where At is a productivity parameter, Lt is the volume of hours worked with human
capital ht, xt (i) is the time devoted to task i, nt is the number of tasks performed
per worker and ρt is the fraction of the workforce in the personnel (human resources)
service.
The worker’s productive time is equal to Tt, hence we also have the constraint:
nt
0 xt(i)di = Tt
(2)
(3)
Tasks are symmetric, i.e. xt (i) = xt, and from (2) we then get:
Tt
Xt = —
nt
and substituting this expression in (1) we obtain the production function:
yt = At ∙ [(1 - Pt) • ht ∙ Tt ∙ Lt]1 α nα
Producing the good implies two types of costs: production costs and coordination
costs (a similar element can be found in Brynjolfsson et al., 1994). Since production
requires physical resources and knowledge about how to combine them, production
costs correspond to traditional costs of transforming inputs into output (physical
- productive - resources expenses) whereas coordination costs correspond to the
costs of combining and managing interactions and dependencies between resources
(tasks and/or workers). In our model, labour is the sole input, therefore production
costs equal the total wage bill and coordination costs depend on the number of tasks
realized per worker (n) and on the fraction of workers in the human resources service
(ρ).
The firm’s profits (given that output is the numeraire) then write:
πt = At ∙ [(1 - Pt) ∙ ht ∙ Tt ∙ Lt] nα - C (n, ρ) - wt ∙ ht ∙ Tt ∙ Lt
where wt is the wage rate per efficiency unit of labour and C (n, ρ) represents coor-
dination costs measured as pure output loss.
The coordination costs function depends on horizontal and vertical coordination
costs as follows9 :
C (n,ρ) = h (n,ρ) ∙ v (P)
d
where h (n, ρ) denotes horizontal coordination costs and v (ρ) denotes vertical coor-
dination costs, while d reflects the extent of coordination costs (a higher d reduces
the importance or magnitude of coordination costs).
9 In a different (hierarchical) context, Garicano and Rossi-Hansberg (2006) formalize the cost
of knowledge acquisition by workers as the cost of solving problems of a given difficulty level
multiplied by the knowledge cost required to solve a given proportion of problems. Our approach
is very different since we focus on horizontal and vertical costs of coordinating workers and tasks
whereas Garicano and Rossi-Hansberg focus on the costs of knowledge acquisition and poblem-
solving in knowledge hierarchies. However, they also assume a multiplicative interaction between
both types of costs.