45 allocation of a labor force our economy presents a

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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.

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