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Title: Theories and concepts of static production economics
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Table of Contents
    Title Page
        Title Page
    Table of Contents
        Page i
        Page ii
    Scope and nature of static agricultural production economics
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
    Static production economics
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
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Full Text


TV-


Theories and Concepts of
STATIC PRODUCTION ECONOMICS


/L- / ,,-(


A Term Paper for
AEC 5U0
22 March 1957







by
Peter B. Hildebrand


N-t






)A-#'69



ai-8


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 nroduct-product
dimension. . .
T) Il i 4- -li


1 LL s o-LUos .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 micro-statics and macro-statics. The

relationship between micro and macro 'dynamics is less well

defined as is that between micro-statics and micro-dynamics

and between macro-statics and macro-dynamics.

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, risk-trend,

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. Risk-trend

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 set-up

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) Non-firm 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 t-eir 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 non-eco-

nomic efficiency. So-called "technical efficiency' which al-

ways has an input-output 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 Diminish-ing 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 of-X1.

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

1T-0l


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.

Iso-product 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 two-variable

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) Iso-cost line.

An iso-cost 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


iso-cost line.






17







Iso-product curve

Xli I


4 x


Ax?, MPPX2(y) =- Xl 'PPx1(y)
1k= MPP (
XX2 -MPFx x(y)
The slope of the tangent of an iso-product curve

dx1 equals IPPx( )


c) The most profitable combination of X1 and X..


If AB is An iso-cost 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', or-the
'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


X-X2


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 2-mLy)
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

2-a


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
< x-7


Xl1



1\1 \ iso-cost

Siso-product

0 X,
the expansion path will be


x11


Siso-product

Siso-cost


0 X2


PXl > Px2, expansion path on X2 axis.







3)














b) The

1)


MPPxl Pxl
P q Px2 -, iso-cost

\\ -iso-product


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 non-eco-

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 non-economic

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)

1-----l------}_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 iso-cost curve will have no effect on the com-

'bination.

Xl|
1 \
i

\ iso-cost
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 non-economic 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 iso-product 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 iso-product 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 b-tc-i-of 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 iso-cost 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 iso-coso 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 iso-product

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 iso-product curves

will, in this limited case, be identical to the iso-cost

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 iso-product curves in tne input-time

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 iso-product curves w;ile A', B', and C' rep-

resent decreasing returns.

In the following figure the diminishing iso-product

curve has been combined with the iso-cost 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 \


- .--iso-product


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 factor-factor 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 factor-product 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~ctiensa-would 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 (one-use)
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 Z-5.

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
i-1
d
TVC Pxi Xi
i-l
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 iso-product 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 dis-economies 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 dis-economy 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
dis-economy is realized when pushing produc-
tion from A to D. When lower-cost-plant
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 dis-economy
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 dis-economies 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 clear-cut distinction between in-
ternal and external conditions.
d) Neo-classical snort run and long-run notions.
In tne neo-classical 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 neo-classical,

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

box-like 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 non-farm 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 non-farm 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
Non-land real estate
fences, tile
Family labor
Hired labor


Classification

non-farm produced durable
farm-produced durable
farm produced expendables

non-farm produced expendables


farm produced durable
non-farm produced durable

farm produced
1. farm produced
2. non-farm 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 -- regional-may be due

partially to technology, credit, etc.

c. between farm and non-farm 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 factor-factor 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









^----- iso-product


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.


iso-cost


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<,-| C-ery-o r Q, i Xk-wU'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 iso-revenue line, and is but one

of many, and if A"-*.." is an iso-cost 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 iso-revenue 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 product-product

dimension.

In the case of a farm that is so short of capital

that even one enterprise is operating in stage one,

the iso-cost curve in the product-product 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 iso-cost 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

product-product dimension. The expansion path

is the locus of tue points of tangency of the

iso-cost and iso-revenue lines.

h. Pseudo iso-cost lines or by-product complementarity.
oF
In the casecomplementary crops such as a

legume and corn, each crop contributes some inputs

to the other. These particular by-products are usu-

ally not taken into account in the iso-cost 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 iso-cost and thus is

termed a pseudo iso-cost curve.

5. Sources of supplementarity and complementarity.








Two sources of complementarity have been

) previously discussed, i.e. by-products 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 iso-cost 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. By-products 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 = 1-3) too much is being charged

against them and negative quasi-rents are received.

If MVPxiyj > Fci, the inputs are worth more than is

being charged for them and positive quasi-rents 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 Quasi-rent


Positive Quasi-rent


No Quasi-rent


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 = (l---d) 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) 7371-8
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 factor-factor with iso-product

lines or product-product 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


iso-product


(. 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 7-APPx,*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 factor-factor 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 iso-cost 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.




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