
the marginal productivity of a task (that is, the increase in output due to an ad-
ditional task) with the marginal cost of a task (that is, the marginal increase in
horizontal coordination costs). Similarly, the optimal demand for productive time
T determined in equation (5) equalizes the marginal product of productive time
with its marginal remuneration.
Equation (6) is the most important condition of the firm’s block, as it provides the
optimality condition for the fraction of labour devoted to coordination tasks. The
left-hand side clearly reflects the loss in production induced by diverting workers
from production. The right-hand side reflects the marginal impact of a larger share
of human resources specialists on the costs of internal coordination. By definition,
this impact is twofold. On the one hand, more workers in the human resources
department implies less workers in production, and less productive workers means a
lower exposure by the firm to the risk of productive mistakes, thereby a lower level
of horizontal costs of tasks coordination. On the other hand however, more workers
in the human resources department drives the vertical component of coordination
costs in the opposite direction by construction. More human resources specialists
means indeed a higher bureaucratic and incentives burden, thereby driving vertical
coordination costs upward (in the case of η>0). For condition (6) to make sense, it
is of course necessary to ensure that the right-hand side is positive at least locally,
which is done in Proposition 2. Note that this needed property amounts to guarantee
that the coordination costs function C(n, ρ) is a decreasing function of ρ, a point
made before.
13