________________________________________________________

"Once the equilibrium has been established in principle, exchange can take place immediately. Production, however, requires a certain lapse of time. We shall resolve the second difficulty purely and simply by ignoring the time element at this point."

(Léon Walras,

Elements of Pure Economics, 1874, p.242)

"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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Contents

(1) Axioms of Production

(2) Technical Efficiency

(3) The Producer's Decision

(4) The Efficiency of the Producer

(5) Marginalism

(6) Competition

(7) Aggregation

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.

A *production plan* y is a "net output" vector in R^{n}.
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 R^{n}. Thus, pｷy = profits as it is equal to total revenue (price ｴ output) minus total cost (price ｴ
input).

Let Y ﾌ R^{n} 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 = [y_{1},
y_{2}], is shown. Note that at that vector, y_{1} is treated as an input
(and hence takes a negative value) and y_{2} 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

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_{-i}^{0}] = {y ﾎ Y ｽ y_{-i} = y_{-i}^{0}_{ }}, i.e. we
are definining the "set" Y[y_{-i}^{0}] 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_{-i}^{0} 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_{-i}^{0}] as a set within Y
of one dimension less than Y. Figure 2 may help: here, all except y_{2} is fixed -
which, in a two-dimensional space, implies that y_{1} is held fixed at y_{1}^{0}
while y_{2} fluctuates - so Y[y_{-2}^{0}] = Y[y_{1}^{0}]
which is the thick line shown in Figure 2 (a subset of Y). Thus, a vector such as y* = [y_{1}^{0},
y_{2}*] is an element of Y[y_{1}^{0}] as well as Y.

Figure 2- Production set with fixed factors, Y[y_{-2}^{0}]

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

Lemma 1: Y[y_{-i}^{0}] 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_{-i}^{0}],
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_{-i}^{0}] in Y. We know y* ﾎ Y, but is y* ﾎ Y[y_{-i}^{0}]
? Actually, yes: as {y^{k}} ｮ y*, then y_{i}^{n}
ｮ y_{i}* (ith element converges). As the other
elements are fixed, then y* = [y_{i}*, y_{-i}^{0}] which is an
element of Y[y_{-i}^{0}] by definition of this set. Thus, the set Y[y_{-i}^{0}]
is closed.ｧ

Lemma 2: Y[y_{-i}^{0}] is bounded, i.e. there is b ﾎ R^{n}such that y < b for all y ﾎ Y[y_{-i}^{0}].

Proof: Suppose not. Then then there is no such b ﾎ
R^{n}. 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_{-i}^{0}] ﾎ Y[y_{-i}^{0}].
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_{-i}^{0}/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_{-i}^{0}/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_{-i}^{0}]
is bounded.ｧ

Let us now define the *efficiency function* F[y_{-i}^{0}]
as:

F[y

_{-i}^{0}] = max y_{i}s.t. y ﾎ Y[y_{-i}^{0}].

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

Lemma 3:if Y is convex, then the efficiency function F[y_{-i}^{0}] is concave.

Proof: Y is convex: take two vectors, y^{1} and y^{2} in Y
and we know that l y^{1} + (1-l
)y^{2} ﾎ Y. Let us take two technologically efficient
points, i.e. let y^{1} = [F[y_{-i}^{1}], y_{-i}^{1}]
and y^{2} = [F[y_{-i}^{2}], y_{-i}^{2}]. By the
convexity of Y, then:

l y

^{1}+ (1-l )y^{2 }=^{ }[l F[y_{-i}^{1}] + (1-l )[F[y_{-i}^{2}], l y_{-i}^{1}+ (1-l )y_{-i}^{2}] ﾎ Y

But by the definition of an efficiency function, F[l
y_{-i}^{1} + (1-l )y_{-i}^{2}] ｳ y_{i} " [y_{i} , l y_{-i}^{1} + (1-l )y_{-i}^{2}]
- this includes y_{i} = l F[y_{-i}^{1}]
+ (1-l )[F[y_{-i}^{2}]. Thus, it must be that
F[l y_{-i}^{1} + (1-l
)y_{-i}^{2}] ｳ l
F[y_{-i}^{1}] + (1-l )[F[y_{-i}^{2}],
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_{-i}^{0}], y_{-i}^{0}] or y* = argmax
F[y_{-i }^{0}]. It is readily seen that F[y_{-i}^{0}],
which we know is concave, is also a continuous function. We already know from our first
two lemmas that Y[y_{-i}^{0}] 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_{-i}^{0}]. By
concavity, y* = [F[y_{-i}^{0}], y_{-i}^{0}] is a maximum,
thus there exists a set of technologically efficient points in Y.ｧ

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 = [p_{1} p_{2}], some point in the unit simplex S ﾌ R^{2}, and the defined the price hyperplane pｷy as a
hyperplane orthogonal to vector p. Obviously, y* = [y_{1}*, y_{2}*] 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 R^{n})
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, p_{2}^{a}/p_{1}^{a} < p_{2}^{b}/p_{2}^{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. y_{1} 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, p^{1}, p^{2} and two
allocations y^{1} , y^{2} ﾎ Y which are the
equivalent profit-maximizing allocations, i.e. y^{1} ﾎ
P (p^{1}) and y^{2} ﾎ
P (p^{2}). As Y is convex, then l
y^{1} + (1-l )y^{2} ﾎ
Y. For ease of notation, define yl = l
y^{1} + (1-l )y^{2}. By the definition of
profit-maximization, p^{1}y^{1} ｳ p^{1}yl , p^{2}y^{2} ｳ p^{2}yl . Thus multiplying the first by l ﾎ (0, 1) and the second by (1-l ), then:
, l p^{1}y^{1} ｳ l p^{1}yl + (1-l
)p^{2}y^{2} ｳ (1-l
)p^{2}y^{2}. Adding pointwise, we see immediately that: , l p^{1}y^{1} + (1-l )p^{2}y^{2}
ｳ (l p^{1} + (1-l )p^{2})yl . We can rewrite this
in terms of profit functions as l p
(p^{1}) + (1-l )p (p^{2})
ｳ p (l p^{1}_{
}+ (1-l )p^{2}). 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

Property 4: p(p) is continuous and P(p) is upper semicontinouous.

Proof: p(p) = py s.t. y ﾎ Y. By definition of inner products and linear functionals, py is jointly continuous in p and y; as Y is a "constant correspondence of p", it is both upper semicontinuous and lower semicontinuous. Applying Berge’s Theorem, then, p(p) is continuous and P(p) is upper semicontinuous.ｧ

Theorem: There exists a set of profit maximizing points P (p).

Proof: p (p) is a continuous function (Property 4) defined over a closed set that is bounded above. A relatively weaker version of the Weierstrass Theorem implies that this function attains a maximum over Y. Thus, profit-maximizing points exist, i.e. P (p) exists.ｧ

**(4) The Efficiency of the Producer**

It is apparent from Figure 3 that the profit-maximizing point y* is also a
technically efficient point. Similarly, it is conceivable that all technically-efficient
points can be achieved as profit-maximizing plans provided one finds the appropriate set
of prices. The following two theorems state this clearly. Note that the phrase *universality
of markets*" merely means that p >> 0.

Theorem: (p-maxｮt.e.) If y* is profit-maximizing at p and there is universality of markets (p >> 0), then y* is technologically efficient.

Proof: Suppose not. Suppose y* is profit maximizing but it is not t.e. Then by definition there is a y ﾎ Y such that y > y*. Premultiplying by p (recall, p >> 0), then py > py*. Thus y* could not have been profit-maximizing. Contradiction.ｧ

Figure 5 illustrates why the assumption of universality of markets is
required. At the given p shown in Figure 5 (where p_{1} = 0) there are several
production vectors y which are profit maximizing (e.g. y^{a} and y^{b}),
yet note that y^{b} is technically efficient whereas y^{a} is not because
y^{b}_{ }> y^{a} (i.e. y^{b} = [y_{1}^{b},
y_{2}^{b}] > [y_{1}^{a}, y_{2}^{a}] = y^{a}).
Thus, y^{a} is a profit-maximizing point but it is not technologically efficient.

Figure 5- Need for Universality of Markets in p -max ｮ t.e. proof

We now turn to the converse theorem:

Theorem: (t.e.ｮ p-max) If y* is technologically-efficient, then there is a p (ｹ 0) such that at this p, y* is profit-maximizing.

Proof: For this, we need the Separating Hyperplane Theorem (SHT).
Define two sets: Y (the regular production set) and the following H(y*) = {y ﾎ R^{n} ｽ y >>
y*} which we wish to separate (cf. Figure 6 for heuristics). They both need to be
non-empty, convex and disjoint. For Y, this is easy: by assumption (1), 0 ﾎ Y, thus it is non-empty. Similarly, by assumption (5), Y is
convex. Finally, by the definition of H(y*) and "technological efficiency" of
y*, Y has no points in common with H(y*).

H(y*) is also non-empty: i.e. take a vector e
> 0. Then y* + e >> y*. Thus, y*+e
ﾎ H(y*). Thus, H(y*) is non-empty. H(y*) is also convex: take
two vectors, y^{1} and y^{2} in H(y*). Thus, y^{1} >> y* and
y^{2} >> y* by definition. Let take l ﾎ (0, 1) and premultiply the inequalities by l
and (1-l ) respectively so that l y^{1}
>> l y* and (1-l )y^{2}
>> (1-l )y*. Add these two together pointwise: l y^{1} + (1-l )y^{2}
>> l y* + (1-l )y* = y*. Thus,
the convex combination of y^{1} and y^{2} is strictly greater than y*.
Thus, l y^{1} + (1-l )y^{2}
ﾎ H(y*), thus H(y*) is convex.

Hence, we have two non-empty, convex and disjoint sets H(y*) and Y. Apply
SHT. There is a pair (p, c) where p ﾎ R^{n} and p (ｹ 0) and c ﾎ R such that:

py ｣ c " y ﾎ Y

py ｳ c " y ﾎ H(y*)

Now, let us define a positive vector e > 0 and a number n > 0 so we can define the vector [y* + (1/n)e ]. Thus, by definition, [y* + (1/n)e ] >> y* and thus, by definition, [y* + (1/n)e ] ﾎ H(y*). From the SHT, then, we know that p[y* + (1/n)e ] ｳ c ｳ py " y ﾎ Y. Taking n ｮ ･ , then we see that py* ｳ py " y ﾎ Y. This is the definition of y* as a profit-maximizing point.ｧ

A diagrammatic illustration (slightly exaggerated for effect) of the theorem can be conceived in Figure 6.

Figure 6- Separating Hyperplane for t.e. ｮ p -max proof

The meaning of these equivalence theorems is not entirely vacuous - particularly in light of the Socialist Calculation debate. It implies, quite simply, that any technically efficient outcome can be achieved in a "capitalist" economies by profit-maximizing behavior - as long as "one gets the prices right". Conversely, it also implies that centrally-planned economies can achieve technically efficient outcomes - provided the planning board "gets the prices right" and requires that the managers of state enterprises maximize profit/minimize cost. These equivalence theorems were central to Abba Lerner (1934) and Oskar Lange (1936), although they expressed them in Paretian context. The Neo-Walrasian version here, due to Tjalling Koopmans (1951), thus lent support to the Lange-Lerner contention.

Let us connect the production idea back to the Paretian notion of differentiable
"marginalism" - more specifically, the idea that output/inputs will be chosen by
producers where the marginal rates of technical substitution (MRTS) are equal to price
ratios. Let us assume universality of markets so that p >> 0. Consider y ﾎ P (p). By our second equivalence
theorem outlined above, y ﾎ P (p) ﾞ y = [F[y_{-i}^{0}], y_{-i}^{0}],
i.e. y is a technically efficient point. Let us consider another technologically efficient
point [y + e e_{j}] = [F[y_{-i}^{0} + e e_{j}], y_{-i}^{0} + e
e_{j}] where e ｹ 0 and j ｹ i. By definition of profit maximization, then, py ｳ p[y + e e_{j}], or:

p

_{i}F[y_{-i}^{0}] + p_{-i}y_{-i}^{0}ｳ p_{i}F[y_{-i}^{0}+ e e_{j}] + p_{-i}[y_{-i}^{0}+ e e_{j}]

where p_{i} is the price of the ith good and p_{-i} is the
vector of price of all goods except i. Rearranging:

p

_{i }[F[y_{-i}^{0}] + F[y_{-i}^{0}+ e e_{j}] ｳ p_{-i}[y_{-i}^{0}_{ }- y_{-i}^{0}+ e e_{j}] = e p_{j}

Thus, dividing by p_{i}:

_{[F[y-i0] + F[y-i0 + e ej] ｳ e pj/pi }

We now wish to divide by e but we have not placed any restrictions on e except that it be non-zero. Thus, if e > 0, this becomes:

[F[y

_{-i}^{0}] + F[y_{-i}^{0}+ e e_{j}]/e ｳ p_{j}/p_{i}

or

[F[y

_{-i}^{0}+ e e_{j}] + F[y_{-i}^{0}]]/e ｣ - p_{j}/p_{i}

whereas if e < 0, then:

[F[y

_{-i}^{0}] + F[y_{-i}^{0}+ e e_{j}]/e ｣ p_{j}/p_{i}

or

[F[y

_{-i}^{0}+ e e_{j}] + F[y_{-i}^{0}]]/e ｳ - p_{j}/p_{i}

Thus, taking the limit, as e ｮ 0, we obtain:

[dF/dy

_{j}]^{+}_{ }｣ -p_{j}/p_{i}

and

[dF/dy

_{j}]^{-}_{ }ｳ -p_{j}/p_{i}

But dF/dy_{j} is merely dy_{i}/dy_{j}. Thus, this
implies that if "differentiable", then:

- dy

_{i}/dy_{j}= -[dF/dy_{j}] = p_{j}/p_{i}

or, simply, the very familiar claim that the marginal rate of technical
substitution (MRTS) between goods i and j (i.e. - dy_{i}/dy_{j}) is equal
to the reciprocal ratio of their prices.

We have so far kept the assumption that Y is convex and thus maintained non-increasing returns. Naturally, if the production set Y is non-convex so as to allow for "increasing returns to scale" or if we also disallowed decreasing returns, then what would happen?

Increasing returns is implied if the production set is non-convex. This
would remove the concavity of the efficiency function but it would not implicate the first
equivalence theorem (profit-maximizing points remain technically efficient). However, it
would affect the existence of profit-maximizing points (or rather, those points and
associated profits would be either infinite or non-positive) and thus the second
equivalence theorem would be completely disabled. This is quite easy to visualize, as in
Figure 7 where we have a non-convex Y (i.e. Y with increasing returns to scale). We see
that the lightly shaded area represents the area of Y which yields positive profits (i.e.
where pｷy > 0 for a given p). Note that y* is *not* profit maximizing at the
given price p as y** yields even greater profits, i.e. py* = b < c = py**. Thus, we can
conceive that in this situation, at a given p, there are no profit-maximizing points in Y
and no limit to the amount of output produced (or rather, the firm will produce until
monopoly).

Figure 7- Production Set with Increasing Returns to Scale

Note that as "increasing returns" are required for any discussion of monopolistic or imperfect competition, then the assumption of convexity of Y implies that we must rule imperfectly competitive situations out. What if there is only a small portion of the production set that is non-convex? We might conceivably, in that case, still be able to define profit-maximizing points, but the associated supply correspondence would be discontinuous. This might be problematic for existence of equilibrium if the demand function passes through that hole in the supply function. However, as conjectured by M.J. Farrell and Jerome Rothenberg (1960) and proved by Ross M. Starr (1969), one can tolerate some amount of non-convexity as the process of aggregation will tend to "convexify" the aggregate production set and thus fill in the holes in the aggregate supply correspondence. For more reflections on incorporating non-convexities and imperfect competition in general equilibrium theory, see Negishi (1961), Arrow (1971) and Arrow and Hahn (1971: Ch.6, 7) and attempts using non-smooth differentials are surveyed in Quinzii (1992) and Brown (1991) and in our discussion of non-convexity.

However, is convexity equivalent to perfect competition? Not quite. The
possible existince of diminishing returns to scale not only is logically troublesome, but
the existence of positive profits may itself imply that entry is not exactly
"free". Many G.E. theorists have tried to make arguments for the retention of
diminishing returns on the basis of goods "private" to the firm or by choosing a
private-ownership economy and thus reallocating that profit to agents. However, others
have chosen to replace the simple convexity assumption outright and assume instead that
production is a "convex cone", i.e. for any y^{1}, y^{2} ﾎ Y and any a ｳ
0, b ｳ 0, it is true that a y^{1} + b y^{2} ﾎ Y. This, in turn, implies "constant returns" as it rules
out both increasing and decreasing returns to scale. This is shown heuristically in Figure
8.

Figure 8- Production Set with Constant Returns to Scale

The assumption of constant returns or that Y is a "convex cone"
essentially implies that the "bounds" of the production set are straight and
thus the profit-maximizing points are either indeterminate (along the boundary of Y, y*
and y** are equally profitable at p) or zero (no production at all). In our diagram, this
implies that there is a *unique* price that can yield a positive profit-maximizing
output yet, at that price, the profit-maximizing plan is actually indeterminate. As is
obvious from Figure 8, any other p would imply that profit-maximizing output is the
origin, 0 ﾎ Y, and thus inactivity. For an analogue, recall
that constant returns to scale implies that, in a supply-and-demand diagram, the supply
curve is horizontal, thus there is a unique p where output is supplied but at this p, the
actual output is actually indeterminate (or rather, in equilibrium, it is determined by
the demand function and thus the consumer decision). Note also that a convex cone
production set necessarily implies zero profits, thus fitting better with Walras's
"perfect competition" idea of "no gains, no losses" to the
entrepreneur.

We should note that the assumptions of divisibility (if y ﾎ Y, then l y ﾎ
Y where l ﾎ [0, 1]) and additivity
(if y^{1}, y^{2} ﾎ Y, then y^{1} + y^{2}
ﾎ Y) of activities in Y are enough to give us a convex cone.

Theorem: If production is divisible and additive, then Y is a convex cone.

Proof: Y is a "convex cone" if for any y^{1}, y^{2}
ﾎ Y and any a ｳ
0, b ｳ 0, it is true that a y^{1} + b y^{2} ﾎ Y. Divisibility implies that ly^{1}
ﾎ Y and (1-l )y^{2} ﾎ Y for l ﾎ
(0, 1). By additivity, l y^{1} + (1-l
)y^{2} ﾎ Y, which implies convexity. Consider now any
particular a ｳ 0. Let n be the
largest integer not exceeding a , then l
= a - n ﾎ (0, 1). By divisibility,
then, if y ﾎ Y, then l y = (a - n)y ﾎ Y. Consider the set of vectors
{y^{i}}^{n}_{i=1} where y^{i} = y. Thus, by additivity, ・/font> _{i=1}^{n} y^{i} = ny ﾎ
Y. Let y^{n+1}_{ }= l y, then by additivity ・/font> _{i=1}^{n+1} y^{i} = ny + l y = a y. As ny ﾎ
Y and l y ﾎ Y, then by additivity a y ﾎ Y. This was true for any a ｳ 0. Thus, Y is a convex cone.ｧ

Let us now reintroduce our firm index, f . Thus we can define for each firm f ﾎ F, a profit function and a supply correspondence:

p

^{f }(p) = max py s.t. y ﾎ Y^{f}P

^{f}(p) = argmax py s.t. Y ﾎ Y^{f}

Thus the supply correspondence is P ^{f}(p)
= {y ﾎ Y^{f} ｽ py >
py’ " y’ ﾎ Y^{f}}.

Now, let us define the "*aggregate production set*" as Y = ・/font> _{f}Y^{f}. We can also define a profit function
and supply correspondence for the aggregate as:

p (p) = max py s.t. y ﾎ Y

P (p) = argmax py s.t. Y ﾎ Y

Now, we would like to know whether "*aggregate supply*",
defined as the result of profit maximization over the "aggregate" production
set, can also be found by summing up the individual supplies from individual firm
profit-maximization. To show this, we first need a lemma - that the profit functions are
equal:

Lemma: p (p) = ・/font>_{f}p^{f}(p)

Proof: Step (i): By definition of the aggregate production set, ・/font> _{f} y^{f} ﾎ Y. By
definition of profit-maximization, p (p) ｳ
py " y ﾎ Y. Thus, p (p) ｳ p(・/font> _{f
}y^{f}) " y^{f} ﾎ
Y^{f}. Thus, p (p) ｳ ・/font> _{f} py^{f} " y^{f}
ﾎ Y^{f}. Thus, p (p) ｳ ・/font> _{f} p
^{f}(p).

Step (ii): for any y ﾎ Y, there is a
collection y^{f }ﾎ Y^{f} such that ・/font> _{f}y^{f} = y. Thus, py = p(・/font>
_{f }y^{f}) = ・/font> _{f }py^{f}.
By definition of profit maximization, p ^{f}(p) ｳ py^{f} " y ﾎ Y^{f}. Thus, ・/font> _{f }p^{f}(p) ｳ ・/font> _{f} py^{f} = py " y ﾎ Y. Thus, ・/font>
_{f }p^{f}(p)
ｳ p (p).

Thus, by (i) & (ii), then p (p) = ・/font> _{f }p^{f}(p).ｧ

We can now turn to the main theorem on the equivalence of the supply corresponence over the aggregate set and the sum of individual supply correspondences over individual sets:

Theorem: P (p) = ・/font>_{f}P^{f}(p)

Proof: Step (i) Consider {y^{f}} where y^{f} ﾎ P ^{f}. Then p(・/font> _{f }y^{f}) = ・/font> _{f}py^{f}
= ・/font> _{f }p^{f}(p) by the definition of P ^{f}(p). But by the lemma we just proved, ・/font> _{f }p^{f}(p) = p (p). By
definition, p (p) = py where y ﾎ P (p). Thus, as p(・/font> _{f }y^{f})
= py where y^{f} ﾎ P ^{f}(p)
and y ﾎ P (p). Then it must be that
・/font> _{f }y^{f} ﾎ P (p), i.e. ・/font> _{f }P_{ }^{f}(p)
ﾌ P (p).

Step (ii) Consider y ﾎ P
(p). Then there is some {y^{f}} ﾎ Y such that ・/font> _{f }y^{f} = y. But py = p(・/font>
_{f }y^{f}) = ・/font> _{f }py^{f}.
By definition of P (p), p (p) = ・/font> _{f }py^{f}. But in our lemma, p
(p) = ・/font> _{f }p^{f}(p), thus ・/font> _{f
}py^{f} = ・/font> _{f }p^{f}(p) which
implies that py^{f} = p ^{f}(p) " f. Thus, y^{f} ﾎ P ^{f}(p) " f. Or, ・/font> _{f }y^{f} ﾎ ・/font> _{f }P_{ }^{f}(p). Thus, y ﾎ ・/font> _{f }P_{ }^{f}(p).
Therefore, P (p) ﾌ ・/font> _{f }P_{ }^{f}(p).

By (i) & (ii), then P (p) = ・/font> _{f }P_{ }^{f}(p).ｧ

Thus the aggregate supply correspondence is the same as the sum of the individual supply correspondences.

Kenneth J. Arrow (1971) "The Firm in General Equilibrium
Theory", in R. Marris and A. Wood, editors, *The Corporate Economy: Growth,
competition and innovative potential*. Cambridge, Mass: Harvard University Press.

Gerard Debreu (1959) *Theory of Value: An axiomatic analysis of economic
equilibrium*. New York: Wiley.

David Gale. (1960) *The Theory of Linear Economic Models*. New York:
McGraw-Hill.

Tjalling C. Koopmans (1951) "Analysis of Production as an Efficient
Combination of Activities", in Koopmans, editor, *Activity Analysis of Production
and Allocation: Proceedings of a conference*. New York: Wiley.

Tjalling C. Koopmans (1957) *Three Essays on the State of Economic
Science*. New York: McGraw-Hill.