# Production Function, Lecture Notes - Economics, Study notes for Economics. The London School of Economics and Political Science (LSE)

## Economics

Description: Production functions
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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
+,yRm:
+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
yRm
+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
xRn
+. 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) xRn
+.
P.1b: y6∈ P(0), y >0.
P.2: Input Disposability. xRn
+, P(x) P(θx), θ1.
P.2.S: Strong Input Disposability. x, x’ Rn
+,xxP(x) P(x’).
P.3: Output Disposability. xRn
+,yP(x) and 0 λ1λyP(x).
P.3.S: Strong Output Disposability. xRn
+,yP(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 y0P(x0).
P.6: Attainability. If y P(x), y 0 and x 0, then θ0, λθ0 such
that θyP(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
+,0θ1, 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
+,0θ
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):xRn
+,y Rm
+,such that x will produce y}(1a)
T={(x, y)Rn+m
+:yP(x),xRn
+}(1b)
T={(x, y)Rn+m
+:xV(y),yRm
+}(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. yP(x) xV(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