Unformatted text preview: Basic Notions of Production Functions
Charles B. Moss
1 University 1 of Florida August 23, 2011 Charles B. Moss (University of Florida) Production Functions August 23, 2011 1 / 35 Outline 1 Overview of Production Economics 2 Overview of the Production Function
Chambers on Production Functions
Production Function Mapping 3 One Product, One Input
Derived Relationships
Estimated Production Functions
Stages of Production 4 Elasticity of Production
APP and MPP 5 One Product, Two Variable Input Relationships
Empirical Production Functions 6 Isoquants, Isoclines, and Ridgelines
FactorFactor Relationships Charles B. Moss (University of Florida) Production Functions August 23, 2011 2 / 35 Overview of Production Economics Overview of Production Economics The single chapter which follows summarizes the scope and method of
agricultural production economics. The treatment does not do justice to
this important ﬁeld of applied economics, which is surpassed by no other
in its accomplishment, its breadth, and its interesting history (Heady p.1). Charles B. Moss (University of Florida) Production Functions August 23, 2011 3 / 35 Overview of Production Economics Agricultural production economics is the oldest and most widespread
specialization in the applied ﬁeld of science know as agricultural
economics; the science of agricultural economics originated as a study in
farm production economics. The ﬁeld has continued to grow until it now
embraces more workers than any other specialization. In the ﬁeld of
agricultural economics the majority of outstanding names has been
associated with some phase of production economics, both in farm and in
national use of resources. Charles B. Moss (University of Florida) Production Functions August 23, 2011 4 / 35 Overview of Production Economics The list in the United States has included such prominent persons as G.F.
Warren, H.C. Taylor, T.N. Carver, W.J. Spillman, L.C. Grey, C.L. Holmes,
J.D. Black, E.C. Young, H. Tolley, M. Ezekiel, A. Boss, O.V. Wells,
H.C.M. Case, S.E. Johnson, T.W. Schultz, G.E Forster, W.W. Wilcox, and
W. G. Murray. The European list includes such well known names as J.
Von Thunen, M. Kitchen, F Aereboe, T. Brinkman, A.W. Ashby, G. Lauer,
R. Cohen and others. All of these scholars have been known for their
ﬁndings or particular methods of analysis. In this list are not only persons
who pioneered the ﬁeld of agricultural economics but also those who
contributed both to theory and to empirical knowledge. The central
objective of their work has been to increase the eﬃciency with which farm
resources are used (Heady p.1). Charles B. Moss (University of Florida) Production Functions August 23, 2011 5 / 35 Overview of Production Economics Production economists who focus their attention on agriculture are
concerned with choice and decisionmaking in the use of the capital, labor,
land, and management resources in the farming industry. The goals of
agricultural production economics are twofold: (1) to provide guidance to
individual farmers in using their resources most eﬃciently, and (2) to
facilitate the most eﬃcient use of resources from the standpoint of the
consuming economy (Heady p.3). Charles B. Moss (University of Florida) Production Functions August 23, 2011 6 / 35 Overview of the Production Function Overview of the Production Function The production function is a technical relationship depicting the
technical transformation of inputs into outputs.
The production function in and of itself is devoid of economic content.
In the development of production functions, we are interested in certain
characteristics that make it possible to construct economic models
based on optimizing behavior. Charles B. Moss (University of Florida) Production Functions August 23, 2011 7 / 35 Overview of the Production Function Chambers on Production Functions Chambers on Production Functions The production function (and indeed all representations of technology) is a
purely technical relationship that is void of economic content. Since
economists are usually interested in studying economic phenomena, the
technical aspects of production are interesting to economists only insofar
as they impinge upon the behavior of economic agents.... Because the
economist has no inherent interest in the production function, if it is
possible to portray and to predict economic behavior accurately without
direct examination of the production function, so much the better. This
principle, which sets the tone for much of the following discussion,
underlies the intense interest that recent developments in duality have
aroused (Chambers p.7). Charles B. Moss (University of Florida) Production Functions August 23, 2011 8 / 35 Overview of the Production Function Chambers on Production Functions The point of these two statements is that economists are not engineers and
have no insights into why technologies take on any particular shape. We
are only interested in those properties that make the production function
that make the production function consistent with optimizing behavior. Charles B. Moss (University of Florida) Production Functions August 23, 2011 9 / 35 Overview of the Production Function Production Function Mapping Production Function Mapping
One way to write the production function is as a function map
+
+
f : Rn → Rm (1) which states that the production function (f ) is a function that maps
n inputs into m outputs.
By convention, we are only interested in positive input bundles that
yield positive output bundles.
The ﬁrst lecture will focus on the production function as a continuous
function as students have probably seen it in previous courses. The
next lecture will develop the concept of the production function more
rigorously. Charles B. Moss (University of Florida) Production Functions August 23, 2011 10 / 35 One Product, One Input One Product, One Variable Factor
A commonly used form of the production function is the ”closed
form” representation where total physical product is depicted as a
function of a vector of inputs
y = f (x ) (2) where y is the scalar (single) output and x is a vector of inputs.
Focusing for a moment on the single output case, we could simplify
the above representation to
y = f ( x 1  x2 ) (3) or we are interested in examining the relationship between x1 and y
given that all the other factors of production are held constant. Charles B. Moss (University of Florida) Production Functions August 23, 2011 11 / 35 One Product, One Input Derived Relationships Derived Relationships
Using this relationship, we want to identify three primary relationships:
Total Physical Product  which is the original production function.
Average Physical Product  deﬁned as the average output per unit of
input. Mathematically,
APP = y
f (x )
=
x
x (4) Marginal Physical Product  deﬁned as the rate of change in the total
physical product at a speciﬁc input level. Mathematically,
MPP = Charles B. Moss (University of Florida) d (TPP )
dy
d f (x )
=
=
= f (x )
dx
dx
dx
Production Functions August 23, 2011 (5) 12 / 35 One Product, One Input Estimated Production Functions Estimated Production Functions 150 Corn 100 50 50 100 150 Nitrogen
Charles B. Moss (University of Florida) Production Functions August 23, 2011 13 / 35 One Product, One Input Estimated Production Functions This set of production functions are taken from Moss and Schmitz
Investing in Precision Agriculture.
This shape is referred to as a sigmoid shape.
The exact functional form in this ﬁgure can be attributed to Zellner,
Arnold, (1951) An Interesting General form for a Production
Function, Econometrica 19, 18889.
The exact mathematical form of the function is:
φ ( v 1 , v2 ) = 3
av1
v1
exp b
−1
v2 (6) The average function sets v2 = 1.0, a = 0.0005433, and b = 0.01794. Charles B. Moss (University of Florida) Production Functions August 23, 2011 14 / 35 One Product, One Input Estimated Production Functions Corn(bushel/acre) 120
100
80
60
40
20
50 100 150 Nitrogen(lbs/acre)
Charles B. Moss (University of Florida) Production Functions August 23, 2011 15 / 35 Corn One Product, One Input 1.0 Estimated Production Functions A 0.5 B
50 100 150 Nitrogen AveragePhysicalProduct Charles B. Moss (University of Florida) MarginalPhysicalProduct Production Functions August 23, 2011 16 / 35 One Product, One Input Stages of Production Stages of Production Stage I: This stage of the production function is deﬁned as that
region where the average physical product is increasing. In this
region, the marginal physical product is greater than the average
physical product. In this region, each additional unit of input yields
relatively more output on average.
Stage II: This stage of the production process corresponds with the
economically feasible region of production. Marginal physical product
is positive and each additional unit of input produces less output on
average.
Stage III: This stage of production implies negative marginal return
on inputs. Charles B. Moss (University of Florida) Production Functions August 23, 2011 17 / 35 One Product, One Input Stages of Production B Corn(bushel/acre) 120
100 A 80
60
40
20
50 100 150 Nitrogen(lbs/acre)
Charles B. Moss (University of Florida) Production Functions August 23, 2011 18 / 35 Elasticity of Production Elasticity of Production Elasticities are often used in economics to produce a unitfree
indicator of the shape of a function. Most are familiar with the
elasticity of consumer demand.
In deﬁning the production function, we are interested in the factor
elasticity.
The factor elasticity is deﬁned as
dy
%∆y
dy x
MPP
y
E=
=
=
=
dx
%∆x
dx y
APP
x Charles B. Moss (University of Florida) Production Functions (7) August 23, 2011 19 / 35 Elasticity of Production FactorElasticity 2.0
1.5
1.0
0.5 50 100 150 Nitrogen(lbs/acre)
Charles B. Moss (University of Florida) Production Functions August 23, 2011 20 / 35 Elasticity of Production APP and MPP APP and MPP There is a speciﬁc relationship between the average physical product and
marginal physical product when the average physical product is
maximized. Mathematically,
d TPP
d x APP
d APP
=
= APP + x
dx
dx
dx
Thus, when the APP is maximized
MPP = d APP
= 0 ⇒ MPP = APP
dx Charles B. Moss (University of Florida) Production Functions (8) (9) August 23, 2011 21 / 35 Elasticity of Production APP and MPP Following through this relationship, we have
d APP
> 0 ⇒ MPP > APP ⇒ E > 1
dx
d APP
= 0 ⇒ MPP = APP ⇒ E = 1
dx (10) d APP
< 0 ⇒ MPP < APP ⇒ E < 1
dx
In addition, we know that if E = 0 ⇔ MPP = 0, TPP is maximum and
E < 0 ⇔ MPP < 0.
Thus, if E > 1 then the production function is in Stage I. If 1 > E > 0,
then the production function is in Stage II. If E < 0, then the production
functio is in Stage III. Charles B. Moss (University of Florida) Production Functions August 23, 2011 22 / 35 One Product, Two Variable Input Relationships One Product, Two Variable Input Relationships Expanding the production, we start by considering the case of two inputs
producing one output. In the general functional mapping notation
+
+
f : R2 → R1
y = f ( x1 , x2 ) Charles B. Moss (University of Florida) Production Functions (11) August 23, 2011 23 / 35 One Product, Two Variable Input Relationships Figure: Two Inputs Charles B. Moss (University of Florida) Production Functions August 23, 2011 24 / 35 One Product, Two Variable Input Relationships These functions still have average physical products and marginal physical
products, but they are conditioned on the level of other inputs
APP1 = y
f ( x1 , x2 )
=
x1
x1 (12) y
f ( x1 , x2 )
APP2 =
=
x2
x2
Similarly, the marginal physical products are deﬁned by the partial
derivatives
MPP1 = ∂y
∂ f ( x1 , x2 )
=
∂ x1
∂ x1 (13) ∂y
∂ f ( x1 , x2 )
MPP2 =
=
∂ x2
∂ x2 Charles B. Moss (University of Florida) Production Functions August 23, 2011 25 / 35 One Product, Two Variable Input Relationships It may be useful at this point to brieﬂy visit the notion of the Taylor
expansion. Taking the secondorder expansion of the production function
yields
0 0
f ( x 1 , x 2 ) = f x1 , x 2 + ∂ f ( x 1 , x2 )
∂ x1
1
dx1
2 Charles B. Moss ∂ 2 f ( x1 , x2 )
2
dx2 2 ∂ x1 ∂ f ( x1 , x2 )
∂ x1 ∂ x2 (University of Florida) ∂ f ( x1 , x2 )
∂ x2 dx
1
+
dx2 ∂ 2 f ( x1 , x2 )
∂ x1 ∂ x2 dx1 ∂ 2 f (x1 , x2 ) dx2
2
∂ x2 Production Functions August 23, 2011 (14) 26 / 35 One Product, Two Variable Input Relationships This approximation is exact in the case of either linear or quadratic
production functions. However, if we focus on a quadratic production
function, it is clear that
∂ f ( x1 , x2 )
∂ f ((x1 , x2 )
dy ≈ f1 =
dx1 + f2 =
dx2
(15)
∂ x1
∂ x2 Charles B. Moss (University of Florida) Production Functions August 23, 2011 27 / 35 One Product, Two Variable Input Relationships Empirical Production Functions Empirical Production Functions
Linear Production Function
y = b 1 x1 + b 2 x2 (16) Quadratic Production Function
1
2
2
A11 x1 + 2A12 x1 x2 + A22 x2
2
x1
A11 A12
1
x1
a2
x1 x 2
+
x2
A21 A22
x2
2 y = a 1 x1 + a 2 x 2 +
y= a1 (17) CobbDouglas Production Function bb
y = Ax1 1 x2 2
Charles B. Moss (University of Florida) Production Functions (18)
August 23, 2011 28 / 35 One Product, Two Variable Input Relationships Empirical Production Functions Transcendental Production Function
a
a
y = Ax1 1 e b1 x1 x2 2 e b2 x2 (19) Constant Elasticity of Substitution (CES) Production Function
−v
g
−
−
y = A bx1 g (1 − b ) x2 g Charles B. Moss (University of Florida) Production Functions (20) August 23, 2011 29 / 35 Isoquants, Isoclines, and Ridgelines Isoquants, Isoclines, and Ridgelines Given the multivariate nature of the production function, it is possible
to deﬁne isoquants, or the relationship that depicts the combinations
of inputs that yield the same output.
Starting from the basic production function
y = f ( x 1 , x 2 ) ⇒ x2 = f ∗ ( x1 , y ) (21) That is we are interested in constructing a functional mapping of x2
based on the level of x1 and y .
Next, we hold the level of output constant and derive the levels of x2
for any level of x1 . Charles B. Moss (University of Florida) Production Functions August 23, 2011 30 / 35 Isoquants, Isoclines, and Ridgelines x2 y3
y2
y1
x1 Charles B. Moss (University of Florida) Production Functions August 23, 2011 31 / 35 Isoquants, Isoclines, and Ridgelines The isoquants are useful in deﬁning the rate of technical substitution or
the rate at which one input must be traded for the other input
dy = f1 dx1 + f2 dx2 = 0 ⇒ Charles B. Moss (University of Florida) Production Functions dx1
f1
=−
dx2
f1 (22) August 23, 2011 32 / 35 Isoquants, Isoclines, and Ridgelines Ridgeline Isocline Charles B. Moss (University of Florida) Production Functions August 23, 2011 33 / 35 Isoquants, Isoclines, and Ridgelines FactorFactor Relationships FactorFactor Relationships
Factor independence: Two factors are independent if the MPP of one
factor is not a function of the MPP of the other factor.
The simplest example of this is a quadratic production function with
A12 = A21 = 0. In this case, the isoquants are circles (or elipses). ∂y = a + A x
1
1
11 1
2
2
x1
y = a 1 x1 + a 2 x2 +
A11 x1 + A22 x2 ⇒
∂y 2
x2 = a2 + A22 x2 (23) Case I: If f12 > 0, then x1 and x2 are technically complementary.
∂y
∂
2
∂y
∂ x1
=
∂ x1 ∂ x2
∂ x2
Charles B. Moss (University of Florida) = f12 > 0 Production Functions (24)
August 23, 2011 34 / 35 Isoquants, Isoclines, and Ridgelines FactorFactor Relationships Case II: If f12 = 0, then x1 and x2 are technically independent.
Case III: If f12 < 0, then x1 and x2 are technically competitive. Charles B. Moss (University of Florida) Production Functions August 23, 2011 35 / 35 ...
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