4.5
Allocation of a Labor Force
Our economy presents a network of interdependent industries.
Each both
produces and consumes goods.
For example, the steel industry consumes
coal to manufacture steel. Reciprocally, the coal industry requires steel to
support its own production processes. Further, each industry may be served
by multiple manufacturing technologies, each of which requires different re-
sources per unit production. For example, one technology for producing steel
starts with iron ore while another makes use of scrap metal. In this section,
we consider a hypothetical economy where labor is the only limiting resource.
We will develop a model to guide how the labor force should be allocated
among industries and technologies.
In our model, each industry produces a single good and may consume
others. There are
M
goods, indexed
i
= 1
, . . . , M
. Each can be produced
by one or more technologies.
There are a total of
N
≥
M
technologies,
indexed
j
= 1
, . . . , N
. Each
j
th technology produces
A
ij
>
0 units of some
i
th good per unit of labor. For each
k
=
i
, this
j
th industry may consume
some amount of good
k
per unit labor, denoted by
A
kj
≤
0.
Note that
this quantity
A
kj
is nonpositive; if it is a negative number, it represents
the quantity of good
k
consumed per unit labor allocated to technology
j
.
The productivity and resource requirements of all technologies are therefore
captured by a matrix
A
∈
M
×
N
in which each column has exactly one
positive entry and each row has at least one positive entry. We will call this
matrix
A
the
production matrix
. Without loss of generality, we will assume
that
A
has linearly independent rows.
Suppose we have a total of one unit of labor to allocate over the next
year.
Let us denote by
x
∈
N
our allocation among the
N
technologies.

c
Benjamin Van Roy and Kahn Mason
103
Hence,
x
≥
0 and
e
T
x
≤
1. Further, the quantity of each of the
M
goods
produced is given by a vector
Ax
.
Now how should we allocate labor? One objective might be to optimize
social welfare. Suppose that the amount society values each unit of each
i
th
good is
c
i
>
0, regardless of the quantity produced. Then, we might define
the social welfare generated by production activities to be
c
T
Ax
. Optimizing
this objective leads to a linear program:
maximize
c
T
Ax
subject to
Ax
≥
0
e
T
x
≤
1
x
≥
0
.
(4.2)
A production matrix
A
is said to be
productive
if there exists a labor
allocation
x
(with
x
≥
0 and
e
T
x
= 1) such that
Ax >
0. In other words,
productivity means that some allocation results in positive quantities of every
good. It turns out that – when the production matrix is productive – only
M
technologies are beneficial, and the choice of
M
technologies is independent of
societal values. This remarkable result is known as the substitution theorem:
Theorem 4.5.1. (substitution)
If a production matrix
A
is productive,
there is a set of
M
technologies such that for any vector
c
of societal values,
social welfare can be maximized by an allocation of labor among only these
M
technologies.