PRODUCTION FUNCTIONS

1. ALTERNATIVE REPRESENTATIONS OF TECHNOLOGY

The technology that is available to a ﬁrm can be represented in a variety of

ways. The most general are those based on correspondences and sets.

1.1. Technology Sets. The technology set for a given production process is de-

ﬁned as

T={(x,y) : x ∈Rn

+,y∈Rm:

+x can produce y}

where x is a vector of inputs and y is a vector of outputs. The set consists of

those combinations of x and y such that y can be produced from the given x.

1.2. The Output Correspondence and the Output Set.

1.2.1. Deﬁnitions. It is often convenient to deﬁne a production correspondence and

the associated output set.

1: The output correspondence P, maps inputs x ∈Rn

+into subsets of outputs,

i.e., P: Rn

+→2Rm

+. A correspondence is different from a function in that a

given domain is mapped into a set as compared to a single real variable

(or number) as in a function.

2: The output set for a given technology, P(x), is the set of all output vectors

y∈Rm

+that are obtainable from the input vector x ∈Rn

+. P(x) is then the

set of all output vectors y ∈Rm

+that are obtainable from the input vector

x∈Rn

+. We often write P(x) for both the set based on a particular value of

x, and the rule (correspondence) that assigns a set to each vector x.

1.2.2. Relationship between P(x) and T(x,y).

P(x)=(y:(x, y )∈T)

1.2.3. Properties of P(x).

P.1a: Inaction and No Free Lunch. 0 ∈P(x) ∀x∈Rn

+.

P.1b: y6∈ P(0), y >0.

P.2: Input Disposability. ∀x∈Rn

+, P(x) ⊆P(θx), θ≥1.

P.2.S: Strong Input Disposability. ∀x, x’ ∈Rn

+,x’≥x⇒P(x) ⊆P(x’).

P.3: Output Disposability. ∀x∈Rn

+,y∈P(x) and 0 ≤λ≤1⇒λy∈P(x).

P.3.S: Strong Output Disposability. ∀x∈Rn

+,y∈P(x) ⇒y’ ∈P(x), 0 ≤y’ ≤

y.

P.4: Boundedness. P(x) is bounded for all x ∈Rn

+.

P.5: T is a closed set P: Rn

+→2Rm

+is a closed correspondence, i.e., if [x`→

x0,y

`→y0and y`∈P(x`), ∀`] then y0∈P(x0).

P.6: Attainability. If y ∈P(x), y ≥0 and x ≥0, then ∀θ≥0, ∃λθ≥0 such

that θy∈P(λθx).

Date: August 29, 2005.

1

2 PRODUCTION FUNCTIONS

P.7: P(x) is convex

P(x) is a convex set for all x ∈Rn

+. This is equivalent to the correspon-

dence V:<n

+→2<m

+being quasiconcave.

P.8: P is quasi-concave.

The correspondence P is quasi-concave on Rn

+which means ∀x, x’ ∈

Rn

+,0≤θ≤1, P(x) ∩P(x’) ⊆P(θx + (1-θ)x’). This is equivalent to V(y)

being a convex set.

P.9: Convexity of T. P is concave on Rn

+which means ∀x, x’ ∈Rn

+,0≤θ≤

1, θP(x)+(1-θ)P(x’) ⊆P(θx+(1-θ)x’)

1.3. The Input Correspondence and Input (Requirement) Set.

1.3.1. Deﬁnitions. Rather than representing a ﬁrm’s technology with the technol-

ogy set T or the production set P(x), it is often convenient to deﬁne an input corre-

spondence and the associated input requirement set.

1: The input correspondence maps outputs y ∈Rm

+into subsets of inputs,

V: Rm

+→2Rn

+. A correspondence is different from a function in that a

given domain is mapped into a set as compared to a single real variable

(or number) as in a function.

2: The input requirement set V(y) of a given technology is the set of all com-

binations of the various inputs x ∈Rn

+that will produce at least the level

of output y ∈Rm

+. V(y) is then the set of all input vectors x ∈Rn

+that

will produce the output vector y ∈Rm

+. We often write V(y) for both the

set based on a particular value of y, and the rule (correspondence) that

assigns a set to each vector y.

1.3.2. Relationship between V(y) and T(x,y).

V(y)=(x:(x, y)∈T)

1.4. Relationships between Representations: V(y), P(x) and T(x,y). The technol-

ogy set can be written in terms of either the input or output correspondence.

T={(x, y):x∈Rn

+,y ∈Rm

+,such that x will produce y}(1a)

T={(x, y)∈Rn+m

+:y∈P(x),x∈Rn

+}(1b)

T={(x, y)∈Rn+m

+:x∈V(y),y∈Rm

+}(1c)

We can summarize the relationships between the input correspondence, the

output correspondence, and the production possibilities set in the following propo-

sition.

Proposition 1. y∈P(x) ⇔x∈V(y) ⇔(x,y) ∈T

2. PRODUCTION FUNCTIONS

2.1. Deﬁnition of a Production Function. To this point we have described the

ﬁrm’s technology in terms of a technology set T(x,y), the input requirement set

V(y) or the output set P(x). For many purposes it is useful to represent the re-

lationship between inputs and outputs using a mathematical function that maps

vectors of inputs into a single measure of output. In the case where there is a single