Modeling in and Economy

Introduction and Definitions

In this section we follow Debreu [?], Chapter 3 where he discusses the production sector of an economy. Having stripped away all legal fictions, we are left with an economic agent whose role is to choose and carry out a process of producing a particular class of . The as- sumptions are made that there is a finite number of possible inputs to the production process, a finite number of possible outputs whose nature depends on the technological knowledge of the society, and a finite number of producers. Both inputs and outputs are associated with a particular time and location of availability which means that a particular model of car rolling off a production line next Friday in Detroit, Michigan is a different output than the same model of car rolling off the same production line the following Wednesday. Likewise, that former output is different from a car available next Friday, but from a production line in Toledo, Ohio.

These finiteness assumptions mean that the input-output space can be taken to be some multi-dimensional Euclidean space, R`, and that the finite number of agents may be indexed by, say, j = 1, . . . , n. The jth agent, a producer, chooses a production plan by choosing a set of inputs of raw materials∗and the set of outputs of produced goods. Each such choice depends, of course, not only on the technical knowledge in the society at large, but on the technologies known by, and available to, such an agent. Such a plan will be called feasible. Of course, for any given agent, there may be several choices of feasible plans available. So to each producer we have associated a set of production plans feasible for that agent. The set of all th ` feasible production plans for the j agent we will denote by Yj ⊂ R . A feasible production plan for the jth agent is then represented by a vector in R`. Most of the components of this vector will be zero, but for those components involving actual inputs and outputs for the particular producer, the inputs are represented as negative numbers and the outputs by ones that are positive.

The total feasible production set, Y of the is represented by the Minkowski n P sum of the individual producers’s feasible production sets, that is, Y = Yj. Given any j=1 n P j j point y ∈ Y , the corresponding total production istherefore just y = y where y ∈ Yj. j=1 Note that the set Y ⊂ Rn`.

Possible Assumptions and Interpretations

In the analysis of the production model here, we must make certain assumptions which may, or may not, have an economic interpretation, but all of which are mathematically useful. ∗The common use of the term “raw material” is usually applied to such materials as oil, iron ore, or labor. Here we simply mean inputs to a particular production process. Thus, what may be produced by one producer may well serve as the raw material for another. Of course, in the description of the total production of the economy, the output of the one is “cancelled out” when it becomes the input of another.

1 The assumptions are made about the properties of either the individual feasible sets Yj which are sometimes inherited by the Minkowski sum Y or as assumptions on the set Y itself. Here is the list of possible assumptions together with some remarks.

1. For each j = 1, . . . , n the set Yj is closed. This is a continuity assumption in the sense that any production plan that can be ap- j n proximated by feasible production plans, is itself feasible. Formally, if y ∈ Yj and if j n j o j o y → y then y ∈ Yj. 1’. The total production set Y is closed. Since each of the sets Yj is assumed closed, we might be tempted at first glance to assume that the Minkowski sum Y was closed as well. Unfortunately this is in general false. In general additional assumptions of compactness must be introduced for the Yj in order to insure that the sum Y is closed (see Lemma 3.1.5).

2. The vector 0 ∈ Yj. This is the possibility of inaction by the jth producer. That agent has the possibility of doing nothing at all, in which case, of course, the profits must be zero. For example, if reach a certain level, some firms may well find it unprofitable to produce anything and shut down production. If prices change, the firm may “fire up” and resume production. In this model, it is irrelevant whether such a firm is considered “new” or not. 2’. The vector 0 ∈ Y . Debreu adds this separately, although, considering the definition of Minkowski sum, it is unnecessary in light of assumption 2. On the other hand, if 0 ∈ Y then 0 ∈ Y 1 for all j = 1, . . . , k, hence Assumption 2 and Assumption 2’ are interchangable.

n` 3. Y ∩ R≥ = {0}. The intersection here represents those total productions for which there are no inputs (since all inputs are represented by negative components). In the literature this assumption is known as No Land of Cockaigne† . More familiarly, “There ain’t no such thing as a free lunch.” 4. Y ∩ (−Y ) ⊂ {0}. This assumption is called irreversibility. If the total production y 6= 0 is feasible, then −y is not. It is difficult to imagine producing an iron ingot and then being able to reverse the process to produce the original ore. In fact, we can say in light of our knowledge of thermodynamics, that it is physically impossible. Likewise, since we “time stamp” output, and time travel is not known to be possible, the irreversibility assumption makes perfect physical as well as economic sense.

5. (Yj + Yj) ⊂ Yj. This assumption is called additivity. It is based on the idea that if two productions yj ,1 and yj ,2 are feasible for the jth agent, then the sum of the two should also be a feasible

†The Land of Cockagne is a land in Midieval myth that describes a country where people live in complete luxury and physical comfort, the sky rains bread and cheese, and nobody has to work.

2 output. Insofar as the Yj involves technical expertise, this assumption seems reasonable. However, it can be criticized on the basis of limited capacity and “time stamping”. Indeed, if under certain conditions a given machine can be used to either stamp out aluminum forms or tin forms, to do both forms at the same time may be reasonably questioned. One should also note that under this assumption, simply by iteration, if yj is feasible, so is k yj, where k is any positive integer. Thus, this assumption implies a type of non-decreasing and, in particular, an unbounded production set.

6. The sets Yj are convex. j ,1 j ,2 This assumption of convexity. Hence, if y , y ∈ Yj then, for all λ ∈ [0, 1] , (1 − j ,1 j ,2 j ,1 j ,2 λ) y + λ y ∈ Yj. Under Assumption 2, we can take y = 0 and so λ y ∈ Yj and ‡ this implies Yj exhibits non-increasing returns to scale . In this situation it therefore rules out increasing returns to scale. 6’. The total production set Y is convex. Here, while under Assumption 6, the set Y is automatically convex, it may well be that Y is convex and some of the Yj are not. Hence this assumption is weaker that the previous one. Again, combining this assumption with Assumption 2’, implies that Y exhibits non-increasing returns to scale.

7 The individual production sets Yj are pointed cones This is a much more restrictive assumption than the assumption of convexity. In par- j j ticular, since for any y ∈ Yj and α > 0 , αy ∈ Yj, this implies constant returns to scale in the production processes. Moreover, Proposition 2.2.5 shows that, under this and Assumption 5, the cones Yj are convex. Hence there is some justification for the convexity assumption.

‡ j j Recall that a given production set Yj exhibits non-increasing returns to scale provided, for any y ∈ Yj , the point α y ∈ Yj for 0 ≤ α < 1, and non-decreasing returns to scale if α > 1.

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