Technical Aspects of Production
One Output, One (Variable) Input Production Function
Reading: Beattie and Taylor, Chapter 2.1 (pp. 9-16)
Notation
Output y
Input xi
What the firm sells or uses as input in another production process
(e.g. crop, livestock)
What the firm buys and/or uses to produce y
Variable inputs:
fertilizer, pesticide, labor
Fixed inputs:
land, human capital
Functional Representation
y  f ( x1 , x2 ,..., xn )
For various analyses we often simplify problem by ignoring fixed inputs, assume all variable
inputs are used at optimal levels, and focus on one variable input x1, to write:
y  f ( x1 | x2 ,..., xn )  f ( x)
Examples without plateaus
Quadratic:
y  a  bx  cx 2
Polynomial:
y  a  bx  cx 2  dx3  ...
Square Root
Cobb-Douglas
y  a  b x  cx  a  bx 0.5  cx  ...
y  axb or y  A  axb
Examples with plateaus
(Negative) Exponential
Exponential-Power
y  y max 1  exp  a  bx  Mitscherlich-Baule
y  ymax 1  exp  a  bx c  generalized Mitscherlich-Baule
von Liebig (LRP or QRP)

y  min y
Hyperbolic
y
Spillman
y  y max 1  ab x
max

, a  bx  cx 2

ax
1  ax / b
Questions to think about/discuss
Where get functional forms? Biological or Ecological theory? Let data speak?
Does an Intercept make sense or not?
Should there be a maximum output/yield?
1
Typical Properties of Production functions
1. Non-Negative Output: y ≥ 0 for all x
[Level]
Strict case: Positive output y > 0 for all x
y(x0) ≥ 0 for all x0 ≥ 0 (strict case: y(x0) > 0 for all x0 > 0)
dy
[Slope]
 f '( x)  0 , at least in region where operate
dx
dy
Strict case: positive slope
 f '( x)  0
dx
Non-differentiable: y(x1) ≥ y(x0) for all x1 ≥ x0 (strict case: y(x1) > y(x0) for all x1 > x0)
Exception: phyotoxic effects, nutrient toxicity, etc.
2. Non-Negative Slope
3. Concavity: as use more input, eventually extra output generated by extra input begins to
decrease: Law of Diminishing Marginal Product
[Curvature]
2
2
d y
d y
 f ''( x)  0 , strict case
 f ''( x)  0
2
dx
dx 2
Non-differentiable:
f ( x0  (1   ) x1 )   f ( x0 )  (1   ) f ( x1 ) for all x0 and x1 and 0 ≤  ≤ 1 (Concave)
f ( x0  (1   ) x1 )   f ( x0 )  (1   ) f ( x1 ) for all x0 and x1 and 0 ≤  ≤ 1 (Strict Case)
Concave, differentiable
Strictly concave
Concave, non-differentiable
linear section: only concave, not strictly concave
Quasi-concavity: less strict than concavity, allows convex regions, so sigmoidal functions
acceptable. Allows biologically relevant cases that will still be economically meaningful
For all f(x0) ≥ = t and f(x1) ≥ = t, f ( x0  (1   ) x1 )  t for 0 ≤  ≤ 1 (quasi-concave)
For all f(x0) ≥ = t and f(x1) ≥ = t, f ( x0  (1   ) x1 )  t for 0 <  < 1 (strict case)
Quasi-concave
Convex section okay
not quasi-concave
convex section make sit non-quasi-concave
2
Technical Terms
Total Product: Another term for output y  f (x) (yield or meat)
Marginal Product: “Output from last unit of input”
Derivative formulation:
MP = dy / dx  f ' ( x)
Discrete formulation:
MP = y / x  f ( x1 )  f ( x 0 ) / x1  x 0
Average Product: AP = y / x  f ( x) / x “Average output per unit of input”
MP at x0 = Slope of the TP function at x = x0
AP at x0 = Slope of the line from origin to TP function at x = x0 (Prove graphically)



AP: Input use efficiency: nutrient use efficiency, water use efficiency, etc.
Numerical Example:
Polynomial:
y  ax  bx 2  cx 3 = 2.5x + 3.5x2 – 0.1x3
MP = dy / dx  f ' ( x) = 2.5 + 7x – 0.3x2
AP = y  f ( x) / x = 2.5 + 3.5x – 0.1x2
Graphical Presentation of “Classical Production Function”
800
700
600
f(x)
500
400
300
200
100
0
0
5
10
15
20
25
30
20
25
30
x
60
40
MP and AP
20
0
-20
MP
0
5
10
15
-40
-60
-80
x
3
AP
Evaluate MP and AP at x = 10
y = 2.5x + 3.5x2 – 0.1x3 = 25 + 350 – 100 = 275
AP = 2.5 + 3.5x – 0.1x2 = 2.5 + 35 – 10 = 27.5
At x = 10, on average 27.5 units of output are generated per unit of input.
Input or Factor Elasticity
Economists like to use elasticities: unitless measures of responsiveness
% change B B / B dB A



% change A A / A dA B
 If A increases by z %, what % does B increase?
 Numerical measure of how responsive B is to changes in A
 Unitless: so does not matter what units used to measure A and B, get same result
If input use increases z%, what % does output increase?
% change y y / y dy x
 input 


% change x x / x dx y
dy x MP
 input 

dx y AP
Implications:
 input > 1
 MP > AP
Output increases more than proportionally with x

MP = AP
 input = 1
Output increases proportionally with x

MP < AP
 input < 1
Output increases less than proportionally with x
dy d ( xy / x) d ( xAP)


dx
dx
dx
Implications:
 When AP rising
 When AP flat
 When AP falling
MP =
Apply Product Rule: MP = AP  x
dAP
dx
MP > AP
MP = AP
MP < AP
Relationship between AP and MP will matter for economics and input use efficiency
(Nutrient use efficiency, water use efficiency, etc.)
We will find that it is only economical to use inputs at levels only where AP is flat or decreasing
(where MP < AP). AP maximum will occur at competitive optimum of zero economic profit, so
if farmers earning above normal rates of return (positive economic profit), then MP < AP, AP
falling, output price increases will decrease input use efficiency, etc.
More on this later when we cover economics of production.
4
One Output, Two (Variable) Inputs
(Beattie and Taylor Section 2.2: pp. 16 ff)
Notation: y  f ( x1 , x2 | x3 ,..., xn )  f ( x1 , x2 ) , again assuming all other inputs fixed or at optimal
levels
Three types of relationships:
a) Factor-Output Relationship (One Output, One Input Production Function))
b) Factor-Factor Relationship (Isoquants)
c) Scale Relationship (Proportional increase of all inputs versus output)
See text Figure 2.3
Figures from 317 Text:
Graphical Intuition for:
One Output, One Input Production Function and “classical” shape
Isoquants as “Level Sets,” i.e. contour lines
Scale Relationship
5
Factor-Output Relationships
Average Product: One for each input
APi  y / xi  f ( x1 , x2 ) / xi
Marginal Product: One for each input
y f ( x1 , x2 )
MPi 

 fi
xi
xi
Note: In general the AP and MP functions are depend on both inputs
Example: Generalized Cobb-Douglas
y  ax1b1 x2b 2
AP1  y / x1  ax1b1 x2b 2 / x1  ax1(b11) x2b 2
y
MP1 
 f1  b1 ax1(b11) x 2b 2  b1 AP1
x1
Text works through some other examples: linear, quadratic, transcendental, CES
Good problem set or exam questions
Factor-Factor Relationships
Isoquant: curve or function showing all combinations of inputs that can produce the same level
of output (level set).
Show 317 figure deriving isoquants graphically from 3-diminensional production surface
Intuition: An isoquant is x2 as a function of x1 and output y: Solve y  f ( x1 , x2 ) for x2
For some functions, can solve for equation for isoquant directly, the
Generalized Cobb-Douglas: y  ax1b1 x2b 2
y
x 2b 2  b1  ya 1 x1b1
ax1
x 
b2 1/ b2
2
1/ b2
2
x y

 ya 1 x1b1

1/ b2
a 1 / b 2 x1b1 / b 2
Numerical example: Isoquant for y = 10 and b1 = b2 = ½ and a = 1
x2  y 1 / b 2 a 1 / b 2 x1b1 / b 2
i.e. isoquant is rectangular hyperbola
x2  100 x11  100 / x1
6
30
25
x2
20
15
10
5
0
0
5
10
20
15
25
30
x1
Negative exponential y  y max 1  exp( a  b1 x1  b2 x2  gives linear isoquant
x2   ln(1  y / ymax ) / b2  (b1 / b2 ) x2 (do math on own)
Can I find one for which you can only get implicit equation for x2 as function of x1?
Marginal Rate of Technical Substitution: MRTS:
Rate at which one input must be substituted for the other input to keep output unchanged.
MRTS12 = Rate at which x1 can be substituted for x2 and leave output unchanged:
x 2 dx 2
MRTS12 =
= Slope of the isoquant < 0

x1 dx1
MRTS12 is negative because isoquant has negative slope
Intuitively: if x2 decreases (a negative change), then x1 must increase to keep output unchanged,
thus MRTS12 is negative.
Often texts/articles multiply MRTS by minus 1 so it is a positive number
Use a graph to show intuition of MRTS as
More rigorous mathematical formulation:
Totally differentiate production function:
x 2
x1
y  f ( x1 , x2 )
7
dy  f1 ( x1 , x2 )dx1  f 2 ( x1 , x2 )dx2  f1dx1  f 2 dx2
Along an isoquant, by definition output does not change, so dy = 0
f1dx1  f 2 dx2  dy  0
dx 2
dx
f
, the slope of isoquant: 2   1
dx1
dx1
f2
dx
f
MP1
Summary:
MRTS12 = 2   1  
dx1
f2
MP2
Notice derivative wrt x1 in numerator, and derivative wrt x2 in denominator
Rearrange to solve for
Generalized Cobb-Douglas:
y  ax1b1 x2b 2
f1  ab1 x1(b11) x2b 2
f 2  ab2 x1b1 x2(b 21)
f1
ab x (b11) x b 2
b
  1 1b1 (b 221)   1 x1(b11)b1 x2b 2(b 21)
f2
b2
ab2 x1 x2
b x
MRTS12 =  1 2
Implies constant MRTS along proportional expansion path
b2 x1
MRTS12 = 
Numerical example: b1 = b2 = ½ and a = 1
f
b x
x
MRTS12 =  1   1 2   2
f2
b2 x1
x1
Mathematics of Curvature
Production Functions should have positive Marginal Products for each input: fi > 0
Restrictions are placed on the curvature of production functions to obtain economically sensible
results for “rational” decision-makers (second order conditions for profit max).
Often assume strict concavity, which in terms of derivatives for two input production function
requires f11  0 , f 22  0 , and f11 f 22  f122  0 .
Most of the production functions presented the first day of class satisfy these properties with
simple restrictions on the parameters: e.g. Cobb-Douglas y  ax b , a > 0, b  1.
I will use this and expect everyone to know how to check the second order conditions.
Graphically, concavity implies a production surface that is a “bubble” or “dome.”
In some situations, less restrictive quasi-concavity is assumed, which in terms of derivatives for
two input production function requires  f 22 f11  2 f1 f 2 f12  f12 f 22  0 .
I will not explain where this comes from, nor will I use this.
8
Graphical intuition is sufficient in this class: quasi-concavity implies “bell-shaped” hill that
flares out at the bottom.
Note:
 Concave functions are also quasi-concave
 Both strict concavity and strict quasi-concavity imply isoquants convex to the origin.
Discussion of Curvature/Convexity of Isoquants:
Text shows derivation of the curvature (second derivative) of an isoquant using the total
differential. You cannot simply take derivative of –MP1/MP2 because must assume y is
held constant. See discussion and footnote, pp. 24-26.
Assuming the standard strictly quasi-concave production function (or more restrictive: strictly
concave production function) implies that isoquants are convex to the origin (the way
they are usually drawn).
Show with graph why the other cases (positive sloping and/or concave isoquants) do not make
economic sense as long as have free disposal of inputs.
Three Cases that matter economically
1) Imperfect Substitutability (isoquants convex to origin)
2) Perfect Substitutability (linear isoquants)
3) No Substitutability (right angle isoquants)
Case 1: Factors can be substituted, but the MRTS is diminishing.
Implies that the “return” to using an input as a substitute decreases, or
as you want to substitute away from one of the inputs, you have to use more and more of
the substituting input.
Case 2: Factors can be perfectly substituted, the MRTS remains constant.
Once put into economic decision making (profit max) situation, leads to corner solutions
where only one of the inputs is used.
Case 3: (Linear) Leontief production function or fixed proportion production function:
y  f ( x1 , x2 )  min{ ax1 , bx2 ) : No input substitutability.
Inputs are used in constant proportions and output follows an expansion path that is a ray
from origin with slope a/b.
Work through example with a = b = 1.
Note: Calculus cannot be used because the function is non-differentiable.
9
Minor Topics Concerning Isoquants:
1) Elasticity of Factor Substitution ()
2) Isoclines and Ridgelines
SKIP/very fast
Elasticity of Factor Substitution ()
SKIP
Measures the substitutability between inputs along the isoquant, which is inversely related to the
curvature of isoquant (remember MRTS measures its slope).
Definition: percent change in factor ratio (x1/x2) over the percent change in RTS:
( x2 / x1 ) d ( x2 / x1 )
%( x2 / x1 )
( x2 / x1 )
( x2 / x1 )
d ( x2 / x1 ) MRTS




MRTS
dMRTS
%MRTS
dMRTS ( x2 / x1 )
MRTS
MRTS
How does the factor ratio change as the isoquant’s slope changes?
Show graphically difference between large and small :
Small change in slope associated with a large change in factor ratio gives a very flat isoquant
(little curvature) with lots of substitutability between inputs: large 
Large change in slope associated with a small change in factor ratio gives a very curved isoquant
with little substitutability between inputs: small 
Linear isoquants: little curvature, lots input substitutability, large 
Right-angle isoquants: lots of curvature, little input substitutability, small 
Text derives the general formula (page 30): quite complex function of first and second partial
derivatives and input levels. We will not do so here.
Note: in general  depends on level of inputs (and thus on output) so that should expect variable
elasticity of factor substitution as you change the input mix and/or output.
However, there is a special class of production functions that exhibit constant elasticity of
substitution (CES). Most common example is the Cobb-Douglas, for which  = 1.
What good is this elasticity ?
How does changing the MRTS affect the input ratio?
We will show that the profit maximizer will equate the MRTS to the input price ratio
–(r1/r2), so  then is the elasticity of the input ratio x1/x2 to the input price ratio r1/r2, or the
responsiveness of relative input use to relative factor prices.
Flat/Linear Isoquants: Very responsive to input price changes: large 
Sharp/Right Angle Isoquants: Unresponsive to input price changes: small 
10
Note: in general  depends on level of inputs (and thus on output) so that should expect variable
elasticity of factor substitution as you change the input mix and/or output.
However, there is a special class of production functions that exhibit Constant Elasticity of
Substitution (CES). Most common example is the Cobb-Douglas, for which  = 1.
Isoclines and Ridge Lines
Isocline: locus of points (here a line) for which isoquants have the same MRTS (i.e. slope).
Traces out input combinations for which isoquant slopes are all the same.
An “expansion path” for points having the same MRTS.
Mathematically: hold MRTS fixed and solve for x2 as function of MRTS and x1
f (x , x )
MRTS12 =  1 1 2
Solve for MRTS and x1
f 2 ( x1 , x2 )
Show graphically what this looks like
Generalized Cobb-Douglas:
y  ax1b1 x2b 2
Last class it was shown
Solve for x2: the isocline is
b1 x 2
b2 x1
b
x2   MRTSx1 2
b1
MRTS12 = 
A ray from the origin
Ridge Line: isocline for MRTS of zero or  (i.e. undefined 1/0).
Defines the economically meaningful portion of the input space (x1,x2)
Basic Intuition:
 Ridge Lines “cut off” the portion of isoquants that bend backward or bend upward
 Needed for production functions that eventually have areas with negative marginal
products (e.g. polynomial).
 “Well behaved” production functions have the axes as their ridge lines.
See the text Figures 2.9 and 2.12.
11
Factor Interdependence: Technical Substitutes and Complements
Difference between input substitutability and technical substitution/complementarity between
inputs.
Input Substitutability:
 Concerns the substitution of inputs when output is held fixed along an isoquant
 Measured by MRTS (and/or Elasticity of Factor Substitution 
 Inputs must be substitutable along a “well-behaved” isoquant
Technical Substitution/Complementarity:
 Concerns the interdependence of input use
 Does not hold output constant
 Measured by changes in marginal products
Technically Competitive/Complementary indicates how increasing the level of one input affects
the marginal product of the other input
1) Technically Competitive: as x1 increases the marginal product of x2 decreases.
2) Technically Complementary: as x1 increases the marginal product of x2 increases.
3) Technically Independent: as x1 increases the marginal product of x2 does not change.
Competitive = Substitutes
Mathematical Condition: Determined by the sign of the cross partial derivative f12
Technically Competitive
f12 < 0
Technically Complementary
f12 > 0
Technically Independent
f12 = 0
Intuition:
What does using more of x1 do to the productivity of x2?
Assume price x1 falls, so use more of it, how will use of x2 respond?



If increasing x1 decreases productivity of x2, then it is “rational” to use less of x2 because it is
less productive but costs the same. Thus as x1 increases, the use of x2 decreases and the
inputs “substitute” for one another.
If increasing x1 increases productivity of x2,, then it is “rational” to use more of x2 because it
is more productive but costs the same. Thus as x1 increases, the use of x2 increases as well
and the inputs “complement” one another.
If increasing x1 does not change productivity of x2, then it is rational to ignore the level of x1
when choosing x2. Thus as x1 increases, the use of x2 does not change because the inputs are
“independent” of one another.
Examples:
Technically Competitive (Substitutes): fertilizer and manure in crop production, tillage and
herbicides for weed control, corn-sorghum-wheat-barley in a feed ration
Technically Complements: corn and soybeans in a feed ration, fertilizer and irrigation water in
crop production
12
Mathematical Condition: Determined by the sign of the cross partial derivative f12
Technically Competitive
f12 < 0
Technically Complementary
f12 > 0
Technically Independent
f12 = 0
Examples:
Cobb-Douglas: y  ax1b1 x2b 2
f1  b1ax1(b11) x2b 2
f12  b1b2 ax1(b11) x2(b 21) > 0
Thus the Cobb-Douglas does not allow inputs to be technically competitive (substitutes), but
rather imposes technical complementarity.
Mitscherlich: y  y max 1  exp( a  b1 x1  b2 x2 
f1  y max b1 exp( a  b1 x1  b2 x2 )
f12   y max b1b2 exp( a  b1 x1  b2 x2 ) < 0
Thus this specification of the Mitscherlich does not allow inputs to be technically
complementary (technical complements), but rather imposes technical substitution.
When estimating production relationships, you should be aware of any properties such as
technical substitution/complementarity that your function specification imposes.
 It may cause econometric problems because you are imposing relationships that the data
cannot support biologically/physically.
 In terms of policy analysis, your assumed production function specification may qualitatively
determine your results before you even do any analysis.
Examples of this sort of specification error:
 Using Cobb-Douglas to estimate feedlot cattle meat production as a function of the pounds of
corn, wheat, and the other inputs and then make ration recommendations as the prices of corn
and wheat change relative to each other.
 Imposes complementarity between corn and wheat when they are substitutes biologically
 Function implies that as price of corn increases, use of wheat will increase as well.
 Using the above Mitscherlich to estimate yield of irrigated corn as a function of fertilizer
(lbs/ac) and water (in/ac) and then make water policy recommendations meant to reduce
nitrogen non-point source pollution.
 Imposes substitutability between N and water when they are complements biologically
 Function implies that as price of N increases, use of water will decrease as well.
This why “Flexible Functional Forms” are popular among economists to estimate production
relationships when there are several inputs, e.g. Translog, Normalized Quadratic.
Quadratic:
y  a1 x1  a 2 x 2  b1 x12  b2 x 22  b3 x1 x 2 , where a1 > 0 a2 > 0, b1 < 0, b2 < 0
13
f1  a1  2b1 x1  b3 x2
f12  b3
Thus sign of b3 determines whether x1 and x2 are substitutes, complements or independent and its
sign can be tested for statistical significance.
Note: Quadratic is a flexible functional form: i.e. second order approximation
Partial Elasticity of Production with respect to a factor:
(Partial Factor Elasticity or Partial Input Elasticity)
How does output respond to a proportional increase in one input, holding all others fixed?
i.e. If input xi increases Z%, what % does output increase, holding all other inputs fixed?
% change y
y / y
dy xi MPi



% change xi xi / xi dxi y
APi
Note: i depends on the level of all other inputs, not just on xi.
i 
For one input-one output production functions, we defined the Input or Factor Elasticity:
x 
% change y y / y dy x MP



% change x x / x dx y AP
i is simply the multivariate version of x.
Returns to Scale
How does output respond to a proportional increase in all inputs?
We will show that this is the “Total Elasticity of Production” and is simply the sum of all the
Partial Elasticities of Production.
What is a “proportional increase”?
Increasing each input by a constant proportion : i.e. x1  x10 , where x10 is the original level.
Basic idea: increase all inputs by same percentage (up or down).
What does a “proportional increase” look like?
Start from the current input allocation point ( x10 , x20 ) and move along the ray defined by that
point and the origin. These rays are what the text calls “Factor Beams”
The equation of a ray from the origin is x2  kx1 . Increasing or decreasing both x1 and x2
proportionally (i.e. multiply both by the same factor ) will move you along this ray.

x2  kx1
x2  kx1
14
Constant Returns to Scale (CRTS)
A production function exhibits CRTS if f (x1 , x2 )  f ( x1 , x2 )    0
i.e. scaling all inputs up 1% increases output by 1% as well
Decreasing Returns to Scale (DRTS)
A production function exhibits DRTS if f (x1 , x2 )  f ( x1 , x2 )   > 1
i.e. scaling all inputs up 1% increases output by less than 1%
Increasing Returns to Scale (IRTS)
A production function exhibits IRTS if f (x1 , x2 )  f ( x1 , x2 )   > 1
i.e. scaling all inputs up 1% increases output by more than 1%
Graphical Intuition
1) The y = f(x) version of this and how to extend it to y  f ( x1 , x2 )
2) Implications in terms of distance between isoquants (cf. Fig 2.25)
Basically: is the production function convex (IRTS), linear (CRTS) or concave (DRTS)?
Think of y  f ( x1 , x2 ) as the per acre or per animal unit production function and then the
“scale” as how many acres or how many animal units you have. As you increase the
acres/animals, what happens to production function per acre or per animal?
Returns to Scale and Partial Input Elasticity: Elasticity of Scale
Think of this  is a “scale input” then calculate the elasticity of output wrt to this scale input, or
% y
|  1
the “Elasticity of Scale”  
%
Note:  = 1 is needed because you want to know the % change in output if all inputs were
increased by the same proportion  from the current level at which  = 1.
%y
dy ( ) 
 1 
 1
%
d y ( )
df (x1 , x2 )
df (x1 , x2 )

1

 1 
 1
d
f (x1 , x2 )
d
f ( x1 , x2 )
First take the derivative of y( )  f (x1 , x2 ) wrt , then substitute in  = 1.

Because the Elasticity of Scale requires that all inputs be scaled up or down by the same
proportion , the Elasticity of Scale  also serves as a measure of the Returns to Scale
 < 1  DRTS
 = 1  CRTS
 > 1  IRTS
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Example: Cobb-Douglas
y( )  f (x1 , x2 )  a(x1 ) b1 (x2 ) b 2  (b1b 2) ax1b1 x2b 2
df (x1 , x 2 )
d(b1b 2) ax1b1 x 2b 2
 1 
 1
d
d
(b1  b2 )(b1b 21) ax1b1 x2b 2  1  (b1  b2 )ax1b1 x2b 2
df (x1 , x 2 )
(b1  b2 )ax1b1 x 2b 2
1

 b1  b2
 1
d
f ( x1 , x 2 )
ax1b1 x 2b 2
Summary:
b1  b2 < 1  DRTS
b1  b2 = 1  CRTS

b1  b2 > 1  IRTS
Alternative way of expressing Elasticity of Scale shows that it is the sum of the Partial Input
Elasticities:
% y
, where % x k is the percent change in any input xk and must be the same for

% x k
all inputs (i.e. all inputs must change by the same percent.
Note that for infinitesimally small changes (i.e. derivatives):
dx
dy
%y 
and %xk  k
y
xk
dx
dx
dx
dx
%y
dy xk

Then  
, where k  1  2 ...  n to ensure equal % changes
%xk dxk y
xk
x1
x2
xn
For the two input case:
Totally differentiate y  f ( x1 , x2 )
dx
dy
 f1 1  f 2
dx2
dx2
Use the equal percent change condition in its derivative form:
dx1 dx 2
dx1 x1



x1
x2
dx2 x2
Substitute this into the first equation:
dx
x
dy
 f1 1  f 2  f1 1  f 2
dx2
dx2
x2
Use this with the definition of  when k = 2
x x
x
x
x
MP1 MP2
dy x2

 f1 1 2  f 2 2  f1 1  f 2 2 

 1   2
dx2 y
x2 y
y
y
y
AP1 AP2
dy  f1dx1  f 2 dx2

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Thus the Scale Elasticity is the sum of the Partial Input Elasticities, or the sum of the Partial
n
n
i 1
i 1
Input Elasticities is the Total Input Elasticity:     i  
n
MPi
fx
 i i
APi i 1 y
Returns to Scale and optimal firm size:
We will (may) show that Returns to Scale determines the optimal size of the firm in a perfectly
competitive market
 CRTS generates zero economic profit
 DRTS generates negative economic profit
 IRTS generates positive economic profit.
Firms expand and contract to maintain CRTS in accordance to the production technology they
utilize. In a competitive economy, the CRTS firm size is the equilibrium size that firms gravitate
towards, but this equilibrium size is always changing as a result of technology changes.
Examples: Farm size, feedlot and confinement buildings, meat packing industry, seed
companies, etc. are all responding to technology changes:
Some production functions impose structure on Returns to Scale. We showed for the CobbDouglas that Returns to Scale is the sum of the exponents and so is constant, no matter what size
the firm is.
This is why often the restriction is imposed that the exponents sum to one, otherwise the model
will imply that the industry will keep having larger and larger firms, or smaller and smaller
firms.
Careful specification/understanding of the Returns to Scale implied by a production function is a
concern for long term policy analysis and Industrial Organization.
Returns to Scale and Homogeneity
A function is homogeneous of degree k if: f (x1 , x2 )  k f ( x1 , x2 ) , where  > 0 and k is a
constant.
Applying this to production functions, the degree of homogeneity is the same as the Scale
n
Elasticity: k      i .
i 1
Functions that are homogeneous of degree 1 are called linearly homogeneous and for production
functions, linear homogeneity implies Constant Returns to Scale
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Method for determining whether a function f ( x1 , x2 ) is homogeneous and if so, its degree of
homogeneity:
1) Multiply each input of f ( x1 , x2 ) by a fixed proportion 
2) Use algebra to try to separate the function into two terms:  k and f ( x1 , x2 )
3) If this can be done, the function is homogeneous of the degree k
Examples:
Quadratic:
f ( x1 , x 2 )  a1 x1  a 2 x 2  b1 x12  b2 x 22  b3 x1 x 2
f (x1 , x 2 )  a1x1  a 2 x 2  b1 (x1 ) 2  b2 (x 2 ) 2  b3 x1x 2
  (a1 x1  a 2 x 2 )  2 (b1 x12  b2 x 22  b3 x1 x 2 )
Not homogenous
Cobb-Douglas:
f ( x1 , x2 )  ax1b1 x2b 2
f (x1 , x2 )  a(x1 ) b1 (x2 ) b 2  (b1b 2) ax1b1 x2b 2
Homogeneous of degree b1  b2
Note: 1st derivatives a function homogeneous of degree k are homogeneous of degree (k – 1)
Implication: For linearly homogeneous production functions (i.e. CRTS), the marginal products
MPi = fi are homogeneous of degree 0.
Homogeneity of degree 0 implies: f (x1 , x2 )  0 f ( x1 , x2 )  f ( x1 , x2 ) , so that increasing all
inputs proportionally does not change the function.
In terms of marginal products, this implies that proportionally increasing all inputs does not
change the MP of the inputs because the MP are proportional functions of input ratios. Because
the MP do not change with proportional increase, the MRTS (-f1/f2) will not change along a ray,
so isoclines are rays from the origin.
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