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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+*, *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 ∈*R*n+ into subsets of outputs,
i.e., P: *R*n+ → 2R

m + . 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 ∈ *R*m+ that are obtainable from the input vector x ∈ *R*n+. P(x) is then the
set of all output vectors y ∈ *R*m+ that are obtainable from the input vector
x ∈*R*n+ . 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 ∈ *R*n+ .
**P.1b: **y 6∈ P(0), y *> *0.
**P.2: **Input Disposability. ∀ x ∈ *R*n+ , P(x) ⊆ P(*θ*x), *θ *≥ 1.
**P.2.S: **Strong Input Disposability. ∀ x, x’ ∈ *R*n+ , x’ ≥ x⇒ P(x) ⊆ P(x’).
**P.3: **Output Disposability. ∀ x ∈ *R*n+ , y ∈ P(x) and 0 ≤ *λ *≤ 1⇒ *λ*y ∈ P(x).
**P.3.S: **Strong Output Disposability. ∀ x ∈ *R*n+ , 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: *R*n+ → 2R

m + is a closed correspondence, i.e., if [x` →

x0, y` → y0 and y` ∈ P(x`), ∀ *`*] then y0 ∈ P(x0).
**P.6: **Attainability. If y ∈ P(x), y ≥ 0 and x ≥ 0, then ∀ *θ *≥ 0, ∃ *λ*θ ≥ 0 such

that *θ*y ∈ P(*λ*θx).

*Date*: August 29, 2005.
1

2 PRODUCTION FUNCTIONS

**P.7: **P(x) is convex
P(x) is a convex set for all x ∈ *R*n+ . 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 *R*n+ which means ∀ x, x’ ∈

*R*n+ , 0 ≤ *θ *≤ 1, P(x) ∩ P(x’) ⊆ P(*θ*x + (1-*θ*)x’). This is equivalent to V(y)
being a convex set.

**P.9: **Convexity of T. P is concave on *R*n+ which means ∀ x, x’ ∈ *R*n+ , 0 ≤ *θ *≤
1, *θ*P(x)+(1-*θ*)P(x’) ⊆ P(*θ*x + (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 ∈ *R*m+ into subsets of inputs,
V: *R*m+ → 2R

n + . 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 ∈ *R*n+ that will produce at least the level
of output y ∈ *R*m+ . V(y) is then the set of all input vectors x ∈ *R*n+ that
will produce the output vector y ∈ *R*m+ . 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 *∈ *R*n+*, y *∈ *R*m+ *, *such that x will produce y} (1a)

*T *= {(*x, y*) ∈ *R*n+m+ : *y *∈ *P *(*x*)*, x *∈ *R*n+} (1b)

*T *= {(*x, y*) ∈ *R*n+m+ : *x *∈ *V *(*y*)*, y *∈ *R*m+} (1c)
We can summarize the relationships between the input correspondence, the

output correspondence, and the production possibilities set in the followingpropo- 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 singlemeasure of output. In the casewhere 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 *(*x*1*, 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 fromwhich 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

*P*f (*x*) = [*y *∈ *R*+ : *f*(*x*) ≥ *y*]*, x *∈ *R*n+ (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. ***P*f (*x*) = *P *(*x*)*, *∀*x *∈ *R*n+*.
*

*Proof. *Let y ∈ Pf (x), x ∈ *R*n+ . 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 *∈ *R*n+
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, x2). A Cobb-Douglas [4] [5] represention of technology
has the following form.

*y *= *Ax*α11 *x
*α2
2

= 5*x
*1
3
1 *x
*

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 *= *α*1 *x *+ *α*2 *x*2 + *α*3 *x*3

= 10*x *+ 20*x*2 − 0*.*60*x*3
(7)

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

PRODUCTION FUNCTIONS 5

FIGURE 2. Cobb-Douglas Production Function

5

10

15

20

x1

5

10

15

20

x2

0

10

20f **H**x1,x2**L
**

FIGURE 3. Contours of a Cobb-Douglas Production Function

5 10 15 20 25 30

5

10

15

20

25

30

Notice that the function first rises at an increasing rate, then increases at a de- creasing rate and then begins to fall until it reaches zero.

6 PRODUCTION FUNCTIONS

FIGURE 4. Cobb-Douglas Production Function with One Input

**x
**

**y
**

FIGURE 5. Cubic Production Function

**x
**

**y
**

**fHxL
**

2.3.4. *Constant elasticity of substitution (CES) production function. *An early alterna-
tive to the Cobb-Douglas production function is the constant elasticity of substi-
tution (CES) production function [1]. While still being quite tractable, with a min-
imum of parameters, it is more flexible than the Cobb-Douglas production func-
tion. For the case of two inputs, the CES production function takes the following
form.

*y *= *A
*[
*δ*1 *x
*

−ρ
1 + *δ*2 *x
*

−ρ 2

]−1 ρ

= 5
[
0*.*6*x*−21 + 0*.*2*x
*

−2 2

]−1 2

(8)

The CES production function in equation 8 is shown in figure 6. The production
contours of the production function in equation 8 are shown in figure 7. If we
change *ρ *to 0.2, the CES contours are as in figure 8.

2.3.5. *Translog production function. *An alternative to the Cobb-Douglas production
function is the translog production function.

PRODUCTION FUNCTIONS 7

FIGURE 6. CES Production Function

5

10

15

20

x1

5

10

15

20

x2

0 25 50 75

100

f **H**x1,x2**L
**

FIGURE 7. CES Production Function Contours - *ρ*=2

2.5 5 7.5 10 12.5 15 17.5 20

2.5

5

7.5

10

12.5

15

17.5

20

*ln y *= *α*1 *ln x*1 + *α*2 *ln x*2 + *β*11 *ln x*21 + *β*12 *ln x*1 *ln x*2 + *β*22 *ln x
*2
2

=
1
3
*ln x*1 +

1
10
*ln x*2 −

2 100

*ln x*21 +
1
10
*ln x*1 *ln x*2 −

2 10

*ln x*22
(9)

The translog can also be written with y as compared to ln y on the left hand side.

8 PRODUCTION FUNCTIONS

FIGURE 8. CES Production Function Contours - *ρ*=0.05

2.5 5 7.5 10 12.5 15 17.5 20

2.5

5

7.5

10

12.5

15

17.5

20

*y *= *A , x*α11 *x
*α2
2 *e
*

β11 ln, x 2 1 + β12 ln x1 lnx2 + β22 lnx

2 2

= *x*1/31 *x
*1/10
2 *e
*

−.02 ln x21 + 0.1 ln x1 ln x2 − 0.2 ln x 2 2

(10)

Figure 9 shows the translog function from equation 9 while figure 10 shows the contours of the translog function.

2.4. **Properties of the Production Function. **We can deduce a set of properties on
f that are equivalent to the properties on P in the sense that if a particular set holds
for P, it implies a particular set on f and vice versa.

2.4.1. *f.1 Essentiality. *f(0) = 0.

2.4.2. *f.1.S Strict essentiality. *f(x1, x2, *. . . *, 0, *. . . *, xn) = 0 for all xi.

2.4.3. *f.2 Monotonicity. *∀ x ∈ *R*n+ , f(*θ*x) ≥ f(x), *θ *≥ 1.

2.4.4. *f.2.S Strict monotonicity. *∀ x, x’ ∈ *R*n+ , if x ≥ x’ then f(x) ≥ f(x’).

2.4.5. *f.3 Upper semi-continuity. *f is upper semi-continuous on *R*n+ .

2.4.6. *f.3.S Continuity. *f is continuous on *R*n+.

2.4.7. *f.4 Attainability. *If f(x) *> *0, f(*λ*x) → +∞ as *λ*→ +∞.

2.4.8. *f.5 Quasi-concavity. *f is quasi-concave on *R*n+.

2.4.9. *f.6 Concavity. *f is concave on *R*n+.

2.5. **Discussion of the Properties of the Production Function.**

PRODUCTION FUNCTIONS 9

FIGURE 9. Translog Production Function

5

10

15

20

x1

5

10

15

20

x2

0

10

20
f **H**x1,x2**L
**

FIGURE 10. Contours of Translog Production Function

2.5 5 7.5 10 12.5 15 17.5 20

2.5

5

7.5

10

12.5

15

17.5

20

2.5.1. *f.1 Essentiality. *f(0) = 0.

10 PRODUCTION FUNCTIONS

This assumption is sometimes called essentiality. It says that with no inputs, there is no output.

2.5.2. *f.1.S Strict essentiality. *f(x1, x2, *. . . *, 0, *. . . *, xn) = 0 for all xi.

This is called strict essentiality and says that some of each input is needed for a positive output. In this case the input requirement set doesn’t touch any axis. Consider as an example of strict essentiality the Cobb-Douglas function.

*y *= *Ax*α11 *x
*α2
2 (11)

Another example is the Generalized Leontief Function with no linear terms

*y *= *β*11*x*1 + 2*β*12 *x*1 *x*2 + *β*22 *x*2 (12)

2.5.3. *f.2 Monotonicity. *∀ x ∈ *R*n+ , f(*θ*x) ≥ f(x), *θ *≥ 1.

This is amonotonicity assumption that sayswith a scalar expansion of x, output cannot fall. There is also a strong version.

2.5.4. *f.2 Strict monotonicity. *∀ x, x’ *in R*n+ , if x ≥ x’ then f(x) ≥ f(x’).

Increasing one input cannot lead to a decrease in output.

2.5.5. *f.3 Upper semi-continuity. *f is upper semi-continuous on *R*n+ .

The graph of the production function may have discontinuities, but at each point of discontinuity the function will be continuous from the right. The prop- erty of upper semi-continuity is a direct result of the fact that the output and input correspondences are closed. In fact, it follows directly from the input sets being closed.

2.5.6. *f.3.S Continuity. *f is continuous on *R*n+.

We often make the assumption that f is continuous so that we can use calculus for analysis. We sometimes additionally assume the f is continuously differen- tiable.

2.5.7. *f.4 Attainability. *If f(x) *> *0, f(*λ*x) → +∞ as *λ*→ +∞.

This axiom states that there is always a way to exceed any specified output rate by increasing inputs enough in a proportional fashion.

2.5.8. *f.5 Quasi-concavity. *f is quasi-concave on *R*n+.

If a function is quasi-concave then

*f*(*x*) ≥ *f*(*x*0) ⇒ *f*(*λx *+ (1− *λ*) *x*0) ≥ *f*(*x*0) (13)
If V(y) is convex then f(x) is quasi-concave because V(y) is an upper contour

set of f. This also follows from quasiconcavity of P(x). Consider for example the traditional three stage production function in figure 11. It is not concave, but it is quasi-concave.

PRODUCTION FUNCTIONS 11

FIGURE 11. Quasi-concave Production Function

**x
**

**y
**

If the function f is quasi-concave the upper contour or isoquants are convex. This is useful in problems of cost minimization as can be seen in figure 12.

FIGURE 12. Convex Lower Boundary of Input Requirement Set

**x1
**

**x2
**

2.5.9. *f.6 Concavity. *f is concave on *R*n+.
If a function is concave then

*f*(*λx *+ (1 − *λ*)*x*′) ≥ *λ f*(*x*) + (1 − *λ*)*f*(*x*′) (14)
Concavity of f follows from P.9 (V.9) or the overall convexity of the output and

input correspondences. This means the level sets are not only convex for a given level of output or input but that the overall correspondence is convex. Contrast the traditional three stage production function with a Cobb-Douglas one. Concavity is implied by the function lying above the chord as can be seen in figure 13 or below the tangent line as in figure 14.

2.6. **Equivalence of Properties of P(x) and f(x). **The properties (f.1 - f.6) on f(x) can
be related to specific properties on P(x) and vice versa. Specifically the following
proposition holds.

12 PRODUCTION FUNCTIONS

FIGURE 13. Concavity Implies that a Chord Lies below the Function

**x
**

**y
**

FIGURE 14. A Concave Function Lies Below the Tangent Line

**x
**

**y
**

**Proposition 4. ***The output correspondence P: R*n+ → 2R+ *satisfies P.1 - P.6 iff the pro-
duction function f : R*n+ → *R*+ *satisfies f.1 - f.4. Furthermore P.8 *⇔ *f.5 and P.9*⇔ *f.6.
*

Proofs of some of the equivalencies between properties of P(x) and f(x).

2.6.1. *P.1 *⇔ *f.1. *P.1a states that 0 ∈ P(x) ∀ x ∈ *R*n+ ; y 6∈ P(0), y *> *0. Let x ∈ *R*n+ so
that 0 ∈ P(x). Then by Proposition 3 f(x) ≥ 0 . Now if y *> *0 then y 6∈ P(0) by P.1b.
So y *> *f(0). But f(x) ≥ 0 by Proposition 3. Thus let y→ 0 to obtain f(0) = 0.

Now assume that f.1 holds. By Proposition 3 it is obvious that 0 ∈ P(x), ∀ x ∈
*R*n+. Now compute P(0) = [y∈R+: f(0) ≥ y ]. This is the empty set unless y = 0. So
if y *> *0, then y 6∈ P(0).

2.6.2. *P.2*⇔ *f.2. *P.2 states that ∀ x ∈*R*n+, P(x)⊆ P(*θ*x), *θ *≥ 1. Consider the definition
of f(x) and f(*θ*x) given by

*f*(*x*) = max [*y *∈ *R*+ : *y *∈ *P *(*x*)] *x *∈ *R*n+
*f*(*θx*) = max [*y *∈ *R*+ : *y *∈ *P *(*θx*)] *x *∈ *R*n+

PRODUCTION FUNCTIONS 13

Nowbecause P(x)⊆ P(*θ*x) for *θ *≥ 1 it is clear that themaximumover the second
set must be larger than the maximum over the first set.

To show the other way remember that if y ∈ P(x) then f(x) ≥ y. Now assume
that f(*θ*x) ≥ f(x) ≥ y. This implies that y ∈ P(*θ*x) which implies that P(x) ⊆ P(*θ*x).

2.6.3. *Definition of f *→ *P.3. *Remember that P.3 states ∀ x ∈ <n+ , y ∈ P(x) and 0 ≤
*λ *≤ 1 ⇒ *λ*y ∈ P(x). So consider an input vector x and an output level y such that
f(x) ≥ y. Then consider a value of *λ *such that 0 ≤ *λ *≤ 1. Given the restrction on *λ*,
*y *≥ *λy*. But by Proposition 3 which follows from the definition of the producion
function in equation 2 and Proposition 2, *λy *∈ *P *(*x*).

2.6.4. *F.3 *→ *P.4. *Recall that P.4 is that P(x) is bounded for all x ∈ Rn+. Let *x *∈ <n+ .
The set

*M *(*x*) = {*u *∈ <n+ : *u *≤ *x*} (15)
is compact as it is closed and bounded. F.3 says that f(x) is upper-semicontinuous,

thus the maximum

*f*(*u*∗) = *max*{*f*(*u*) : *u *∈ *M *(*x*)} ≥ *f*(*x*) (16)
*u*∗ ∈ *M *(*x*) exists. The closed interval [0,f(x)] is a subset of the closed interval

[0,f(*u*∗], i.e, [0*, f*(*x*)] ⊆ [0*, f*(*u*∗)]. Therefore P(x) = [0,f(x)] is bounded.

2.6.5. *P.4*→ *F.3. *Recall that f(x) is upper semi-continuous at*x*0 iff lim supn → ∞ *f*(*x*n) ≤ *f*(*x*0) for
all sequences *x*n → *x*0.

Consider a sequence {*x*n} → *x*0. Let *y*n ≡ *f*(*x*n). Now suppose that lim supn → ∞ *y*n = *ȳ *≥ *f*(*x*0).
Then {*y*n} → *y*0 ≥ *f*(*x*0) because the maximum value of the sequence {*y*n} is
greater than *f*(*x*0). Because *P *: <n+ → 2<+ is a closed correspondence, *y*0 ∈
[0*, f*(*x*0)] and *y*0 ≤ *f*(*x*0), a contradiction. Thus lim supn → ∞ *f*(*x*n) ≤ *f*(*x*0).

2.6.6. *Other equivalencies. *One can show that the following equivalences also hold.
**a: **P.6⇒ f.4
**b: **P.7 follows from the definition of P(x) in terms of f in equation 4.

2.7. **Marginal and Average Measures of Production.
**

2.7.1. *Marginal product (MP). *The firm is often interested in the effect of additional
inputs on the level of output. For example, the field supervisor of an irrigated crop
maywant to know howmuch crop yield will rise with an additional application of
water during a particular period or a district managermaywant to knowwhatwill
happen to total sales if she adds another salesperson and rearranges the assigned
areas. For small changes in input levels this output response is measured by the
marginal product of the input in question (abbreviated MP or MPP for marginal
physical product). In discrete terms the marginal product of the ith input is given
as

*MP*i =
∆*y
*∆*x*i

=
*y*2 − *y*1

*x*2i − *x*1i
(17)

where y2 and x2 are the level of output and input after the change in the input level and y1 and x1 are the levels before the change in input use. For small changes

14 PRODUCTION FUNCTIONS

in xi the marginal physical product is given by the partial derivative of f(x) with respect to xi, i.e.,

*MP*i =
*∂f *(*x*)
*∂x*i

=
*∂y
*

*∂x*i
(18)

This is the incremental change in f(x) as xi is changed holding all other inputs levels fixed. Values of the discrete marginal product for the production function in equation 19 are contained in table 2.7.1.

*y *= 10*x *+ 20*x*2 − 0*.*60*x*3 (19)
For example the marginal product in going from 4 units of input to 5 units is

given by

*MP*i =
∆*y
*∆*x*i

=
475 − 321*.*6

5 − 4 = 153*.*40

The production function in equation 19 is shown in figure 15.

FIGURE 15. Cubic Production Function

**10 20 30
x
**

**1000
**

**2000
**

**3000
**

**4000
y
**

**fHxL=10x+20x2-0.6x3**

PRODUCTION FUNCTIONS 15

TABLE 1. Tabular representation of y = 10 x + 20 x2 − 0*.*60 x3

Input (x) Output (y)

Average Product y/x

Discrete Marginal Product

∆y ∆x

Marginal Product

0.00 0.00 10.00 1.00 29.40 29.40 29.40 48.20 2.00 95.20 47.60 65.80 82.80 3.00 193.80 64.60 98.60 113.80 4.00 321.60 80.40 127.80 141.20 5.00 475.00 95.00 153.40 165.00 6.00 650.40 108.40 175.40 185.20 7.00 844.20 120.60 193.80 201.80 8.00 1052.80 131.60 208.60 214.80 9.00 1272.60 141.40 219.80 224.20 10.0 1500.00 150.00 227.4 230.0 11.0 1731.40 157.40 231.4 232.2 12.0 1963.20 163.60 231.8 230.8 13.0 2191.80 168.60 228.6 225.8 14.0 2413.60 172.40 221.8 217.2 15.0 2625.00 175.00 211.4 205.0 16.0 2822.40 176.40 197.4 189.2 17.0 3002.20 176.60 179.8 169.8 18.0 3160.80 175.60 158.6 146.8 19.0 3294.60 173.40 133.8 120.2 20.0 3400.00 170.00 105.4 90.0

We can compute themarginal product of the production function given in equa- tion 19 using the derivative as follows

dy dx

= 10 + 40 x − 1*.*80 x2 (20)

At x = 4 this gives 141.2 while at x = 5 this gives 165.0. The marginal product function for the production function in equation 19 is shown in figure 16.

Notice that it rises at first and then falls as the production function’s rate of increase falls. Although we typically do not show the production function and marginal product in the same diagram (because of differences in scale of the ver- tical axis), figure 17 shows both measures in the same picture to help visualize the relationships between the production function and marginal product.

2.7.2. *Average product (AP). *The marginal product measures productivity of the
ith input at a given point on the production function. An average measure of the
relationship between outputs and inputs is given by the average product which is
just the level of output divided by the level of one of the inputs. Specifically the
average product of the ith input is

16 PRODUCTION FUNCTIONS

FIGURE 16. Marginal Product

**5 10 15 20 25 30
x
**

**-200
**

**-100
**

**100
**

**200
**

**MP
**

**f’HxL=40-0.36x
**

FIGURE 17. Production and Marginal Product

**10 20 30
x
**

**1000
**

**2000
**

**3000
**

**4000
y
**

*AP*i =
*f *(*x*)
*x*i

=
*y
*

*x*i
(21)

For the production function in equation 19 the average product at x=5 is 475/5 = 95. Figure 18 shows the average and marginal products for the production func- tion in equation 19. Notice that the marginal product curve is above the average product curve when the average product curve is rising. The two curves intersect where the average product reaches its maximum.

We can show thatMP =AP at themaximumpoint ofAP by taking the derivative of APi with respect to xi as follows.

PRODUCTION FUNCTIONS 17

FIGURE 18. Average and Marginal Product

**5 10 15 20 25 30
x
**

**-200
**

**-100
**

**100
**

**200
**

**MP AP
**

**f’HxL=40-0.36x
fHxLx= 10 x + 20 x2 - 0.6€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€€
**

**x
**

*∂
*(

f(x) xi

)

*∂ x*i
=
*x*i

∂ f ∂ xi

− *f*(*x*)
*x*2i

=
1
*x*i

(
*∂ f
*

*∂ x*i
− *f*(*x*)

*x*i

)

=
1
*x*i

(*MP*i − *AP*i )

(22)

If we set the last expression in equation 22 equal to zero we obtain

*MP*i = *AP*i (23)
We can represent MP and AP on a production function graph as slopes. The

slope of a ray from the origin to a point on f(x) measures average product at that point. The slope of a tangent to f(x) at a point measures the marginal product at that point. This is demonstrated in figure 19.

2.7.3. *Elasticity of ouput. *The elasticity of output for a production function is given
by

i =
*∂ f
*

*∂ x*i

*x*i
*y
*

(24)

3. ECONOMIES OF SCALE

3.1. **Definitions. **Consider the production function given by

*y *= *f *(*x*1*, x*2*, ... x*n) = *f *(*x*) (25)
where y is output and x is the vector of inputs x1...xn. The rate at which the

amount of output, y, increases as all inputs are increased proportionately is called
the degree of returns to scale for the production function f(x). The function *f *is
said to exhibit **nonincreasing returns to scale **if for all x ∈ *R*n+ , *λ *≥ 1, and *µ *≤ 1,

*f *(*λx*) ≤ *λf*(*x*) and *µf*(*x*) ≤ *f *(*µx*) (26)

18 PRODUCTION FUNCTIONS

FIGURE 19. Average and Marginal Product as Slopes

**x
**

**y
**

**Slope = Average Product
**

**Slope = Marginal Product
fHxL
**

Thus the function increases less than proportionately as all inputs x are in-
creased in the same proportion, and it decreases less than proportionately as all
x decrease in the same proportion. When inputs all increase by the same pro-
portion we say that they increase along a ray. In a similar fashion, we say that *f
*exhibits **nondecreasing returns to scale **if for all x ∈ *R*n+ , *λ *≥ 1, and 0 *< µ *≤ 1.

*f *(*λx*) ≥ *λf*(*x*) and *µf*(*x*) ≥ *f *(*µx*) (27)

The function *f *exhibits **constant returns to scale **if for all x ∈ *R*n+ and *θ > *0.

*f *(*θ x*) = *θf*(*x*) (28)

This global definition of returns to scale is often supplemented by a local one that yields a specific numerical magnitude. This measure of returns to scale will be different depending on the levels of inputs and outputs at the point where it is measured. The elasticity of scale (Ferguson 1971) is implicitly defined by

* *=
*∂ *ln *f *(*λx*)
*∂ *ln *λ
*

| *λ *= 1 (29)

This simply explains how output changes as inputs are changed in fixed pro-
portions (along a ray through the origin). Intuitively, this measures how changes
in inputs are scaled into output changes. For one input, the **elasticity of scale is
**

* *=
*∂f*(*x*)
*∂x
*

*x
*

*f*(*x*)
(30)

We can show that the expressions in equations 29 and 30 are the same as follows.

PRODUCTION FUNCTIONS 19

*∂ *ln *f*(*λx*)
*∂ *ln *λ
*

|λ=1 =
*∂f*(*λx*)
*∂λ
*

*λ
*

*f*(*λx*)
|λ=1

=
(
*∂f
*

*∂λx
x
*

)
*λ
*

*f*(*λx*)
|λ=1

=
*∂f
*

*∂x
*

1
*λ
*

· *x λ
f*(*λx*)

|λ=1

=
*∂f
*

*∂x
*·

*x
*

*f*(*λx*)
|λ=1

=
*∂f
*

*∂x
*

*x
*

*f*(*x*)

(31)

So the elasticity of scale is simply the elasticity of the marginal product of x, i.e.

* *=
*∂f *(*x*)
*∂x
*

*x
*

*f*(*x*)
=

*∂y
*

*∂x
*

*x
*

*y
*=

*∂ln y
*

*∂ln x
*(32)

In the case of multiple inputs, the elasticity of scale can also be represented as

* *=
∑

n i=1

*∂f
*

*∂x*i

*x*i
*y
*

= n∑

i=1

∂f ∂xi y xi

= n∑

i=1

*MP*i
*AP*i

(33)

This can be shown as followswhere x is now an *n *element vector:

*∂ *ln *f*(*λx*)
*∂ *ln *λ
*

|λ=1 =
*∂f*(*λx*)
*∂λ
*

*λ
*

*f*(*λx*)

= n∑

i=1

(
*∂f
*

*∂λx*i
*x*i

)
*λ
*

*f*(*λx*)
|λ=1

= n∑

i=1

*∂f
*

*∂x*i

1
*λ
*

· *x*i
*λ
*

*f*(*λx*)
|λ=1

= n∑

i=1

*∂f
*

*∂x*i

*x*i
*f*(*λx*)

|λ=1

= n∑

i=1

*∂f*(*x*)
*∂x*i

*x*i
*f*(*x*)

= n∑

i=1

*MP*i
*x*i
*y
*

(34)

20 PRODUCTION FUNCTIONS

Thus elasticity of scale is the sum of the output elasticities for each input. If *
*is less than one, then the technology is said to exhibit decreasing returns to scale
and isoquants spread out as output rises; if it is equal to one, then the technol-
ogy exhibits constant returns to scale and isoquants are evenly spaced; and if * *is
greater than one, the technology exhibits increasing returns to scale and the iso-
quants bunch as output expands. The returns to scale from increasing all of the
inputs is thus the average marginal increase in output from all inputs, where each
input is weighted by the relative size of that input compared to output. With de-
creasing returns to scale, the last expression in equation 33 implies thatMPi *<*APi
for all i.

3.2. **Implications of Various Types of Returns to Scale. **If a technology exhibits
constant returns to scale then the firm can expand operations proportionately. If
the firm can produce 5 units of output with a profit per unit of $20, then by dou-
bling the inputs and producing 10 units the firm will have a profit of $40. Thus
the firm can always make more profits by expanding. If the firm has increasing
returns to scale, then by doubling inputs it will have more than double the output.
Thus if it makes $20 with 5 units it will make more than $40 with 10 units etc. This
assumes in all cases that the firm is increasing inputs in a proportional manner.
If the firm can reduce the cost of an increased output by increasing inputs in a
manner that is not proportional to the original inputs, then its increased economic
returns may be larger than that implied by its scale coefficient.

3.3. **Multiproduct Returns to Scale. **Most firms do not produce a single product,
but rather, a number of related products. For example it is common for farms to
produce two or more crops, such as corn and soybeans, barley and alfalfa hay,
wheat and dry beans, etc. A flour miller may produce several types of flour and a
retailer such as Walmart carries a large number of products. A firm that produces
several different products is called a multiproduct firm. Consider the production
possibility set of the multi-product firm

T = {(x*, *y) : x ∈ Rn+*, *y ∈ Rm :+ x can produce y}

where y and x are vectors of outputs and inputs, respectively. We define the mul- tiproduct elasticity of scale by

m = sup{*r *: *there exists a δ > *1 *such that *(*λx, λ y*r) ∈ *T for *1 ≤ *λ *≤ *δ*} (35)

For our purposes we can regard the sup as a maximum. The constant of pro-
portion is greater than or equal to 1. This gives the maximumproportional growth
rate of outputs along a ray, as all inputs are expanded proportionally [2]. The idea
is that we expand inputs by some proportion and see how much outputs can pro-
portionately expand and still be in the production set. If r = 1, then we have con-
stant returns to scale. If r *< *1 then we have decreasing returns, and if r *> *1, we
have increasing returns to scale.

PRODUCTION FUNCTIONS 21

4. RATE OF TECHNICAL SUBSTITUTION

The rate of technical substitution (RTS) measures the extent to which one input substitutes for another input, holding all other inputs constant. The rate of techni- cal substitution is also called the marginal technical rate of substitution or just the marginal rate of substitution.

4.1. **Definition of RTS. **Consider a production function given by

*y *= *f*(*x*1*, x*2 *. . . , x*n) (36)
If the implicit function theorem holds then

*φ*(*y, x*1*, x *2*, . . . , x*n) = *y *− *f*(*x*1*, x*2 *. . . , x*n) = 0 (37)
is continuously differentiable and the Jacobian matrix has rank 1. i.e.,

*∂φ
*

*∂x*j
=

*∂f
*

*∂x*j
6= 0 (38)

Given that the implicit function theorem holds, we can solve equation 38 for xk as a function of y and the other x’s i.e.

*x*∗k = *ψ*k(*x*1*, x*2*, . . . , x*k−1*, x*k+1*, . . . , y *) (39)
Thus it will be true that

*φ*(*y, x*1*, x*2*, . . . , x*k−1*, x*∗k*, x*k+1*, . . . , x*n ) ≡ 0 (40)
or that

*y *≡ *f*(*x*1*, x*2*, . . . , x*k−1*, x*∗k*, x*k+1*, . . . , x*n ) (41)
Differentiating the identity in equation 41 with respect to xj will give

0 =
*∂f
*

*∂x*j
+

*∂f
*

*∂x*k

*∂x*k
*∂x*j

(42)

or

*∂x*k
*∂x*j

= − ∂f

∂xj ∂f ∂xk

= *RTS *= *MRS *(43)

Or we can obtain this directly as

*∂φ*(*y, x*1*, x*2*, . . . , x*k−1*, ψ*k*, x*k+1*, . . . , x*n)
*∂x*k

*∂ψ*k
*∂x*j

=
−*∂φ*(*y, x*1*, x*2*, . . . , x*k−1*, ψ*k*, x*k+1*, . . . , x*n)

*∂x*j

⇒ *∂φ
∂x*k

*∂x*k
*∂x*j

= − *∂φ
∂x*j

⇒ *∂x*k
*∂x*j

= − ∂f

∂xj ∂f ∂xk

= *MRS
*

(44) The above expression represents the slope of the projection of the boundary of

the input requirement (or level/contour) set V(y) into xk, xj space. With two in- puts, this is, of course, just the slope of the boundary. Its slope is negative. It is

22 PRODUCTION FUNCTIONS

convex because V(y) is convex. Because it is convex, there will be a diminishing rate of technical substitution. Figure 20 shows the rate of technical substitution.

FIGURE 20. Rate of Technical Substitution

**x1
**

**x2
**

**RTS =
¶x2
€€€€€€€€€
¶x1
**

**Dx2
**

**Dx1
**

4.2. **Example Computation of RTS.
**

4.2.1. *Cobb-Douglas. *Consider the following Cobb-Douglas production function

*y *= 5*x
*1
3
1 *x
*

1 4 2 (45)

The partial derivative of y with respect to x1 is

*∂y
*

*∂x*1
=

5
3
*x
*

−2 3

1 *x
*1
4
2 (46)

The partial derivative of y with respect to x2 is

*∂y
*

*∂x*2
=

5
4
*x
*

1
3
1 *x
*

−3 4

2 (47)

The rate of technical substitution is

*∂x*2
*∂x*1

= − ∂f

∂x1 ∂f ∂x2

= −
5
3 *x
*

−2 3

1 *x
*1
4
2

5
4
*x
*

1
3
1 *x
*

−3 4

2

= − 5 3 5 4

*x*2
*x*1

= − 4
3
*x*2
*x*1

(48)

PRODUCTION FUNCTIONS 23

4.2.2. *CES. *Consider the following CES production function

*y *= 5
[
0*.*6*x*−21 + 0*.*2*x
*

−2 2

]−1 2 (49)

The partial derivative of y with respect to x1 is

*∂y
*

*∂x*1
=

−5
2
[
0*.*6*x*−21 + 0*.*2*x
*

−2 2

]−3
2 (−6*/*5)*x*−31 (50)

The partial derivative of y with respect to x2 is

*∂y
*

*∂x*2
=

−5
2
[
0*.*6*x*−21 + 0*.*2*x
*

−2 2

]−3
2 (−2*/*5)*x*−32 (51)

The rate of technical substitution is

*∂x*2
*∂x*1

= − ∂f∂x1

∂f ∂x2

= − −5 2

[
0*.*6*x*−21 + 0*.*2*x
*

−2 2

]−3
2 (−6*/*5)*x*−31

−5 2

[
0*.*6*x*−21 + 0*.*2*x
*

−2 2

]−3
2 (−2*/*5)*x*−32

= − 2 5 6 5

*x*−31
*x*−32

= − 3 *x
*3
2

*x*31

(52)

5. ELASTICITY OF SUBSTITUTION

The elasticity of substitution is a unitlessmeasure of how various inputs substi- tute for each other. For example, how does capital substitute for labor, how does low skilled labor substitute for high skilled labor, how do pesticides substitute for tillage, how does ethanol substitute for gasoline. The elasticity of substituion at- tempts to measure the curvature of the lower boundary of the input requirement set. The most commonly used measure of the elasticity of substitution based on the slope of an isoquant is due toHicks [13, p. 117, 244-245], [17, p.330]. He defines the elasticity of substitution between x2 and x1 as follows.

*σ *=
d(x2*/*x1)
d(f1*/*f2)

· (f1*/*f2)
(x2*/*x1)

(53)

This is the percentage change in the input ratio induced by a one percent change in the RTS.

5.1. **Geometric Intuition. **We can better understand this definition by an appeal
to geometry. Consider figure 21. First consider the factor ratio of x2 to x1. Along
the ray labeled *a*, the ratio x2x1 is given by the tangent of the angle *θ*. Along the ray
labeled *b*, the ratio x2x1 is given by the tangent of the angle *φ*. For example at point

d, x 0 2

x01
= tan *φ *along the ray *b*.

Now consider the ratio of the slopes of the input requirement boundary at two
different points. Figures 22 and 23 show these slopes. The slope of the curve at
point *c *is equal to minus the tangent of the angle *γ *in figure 22. The slope of the
curve at point *d *is equal to minus the tangent of the angle *δ *in figure 23.

24 PRODUCTION FUNCTIONS

FIGURE 21. Elasticity of Substitution (Factor Ratios)

Consider the right triangle formed by drawing a vertical line from the inter-
section of the two tangent lines and the x1 axis. The base is the x1 axis and the
hypotenuse is the tangent line between the other two sides. This is shown in fig-
ure 24. Angle *α*measures the third angle in the smaller triangle. Angle *α *+ angle
*γ *equal 90 degrees. In figure 25, the angle between the vertical line and the tan-
gent at point d is represented by *α *+ *β*. Angle *β *+ angle *α*+ angle *δ *also equals 90
degrees, so that *α *+ *γ *= *α *+*β *+ *δ *or *γ *= *β *+ *δ*. This then means that *β*, the angle
between the two tangent lines is equal to *γ *- *δ*.

FIGURE 22. Elasticity of Substitution (Angle of Tangent at Point c)

PRODUCTION FUNCTIONS 25

FIGURE 23. Elasticity of Substitution (Angle of Tangent at Point d)

FIGURE 24. Elasticity of Substitution (Angle Tangent at Point d)

If we combine the information in figures 22, 23, 24 and 25 into figure 26, we can measure the elasticity of substitution. Remember that it is given by

*σ *=
d(x2*/*x1)
d(f1*/*f2)

· (f1*/*f2)
(x2*/*x1)

The change in x2x1 is given by the angle *ξ *which is *θ *- *φ*. The marginal rate of
technical substitution is given by the slope of the boundary of V(y). At point *d *the
slope is given by *δ *while at *c *it is given by *γ*. The change in this slope is *β*. The
rate of technical substitution is computed as