Wage Differences in a Firm and the Substitutability
of Labor Inputs
(1) Objectives: Our goal
is to develop a model of a hierarchically structured firm in which the skewness of the internal wage distribution is explained by
the substitutability of the labor inputs.
(2) Background: Mostly,
in a hierarchically structured firm not all wages are equal. Within a
particular hierarchical level there can be horizontal wage differences because
of differences in the importance of the tasks done, or the scarcity of the labor inputs that are able to perform the different tasks.
Besides these wage differences within levels there are also vertical wage
differences between different firm levels. This paper studies this second kind
of wage differences.
(3) Method: A
hierarchically structured firm is described by a set of employees (workers and
coordinators), a hierarchical coordination structure, and a production
technology. The coordination structure is represented by a directed graph
having a tree structure with a unique position at the top. This implies that
there are positions that have no subordinates. These positions constitute the
lowest level of the firm structure and are occupied by the workers of the firm.
As done often in the firm literature, it is assumed that the production process
is carried out by these workers in the lowest firm level. The employees in
higher levels are managers or coordinators who organize and coordinate the
production process. The production process is described by a supermodular cooperative production game that is defined on
the workers in the firm. The importance of coordination in supermodular
production technologies has been stressed at various places in the literature.
The wages of the workers
and the coordinators on the different levels of the coordination structure are
determined by a cooperative solution that is based on the permission value
being a solution concept for cooperative games with a permission structure. An
important consequence from using such a wage function is that the remuneration
of firm employees not only depends on total production that can be generated by
the fully employed firm, but also on the production capabilities of subsets of
employees, i.e., on the production that can be generated by the firm if some
positions are vacant. The use of this wage function is motivated by discussing
some properties, and giving an axiomatic characterization.
(4) Results: It turns
out that according to the wage function that is used the wage of an employee
never exceeds the wage of its superior coordinators. On the other hand, the
wage of a coordinator never exceeds the sum of the wages of its direct
subordinates. It is also shown that these bounds are sharp in the sense that
there are production technologies for which these bounds are reached. In the
special case that the workers are identical in the production process and the
firm has constant span of control (i.e., every coordinator has the same number
of direct subordinates,) this means that the ratio between the wage of a
coordinator and each of its direct subordinates lies between one and the span
of control.
After discussing the
general model, constant elasticity of substitution (CES) production
technologies are used to illustrate this result. Moreover, it is shown that the ratio between
the wage of a coordinator and each of its direct subordinates increases if the
elasticity of substitution of the labor inputs
increases. It reaches the upper bound (span of control) for linear technologies
with substitutable labor inputs. It reaches the lower
bound (one) for Cobb-Douglas technologies with indispensable labor inputs.
(5) Discussion: The
lower- and upper bounds for the ratio between the wages of superiors and their
direct subordinates that we found here is often assumed in the literature on
the firm. The purpose of this paper is to gain insight in the internal wage
distribution in a firm of given size. Although in this paper
the size of the firm is exogenously given the model developed here can be used
to endogenously determine the optimal size of the firm. This can be done
by letting the top-position represent the governor, and thus profit is given by
the remuneration assigned to the top-position. The governor then chooses the
firm size that maximizes profit under some worker-participation constraint
(meaning that the worker wages should be at least equal to their reservation
wage).
In the model presented
in this paper there is no uncertainty and information problems do not arise.
Thus, issues like moral hazard do not play a role. In this sense the model
presented here is complementary to the literature on principal-agent theory.
The model also complements the literature on incomplete contracts which tries
to explain the distribution of residual rights concerning the control over non-contractable assets. Control over assets is comparable to
control over subordinate positions in the model of this paper. However, we
assume all 'assets' to be contractable.
Other complementary studies
that can be found in the literature relate wage differences to marginal productivity
of laborers, or to technical progress and education.