TV
Theories and Concepts of
STATIC PRODUCTION ECONOMICS
/L / ,,(
A Term Paper for
AEC 5U0
22 March 1957
by
Peter B. Hildebrand
Nt
)A#'69
ai8
Most of the material
contained herein was
taken as class notes
in the lectures giv
en by
Dr. ulenn L. Johnson
TABLE OF CONTENTS
Page
I. Scope and Nature of Static Production
Economics. . . . . . 1
A. Locating in the scientific world . 1
B. Procedures in economics . 3
Problem solving . . 3
Role of classification. .
C. Assumptions of static economics. 5
Those which make the system
stated . . . 5
Those which eliminate random
variables . . . 6
Those concerning motivations 7
E. Efficiency. . . . . 7
F. Value systems. . . . . 7
II. Static Production Economics . . 8
A. The concept of a function . . 8
Derivatives . . . 8
Symbols. . . . . 8
B. Single variable functions  no
fixed inputs . . . . 8
C. Single variable functions with
fixed inputs . . . 9
Law of diminishing returns . 9
Definition of terms. . 10
Stages of production . . 10
Optimum level of production 11
Application of maximizing
principle . . 12
Long run vs short run equi
librium . . 13
D. Two variable input functions. . 14
Geometric presentation 
algebraic notation . 14
Stages of production . . 15
E. Applications of economizing
principle . . . 17
Most profitable combination
of factors . . 17
Most profitable amount to use 19
TABLE OF CONTENTS  Continued
Page
F. Extremes of perfect substitutability ,
and complementarity . . . .
Perfect substitutes . .
Perfect complements . .
G. MVP vs VMP and IFC vs Px . .
H. N equations for the N unknowns
Two inputs . .
Multiple inputs . . .
I. Definition of a fixed asset . .
One factor . . .
Two factors. . . . .
J. The problem of a time unit. .
K. Indivisible inputs . . . .
L. Aggregating inputs . . . .
M. Cost concepts and supply functions .
Cost concepts . . .
Envelope curves . . .
Supply functions . . .
Industry supply responses .
N. Demand for factors of production.
The case for perfect substitutes.
The case for perfect complements.
` Enterorise combinations. . .
Assumpntions. . . .
Optimum combinations. ..
Characteristics of curves in
the nroductproduct
dimension. . .
T) Il i 4 li
1 LL s oLUos .l1et * *
Sources of supplementarity and
complementarity. .
impact or product complementarity
on supply. .. . .
F. vertical and horizontal enterprise
combinations . . .
Q. Five fundamental equations of static
economics . . . .
~. Euler's theorem . . . .
S. Inconsistency between perfect com
petition and ultimate long run
concepts. . . . . .
T. Results from static theory what
it can do . . . .
U. what static theory cannot do . .
V. Supplement . . . .
Stages of production in factor
factor dfimnsion ..
Algebraic proof concerning
envelope curves. . .
Specialization and aggregate
supply response. . .
7,
77 , *wt '
78
79
I. Scope and nature of static agricultural production economics.
A. Locating static agricultural production economics in the
scientific world.
As a science it is the study of what is rather than
what ought to be. Under the assumption that the producing
unit is striving for maximum profit, static production eco
nomics studies alternatives of combining labor, land, and
capital, taking into consideration the technical sciences
(technology), and uses the theories and concepts of economics.
Labor T Force, fraud, SP
e strategic position o s
c c y Ethics
h ic
n Political Science o h Philisophical
meas L o Political Economics 1 o value theory ends
1 o 1
o go Religio
g y g
Capital y Economics y
Under the broad classification of economics, one can
make two major divisions  production and consumption. In
agricultural economies there are five applied areas which
taken collectively show the areas of emphasis under the two
major divisions. The five applied areas are: farm manage
*ment, marketing, land economics, policy, and price analysis.
Farm management and land economics place major emphasis on
production economics, and marketing places major emphasis
on consumption economics. Policy and price analysis are
about equally divided
It should be noted that production economics is not
synonymous with farm management. Not only is farm management
also concerned with consumption economics but in addition,
all five applied phases of agricultural economics place some
emphasis on production economics.
Economics
Applied Areas Production Consumption
Farm Management
Marketing
Land Economics /_///
Policy
Price Analysis
Both micro economics and macro economics are divided
into static theory and dynamic theory. In the realm of
static theory there is a rather firm and well defined re
lationship between microstatics and macrostatics. The
relationship between micro and macro 'dynamics is less well
defined as is that between microstatics and microdynamics
and between macrostatics and macrodynamics.
Depending upon the source, there are a varied number
of categories of economic theory used in classifying static
to dynamic theories. That which seems to be the easiest to
understand and yet concise and complete is presented by
Johnson.* The categories are static, trend, risktrend,
and dynamic.
The static theories assume perfect knowledge and constant,
exact relationships between variables. The trend theories
assume perfect knowledge and involve no probability distri
* Journal of Farm Economics, vol. 32, p. 11)0.
2
butions, but handle problems concerning changes in relation
ships between variables which occur over time. Risktrend
theories handle all problems dealt with in the first two
categories but in addition, handle problems involving unchang
ing probability distributions. Perfect knowledge concerning
the stable probability distributions is assumed.
The dynamic theories handle problems of changing proba
bility distributions and in addition do not assume perfect
knowledge. Subjective probability distributions which change
with the learning process are permitted.
B. Procedures in economics.
1. Problem solving.
When one realizes that there is a difference exist
ing between 'what is' and 'what ought to be', then he real
izes a problem exists. Whitney* lists this as the first
step in problem solving  awareness of the problem.
Though both the concept of 'what is' and the concept of
'what ought to be' may be subjective, the beliefs or
concepts of 'what is' should be objective.
The second step in problem solving is to define
the difficulty or problem, and the third step is to
arrive at a tentative solution to the problem. The
fourth step  mental elaboration of the problem  may
make it necessary to cover steps two through four several
times. Theory plays an important role in steps two
through four. Theory helps to define and elaborate
the facts concerning the problem and, indeed, may define
some solutions. Many ramifications of the solution may
The Elements of Research, p. 1.
3
be defined by theory.
Believing in the solution or having faith in it
is necessary before experimental or factual verifica
tion (the sixth step) is undertaken. If the solution
is verified then the last step  application  can
be achieved. Theory is necessary in experimentation
in that it will allow one to test secondary variables
when it is not possible to test primary variables.
Theory is also needed to understand and interpret the
results of the experiment.
2. The role of classification.
Since the knowledge that an individual is capable
of attaining is finite in scope and since the universe
is composed of an infinite varietyphenomena, it is
necessary to classify these phenomena in order for an
individual to begin to comprehend the world in which
he lives. Classifications are constructed on the as
sumption that there is similarity among "things". _, U
The classification of variables. V'Variables
are classified into three groups: those to be studied,
those regarded as fixed, and random variables. A
problem would be set up in the following manner. A
dependent variable would be studied as affected by
certain independent variables under certain fixed con
ditions but subject to random variables or unexplained
residuals. It is desired taat the random variables
or unexplained residual do not bias the results. The
impact of the random variables on the dependent variables
will be governed by the accuracy of the fixed condi
4
tions. These fixed factors should duplicate actual
conditions. In stating the functions of the variables
it is necessary to have the same number of relationships
or equations as variables. That is, for N unknowns it
is necessary to have N equations. In addition to hav
ing N equations, these equations must be independent
of eacll other (one cannot be a multiple of another),
they must be consistent, and the number of equations
must be exactly N.
C. Assumptions of static economics.*
The term static economic theory has a variety of meanings.
Hence, when one tries to outline the assumptions'underlying
static economics a rather specific definition of static eco
nomics is required. The theory usually considered when the
word static is used is a theory of a given number of exact
relationships among the same given number of economic varia
bles. An exact relationship, as used herein, is one which has
a standard deviation of zero. In a theory of exact static
relationships, the magnitudes of certain variables can and
are permitted to change as the theory is used to explain
changes which occur when the value of one or of a set of va
riables is changed.
What, then, are the assumptions which can be made to
secure static equilibrium theory in its usual form? The
assumptions fall into three categories: (1) those which make
the system static with respect to: (a) production functions,
(b) consumption functions, (c) institutions; (2) those which
eliminate random elements, and (3) those concerning motiva
tions.
1. Those which make the system static.
3
 Class Outline
a) Assumptions which fix the production functions of the
economy: ,
The state of the arts is assumed constant, i.e.
the total production of any given set of productive
factors remains fixed.
b) Assumptions which fix the utility functions of the
economy:
1) Tastes, habits, customs (i.e., everything affect
ing utility functions) are assumed fixed.
S2) The ownership pattern for resources and, hence,
the equilibrium distribution of private real
incomes is assumed fixed.
3) Population is assumed constant.
h) Utility functions are independent among people
i.e. jealousy and "copying" of tastes and value
systems are absent.
c) Assumptions which specify the institutional setup
of the economy:
1) Government is assumed fixed.
2) It is assumed that goods and services are sold in
a market where both producing and consuming indi
viduals and groups can make their choices free of
force or coercion but with consumers subject, how
ever, to limitations imposed by their real incomes.
3) Nonfirm groups are assumed fixed.
2. Those which eliminate random elements.
a) It is assumed that persons and groups making up the
economy possess perfect knowledge. This assumption
implies perfect foresight.
b) It is assumed that the persons and groups making up
the economy are rational.
6
3) Those concerning motivations.
a) Consumer units (or households) are assumed to be
motivated to maximize the satisfactions derivable
from teir real incomes.
b) Producer units (or firms) are assumed to be motivated
to maximize money profits.
(Note that the above assumptions do not limit the subject
to either perfect competition or to continuous functions)
D. Efficiency.
Efficiency is defined as using as little of those things
which have alternative uses as possible to get the desired prod
uct, regardless of how that product is measured.
Ratio = useful output ( ends
useful input means/
There is no reliable difference between economic and noneco
nomic efficiency. Socalled "technical efficiency' which al
ways has an inputoutput ratio of one,is actually only a par
tial measure of total efficiency, :That is, if total efficiency
is a measure of useful output/useful input, all mea.ngIfu meas
ures of efficiency therefore, will involve some sort of value
judgement. Efficiency, in itself, should not be regarded as
an ultimate goal, but should be instrumental  a means  in
reaching a higher goal.
E. Value systems.
A scientist should adhere to the following values: truah
about the world, objectivity, and factual honesty. A scientist
is usually guided by an ii directed source of value. There
are three sources of value: traditional, in prdirected, and
otner directed. The ir directed values are those acquired
from the learning process such as religious values. They differ
from traditional values in that they are more cultural and prob
7
ably more rigid. Traditional values are acquired directly from
the environment while inrerdirected values may come from other
sources. Other directed values are dynamic and can be changed.
This is the fastest growing source of value today  more and
more people are in this class.
II. Static Production Economics
A. The concept of a function. Y = f(X1X2 Xy".. n)
1. Derivatives.
Y = slope of the function between two points.
6x
A 0 = d, i.e. as 4x 0 = dx
slope of a tangent
Y = a+bx, =d
b
y. partial derivative (only one variable changes)
aX, 6 Y d[f(XjX21)
total derivative (all independent variables change)
dy d [f(Ixj &.
x1 l dx J Xaxl x2 Xl
2. Symbols
Y = useful product
X = useful inputs, (i = l*...n)
Y = f(Xl....Xd Xd1....Xn)
variable fixed (quantity and quality)
B. Single variable functions  no fixed inputs.
Y = f(X1) (no real basis)
Y (?)
O /
/
/
/
/
/
/
/
/
0 KX,
C. Single variable functions with fixed inputs.
Y = f(X1 X2 a)
Y
TPP(y
o Xl Jl
0 XI X,
1. Law of Diminishing Returns.
The 'Law of Diminishing Returns' has no theoretical
background, but is based on empirical evidence. It
should be considered as a basic law of nature. The
law can be stated in terms of productivity, but it is
necessary to differentiate between total, average, and
marginal productivity. The law states; As variable
inputs are added, the application of one or more vari
able inputs to specified fixed inputs in producing a
product:
a) .total product first increases at an increas
ing rate, tien increases at a decreasing rate
up to a maximum where it will then tend to
decrease.
b) average product will be positive and increase
to a maximum, then decrease; but will remain
positive so long as total product is positive.
c) marginal product will be positive and in
crease to a maximum and then will decrease.
It will be positive so long as total product
is increasing.
From a) above, it can be seen that the law should
possibly be stated as eventually diminishing productiv
ity.
2. Definition of terms.
PFM
Marginal physical product (MPP) is the addition
to total product from the use of the last increment
FFT
of variable input. Total physical product T1PP) is
tne total amount of product and equals the sum of the
marginal physical products. Average physical product
(APP) is equal to TPP divided by the amount of the va
riable input.
TPP= MPP
TPP = 2MPP App
Xl X1
3. Stages of production.
Y
I I
TPP
I I
I I
APP
0 MPP Xl 2
Stage I is that portion up to the point of maximum
APP. Stage II is that portion from maximum APP to the
point where MPP = 0. Stage III covers the portion of
negative MPP. Stage II is the only rational stage of
production.
' 10,
4. Optimum level
Assume:
SY
0
of production (in stage II)
Py / f (Y), Px / f (X)
TVP
I
I
N VP
MV I
MVP = VMP
TVP(y)
= AVPx y)
 VPX1(y)
if P / f(y)
 Px = MFCxl
(marginal
factor cost)
SX,
R represents
Ei (y) =
the most profitable use ofX1.
PX1 = MFCx
77 = profit
Py'Y = gross income = TVP
PxX1 total variable cost of X1
FC = fixed cost
= Py Pxl , FC
TPP Py
,PP Py
APP Py
TIC
  {1XIFC
0
MVP
Profit function  point of maximum profit
1T0l
7f= Py Y Pxl X
 FC
maximum profit where d7/ =
dx1
dxl
= Py dy
dx1
MPP
Px l o
PY MFx1(Y) = WPx1(y)
VPX(y) 
Px1 = 0
Under the assumption that Px1 "f(x1), the point of
maximum profit is where
MVPl1 (y)
= PX1 or MVPx(y)
Px1
5. Application of maximizing principle.
TPP
/FCF
1 ^ M~VPF(H)
L  ij__..._~~
u a1 leea con
sumed
A = weight to sell hogs.
B amount of feed fed.
In experiments where the application of the variable
does not start at zero, it is impossible to determine APP
and, therefore, where stage II begins. It is still
possible to determine MPP even in this case.
6. Long run vs. short run equilibrium.
Y
AVP
profit ...
(short // .~' /!
run) // /
L = long run S = short run
MVP = AVP = MFC MVP = MFC
1.
weight
of hogs
I
Py = f(y)
__* ft _ 1 ^ _ ^
In the short run, there may be a profit when op
erating where MVP = MFC. Over time, the MFC and/or
AVP will change bringing the firm into long run equilib
rium. At the point of long run equilibrium, there is
no profit and the firm has reached maximum efficiency.
Production will be at point of maximum AVP.
D. Two variable input functions.
1. geometric presentation  algebraic notation.
Isoproduct curves are the locus of all combinations
of Xi and 12 which will product a given amount of Y.
Y = fXX2 I X3....Xn)
0 X2
Y = f(Xi1 X2....Xn) is a sub function of
Y f(XlX2 X3....n). The level at wnich X2 is fixed
affects the Y value.
x0 Y r 1
0 L Y 0
X = b
X2= a
V
2 Al
2. Stages of production.
Sa
III
III
MPPxl <0
MPPx22 D
MPPXl>
MPP < 0
MpX 2
3. Influence of fixed factors on two variable input func
tions.
MPPx2
a) Fixed factors
0
21
15
**Xn)
1I
Diminishing
returns
III
_1__~~1
b) No fixed factors  Y f(X1X2)
K (constant
\ returns
N\.i to scale
i >00 IA=B=C=D
HD
diminishing
returns to
scale
' "" 1 0 0
2
c) Expanding both inputs simultaneously
S(1)
A\X
dy = d
I 
x(3)
SX2
SMPP(y)
x2(Y)
Y + S
" x2
dx2 (2)
y
axl
1 MPP and associated
xl and associated X2
Y + 
xl _x
F... h dividig zgua+t ^( 2) obh b5 dr .
When both X1 and X2 vary, it is necessary to know
how tiey change in relation to each other. That is,
6y
x",t
i
it becomes necessary in the above example to de
termine dx
E. Application of the economizing :rinciple to twovariable
input functions.
There are two questions of primary importance in stud
ying the optimum combination of two inputs.
1. The proper combination of X1 and XZ to use.
2. The proper amount of (X1 and X2) to use.
Proper refers to the most profitable use.
1. The most profitable combination of factors.
a) Isocost line.
An isocost line is the locus of all points
which equal a given cost.
X1
10 Pxl = 1
Px =
5
0 5 10 X2
If the total cost of the two factors = C,
then o10 = C = P X1X t Px2, and 1 = PX *
Pxl F
C is constant and equals the intersection with
xl
the X axis. then, is the slope of the
isocost line.
17
Isoproduct curve
Xli I
4 x
Ax?, MPPX2(y) = Xl 'PPx1(y)
1k= MPP (
XX2 MPFx x(y)
The slope of the tangent of an isoproduct curve
dx1 equals IPPx( )
c) The most profitable combination of X1 and X..
If AB is An isocost line and CD is an iso
product curve, point M represents that combination
of X1 and X2 for which a given cost will produce
the maximum product. The slopes of the two curves
at M are equated by the following equation:
PX2 _MP (y
_Px1 =Px(y)
Since AB is tangent to CD at M and Px X x MPPx
Pxl AX2 M
this represents the least cost combination (LCC)
for producing a product (y). /, X = A b
2. The most profitable amount of (X1 and X2 ) to use.
a) Line of least cost combination (AB).
X, B
0 XI
AB,the line of LCCis also called the 'scale
line', the 'line of optimum proportions', orthe
'expansion path'. Actually, it is the locus of
all the points where increasing amounts of Y are
produced at minimum cost given the ratio Px,
. ___ . x2
and the production surface where Y = f(X IX3 3'.** n).
b) The most profitable amount of (Xland X2) to use.
Since the line of LCC is never vertical or
horizontal (unless either X1 or X2 is free), it
involves the expansion of both inputs simultane
ously. Therefore, when viewed as a section of the
surface, the LCC line and horizontal axis involve
combinations of X1 and X2, which act as a single
variable input (XlX2).
Y
LCC
0 (XlX2 X3 Xn)
There are four alternative ways of combining
(XX2) (1) It can be thought of as "batches",
combined in'the LCC.
(2) It can be thought of as units of X1
and associated X2 in which case X1
would appear on the horizontal axis.
(3) It can be thought of as units of X2
and associated X1.
(U) (XlXl) can be combined in a common
denominator as dollars worth of (X1X2)
in LCC, or TDN, etc.
c) Determining the most profitable output when (X1X1)
is considered in "batches".
MFC = PXI
X1X
XX2
WPxlxe (y) = MPPxlx (y) *
d) Algebraic method.
7/ =  Y PXX1 PXX2 FC
constant X2
(1) 3~ Py d P =0
a X1 dxl
(at maximum profit)
constant X1
(2) b = Py dy Px O = 0 where d=
xb dx2 dx1
and dy = f(X1X2).
dx2
Since tnere is interaction between tr.ese two equa
tions, they have to be solved simultaneously. Tne
simultaneous solution gives the most profitable
point on the LCC curve.
From equation: (i):
(3) Py d = Py MPP 1(y) = Pxl
From equation (2):
(4) Py j = Py MPP x(y)) x2
dX2
Combining (3) and (4):
(5) Py o omxi (p fit) PMx O/MVP
7 t pyn fXy) 2im fi .X2
at the point of maximum pTofit.
f'(XlX2)
f) X 2mLy)
PX2
e) Application.
Phogs = Py 4 10/cwt
P p = P
corn x
MPPx = I0
MPP2 = 12
From equation (5)
.10 50 .10 15
1.5 = 3
T 2 4
MVP x > MVPx and P 1
x1 x'rX
In this case it will pay to feed more of X1
which will probably decrease the amount of Xt.
The MVP will decrease and MVP will increase,
eventually becoming equal at the point of maximum
profit.
F. The extremes of perfect substitutability and perfect comple
mentarity.
1. Perfect substitutes.
f (XX2)
X1 Y
\ X2=b
2a
0 a
X. 0
Y = f(XiX2IX3* .Xn)
0 a b c X2
a) Expansion path.
1) MPPX1 PXl
Px2 > Px2
Since Px1 < PxL,
the X1 axis.
2) IMPxl Pxl
< x7
Xl1
1\1 \ isocost
Sisoproduct
0 X,
the expansion path will be
x11
Sisoproduct
Sisocost
0 X2
PXl > Px2, expansion path on X2 axis.
3)
b) The
1)
MPPxl Pxl
P q Px2 , isocost
\\ isoproduct
0 X2
In this case Pxl = Px2 so the whole surface
is an "expansion surface". The combination
of X1 and X2 would be determined by noneco
nomic factors.
most profitable output. Y = f(XIX2jX3 Xn)
If MPPxi PX ', the most profitable outpu
iI2PPx2 Px2
will depend only on Xl.
MFCx1 (y)
0 X1
z) If MPPxxl Pxl the most profitable output
MPPx2 Px2
will depend on X2 alone.
3) If MPPx: there is an output at which
the amount of X1 and X2 used at the "point"
of maximum profit, will depend on noneconomic
factors. In this case Y will be fixed, but
the combination of X1 and X2 will be variable.
c) The most profitable output. Y = fXIX2)
Where tnere are no fixed inputs and MPP_ PX1 ,
otPx2 = Px2
not only is there no economic optimum combi
t
nation of X1 and X2 but, there is no optimum catput.
Even when MPPxl Pxl there is no economic limit
MP'? / Px2
to output. In the latter case, however, an optimum
combination of either X1 or X2 can be determined.
Y
(constant returns
/ to scale)
0
2. Perfect complements.
a) Y = f(XlX2)
 MVP = VMP
IMFor (X2
Xi or X2
no fixed factors
I 
I I _.
0 a X2
The production surface is composed of two sides
of a four sided pyramid with no peak.
b) When the function includes fixed factors, i.e.
S= f(XiX2 X3....Xn), the law of diminishing total
 x= a
 X= a
returns will hold. If additional amounts of input
do not interfere with production, the production
surface will be an L shaped ridge.
S i I I
I I I
i ui
I II I I
+ 0
I I
I I sO
  ioe (maximum)
1l}_jO
    ^ "0
'30
4 I
o I 1
0 a b c d e X
A section taken at X2 = e will have the normal
looking TPP curve. Sections at X2 = a,ib will reach
a maximum and then level off. Sections at XZ = c,d
will pass maximum TPP before leveling off.
YI
S X2=c
/
/ X2=a
X=d
c) Combination of factors.
wnen X1 and X2 are perfectly complemtary and
neither is fixed, there is only one possible choice
of combination of X1 and X2. This combination is
the ridge line formed by the corners on the contour.
The ridge line will slope depending on the rate of
combination. If zX2 combine with 1X1, the ridge
line will have a slope of one half. The slope of
the isocost curve will have no effect on the com
'bination.
Xl
1 \
i
\ isocost
ridge line
30 \L f
10
0 20 40 60 X
d) Optimum putput.
Since the combination is fixed, X1 and X2
will normally be thought of as a single input,
e.g. one pair of s:oes rather than one left and
one right shoe. The point of maximum profit will
be on the ridge line if X3.****Xn are fixed. If
there are no fixed factors, amount of product
Swill depend on noneconomic factors.
27
Y xPx.l(y) = ridge line
S IVPx1(y)
I
Xaxeky)
O X112 IX3. Xn
3) The apparent desire to assume perfect complementarity
and substitutability in the history of agricultural
research.
If Y = f(X1...*Xdj Xdtl""*Xn) and all pairs of
X's are either perfect substitutes or perfect comple
ments such that at least one portion of the complement
is fixed, there would be no need for economics. The
cheapest of substitutes would be used and the comple
ment would be used in correct proportion (along the
ridge line) to the point where one is exhausted. This
assumption is integrated in traditional budgeting, in
Morrison's, "Feeds and Feeding", and in linear program
ming.
U. MVP vs. VMP and MFC vs. Pxi.
All four deal with dollars as a function of input. MVP
and VMP deal with income as a function of input, and MFC
and PXi deal with cost as a function of input.
VMP (value of the marginal product) is exactly what it
says. It is the dollar value of the marginal product, or
Py *MPPx(y). MVP is the increase or decrease in gross
income creditable to the last unit of input.
VMP = Py xPPi(y) = Py &
E xi
MVP Y P + PY Y
VMP
When Py f(y) and Y is homogeneous (no quality change),
MVP VMP since Y P 0. However, when Py = f(y)
or Y is not homogeneous, then MVP #VIP and it is nec
essary to use MVP when determining T_ Thus: when
Sxi
Py= f(y) and f = Py Y Px Xi
(1) i x.Y P
when: Y = original output
Py = new price
Pxi f(xi)
MFC is the increase or decrease in total variable
cost resulting from X 1l more unit of Xi). When
Pxi f(xi) and Xi is homogeneous, MFC = Pxi. However,
when Pxi = f(xi) or Xi is not homogeneous, then
MFC = Xi Px+ Pxi 6 Xi = Xi b Pxi + Pxi
where Xi = original input
Pxi = new price
6 Pxi = change in price
Hence, when:
Py f f(y)
Pxi = f(ti)
= Y Py XiPxi
77= Py Y Px)
a xi a xi 5~Xi
0 XXi
H. N equations for the N unknowns.
1. Two inputs.
Profit (1/") and output (Y) are always classed as
unknowns.
(a) Y = f(XlX2 3...Xn)
(b) = Y Py X1'Px1 X2 Px2 FC
Equations (a) and (b) remain constant so long as there
are only two variable inputs. Under perfect competition
there are two additional unknowns, X1 and XZ, since all
prices are constant. The four equations under this as
sumption are
(a) and (b) from above, plus
(c) ]I77 P Py .L Y_ P xy MPPl(y) Px1
(d) Py MPPx2(y) Px VhMPXi(y Px
a Xz 
Maximum profit occurs where
= VMPxi PxI = 0 and/or VMPx = Px2
The 7I equation can be removed, as can the P L,
SXi
~ /1 terms, from the above equations, leaving 3 equa
tions in 3 unknowns, Y, X1, X2. Likewise the Y equa
tions can be removed leaving 2 equations in 2 unknowns.
VMPxl Pxl = 0
VMPX2 Px2 = 0
If X2 is fixed, one equation remains. Solving VMPxl Pxl = 0
gives point of maximum profit in terms of X1 That is,
at the point where Py MPPxly) PXl
or
rVMP = Pxl the optimum has been
reached.
When, in addition to equations (a) (d), (e) P =f(y)
is included, there are five equations in five unknowns.
(b) P= Y Py XPxl XPxe FC
(c) then becomes:
= Py * Y PX1 MVPx= Px(y) Px1
and
(d) b_ = MVPx(y) Px
SX 2
It can be seen from kc) above that
MVPXl(y) VMP + new value of original output
MVPx(y) = VMP + Y P
bXl
An example will serve to illustrate:
Let Y = 200# Py = O20
Y'= 210# Py'. 119
Px1 = $.25
where Y' and Py' are the results of adding the last
unit of X1.
VMPxl(y) = Y Py' 10 19S = o1.90
bvk Y = ( 1.00) ( 200 ) = 42.00
aX,
VPx1(y) = $1.90 2.00 = 100
=.2 MVPxI(y) PXl
Xl
= 1.10 .Z>
< v.3>
So ^7? = $.35
6Xl
The results of adding the last unit of input yield a
loss. The operation is being carried beyond the point
of highest profit. The fact that this operation is
not at maximum profit, however, is easy to see since
AVPx1(y) A Pxl*
when (f) Pxl = g(Xl) and kg) PxZ = htLX) are added
there are seven equations in seven unknowns. Equation
(c) becomes = MPxl(y) MFxl(y) and (d) becomes
zZ = MVPx (y) MFCxe .y)
c x2
<. Multiple inputs.
Y = f *X'***XgXg +l***Xn)
_= Py Y PxiXi FC
i = TR TVC FC
Py f)y)
P = gl(X1)
Px = gzkX?)
*
Pxg = gg g)
y =P MVPx2(y) MFCxl(y)
xxl
,12 MVPx2(y)  Cx2(y)
SMVIPxg((y) MFCxgey)
Sax
If there are g variable inputs and perfect compe
tition is not assumed or products and factors are not
homogeneous, there will be 2g + 3 unknowns and 2g + 3
equations.
At optimum output or maximum profit
MVPxl '"*g MFC**"*xg = 0 and
MVPx1(y) MVPx2() .... MVPx (y) = 1
MFCxl(y) MFCx( MFxg(y)
I. Definition of a fixed asset.
1. One factor.
An asset should be considered to be fixed if its
acquisition cost > MVP > salvage value. That is, it
is fixed if its value in use is such that it is not
worth purchasing or producing more of it, but too
valuable to sell. "It is fixed because it is not
worth varying."
The acquisition cost (Ca) is the MFC if it is to
be purchased, or the MC of producing more of it if it
is to be produced at home.
Ca = MFC or Ua = MC
Acquisition cost is the cost of adding one more unit
of the asset. Salvage value (Vs) is the amount which
would be realized if one unit of the asset were dis
posed of either on the market or within the business.
If the MVP of the asset is greater than the ac
quisition cost, the value of the asset is equal to the
Ca, and no more. Likewise, if MVP of the asset is less
than salvage value the value of the asset equals Vs
and no less, since this is the amount it would be
worth if disposed of. If, however, Ca MVP > Vs
of the asset, then the value of the asset is its MVP.
Ca
IPl(y)
0 a b c 1
In the above figure, three situations exist.
The quantity Ob of X1 is fixed since Ca > MVP> Vs.
At the quantity Oa, more of X1 should be purchased or
produced; at Oc, some should be disposed of. The
quantity Oa would become Oa' and Oc would become
Oc' on the following figure.
Vs
Ca
MVP~Y)
0 a a' c' c
X1
Any amount of X1 on hand between a' and c', in
the above figure, is fixed because it does not pay to
change the amount one way or another. If MVPx1(y)f(XlX),
then a variation in X, may change the MVPxl. If the
new MVPxI ) Ca or MVPxl Vs, then X1 is no longer
fixed. In addition to the above condition, the asset
may change from a fixed to a variable category if
either Ca decreases or Vs increases. An increase or
decrease in prices, however, may change a variable
asset into a fixed asset.
2. Two factors.
In the following figure, the broken lines rep
resent isoproduct curves. If X2 is held fixed at
(a) and X1 is varied from zero, the MVP x !P ll first
increase and then decrease. When the MVPxl is de
creasing it will first be greater than acquisition
cost, then the Ca > MVPx1 > Vs and finally, will be
less than salvage value. Let point M be where MVPXl = Ca
and N be where MVPxl = Vs. Line TMS then represents
the locus of points where MVPxl = Ca of X1. Likewise,
line PNR represents the locus of all points where
MVPX1 = Vs of X1. A similar argument can be used to
establish the loci of points where MVPx = Ca of X2
and MVPx2 = Vs of XZ.
1
/ i
/ ( I I I
( I I
PI \
I I
S\ R
T \ 
\ 'M S
"'~ ^
0 a X
(In the following figure the isoproduct lines are omit
ted for simplicity.)
vrx a MVypx2 = Vs
Vs > MVPX,
j /* >> AM
k >Vi MVPxl = Vs
Ca >MVP,, > V5
.MVP = Ca
M vPw > Ca
0 lX
In the center area of the above figure, both
inputs are fixed since MVP is between Ca and Vs for
both. In all other quadrants, at least one of the
factors is not fixed. There will be a tendency to
move toward the center area from all other areas.
X1
VII VII
IV
c VI
\ IXII
\ III
0 X2
For example, in area I, movement toward the center
would occur through purchase or production of more of
both X1 and X2. In quadrant IX, some of both would
be disposed of. In area VII, some of X1 would be sold
and some more X2 acquired. In area IV, X1 would be
held constant since it is fixed and more X, would be.
purchased. Similar arguments follow in all areas
except V.
Barring price changes, only in area I can move
ment be made to point of maximum profit, which is
the point indicated by the arrow (A) in the quadrant.
At this point MVP = Ca of both X1 and X2. If prices
are perfect Ca = MFC and consequently, at this point
MVP = MFC. (Point A below)
IV \ V
~.\  
I \ \ II
0 X
Factors in quadrant I can be snifted into area
V by a drop in price of the product (indicated by
broken lines). The drop in Py will lower MVP of the
factors. Commonly, the price of farm products drops
more rapidly than tie price of farm inputs. Since
the factors become fixed they:will continue to be used.
J. The problem of a time unit.
When considering a production function in an input
time dimension, modification of terminology is necessary.
The variable in uts (Xl*.. Xd) will be considered as batches
of inputs per unit of time. One btciof inputs can be
used for eight hours for the same cost as eight batches
for one hour, or four batc.Les for two .lours. For example,
if a tractor and an operator and gas, etc. is a variable
input, it costs tiie same to run one tractor for eight hours
as eight tractors for one hour or four tractors for two
hours. If the cost per tractor for one nour is a5, all
the above combinations will cost $40. Drawn in an input
time dimension, an isocost curve will be a rectangular
hyperbola since, in effect, we are merely changing the
dimensions of a rectangle of constant area. See the fol
lowing figure.
Batches A' b C D
of inputs
X1i... Xd)
7 r \ \
S4zO
I ,
Time hours)
Time (hours)
I 9 /0o
4'
3
r
I I .
I
rl
I..... I i
Curves A, B, C, D are isocoso curves, the value of each
curve being given to the right.
Consider now, the time scale in lours per day. If,
at the end of 8 nours, the o erator of the tractor receives
overtime pay, a new rectangular hyperbola is introduced
at this point. Less of t.ie variable input batches can be
used each our at the same cost as before the overtime.
If t e in .ut batches are divisible this will result in a
smoot. curve. Curve B' in tre preceding figure is calcu
lated wien overtime costs an additional $1 or 46 total
cost per hour per input,
If there are no fixed factors or if the variable inputs
are working in a plant of unlimited size, the isoproduct
curve will also be a rectangular nyperbola. Let the trac
tors be em; loyed on a farm so large tnau ..one of Inem _et
in tne way of another. If Y = product) is acres covered
by any one operation in a day, we will get the same product
with one tractor for 8 hours or 8 tractors for one hour,
assuming tie inputs are identical. The isoproduct curves
will, in this limited case, be identical to the isocost
curves. Any given number of inputs will be used per day
to the point where overtime is paid. Beyond this point
costs will increase relative to production, so no inputs
will be used over tnis time limit unless there is some
other incentive for production.
when a limit to plant size is imposed so that an
unlimited number of tractors cannot be applied without
slowing down the progress of the others, a decreasing
return will be obtained. If time is held constant and
batches of inputs are varied, a normal looking production
curve will be realized.
Y
t=z4
0 X .... XdXdl .... Xn)
In this case, tne isoproduct curves in tne inputtime
dimension will not be hyperbolic, but will curve back as
inputs are varied.
Batches A At' B' C C'
of inputs \ \ '
(X1 d X \ d)
Time (hours per day)
The curves A, B, and C in the preceding figure are the
hyperbolic isoproduct curves w;ile A', B', and C' rep
resent decreasing returns.
In the following figure the diminishing isoproduct
curve has been combined with the isocost curve showing
overtime payment at 8 ours. Constant returns will be
realized when applying a constant number of inputs, in
creasing lengths of time.
8 \
 .isoproduct
4 If 6 7 Op s /e / f.2
Time Cnours per day)
Decreasing returns will hold as inputs are increased for
a given length of time. Production will continue to the
point where overtime is paid.
K. Indivisible inputs.
Special problems arive when one or more factors of
the production process are indivisible. An indivisible
input is one which cannot be infinitely divided. Examples
would be a tractor or truck or most power equipment. One
aatcnes
of inputb
Xl'*" *d)
>
method of eluding the problem of indivisibility in the
production function is to consider the services of the
input rather than the input as an entity. For example,
the length of time a tractor is used in a production
process can be infinitely divided. If, however, the
input is considered as an entity, special production func
tions have to be derived. The input can be considered as
either having width, or being a point. If it is considered
to have width, the factorfactor figure would be as follows,
were X1 and X2 are perfect complements and X2 is indi
visible, and there are no fixed factors.
Y! Y
Ii  
0 42 0 Xkjxi
If Y = fXlX2 X3... Xn) the factorproduct dimension would
be as follows:
Y Y
0 X, I 0 X.I X
A more realistic production function would not give the
indivisible factor any width. The production "surface"
would not then be a surface but would be a series of lines
and points.
Y = f(XX)
/
I
0 ~1 X 0
o x21xx
The production E~ctiensawould in general be as follows,
below and on page 45.
Perfect Complements Perfect Substitutes general Case
4\ \
\ \ \ \/ /
*\ \ ',\ K \
N N* \\ \, \\
....... \ \\ _
______________ \ ^.. ,^ *
L. Problems of aggregating inputs into input categories.
The complexity of inputs in an agricultural production
function necessitates the grouping of them into input cat
egories. Even a simple input such as corn in a feed ration
is a group of nutrient inputs, i.e. protein, carbohydrate,
mineral, vitamins, etc. The problem of aggregation becomes
more complex as corn is aggregated into a concentrate group,
and concentrates into a feed aggregate. The following is
a set of rules w ich should act as a guide when grouping
inputs.
1. Substitutes should be grouped together.
2. Complements should be grouped together.
3. Complements and substitutes can be grouped
together if they are either substitutes or
complements for each other.
4. Expenditures and investments should never
be grouped in a category. There is a great
0
Perfect Complements
Perfect Substitutes
General case
Y = f(Y12) = f(x1x ...X ) Y = f X ) Y = f(Xx1x 3...xn) Y = f (X1Xx) Y = fXiX2 X3..
Y1
Xl IX2
x I2
X1 X2
X1IX2
Y
l
*
12P1
I
.
S j S S
.
*
211 X1 X21
X1 X2
* xX
12
i/
X1 X2
x1 X2
X1 X2
SX2
X: X'
xl x2
x1 X2
//
difference in the returns expected on these
items each year.
2, = Z1 (group of complements) + ar2 (group of
substitutes) if i1 and ; 2 are complementary or
substitutes for each other.
A problem associated with aggregating is the measure
ment of inputs. Complements are measured in sets and sub
stitutes in some least cost denominator such as TDN or
dollars. It is generally less desirable to measure sub
stitutes in dollars.
What expenses should be included as inputs? An expense
should not be included as an input if no output is measured
for that expense. The number of expenses which are included
as inputs will depend on the accuracy of the accounting
system. For example, consider the situation where $150 is
spent on a barn for paint.
Y f(Z1 ....n)
3 = buildings (durable asset)
SZ = productive cash expense (oneuse)
The manager has to decide whether or not the paint is
an input. He faces two choices.
1. He can ignore the cost of the paint by not
including it as an input, If the inventory
is not accurate enough to measure the in
creased value of the building after paint
ing, this s;ould be his choice. In this
case the MVP would have to be high enough
to cover depreciation,, taxes, etc.
2. If the inventory is so accurate that the
paint will show up as increased value to
the farm, then the cost of the paint can be
put in input category Z5.
Other examples of this problem would include such things
as new tractor tires, fence repair, concrete runways, etc.
M. Cost concepts and supply functions.
1. Cost concepts relation to production functions.
TC = total cost
TVC total variable cost
TFC = total fixed cost
ATO = average total cost
AVC = average variable cost
MC = marginal cost
AFCy average fixed cost
AFCx = average factor cost
MFC marginal factor cost
If Y f(X1....XdXdtl ... Xn)
TC PxiXi
i1
d
TVC Pxi Xi
il
n
TFC = E PxiXi where Pxi = Pxi(y)
i=d*l
If Y = f(X1JX2*.Xn)
n
ATC = T = > PxiXi
Y i=l
APPxl X
AVC = == PXI =P AFOx
XY APPYX1X APPFX
perfect all cases
competition
iC j x  
MPP a
&x1 ^f^
If Y f(Xl...e XXd+l'*Xn)
d P, ,
AVC = L PxiXi i AFCxiXi = AFC.
i=1 APPxiXi 3
APPxi Xi
The following t ree figures show the relationship between
the production and cost concepts as they are related to
production functions.
X1 /
0 X\
Il!
Figure I
PPxlxein
APP
0 MPP X1X2
Figure II
LCC
Figure III
In figure I, Y is represented by the isoproduct contours.
Contours A, B, and C correspond to A, B, and C quantities
of Y respectively in figure II. A similar situation exists
between quantities of Y in figures II and III.
Maximum MPP corresponds to minimum MC, since where
MPP is maximum, JC is at a minimum. At the output where
MPP
APP = MPP, MC = AVS 'When MPP = O, MC = .
2. Envelope curves.
a) Description.
Figure II below, is derived from figure I.
The curve in figure II labeled ATCz1 is derived
from the function Y = f(XllX2=a). The curve
ATCx1x2 is from the function Y = f(X1X2) when
X1 and 12 are combined in LCC.
X /l
0 X2
Figure I
MC AXCC ATC lx2
B Cl / MClX
o y
Figure II
The curve BC in figure I is the expansion path
for any given ratio of prices of X1 and X2. Point C
(corresponding to C in figure II) represents the point
of minimum ATCxo. Since point B is the least cost
combination for any X1 when X2 = a, any point on line
AB, other than B will represent a greater cost than
at B. At B the cost is equal to the cost on the scale
line, so at B, ATCxl = ATCxlx2 and at all other points
on AB, ATCxl > ATCx1x2. The curve ATCxlx2 is
an envelope curve to ATCxl.
If there are no fixed costs, ATC = AVC and
where MC = AVC, MPP = APP. At point C then,
MPPxl = APPxl and MPPxz = APPx.,. Moving to the
left of C, PPxl > APPx1 and LMPPx) > APPx2, the
same being true at point B. Since, then, at point
B, :Pr'xI > APPx, G xl must be less than ATCxl = AVGxl
and ATCxl must still be decreasing. The point of
tangency of ATCx1 and ATCx2 kB) must not be the
low point of ATCx1. Notice that this would hold
even if ATCxl / AVCx.,
b) Lengths of run.
In general, tre fewer the variables tne
shorter the run. If Y = fiX1....A?), flX X"....X) f
is shorter run than f(XlX2Xj3XhX5). Figure III is "
constructed following the terminology of the
above section.
x qjCzxlx2 AT AT
/ x2X3 xl..*
A'I'CX...X
Y = flX1X2XjX X)
Figure III
The family of envelope curves represents
varying lengths of run. Only when all factors
are variable is tnere truly constant returns to
scale. The curve ATCxl.... x5 represents such a
curve.
In reality there are an infinite number
of lengths of run. When dealing with cost curves
it is necessary to determine from which production
function the curves are derived. In addition
to knowing which factors are fixed it is neces
sary to know at what levels tney are fixed.
c) Economies and diseconomies to scale.
1) Internal.
When discussing economies to scale it
is necessary to designate which cost curve
is relevant. It is possible to have eco
nomies on one curve and a diseconomy on a
related curve of the same function. For
example, ATC still drops when AVC increases
and ATC > MIC AVC. Marginal costs increase
when both ATC and ACG are decreasing. It is
also necessary to determine level of output.
$ AT TN N
$g
0 s i
The proceeding discussion refers to
the above figure. In plant IJ, an internal
diseconomy is realized when pushing produc
tion from A to D. When lowercostplant
N is added, an internal economy to scale
is realized if production in the second
plant is at B level. However, if plant N
is pushed toward capacity, a diseconomy
is realized over all three previous levels
of production.
2) External.
In the case were Pxi = f(Xi) and
Py = g (y) the firm may realize external
economies and diseconomies to scale. Cost
curves may increase or decrease as produc
tion increases. The result cannot clearly
be termed an "external" phenomenon, however.
For example, the manager, by increasing pro
duction within the plant, accounts for at
least part of the result. It is difficult
to make a clearcut distinction between in
ternal and external conditions.
d) Neoclassical snort run and longrun notions.
In tne neoclassical literature the snort
run was considered to exist any time Py > AVC
and AVC ATC. This is an oversimplification
and can lead to erroneous conclusions.
When AFC is based on salvage value of the
IV
^v
assets, adjustment will commence when Py = ATC
since ATC AVC AFC. As the price of Y decreases
the MVP of tne fixed assets will approach their
salvage value. When Py = ATC, the MVP of the
fixed assets will equal tieir salvage value.
Any further drop in Py will lead to tne disposal
of some of tne fixed factors. Adjustment will
take place before Py = AVC, the neoclassical,
long run production equilibrium.
3. Supply functions
Perfect prices are defined as the condition where
Ca = Vs. In the case wren prices are perfect, the
product supply curve for the industry is the aggregate
of the firms MC curve where MC > AVC. The industry
supply curve thus defined is in partial equilibrium.
It is in partial equilibrium because wnen the factor
prices change, tne MC curves for each firm will change
and consequently, so will tire industry supply curve.
In tne more general case wnen the perfect price
/ assumption is removed, supply is not perfectly revers
ible. That is, tne supply curve of the firm does not
follow the single MC curve. A drop in Py will cause
the MVP 's to decrease and the variable factors which
are not perfectly priced can become fixed. Conse
quently, another, shorter run, cost function becomes
relevant. If Py decreases sufficiently, the MVP of
t t e fixed factors will drop to Vs and some will be
sold or disposed of. The firm is then operating
on a longer run, but lower, cost curve. An in
crease in Py will again fix some of the factors
as MVP becomes greater than Vs. Again, a shorter
iun set of cost curves becomes relevant. This
boxlike supply response area is shown in the
following figures.
Px2 = Ca
Px2 = VS
7.
a b
Figure I
(Points t, F, G, H correspond on the two figures.
The argument commences at point E and goes in
succession through point H. The supply response
area is defined in dotted lines in the second
figure.)
x? b
Px=Ca
x2
Figure II
The elasticity of the firm's supply curve depends
on the number of factors which are variable. The
higher the number of variable factors, the greater
the elasticity of the supply curve; i.e. the less
the slooe of the MC curve.
Y I = X X ....Xn)
S Y= f(lX8I3..Xn)
S Y fX1X2X3X....Xn)
~
SIndustry supply responses and the notion of fixed assets.
Some factors of production are fixed for a farm as
as a whole, but are variable between enterprises on the
farm. If the importance and number of these factors
is large enough, the industry can be very fluid in
response to relative price changes yet still have an
inelastic total sulply curve.
The factors will be distributed between enter
prises such that the MVP of each factor equals its
opportunity cost. The opportunity cost of X1 in the
production of Y1 = MVPxl(2). Optimum combination
will be where MVPxit.) = MVPxiy.z) or
MVPx1J1 MVPx2 .... MVPxiy1
MVP VP VP
Xly2 x~ Y 2 "xiyZ
An increase in Pyl relative t6 Py2 will have the effect
of shifting some of the factors from the production of
Y2 to the production of Y1.
MVP 'y
Total X1 used in production of Y1 and Y2
57
When determining industry supply responses, it
is necessary to classify the factors of production.
The flows of the categories of factors between farm
and nonfarm sectors and the use.of the factors coupled
with other strategic information must be determined
before supply response can be found. The factors
should be classified according to whether they are
durable or expendable and whether they are farm produced
or nonfarm produced inputs. The categories chosen
should contain homogeneous inputs and should have similar
behaviors with respect to (1) salvage value and (2)
cost of acquisition.
Categories of inputs.
Category
Machinery
Livestock
Production expenses
feed, seed
Production Expenses
gas, oil, commer
cial fertilizer
Land
Nonland real estate
fences, tile
Family labor
Hired labor
Classification
nonfarm produced durable
farmproduced durable
farm produced expendables
nonfarm produced expendables
farm produced durable
nonfarm produced durable
farm produced
1. farm produced
2. nonfarm produced
Problems involved in the process of determining supply
response.
1. Areas of multiple enterprise farming.
2. Aggregating different outputs into a measure
of agricultural production.
3. Flows and use of input categories in relation
to the MVP over a business cycle, and the
relation of MVP to salvage value and acqui
sition cost.
4. Technological advance;
a. in the production of farm products,
b. in marketing and transportation of farm
products.
5. Specialization
a. made possible by MVP > Caor Vs > MVP.
b. made possible by changes in equity or
risk.
c. made possible by technology.
6. Types of specialization;
a. within farms.
b. between areas  regionalmay be due
partially to technology, credit, etc.
c. between farm and nonfarm sectors.
ifor example: egg marketing.)
d. between farms  not too important.
N. Demand for factors of production.
The demand curve of the firm for factor Xi is the
MVPxi where AVPxi > MVPxi > 0. This is the case since
the firm will purchase the factor so long as Pxi MVPxi'
The firm will adjust production before PXi increases to
the point where MVPx AVPxi since at this point, AVC
will have risen so far it no longer pays to produce.
MVP
sAVP
Demand curve x y
for Xi
The relative demand curve for two factors can be
computed from a factorfactor production surface. The
LCC's for various price relationships are laid off and the
locus of high profi. points (HPP) determined. The relative
demand for tne two factors can then be read from the graph.
X1 HPP's X
//
l HPP's
^ isoproduct
0 X, 0 4
Figure I Figure II
Factors X1 and A2 in figure I are relative complements.
An increase in the price of X1 causes less of both factors
to be used. In figure II, the factors are more early
substitutes. More of X4 is used as Px3 increases. In
both cases, less of the factor of rising cost is used.
The demand curves for X1 and X3 can be determined inde
pendently.
PX
px3
K
X1
K Th.  A
I I
x3
1. The case for perfect substitutes.
isocost
C
H .
._... . ... ... \ .... .....
C 02 O
The amount of X2 used is either unaffected or in
creased by an increase in Px1.. At B, PX1 PxMV
At C, MVP
A xP
At C, Pxl > PxY so no Xi is used.
X1
Pxl
0
A<
L.
i \
i
i...":
? \\
\
Xi
2. The case for perfect complements.
/ 1P
S.\
c*~* .~\ A_____
/
C
A
xi
O X 0
The amount of X2 used decreases with an increase
in .the price of X1.
3. Aggregating the demand curves.
Demand curves for each firm for factors of produc
tion are compounded in the same manner the supply
curves were compounded.
0. enterprise combinations.
1. Assumptions.
It is assumed that some inputs are fixed for
each enterprise such as corn planters and pickers
and corn cribs in the production of corn. These
will be represented in the production function by the
usual method for listing fixed factors, i.e.
****.. xg+ 1 *....). Some factors are fixed for
the farm as a whole but variable between enterprises.
These will be shown in the production function just
prior to the fixed inputs. Some inputs are vari
1
able for the farm as a whole. For these Vs = Cs or
very nearly so. These factors (X1***Xd) will be
shown first in the production function. The com
plete functions will be:
for corn Y1 f(Xl" d' d + '1 .. g  g + 1".Xim)
for alfalfa YZ= g(X1..."Xd Xd + l*"Xg I Xm + 1*xr)
2. Optimum combination.
In the case of horizontal integration we shall
use as an example, corn and alfalfa. Each product
can be represented by a production function with
the factors combined in LCC. As above, Y = corn
and kY= alfalfa.
Y11 Y2 1
B
 A /
L' LCC / '
/
S/ IJ
.L... /..
o (Xl'Xd, Xdtl.Xg) o Xl*"Xd, Xd+ ..Xg)
we shall begin by considering a farm which is
all in corn and on which we shall decide whether
or not some land should be put in alfalfa. As
more and more Xl* 'Xg are added to the corn
T^[j LI7~. i^.i<, Ceryo r Q, i XkwU'k
U+1 ,,, L.. aA U* a,i. ., l'*l <
(actually only (X*Xd) are added, with all X u X e
(Xd+l".Xg) on the farm being used for corn but
available for alfalfa) production will move into
stage III, say at point A on YI. No alfalfa is
being produced, see point A' on Y2.
When some of (X1'Xg) are shifted from corn
to alfalfa production, Y1 greatly increases and
Y2 only slightly (see points B and B'). Eventually,
when more factors are shifted, say to point C,
Y1 will cease to increase and more shifting of
factors will cause Y1 to decrease. But as Y1 de
creases, Y. is increasing. Clearly there must be
some optimum or most profitable combination of Y1
and Y2. This is illustrated on the following figure
(the lettered points correspond to those on the two
proceeding figures).
YE M
o Y
If MN is an isorevenue line, and is but one
of many, and if A"*.." is an isocost line, then
clearly point D" represents the most profitable
combination of Y1 and Y2 given (X1** Xg). At
point D", R of Y1 and S of Ye are produced. If
the relative prices of Y1 and Y2 were to change,
a shift in the slope of the isorevenue line, would
lead to a different optimum combination of Y1 and
Y;. At D" Pyl MPPxil Py2 MPPxiY2
Pxi Pxi
when i = (1l*.d)
and Py MPPx jl = VPxjy MVPxy
PY2 MPPxjy2 opportunity WPx y
cost of Xj(,) xjy
when j = (d + 1*.*g).
Stated in words, when two enterprises are combined
j at maximum profit, for tXl1'*Xd) the MVP's of the
inputs for both products are equated with the prices
of the inputs and for kXd 1.*.Xg) the MVP's are
equated with the opportunity cost of the input = MVP
in the production of the alternate product.
3. Characteristics of curves in the productproduct
dimension.
In the case of a farm that is so short of capital
that even one enterprise is operating in stage one,
the isocost curve in the productproduct dimension
will be curved toward the origin. This is true
beciate of the shape of the production function
in stage I.
/ /
~ xlxg~~
X1. Xg
As increasing amounts of capital become avail
able to the farm, the curves will intersect the
axes farther from the origin and will tend to
curve outward from the origin.
y1
\ \ I
0 Y2
The isocost curves will not cross over as does
the broken line above unless the firm moves into
stage III. An over mechanized farm may have tais
h
type of curve.
There are two types of farms which might be
organized in such a manner that both (or all)
enterprises are in stage I. The first type is
the subsistence farm where enterprises are added,
not for economic reasons, but in order to obtain
a more balanced diet. The second type of farm
is one where so much capital is tied up in fixed
assets that the farmer has none left, and no
credit, for operating. Thus, the farm is unable
to reach the stage II level of production in
either (or any) enterprise.
An expansion path can be constructed in the
productproduct dimension. The expansion path
is the locus of tue points of tangency of the
isocost and isorevenue lines.
h. Pseudo isocost lines or byproduct complementarity.
oF
In the casecomplementary crops such as a
legume and corn, each crop contributes some inputs
to the other. These particular byproducts are usu
ally not taken into account in the isocost lines
previously considered. The nitrogen added to the
soil by alfalfa, for example, is definitely an in
put in corn production and should be charged against
the corn. When this is not done, the normal iso
cost curve is not a true isocost and thus is
termed a pseudo isocost curve.
5. Sources of supplementarity and complementarity.
Two sources of complementarity have been
) previously discussed, i.e. byproducts and tne law
of diminishing returns. A third source should
be covered. Inputs unat are fixed for a firm
but that could be varied between enterprises are
a source of complentarity. If the inputs in
question are used only for one enterprise, their
output could be increased by using them in one
or more other enterprises. The additional product
would be forthcoming from less than proportional
added expense for inputs. The arvas of complemen
tarity and supplementarity are demonstrated on
the following figure.
11
0 Y2
The curve ABCD is an isocost curve. In the ranges
SAB and CD, the curve represents complementarity of
products. In the range BC, the products are up (Lf"
piemebeay. Optimum combination will occur in the
range of pe nt ey were the products are
competing for the given inputs.
6. Impact of product complementarity on firm and in
dustry supply curves.
The supply curves will show responses that
are not accounted for by an increase in inputs if
the products are combined in the range of complemen
tarity. That is, the firms MC curves will shift
to the right with a less than proportional use of
inputs.
P. The theory of vertical and horizontal enterprise combina
tions in the presence of fixed assets and in view of op
portunity costs.
Consider the three products,
Y1= milk \as an example)
Y2 hay \as an example)
Y3= grain kas an example)
Hay and grain are horizontally integrated and Y1 is
integrated vertically with Y2 and YI. All Y2 and Y3
are used in the production of Yl, Y1 being the only
product sold. Byproducts are not considered. Inputs
are considered perfectly priced and the product is per
fectly priced and homogeneous. Inputs fixed for the
firm will be priced according to their opportunity cost.
The three production functions are:
Y = f(Xl...Xd, Xd+l'Xg, Y, Y3 Xg+l"'..Y)
Y2 = f'(Xl...Xd, Xd+l..XgI X(+1. Xr)
Y3= f"(Xl...Xd, Xd+l..g I Xr..Xt)
The fixed factors in each function are those used uniquely
for the production of that one product, and for each,
Ca > MVP > Vs. The inputs (Xdel'"Xg) are fixed for
the firm but variable between the enterprises, an ex
ample being labor produced on the farm. For the inputs
(X1..XXd) Ca = Vs and they are variable between enter
prises. The profit equation is:
77= gross income total variable costs total
fixed costs
where
gross income = J PyjYj (in this example j = 1)
TVC = d 3
TC Xiyj Pxi + 2 Ci(MPPxiYl'PYl)
i=l j1 i=d+.l
d 3
The notation z2" Xiyj Pxi represents the costs of
i=l j=l
the inputs (X1...Xd) used in the production of all three
products Yj(j = 1 3). The term Ci = Xi(i = d+l...g).
Since, in this equation it is assumed that the MVP's of
all Xi(i dl"*..g) are equal to tneir opportunity costs,
g g
5 Ci(MPPxiYl.Py7) can be written 21. Cii(MPPPY2MPxiY2)Py.1
i=d+l i=d+l
This expression evaluates the opportunity cost of Xi(i=dl***.g)
considered from the stand point of using it in the pro
duction of Y2. The first expression is the opportunity
cost of using the inputs in Y1. A tuird form of the
expression, which represents the opportunity cost of
g
Xi in Y3 is: :LL Ci(MfPPy3J IMPPxiY3)Pyl.
i=d three of these expressions are equal and a one of
All three of these expressions are equal and any one of
them will give the same results. This follows because
all the opportunity costs are.equal.
t
Fixed cost 2Z XiFci where Fci = unit fixed charges
i=g+l
for the Xi = MVPxiyj and such that Ca > Fci > Vs.
If Fci > MVPxiyj(j = 13) too much is being charged
against them and negative quasirents are received.
If MVPxiyj > Fci, the inputs are worth more than is
being charged for them and positive quasirents result.
When Fci " VPxiYj, no rents are received and MC. = Maj = ATCyj.
.,Mc ATC
^T. 7'77 _T7 ... M= AR
KY"/./" l i
a
MC T
/ TC
, /
MRA
Negative Quasirent
Positive Quasirent
No Quasirent
The complte profit equation is:
i a = PYJ  ~1 Xiyj Pxi  CikdPPxiY Pyl) FC
j=1 i=l j=l i=d+l
where Ci = Xiki=dl..**g)
or
g
;.Ci xPPiy2 MPPy2) Py1
i=dtl
or
(C i ci(Pxiyr*MPPy3yl) Py
i=d+l
.Ml=AR
Fuocrwav s
In addition to the three productionA and the profit
equation there are 3g equations of the following form
necessary to define the optimum amounts of inputs to
use.
1. d equations of the form:
l = 'Piyl P PXi O
= MVPxi Pxi = 0
where i = .l***d)
which give the optimum amounts of Xi to
use in the production of Y1 directly.
2. 2d equations of the form:
=MPPx~ ( MP P Pyyj yl) Pl Pi = 0
0 xiYj
= MVPxiYjyi Pxi = 0
where i = (ld) and j = (.23) which
gives the optimum amount of Xi to use
in the production of Yj to use in the
production of Y1.
3. (g d) equations of the form:
6 / = MVPxiI  (MPPxj.PPyj ) P = 0
1xiy) 73718
XiYi l
where i = (d+l**g)
j = (2 or 3)
and (MPPxiyMPPyyj y) Pyl is the opportu
nity cost of Xi (in producing either Y2
of Y3). These'equations give the optimum
amount of the inputs fixed for the farm
but variable between enterprises to use
in the production of Y1.
4. (g d) equations of the form:
~ = MPPXiy'MPPy2l) Pyl (MPPx i*MPPY3Y)Py1=
or
(MPxiy'Py1)
which equates all MVP's and opportunity
costs.
5. (g d) equations of the form:
Ci iyI + Xy2 + xi73
i = (d+l...g)
which is the restriction that all inputs
of the second type (Xd+.l"Xg) are included
in Ci.
6. Altogether, there are j sets of (g d)
equations, 3 sets of d equations plus
the initial U.
3d + 3(g d) = jg + 4
There are, therefore jg + 4 equations
and 3g + 4 unknowns in the system. The
unknowns are Y1, Y2, Y3, /7', and the
amounts of the g different Xi's devoted
to producing each of the 3 products.
The optimum combination of Y2 and Y3 in producing
Y1 can be represented graphically. The figure can be
considered either as factorfactor with isoproduct
lines or productproduct with opportunities lines.
In either case, the outcome will not insure the equal
ities of the MVP's with the opportunity costs of the
inputs.
Y2
L. ._ i i.,
i B
S..
path
isoproduct
(. Five fundamental equations of static economics.
There are five fundamental conditions, wnich if
fulfilled, assure optimum allocation of resources in
the economy. These conditions can be stated in equation
form.
1. When all MVP's are equated with the MFC's of the
factor for each product, the firm will maximize
profits. For each firm using n factors to prod
uce m products:
1 = Vny MVPx MVPXy MVPi
: = Qn * ""= Yj
2. Equating MVP's of the factors with the opportunity
costs of the factors will assure maximum income
for resource owners. For each owner of each of
n different sources of productive services being
sold to
Yj
firms using the Xi to produce m products,
MVPxiY
"a =I
"R~i~1
where i = (l1*n), j = l.*..m), C 0 and Mxiyj4 =
revenue from selling the last unit to produce another]
product.
3. When each household equates the marginal utility
from each product with the cost of the product, it
assures the household maximum utility. For each
household: MUy1 MUyj
MPC~l MPCj
where MPC is the marginal product cost, or cost
of the last unit of the product.
The two remaining equations are restrictions which define
fixed assets for the firm and the household.
h. For each fixed asset for each firm producing j
products: MFCGxy MVPxiyj M_ IRX
5. For each fixed asset for each household;
MUMPCyj MU, j. MUMP
where MUMpFY = marginal utility from buying a new
y; MUMRy. = MU from selling one unit of yj.
R. Euler's Theorem.
This theorem states that when the elasticity of
production is equal to one (MPP = APP) and when each
factor is imputed its MPP total product will be exactly
exhausted. The theorem proves that when these conditions
are met, total product can be distributed to the factors
r
such that the total amount or total value of the product
is just exhausted. The necessary condition is that
APP = MPP. At this point, if each factor is imputed
its MPP, thentotal product attributable to each factor
is MPPxi~i. Total product equals 7APPx,*Xi. When
the necessary condition is fulfilled, EEMPPxiXi = EAPPxi *Xi and
total product is exhausted.
S. Inconsistency between perfect competition and ultimate
long run concepts.
When the conditions of perfect competition hold,
there is no optimum size of firm in the long run short
of one firm per industry. This follows because under
perfect competition in the ultimate long run no factors
are fixed, when no factors are fixed, a firm will ex
pand indefinitely so long as MVP MFC which occurs up
to tne point when the firm is no longer faced ?ritu a
perfectly elastic demand curve. Conditions of monopoly
would then occur wnicn is per se inconsistent witn per
fect competition.
Kaldor* has stated that in the long run, management
becomes fixed. In tnis case, then, taerm would De an
equilibrium position for the firm wnich would be at
less output than in a monopolistic position.
T. Results from Static Theory  what it can do.
1. Maximize dollar and real income from services for
all resource owners given the resource distribution.
2. Maximize the profits (=0) from the resources it uses.
Kaldor, "The Equilibrium of the Firm", The Economic
Journal XLIV, (194J).
3. Implies optimum size of firm in any but the ultimate
long run.
4. Determine the optimum output of products by each
firm and for the economy as a whole.
5. Determine relative prices of factors and products.
6. Determine the optimum consumption pattern for prod
ucts among consumers.
U. What Static Theory Cannot Do.
1. Does not determine size of firm in ultimate long
run.
2. Does not determine absolute prices.
3. iaplain technology.
4. Does not say anything about wants, preferences,
tastes, value system, and advertising.
5. Does not say anything about management; how it
earns income, or what it does.
6. Lxplain initial ownership pattern.
7. Makes no welfare or efficiency statements.
V. Supplement
1. Stages of production in factorfactor dimension.
This discussion centers around the following figure I,
found on the following page.
Line AEH is the locus of points where MPPx2 =0.
Line DBH is the locus of points where MPPX1 = 0.
Line AGBC is the locus of points where MPPx = APPx
Line DGEF is the locus of points where MPPXl = APP x
The area outside OAHD is stage III by definition.
The area inside OAHD is stage I by definition.
The area inside QAGD is stage I by definition.
The area AHDG remains as stage II.
Only in the area EHBG are both factors in stage II.
X1
1x2
Figure I
2. Algebraic proof concerning envelope curves.
Let ATC = fi(Xi k2. Xijp ATCx
[ATC = 1XX2 X3' *Xn) = ATCx1x
I~I_
ATC
Y
Slope (first derivative)
ATCx1 = XPx = T y
y2 y2
If slope ATCxl f slope AfCxlx2 where s
then they are not tangent at minimum Al
If 0 X1Px1 = X1P X2Px2
Y2 y2 y2
then XiP Pxl 22
Y2 y2 Y2
but then X2Px2
y2
Y2
Lope ATCX = 0,
TCo .
 0
which is possible only if either Px2 = 0 or no X2
is used in production, i.e. either Px 0 or X2 = 0
wnich is clearly not the case.
3. Specialization and aggregate supply response.
Take two firms both producing Y1 and Y2.
Each firm is producing at R.
FIRM B
M AY2 N
Figure I. Figure II.
Invert figure II so that the origin kN) is located
at N in figure III and such that the R's concide.
YI N'
By,
M W Y2
AYA Figure II.
Since the curves are isocost curves, firm A could
B
y^
FIM A
produce at S (all Y1) and firm B at S (only Y2)
with no change in total input. To show this, shift
N in figure III to N'. Both firms are then operating
at S. This specialization has increased Y1 by the
amount VS and Y2 by WT with no corresponding in
crease in input. An increasein output of this
type cannot be explained on the basis of increased
inputs. Some conditions which may have prevented
the sAift to specialization are:
A. Capital rationing caused by imperfect fore
sight.
B. Interrelationships between firm and house
hold, i.e. profit maximization may not be
the only motive.
C. Ignorance about the possibilities or op
portunities lines.
D. Technology.
