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

PRODUCTION FUNCTIONS 3

output it is sometimes useful to represent the technology of the ﬁrm with a math-

ematical function that gives the maximum output attainable from a given vector

of inputs. This function is called a production function and is deﬁned as

f(x) = max

y[y:(x, y)∈T]

= max

y[y:x∈V(y)]

= max

y∈P(x)[y]

(2)

Once the optimization is carried out we have a numerically valued function of

the form

y=f(x1,x

2,..., x

n)(3)

Graphically we can represent the production function in two dimensions as in

ﬁgure 1.

FIGURE 1. Production Function

x

y

fHxL

In the case where there is one output, one can also think of the production func-

tion as the boundary of P(x), i.e., f(x) = Eff P(x).

2.2. Existence and the Induced Production Correspondence. Does the produc-

tion function exist. If it exists, is the output correspondence induced by it the same

as the original output correspondence from which f was derived? What properties

does f(x) inherit from P(x)?

a: To show that production function exists and is well deﬁned, let x ∈Rn

+.

By axiom P.1a, P(x) 6=∅. By axioms P.4 and P.5, P(x) is compact. Thus P(x)

contains a maximal element and f(x) is well deﬁned. NOte that only these

three of the axioms on P are needed to deﬁne the production function.

b: The output correspondence induced by f(x) is deﬁned as follows

Pf(x)=[y∈R+:f(x)≥y],x∈Rn

+(4)

4 PRODUCTION FUNCTIONS

This gives all output levels y that can be produced by the input vector

x. We can show that this induced correspondence is equivalent to the

output correspondence that produced f(x). We state this in a proposition.

Proposition 2. Pf(x)=P(x),∀x∈Rn

+.

Proof. Let y ∈Pf(x), x ∈Rn

+. By deﬁnition, y ≤f(x). This means that y ≤

max {z: z ∈P(x)}. Then by P.3.S, y ∈P(x). Now show the other way. Let y

∈P(x). By the deﬁnition of f, y ≤max {z: z ∈P(x)}= f(x). Thus y ∈Pf(x).

Properties P.1a, P.3, P.4 and P.5 are sufﬁcient to yield the induced pro-

duction correspondence.

2.2.1. Relationship between P(x) and f(x). We can summarize the relation-

ship between P and f with the following proposition:

Proposition 3. y∈P(x) ⇔f(x) ≥y, ∀x∈Rn

+

2.3. Examples of Production Functions.

2.3.1. Production function for corn. Consider the production technology for corn on

a per acre basis. The inputs might include one acre of land and various amounts

of other inputs such as tillage operations made up of tractor and implement use,

labor, seed, herbicides, pesticides, fertilizer, harvesting operations made up of dif-

ferent combinations of equipment use, etc. If all but the fertilizer are held ﬁxed,

we can consider a graph of the production relationship between fertilizer and corn

yield. In this case the production function might be written as

y = f (land,tillage,labor,seed,fertilizer, ...)(5)

2.3.2. Cobb-Douglas production function. Consider a production function with two

inputs given by y = f(x1,x

2). A Cobb-Douglas [4] [5] represention of technology

has the following form.

y=Axα1

1xα2

2

=5x

1

3

1x

1

4

2

(6)

Figure 2 is a graph of this production function.

Figure 3 shows the contours of this function.

With a single output and input, a Cobb-Douglas production function has the

shape shown in ﬁgure 4.

2.3.3. Polynomial production function. We often approximate a production function

using polynomials. For the case of a single input, a cubic production function

would take the following form.

y=α1x+α2x2+α3x3

=10x+20x2−0.60 x3(7)

The cubic production function in equation 7 is shown in ﬁgure 5.