"The firm fits into general equilibrium theory as a balloon fits into an envelope: flattened out! Try with a blown-up balloon: the envelope may tear, or fly away: at best, it will be hard to seal and impossible to mail....Instead, burst the balloon flat, and everything becomes easy. Similarly with the firm and general equilibrium - though the analogy requires a word of explanation."

(Jacques H. Drèze, "Uncertainty and the Firm in General Equilibrium Theory", Economic Journal, 1985, p.1)

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(1) Axioms of Production
(2) Technical Efficiency
(3) The Producer's Decision
(4) The Efficiency of the Producer
(5) Marginalism
(6) Competition
(7) Aggregation

Selected References

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Here we shall consider the Neo-Walrasian theory of production developed by Tjalling C. Koopmans (1951) among others at the Cowles Commission during the 1940s and 1950s which relies heavily on use of activity analysis and convexity theory, notably the Separating Hyperplane Theorem. The more traditional Neoclassical theory of production is surveyed elsewhere.

(1) Axioms of Production

A production plan y is a "net output" vector in Rⁿ. Thus, a good i in the vector y is an "output" if y_i is a positive element and a "factor" if it is a negative element. Let p be a price vector in Rⁿ. Thus, pｷy = profits as it is equal to total revenue (price ｴ output) minus total cost (price ｴ input).

Let Y ﾌ Rⁿ denote the production set and let us impose the following six axioms upon it:

Axioms of Production

(1) 0 ﾎ Y - the possibility of inactivity

(2) Y ﾇ W ⁺ = {0} - no Land of Cockaigne (i.e. no outputs without inputs)

(3) Y ﾇ (-Y) = {0} - irreversibility of production

(4) W ^- ﾌ Y - free disposibility

(5) Y convex - implies non-increasing returns to scale (together with (1))

(6) Y closed

An example of a production set Y in two dimensional space is shown in Figure 1 where a particular production plan, the net production vector y = [y₁, y₂], is shown. Note that at that vector, y₁ is treated as an input (and hence takes a negative value) and y₂ as an output (and thus has positive value). Note how the axioms above imply that Y is something like that shown below (although an interesting alternative is where the boundaries are straight line, i.e. the case of constant returns to scale.).

Figure 1 - Production Set

Note that if we held drew Y for only two outputs holding all other outputs and all inputs constant, then we would effectively have the production possibilities frontier and the area below it so familiar from microeconomics principles. Similarly, if we drew Y for only two inputs and held all other inputs and all outputs constant, we would effectively have an isoquant and the area above it. All the analysis which follows, thus, has its analogues in the familiar Paretian isoquant and PPF worlds. Thus, this axiomatic, set-theoretic Y represents a generalization of these familiar concepts without assumptions about anything being held constant, or differentiability, etc. This generalization was the great advance provided by the activity analysis research at the Cowles Commission during the 1940s and 1950s

(2) Technical Efficiency

We now turn to technical aspects of production, specifically attempting to outline what is meant by technological efficiency in this activity analysis context. We define this as follows:

Technological Efficient Plan: a production plan y* ﾎ Y is "technologically efficient" (t.e.) if there is no y ﾎ Y such that y > y*.

In Figure 1, the set of technologically efficient points is evidently the upper boundary of the production set Y. The particular y shown in the diagram, however, is not technologically efficient.

To prove the existence of technologically efficient points in the set Y, we need three lemmas and the working definition of a new concept, a subset of Y which we shall call Y[y_-i⁰] = {y ﾎ Y ｽ y_-i = y_-i⁰ }, i.e. we are definining the "set" Y[y_-i⁰] as a set of vectors in Y such that all elements in those vectors other than the ith element is fixed at some given level. Thus, the vector y_-i is defined as the vector with all elements but the ith and y_-i⁰ is the set of "fixed values" for these other elements. Thus only the ith element, y_i is free.

We can think of this set, Y[y_-i⁰] as a set within Y of one dimension less than Y. Figure 2 may help: here, all except y₂ is fixed - which, in a two-dimensional space, implies that y₁ is held fixed at y₁⁰ while y₂ fluctuates - so Y[y_-2⁰] = Y[y₁⁰] which is the thick line shown in Figure 2 (a subset of Y). Thus, a vector such as y* = [y₁⁰, y₂*] is an element of Y[y₁⁰] as well as Y.

Figure 2 - Production set with fixed factors, Y[y_-2⁰]

Here are three lemmas we need regarding the production set with fixed factors:

Lemma 1: Y[y_-i⁰] is closed.

Proof: Suppose we take a convergent sequence {y^k} ｮ y*. As Y is closed, then y* ﾎ Y. Let us suppose that y^k = [y_i^k, y_-i⁰], i.e. all vectors in the sequence have the same value for the elements other than i, i.e. only the ith varies in the sequence. Thus, the sequence {y^k} is a movement "along the line" Y[y_-i⁰] in Y. We know y* ﾎ Y, but is y* ﾎ Y[y_-i⁰] ? Actually, yes: as {y^k} ｮ y*, then y_iⁿ ｮ y_i* (ith element converges). As the other elements are fixed, then y* = [y_i*, y_-i⁰] which is an element of Y[y_-i⁰] by definition of this set. Thus, the set Y[y_-i⁰] is closed.ｧ

Lemma 2: Y[y_-i⁰] is bounded, i.e. there is b ﾎ Rⁿ such that y < b for all y ﾎ Y[y_-i⁰].

Proof: Suppose not. Then then there is no such b ﾎ Rⁿ. Or, for any m ﾎ R, there is a y_i^m such that y_i^m > b and y^m = [y_i^m, y_-i⁰] ﾎ Y[y_-i⁰]. We can presume y_i^m is an integer, thus 1/y_i^m ﾎ [0, 1]. Let us then define divide the vector y^m by this integer: (1/y_i^m)y^m = [1, y_-i⁰/y_i^m]. This is a strange vector, but it is actually in Y. How do we know? Recall that Y is convex (Axiom 5) and that 0 ﾎ Y (Axiom 1). Thus, as (1/y_i^m) ﾎ [0, 1], then we can consider (1/y_i^m)y^m as the convex-combination between two points in Y: y^m and 0, i.e. (1/y_i^m)y^m = (1/y_i^m)y^m + (1- 1/y_i^m)0. By the convexity of Y, then (1/y_i^m)y^m ﾎ Y. As we presumed that y_i^m > b for any number b, then we can take y_i^m to infinity. Doing so, then the vector (1/y_i^m)y^m = [1, y_-i⁰/y_i^m] ｮ [1, 0] as y_i^m ｮ ･ . However, this vector [1, 0] ﾎ Y is strictly in the positive orthant. By Axiom (2) on Y, we know this cannot be. Contradiction. Thus, Y[y_-i⁰] is bounded.ｧ

Let us now define the efficiency function F[y_-i⁰] as:

F[y_-i⁰] = max y_i s.t. y ﾎ Y[y_-i⁰].

Thus, F[y_-i⁰] is the set of greatest ith elements in vectors y of subset Y[y_-i⁰] (see Figure 2 for intuition). A "technologically efficient" vector y* can thus be defined as y* = [F[y_-i⁰], y_-i⁰] - or, equivalently, y* = argmax F[y_-i ⁰] (i.e. the arguments, y, that maximize the efficiency function, F[y_-i⁰]). Let us proceed to the next lemma:

Lemma 3: if Y is convex, then the efficiency function F[y_-i⁰] is concave.

Proof: Y is convex: take two vectors, y¹ and y² in Y and we know that l y¹ + (1-l )y² ﾎ Y. Let us take two technologically efficient points, i.e. let y¹ = [F[y_-i¹], y_-i¹] and y² = [F[y_-i²], y_-i²]. By the convexity of Y, then:

l y¹ + (1-l )y² = [l F[y_-i¹] + (1-l )[F[y_-i²], l y_-i¹ + (1-l )y_-i²] ﾎ Y

But by the definition of an efficiency function, F[l y_-i¹ + (1-l )y_-i²] ｳ y_i " [y_i , l y_-i¹ + (1-l )y_-i²] - this includes y_i = l F[y_-i¹] + (1-l )[F[y_-i²]. Thus, it must be that F[l y_-i¹ + (1-l )y_-i²] ｳ l F[y_-i¹] + (1-l )[F[y_-i²], thus the efficiency function is concave.ｧ

We can now turn to the main theorem:

Theorem: (Existence) There exists a set of technologically efficient points in Y.

Proof: Recall that a "technologically efficient" vector y* is defined as y* = [F[y_-i⁰], y_-i⁰] or y* = argmax F[y_-i ⁰]. It is readily seen that F[y_-i⁰], which we know is concave, is also a continuous function. We already know from our first two lemmas that Y[y_-i⁰] is closed and bounded and thus compact. Therefore, by the Weierstrass Theorem, the efficiency function is continuously defined over a compact set and thus it attains an extremal in Y[y_-i⁰]. By concavity, y* = [F[y_-i⁰], y_-i⁰] is a maximum, thus there exists a set of technologically efficient points in Y.ｧ

(3) The Producer's Decision

Let us now add some behavior to this and consider the issue of the profit-maximizing decision of the producer. By definition:

Profit-Maximizing Plan: production plan y* is "profit-maximizing" for a given p if py* > py for all y ﾎ Y.

This is illustrated in Figure 3 where we have taken a particular price vector p = [p₁ p₂], some point in the unit simplex S ﾌ R², and the defined the price hyperplane pｷy as a hyperplane orthogonal to vector p. Obviously, y* = [y₁*, y₂*] is the profit-maximizing point as pｷy* > 0 The lightly-shaded area in Figure 3 is the set of feasible plans which yield positive profits at prices p.

Figure 3 - Profit-Maximizing Plan

We can define a "profit-function" as

p(p) = max py s.t. y ﾎ Y

i.e. it is a mapping from prices (p ﾎ Rⁿ) to some real number ("maximum profit"). (see here for the Paretian equivalent). The set of profit-maximizing points can be defined as:

P(p) = argmax p (p) = argmax py s.t. y ﾎ Y

i.e. the arguments, y ﾎ Y, that maximize profits. Equivalently, P (p) = {y ﾎ Y ｽ py > py’ " y’ ﾎ Y}. Thus, p (p) = pP (p). We can refer to P (p) as the "supply correspondence". An illustration of the principle follows for two prices in Figure 4.

Figure 4 - Profit-Maximizing Function

In Figure 4, we have two sets of prices, p^a and p^b which. yield profit-maximizing activities y^a = ﾕ (p^a) and y^b = ﾕ (p^b). Now, we can tell that, in relative terms, p₂^a/p₁^a < p₂^b/p₂^b thus the price of the output (good 2) is higher at b. This implies that more of good 2 will be supplied at p^b than at p^a and more of good 1 will be demanded as input (i.e. y₁ is more negative at ﾕ (p^b)). Thus, the supply correspondence ﾕ (p) implies that as price of good 2 rises relative to good 1, more of good 2 will be produced, i.e. "supplied"; and more of good 1 will be demanded.

The following properties can be easily shown (see here for the Paretian equivalents)

Property 1: p(p) is homogeneous of degree 1 in prices.

Proof: p (p) = max py s.t. y ﾎ Y. Let a ｳ 0. Thus, p (a p) = max a py s.t. y ﾎ Y = a [max py] s.t. y ﾎ Y = a p (p). Thus, p (a p) = a p (p).ｧ

Property 2: p(p) is a convex function.

Proof: Consider two prices, p¹, p² and two allocations y¹ , y² ﾎ Y which are the equivalent profit-maximizing allocations, i.e. y¹ ﾎ P (p¹) and y² ﾎ P (p²). As Y is convex, then l y¹ + (1-l )y² ﾎ Y. For ease of notation, define yl = l y¹ + (1-l )y². By the definition of profit-maximization, p¹y¹ ｳ p¹yl , p²y² ｳ p²yl . Thus multiplying the first by l ﾎ (0, 1) and the second by (1-l ), then: , l p¹y¹ ｳ l p¹yl + (1-l )p²y² ｳ (1-l )p²y². Adding pointwise, we see immediately that: , l p¹y¹ + (1-l )p²y² ｳ (l p¹ + (1-l )p²)yl . We can rewrite this in terms of profit functions as l p (p¹) + (1-l )p (p²) ｳ p (l p¹ + (1-l )p²). Thus the profit function is convex.ｧ

Property 3: (Law of Supply) "Supply of Output i" increases with price of that output, "Demand for Factor i" decreases with the price of that factor.

Proof: "Output supply" and "factor demand" are merely the profit-maximizing points, y ﾎ P (p). Convexity of Y gave us convexity of the profit function (Property 2) and now we want to show that convexity of profit function yields an "upward-sloping" output supply curve and a "downward-sloping" factor demand curve. For simplicity, then, we shall assume twice differentiability of profit functions. Now, recall that p (p) = py. Thus, differentiating with respect to the ith element of the price vector, dp (p)/dp_i = y_i + ・/font> _i p_i(dy_i/dp_i). By the "envelope theorem", the term on the right collapses to zero so that we obtain the familiar "Shephard’s Lemma": dp (p)/dp_i = y_i where y_i is the ith element of y. Differentiating the profit function again, dp (p)/dp_i = dy_i/dp_i. Thus, by convexity of the profit function, dp (p)/dp_i ｳ 0 and so dy_i/dp_i ｳ 0. If y_i is positive (i.e. an output) this translates to "supply of output i" increases as p_i increases. If y_i is negative (i.e. a factor) this translates to "demand for factor i" decreases as the price of that factor increases.ｧ