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
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:xV(y)]
= max
yP(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)=[yR+:f(x)y],xRn
+(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),xRn
+.
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. yP(x) f(x) y, xRn
+
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+20x20.60 x3(7)
The cubic production function in equation 7 is shown in figure 5.
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