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Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data Deﬁnition and Properties of the Production
Function: Lecture II
Charles B. Moss August 25, 2011 Charles B. Moss Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data Overview of the Production Function
A Brief Brush with Duality
CobbDouglas Cost Minimization
Lagrangian for the CobbDouglas
Solution to CobbDouglas
CobbDouglas Cost Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data
Charles B. Moss Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data A Brief Brush with Duality
CobbDouglas Cost Minimization
Lagrangian for the CobbDouglas
Solution to CobbDouglas
CobbDouglas Cost Function Overview of the Production Function 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 (Chambers p. 7). Charles B. Moss Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data A Brief Brush with Duality
CobbDouglas Cost Minimization
Lagrangian for the CobbDouglas
Solution to CobbDouglas
CobbDouglas Cost Function 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 Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data A Brief Brush with Duality
CobbDouglas Cost Minimization
Lagrangian for the CobbDouglas
Solution to CobbDouglas
CobbDouglas Cost Function A Brief Brush with Duality
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 useful in economic analysis, or those
properties that make the system solvable. One approach would be to estimate a production function, say
a CobbDouglas production function in two relevant inputs
αβ
y = x 1 x2
Charles B. Moss (1) Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data A Brief Brush with Duality
CobbDouglas Cost Minimization
Lagrangian for the CobbDouglas
Solution to CobbDouglas
CobbDouglas Cost Function CobbDouglas Cost Minimization Given this production function, we could derive a cost
function by minimizing the cost of the two inputs subject to
some level of production
min w1 x1 + w2 x2 x 1 ,x 2 αβ
s . t . y = x 1 x2 Charles B. Moss (2) Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data A Brief Brush with Duality
CobbDouglas Cost Minimization
Lagrangian for the CobbDouglas
Solution to CobbDouglas
CobbDouglas Cost Function Lagrangian for the CobbDouglas
Yielding the Lagrangian
αβ
L = w 1 x1 + w 2 x2 + λ y − x 1 x2
x αx β
∂L
= w1 + λ 1 2 = 0
∂ x1
x1
αβ
x1 x 2
∂L
= w2 + λ
=0
∂ x2
x2
∂L
αβ
= y − x 1 x2 = 0
∂λ Charles B. Moss (3) Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data A Brief Brush with Duality
CobbDouglas Cost Minimization
Lagrangian for the CobbDouglas
Solution to CobbDouglas
CobbDouglas Cost Function Solution to CobbDouglas
∂L
w1
x2
w2
∂ x1
⇒
=
⇒ x1 =
x2
∂L
w2
x1
w1
∂ x2
∂L
⇒y−
∂λ w2
x2
w1 α β
x2 =0⇒ ∗
x1 ( w 1 , w 2 , y ) ∗
x2 ( w 1 , w 2 , y ) =y Charles B. Moss 1
α+β w2
w1 =y β
α+β (4) 1
α+β w1
w2 α
α+β (5)
(6) Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data A Brief Brush with Duality
CobbDouglas Cost Minimization
Lagrangian for the CobbDouglas
Solution to CobbDouglas
CobbDouglas Cost Function CobbDouglas Cost Function C ( w 1 , w 2 , y ) = w1 y 1
α +β w2
w1 β
α+β β
α+β + w2 y
α
α+β w1
w2 α
α+β 1
w
w
= y α+β 2 α + 1 β w1α+β
w2α+β Charles B. Moss 1
α+β (7) Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data A Brief Brush with Duality
CobbDouglas Cost Minimization
Lagrangian for the CobbDouglas
Solution to CobbDouglas
CobbDouglas Cost Function Thus, in the end, we are left with a cost function that relates
input prices and output levels to the cost of production based
on the economic assumption of optimizing behavior.
Following Chambers critique, recent trends in economics skip
the ﬁrst stage of this analysis by assuming that producers
know the general shape of the production function and select
inputs optimally. Thus, economists only need to estimate the
economic behavior in the cost function.
Following this approach, economists only need to know things
about the production function that aﬀect the feasibility and
nature of this optimizing behavior.
In addition, production economics is typically linked to
Sheppards Lemma that guarantees that we can recover the
optimal input demand curves from this optimizing behavior.
Charles B. Moss Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data Production Function Deﬁned
Following our previous discussion, we then deﬁne a production
function as a mathematical mapping function
+
+
f : R n → rm However, we will now write it in implicit functional form
Y (z ) = 0 (8) (9) This notation is sometimes referred to as a netput notation
where we do not diﬀerentiate inputs or outputs
Y (y , x ) = 0
Charles B. Moss (10) Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data Following the mapping notation, we typically exclude the
possibility of negative outputs or inputs, but this is simply a
convention. In addition, we typically exclude inputs that are
not economically scarce such as sunlight. Finally, I like to refer to the production function as an
envelope implying that the production function characterizes
the maximum amount of output that can be obtained from
any combination of inputs. Charles B. Moss Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data Properties of the Production Function
Monotonicity and Strict Monotonicity
Quasiconcavity and Concavity
V(y ) = { x : f (x ) y } is aconvex set (quasiconcave).
≥
f θx 0 + (1 − θ) x ∗ ≥ θf x 0 + (1 − θ) f (x ∗ ) for any
0 ≤ θ ≤ 1 (Concave). Weakly essential and strictly essential inputs
If x ≥ x , then f (x ) ≥ f (x ) (monotonicity).
If x > x , then f (x ) > f (x ) (strict monotonicity). f (0n ) = 0, where 0n is the null vector (weakly essential).
f (x1 , · · · xi −1 , 0, xi +1 , · · · xn ) = 0 for all xi (strict essential). The set V (y ) is closed and nonempty for all y > 0.
f (x ) is ﬁnite, nonnegative, real valued, and single valued for
all nonnegative and ﬁnite x .
Charles B. Moss Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data Continuity
f (x ) is everywhere continuous, and
f (x ) is everywhere twicecontinously diﬀerentiable. Charles B. Moss Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data Properties (1a) and (1b) require the production function to be
nondecreasing in inputs, or that the marginal products be
nonnegative.
In essence, these assumptions rule out stage III of the
production process, or imply some kind of assumption of
freedisposal.
One traditional assumption in this regard is that since it is
irrational to operate in stage III, no producer will choose to
operate there. Thus, if we take a dual approach (as developed
above) stage III is irrelevant. Charles B. Moss Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data Properties (2a) and (2b) revolve around the notion of
isoquants or as redeveloped here input requirement sets.
The input requirement set is deﬁned as that set of inputs
required to produce at least a given level of outputs, V (y ).
Other notation used to note the same concept are the level set.
Strictly speaking, assumption (2a) implies that we observe a
diminishing rate of technical substitution, or that the isoquants
are negatively sloping and convex with respect to the origin. Charles B. Moss Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data x1 Vy
Charles B. Moss Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data Assumption (2b) is both a stronger version of assumption (2a)
and an extension. For example, if we choose both points to be
on the same input requirement set, then the graphical
depiction is simply Charles B. Moss Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data x1
f x0 1 x0 f x0 1 f x0 Vy
Charles B. Moss Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data f we assume that the inputs are on two diﬀerent input
requirement sets, then
f θx 0 + (1 − θ) x ∗ ≥ θ f x 0 − f (x ∗ ) + f (x ∗ )
∂ f (x ∗ ) 0
f θx 0 + (1 − θ) x ∗ ≥ θ
x − x ∗ + f (x ∗ )
∂x
(11) Clearly, letting θ approach zero yields f (x ) approaches f (x ∗ ),
however, because of the inequality, the lefthand side is less
than the right hand side. Therefore, the marginal productivity
is nonincreasing and, given a strict inequality, is decreasing.
As noted by Chambers, this is an example of the law of
diminishing marginal productivity that is actually assumed.
Chambers oﬀers a similar proof on page 12, learn it.
Charles B. Moss Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data The notion of weakly and strictly essential inputs is apparent.
The assumption of weakly essential inputs says that you
cannot produce something out of nothing. Maybe a better way
to put this is that if you can produce something without using
any scarce resources, there is not an economic problem.
The assumption of strictly essential inputs is that in order to
produce a positive quantity of outputs, you must use a positive
quantity of all resources.
Diﬀerent production functions have diﬀerent assumptions on
essential inputs. It is clear that the CobbDouglas form is an
example of strictly essential resources. Charles B. Moss Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data The remaining assumptions are fairly technical assumptions
for analysis.
First, we assume that the input requirement set is closed and
bounded. This implies that functional values for the input
requirement set exist for all output levels (this is similar to the
lexicographic preference structure from demand theory).
Also, it is important that the production function be ﬁnite
(bounded) and realvalued (no imaginary solutions). The
notion that the production function is a single valued map
simply implies that any combination of inputs implies one and
only one level of output. Charles B. Moss Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data The assumption of continuous function levels, and ﬁrst and
second derivatives allows for a statement of the law of variable
proportions. The law of variable proportions is essentially restatement of
the law of diminishing marginal returns. The law of variable proportions states that if one input is
successively increase at a constant rate with all other inputs
held constant, the resulting additional product will ﬁrst
increase and then decrease. This discussion actually follows our discussion of the factor
elasticity from last lecture Charles B. Moss Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data dy
%∆y
dyx
MPP
y
E=
=
=
=
dx
%∆x
dxy
APP
x
d TPP
d x APP
d APP
MPP =
=
= APP + x
dx
dx
dx Charles B. Moss (12) Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data Working the last expression backward, we derive
d APP
1
= (MPP − APP )
dx
x
∂ (AP )i
1 ∂f
y
=
−
∂ xi
x i ∂ xi
xi Charles B. Moss (13) Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data Elasticity of Scale The law of variable proportions was related to how output
changed as you increased one input. Next, we want to
consider how output changes as you increase all inputs. In economic jargon, this is referred to as the elasticity of scale
and is deﬁned as
∂ ln [f (λx )]
=
∂ ln (λ) λ=1
Charles B. Moss (14) Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data x2 x2 xx Charles B. Moss Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data The elasticity of scale takes on three important values:
If the elasticity of scale is equal to 1, then the production
surface can be characterized by constant returns to scale.
Doubling all inputs doubles the output.
If the elasticity of scale is greater than 1, then the production
surface can be characterized by increasing returns to scale.
Doubling all inputs more than doubles the output.
Finally, if the elasticity of scale is less than 1, then the
production surface can be characterized by decreasing returns
to scale. Doubling all inputs does not double the output. Note the equivalence of this concept to the deﬁnition of
homogeneity of degree k
λk f ( x ) = f ( λx )
Charles B. Moss (15) Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data For computation purposes
n
n
∂ f xi
∂ ln [f (λx )]
=
=
i
∂ ln (λ) λ=1
∂ xi y
i =1 Charles B. Moss (16) i =1 Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data Tobacco Production Data
Item
Gross Value of Production
Cash Expense
Seed and plant bed
Fertilizer
Chemicals
Custom Operations
Fuel, Lube, and Electricity
Repairs
Hired Labor
Market expenses
Other variable costs
Total variable costs
Price ($/lb, $/cwt)
Yield (lbs/acre, cwt/acre) Cost ($s/acre)
2003
2004
3,800.64 3,862.58 Price ($/cwt)
2003
2004
197.95 198.59 110.34
333.81
99.47
13.80
95.28
82.05
574.81
56.84
22.22
1,388.62
1.98
1,920 5.75
17.39
5.18
0.72
4.96
4.27
29.94
2.96
1.16
72.33
197.95
19.20 Charles B. Moss 115.51
356.58
98.65
13.91
110.93
83.89
624.96
57.12
23.65
1,485.20
1.99
1,945 5.94
18.33
5.07
0.72
5.70
4.31
32.13
2.94
1.22
76.36
198.59
19.45 Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data Input Prices Item
2001
2002
2003
2004 Blended
Fertilizer
($/ton)
184
186
192
207 Diesel
($/gallon)
1.08
0.96
1.30
1.30 Charles B. Moss CopperSulfate
($/lb)
1.15
1.16
1.20
1.30 Farm
Labor
($/hour)
7.22
7.13
7.56
7.63 Deﬁnition and Properties of the Production Function: Lecture Outline
Overview of the Production Function
Production Function Deﬁned
Properties of the Production Function
Law of Variable Proportions
Elasticity of Scale
Tobacco Production Data Homework Assignment
Use the data to estimate a production function for tobacco
(using GaussSeidel).
Plot the isoquants for fertilizer and labor.
Are these inputs complements or substitutes?
Is this conclusion dependent on the production function used? Plot the ridgelines. Charles B. Moss Deﬁnition and Properties of the Production Function: Lecture ...
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This note was uploaded on 02/01/2012 for the course AEB 6184 taught by Professor Staff during the Fall '09 term at University of Florida.
 Fall '09
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