Unit 1.

Overview of agricultural production economics and the production function

Agricultural production economics is concerned primarily with economic theory as it relates to
the producer of agricultural commodities. Some major concern in agricultural production
economics include the following.

1). Goals and objectives of the farm manager- agricultural economists often assume that the
objective of any farm manager is that of maximizing profits, a measure of which is the difference
between returns the sale of crop and livestock less the cost of producing these commodities.
However, individual farmers have unique goals. A farmer might be more interested in; risk
minimization through diversification, attain food self-sufficiency or maximization of utility etc.

2). Choice of outputs and amounts to be produced- a farm manager faces with an array of options
with regards to what to produce given available land, labor, machinery and equipment. The
manager must not only decide how much of each particular commodity to be produced but also
how available resources are to be allocated among alternative commodities. The farmer might
be interested in the maximizing profit but may have other goals as well.

3). Allocation of resources among out puts- once decisions have been made with regard to what
commodity or commodities are to be produced the farmer must decide how his her available
resource are to be allocated among outputs. A simple question to be answered is which filed to
be used for the production of each crop?. But the questions quickly become far more complex.
The amount of farm labor and machinery on each farm is limited. Labor and machinery time
must be allocated to each crop and livestock activity, consistent with the farmers overall
objective.

4). Assumption of risk and uncertainty- models in production economics frequently assume that
the manager knows with certainty the applicable production function (for example, the yield that
would result for a crop if a particular amount of fertilizer were applied) and the prices both of
inputs to be purchased and outputs to be sold. However, in agriculture, the assumption of
knowledge with respect to the production function is almost never met. Weather is, of course the
key variable, but nature presents other challenges. Prices are variable over time.

5). The competitive economic environment in which the farm firm operates- economists often cite
farming as the closest real-world example of the traditional model of pure competition. But the
comparative environment under which a farmer operates depends heavily on the particular
commodity being produced.

Unit 2. Production function

A production function describes the technical relationship that transforms inputs (resources) in to
outputs (commodities). Agricultural production function can be presented in algebraic, tabular,
graphical forms.

Suppose Y= f(X)- algebraic form

Where;

Y= output
1
X= input

Suppose the simple function Y= αX

For each value of X a unique and single value of Y is assigned. This is presented in tabular form
as follows.

Yield of maize
Nitrogen/ ha Qt/ha
0 50
40 75
80 105
120 115
160 123
200 128
240 124

Suppose instead that the relationship between the amount of nitrogen that is applied and maize
yield is described as;

Y= 0.75X + 0.0042X2 – 0.000023X3

Y= Maize yield (total physical product) in Qt/ ha

X= Nitrogen applied in Qt per ha

2
A production function thus represents the relationship that exist between inputs and outputs. For
each level of input use, the function assigns a unique output level. When a zero level of input is
used, output might be zero or in some instances, output might be produced without the input.

2.1 Production with one variable input

2.1.1 Fixed versus variable inputs and the length of run

The general form of the production function is

Y= f(X1/ X2, X3, X4, X5, X6, Xn)

X1 treated as a variable input

X2….Xn assumed to be constant at some fixed level

Variable input- is thought of as an input that the farm manager can control or for which he/she
can alter the level of use. This mean that the farmer has sufficient time to adjust the amount of
input being used.

Fixed input- an input which for some reason the farmer has no control over the amount
available.

The categorization of inputs as either fixed or variable is closely intertwined with the concept of
time. Here Long run is time of sufficient length such that all inputs to the production function
can be treated as variable. Short run is a period of time so short that none of the inputs are
variable.

2.1.2. The Law of Diminishing Returns (LDR)

The law of diminishing returns states that as units of a variable inputs are added to units of one
or more fixed inputs, after a point, each incremental unit of the variable input produces less and
less additional output. As units of variable inputs are added to units of the fixed inputs, the
proportions change between fixed and variable inputs. The LDR has sometimes been referred to
as the law of variable proportion. Refers to the rate of change in the slope of production function.

3
 Y= 2x Y= X2

LDR doesn’t hold b/c each incremental input

LDR doesn’t hold b/c each incremental produces more and more additional output.
input produces exactly same incremental Quadratic function
output. Linear function

Y= X0.5

LDR holds, b/c each incremental units of X produces less

and less additional Y.

2.1.3.Marginal and Average Physical product

Marginal physical product (MPP)- refers to the change in output associated with an
incremental change in the use of an input. MPP representing the incremental change in TPP can
be either positive or negative.

Average physical product (APP)- defined as the ratio of output to input. APP= Y/X. for any level of
input use APP represents the average amount of output per unit of X being used.

MPP and APP for Maize response to Fertilizer

Nitrogen/ ha Yield of maize Qt/ha X Y MPPx APPx
0 50 - - 0 50
40 75 40 25 0.625 1.88
80 105 40 30 0.75 1.31
120 115 40 10 0.25 0.96
160 123 40 8 0.2 0.77
200 128 40 5 0.125 0.64
240 124 40 -4 -0.1 0.52

4
The first derivative of output (y) with respect to input (x) dY/dX represent the exact slope of the
production function at particular point.

D. dY

dX

o A X

Suppose Y= f(X),

5
Hence, the first derivative dY/dX is a function that represents the slope, or rate of change in the
original production function and some times written as;

dY/dX= f’(x) or f1. this can be expressed also ; dY/dX= f’(x)= f1= dTPP/dx= MPP

 All expression refer to the rate of change in the original production or TPP function.

For Y= 2X

dY/dX= dTPP/dx= MPP= 2

if Y= 50+ 5.93x0.5

dy/dx= 2.965x-0.5

if y= bxn
dy/dx= nbx
n-1

APP, MPP for y= 0.75x +.0042x2 -0.000023x3

X
Y (Maize APP of X,
(Nitrogen MPP of X, dy/dx
yield) TPP y/x
)
0 0 - 0.75
20 16.496 0.82 0.89
40 35.248 0.88 0.98
60 55.152 0.92 1.01
80 75.104 0.94 0.98
100 94 0.94 0.90
120 110.736 0.92 0.76
140 124.208 0.89 0.57
160 133.312 0.83 0.33
180 136.944 0.76 0.03
200 134 0.67 -0.33
220 123.376 0.56 -0.74
240 103.968 0.43 -1.21

APPx= 0.75+0.0042x-0.000023x2

MPPx= 0.75+0.0084x-0.000069x2

6
 At x= 60units, MPPx is at a maximum, called inflection point that marks the end of
increasing marginal return and the start of diminishing marginal returns.

 At x= 80units, MPPx = APPx, here APPx is at a maximum

 At x= 180units, MPPx= 0, TPPx reaches at a maximum after which output declines.

7
Stages of Production: Neoclassical Production Function (market is perfect, transaction cost is
zero)

TPP

Stage
II Stage III

Stage I

O A B C

APP

MPP

Stage I. in the graph ranges OB

 APPx is increasing in this region

 MPPx is greater than APPx in this region

 Each additional unit of input yields relatively more output on average

 Out put increases at increasing rate

 Often called as irrational stages of production. Irrational behavior describe goal in

inconsistent with the maximization of net returns or profit.

 Ep> 1

 Ends at the point where APPx reaches its maximum and equal MPPx
8
Stage II. In the graph ranges BC

 Starts from the point where APPx reaches its maximum and equal MPPx, Ep=1

 Corresponds with the economically feasible region of production.

 Output increases at a decreasing rate.

 The Law of Diminishing Return comes into effect in this stage. 0<Ep<1

 MPPx is positive and each additional unit of input produces less output on average.

 Ends at MPP=0

Stage III. Region beyond C

 Imply negative marginal return on input.

 Irrational to produce in this region.

 MPP decline and negative

Remark: If APP is increasing and therefore has a positive slope, MPP must be greater than APP.
If APP is decreasing and therefore has a negative slope, MPP must be less than APP. If APP has
zero slope, such as would be the case where it is maximum, MPP and APP must be equal.

2.1.4.Sign, Slope and Curvature

By repeatedly differentiating a production function, it is possible to determine accurately the
shape of the corresponding MPP function.

Consider the production function

Y= f(x)

 The first derivative represents the corresponding MPP function. dy/dy= MPP. Insert a
value for x in to a function f’(x) if f’(x) or MPP is positive incremental unit of input
produce additional output. Positive f’(x) indicate the underlying production function has a
positive slope and has not yet achieved a maximum. If f’(x) is zero, the production
function (PF) is likely either constant or at its maximum. If f’(x) is negative, the PF is
down sloping, having already achieved its maximum.

 The sign on the second derivative of the PF is used to determine if the TPP is at a
maximum or a minimum. If f’(x) is positive and f’’(x) is negative, shows point of
maximum TPP.

 If the first derivative of the TPP function is zero and the second derivative is negative, the
production function is at its maximum point.

 If f’(x) is zero and f’’(x) is positive, the PF is at its minimum point.

 If both f’ and f’’ are zero, the function is at an inflection point.

9
  If f’ is zero and f’’ doesn’t exist, the PF is constant.

 If both the first and second derivatives are zero, the function is at an inflection point, or
changing from convex to the horizontal axis to concave to the horizontal axis.

 The second derivative of a function is obtained by again differentiating the production

function.

D2y/ d2x

 The second derivative of the MPP function represents the curvature of MPP and is the
third derivative of the original production (or TPP) function.

 The second derivative of the MPP function represents the curvature of MPP and is the 3 rd
derivative of the original PF (TPP). The sign of f’’’(x) for a particular value of x indicates
the rate of change in MPP at that particular point. If f’’’(x) is positive, MPP is increasing
at an increasing rate or decreasing at a decreasing rate. a negative sign indicates MPP is
increasing at a decreasing rate or decreasing at increasing rate. if f’’’(x) is zero MPP has
a constant slope with no curvature.

Suppose y= 50+5.93x0.5

Nitrogen Yield mpp mpp' mpp''

0 50.00 - 0 -
1 55.93 2.97 -1.4825 2.22
-
2 58.39 2.10 0.39
0.52414
-
3 60.27 1.71 0.14
0.28531
-
4 61.86 1.48 0.07
0.18531
5 63.26 1.33 -0.1326 0.04
-
6 64.53 1.21 0.03
0.10087
-
7 65.69 1.12 0.02
0.08005
-
8 66.77 1.05 0.01
0.06552
-
9 67.79 0.99 0.01
0.05491
-
10 68.75 0.94 0.01
0.04688

 Y’= MPP= 2.965x-0.5 > 0 MPP is always positive for any positive level of input, TPP not
reached maximum

10
  Y’’= MPP’= -1.48x-1.5 <0- negative, MPP slopes down ward. Each additional input
produce less and less additional maize. The low of diminishing return holds for the PF
throughout the stage.

 Y’’’= MPP’’= 2.22x-2.5> 0. The MPP function is decreasing at a decreasing rate.

 APP= Y/x= 50/x + 5.93x-0.5 >0, yield produced per unit of input is always positive

 APP’= -50x-2 -2.97x-1.5 < 0; APP slopes downward, as the use of input increase the
average product per unit of input declines

 APP’’= 100x-3 + 4.49x-2.5 > 0; APP is also decreasing at a decreasing rate. as the use of
input increase the APP per unit of input decreases but at a decreasing rate.

2.1.5.A Single-Input Production Elasticity

Reflect relationship between two variables. Elasticity is a number that represent the ratio of two
percentages. Elasticity is unit less.

Elasticity of production- is the percentage change in output divided by the percentage change in
input, as the level of input use is changed.

Where;

Y’ original level of output, y’’ new amount of output, x’ original level of input and x’’ new level
of input. x and y represent mid values between the old and new or mean of the observation.

-∞ ≤Ep≤∞

If Ep ≤ -∞ %age increase in input resulted into a decline in output.

If 0 ≤Ep≤1 the percentage increase in output is less than the percentage increase in input. Less
response in terms of increase in output.

If Ep> 1 out put strongly respond to increase in output

 Elasticity of production can also be defined in terms of relationship between MPP and
APP.

11
 Large elasticity of production indicates that MPP is very large relative to APP. Output
occurring from the last incremental input is very great relative to average output obtained
from all units of input. Help to derive the law of diminishing return.

MPP
X Y
APP of X, of X, Based on Based on
(Nitrogen (Maize f'''(x) f'''(x) EP
y/x dy/dx EP EP
) yield)
= f'(x)
-
indetermin
0 0 0.75 0.0084 0.00013 -
e
8
Increase
-
0.0056 1.08 MPP>AP at
20 16.496 0.82 0.89 0.00013
4 0 P increasin
8
g rate
Increase
-
0.0028 1.10 MPP>AP at
40 35.248 0.88 0.98 0.00013
8 7 P increasin
8
g rate
Increase
-
0.0001 1.09 MPP>AP at
60 55.152 0.92 1.01 0.00013
2 4 P increasin
8
g rate
Increase
- -
1.04 MPP>AP at
80 75.104 0.94 0.98 0.0026 0.00013
4 P increasin
4 8
g rate
Increase
-
0.95 APP<MP at
100 94 0.94 0.90 -0.0054 0.00013
7 P decreasin
8
g rate
Increase
- -
110.73 0.82 APP<MP at
120 0.92 0.76 0.0081 0.00013
6 8 P decreasin
6 8
g rate
Increase
- -
124.20 0.64 APP<MP at
140 0.89 0.57 0.0109 0.00013
8 7 P decreasin
2 8
g rate
Increase
- -
133.31 0.39 APP<MP at
160 0.83 0.33 0.0136 0.00013
2 3 P decreasin
8 8
g rate
180 136.94 0.76 0.03 - - 0.03 APP<MP Increase
4 0.0164 0.00013 5 P at
12
 decreasin
4 8
g rate
- -
200 134 0.67 -0.33 -0.0192 0.00013 0.49 MPP<0
8 3
- - -
123.37
220 0.56 -0.74 0.0219 0.00013 1.32 MPP<1
6
6 8 2
- - -
103.96
240 0.43 -1.21 0.0247 0.00013 2.78 MPP<2
8
2 8 9

MPP
or
APP

APP

Ep>1 0<Ep<1 EP<0

MPP, APP & Ep MPP X

2.1.6.Profit Maximization with one Input and One Output

Assuming the market is competitive, the total income earned by farm operator from total output
(TPP) is given as follows;

TVP= Po*TPP

Where;

TVP= total value product

TPP= total physical product

Po= Market price of output assumed to be constant

Similarly, the cost of input purchased to carry out production of output is given as;

TFC= TRC= Pi* X

Where; TFC= total factor cost

TRC= Total resource cost

13
Pi= price of input

X= total input used in the production of output

Hence, the difference between TVP and TFC is represented by Profit;

Ï= TVP – TFC

Ï= [Po * TPP] – [Pi*X]

Value of the Marginal Product (VMP) - value of the incremental unit of output resulting from
an additional unit of X, when output sold for a constant market price Po.

VMP= dTVP/dX= Po* MPP

Marginal Factor Cost (MFC)- sometimes called marginal resource cost. Increase in the cost of
input associated with the purchase of an additional unit of the input. If input price is assumed to
be constant then MFC= Pi.

MFC= Pi*dXi/ dXi= Pi

Equating VMP and MFC- at the point where VMP and MFC are equal, the firm/ farmer
maximizes profit.

Po*MPP= VMP= MFC= Pi

MPP= Pi/Po- profit maximizing point

Divide both sides by APP;

MPP/APP= (Pi/Po)/ APP

= (Pi/Po)/ (Y/X)

=( Pi*X)/Po* y

MPP/APP= (total Factor cost)/ (Total value product/ Revenue)

Hence, a farmer maximize profit when Ep= the ratio of TFC and Revenue.

MPP
X Y
APP of X, of X, MFC=
(Nitrogen (Maize Po Pi EP VMP
y/x dy/dx= Pi
) yield)
f'(x)
0 0 indetermine 0.75 4 0.15 - 3 0.15
20 16.496 0.82 0.89 4 0.15 1.080 3.562 0.15
40 35.248 0.88 0.98 4 0.15 1.107 3.902 0.15
60 55.152 0.92 1.01 4 0.15 1.094 4.022 0.15
80 75.104 0.94 0.98 4 0.15 1.044 3.922 0.15
100 94 0.94 0.90 4 0.15 0.957 3.6 0.15

14
 120 110.736 0.92 0.76 4 0.15
0.828 3.058 0.15
140 124.208 0.89 0.57 4 0.15
0.647 2.294 0.15
160 133.312 0.83 0.33 4 0.15
0.393 1.31 0.15
180 136.944 0.76 0.03 4 0.15
0.035 0.106 0.15
-
200 134 0.67 -0.33 4 0.15 -1.32 0.15
0.493
- -
220 123.376 0.56 -0.74 4 0.15 0.15
1.322 2.966
- -
240 103.968 0.43 -1.21 4 0.15 0.15
2.789 4.834

At input 180unit, VMP= 0.106 while MFC= 0.15, this suggest that the last unit of input that was
used returns less than its cost. Hence the profit maximizing level of input is slightly less than
180units. If the input is not free, the profit maximizing level of input use will always be
somewhat less than the level of input use that maximizes the TPP.

Calculating exact level of input use to maximize output or profits: if output is at its maximum,
the MPP of the function must be equal to zero. The last unit of input use resulted in no change in
the output level and requires that MPP=dY/dX=0 at the point of output maximization.

General condition for profit maximization- given the TVP, TFC profit is defined as the
difference between TVP and TFC.

Profit(P)= TVP-TFC= Revenue (r)-Cost (c)= b(x)-g(x)

1st order condition for profit maximization require that

dP/dx= b’(x)-g’(x)= 0

= dr/dx-dc/dx=0

= dTVP/dx-dTFC/dx=0

=VMP-MFC=0

VMP= MFC

VMP/MFC= 1

2nd the second derivative test is often used to ensure that profits are maximum, not
minimum at this point. The second derivative test require that

d2P/dX2= b’’(x)-g’’(x)< 0

b’’(x) < g’’(x)< 0

d2TVP/dX2 < d2TFC/dX2

15
 dVMP/dX < dMFC/dX- the slope of VMP function must be less
than the slope of MFC. This condition is met if VMP slopes down ward
and MFC is constant.

2.1.7.Necessary and Sufficient Conditions

The terms necessary and sufficient are used to describe conditions relating to the maximization
or minimization of a function.

i. The necessary condition for the maximization of profit function for a given output is
that the slope of a function be equal to zero. i.e dP/dX= 0.

ii. The sufficient condition for the maximization of profit function for a given output is
that the rate of change in the slope or the second derivative of profit function be
negative.

Both conditions should be met for the event to occur always.

2.1.8.Costs, Returns and Profits on the Output side

The total cost for input or factor of production is the constant price of the input multiplied by the
quantity that is used. However, the cost of production might be also defined not in terms of the
use of the input but in terms of the output.

Variable cost (VC) cost of production that vary with the level of output produced by a farmer.

Fixed cost (FC) costs that must be incurred by a farmer whether or not production takes place.
Eg land rent, depreciation on machinery, building, equipment

The proportion of fixed to variable costs increase as the length of time is shortened and decrease
as the length of time increases.

Long run is a period of time of sufficient length such that the size of plant in the case of farming
can be altered. Production takes place on a short run average cost curve(SRAC) that is U-shaped,
with the manager equating MR (the price of output in the purely competitive model) with short
run marginal cost (SRMC). There exist SRAC and SRMC curve corresponding to the size of
particular plant (farm). Farmers can buy and sell land, machinery, equipment. Long run average cost
can be derived by drawing an envelop curve which comes tangent to each short-run average cost curve.

Short and Long run average and marginal cost with

envelop Long-run average cost

Birr SRMC1
SRACn
SRAC1
LRAC
SRAC2 SRMC2 SRMC3 SRMCn
SRAC3

16
 Y

Variable costs (VC) are normally expressed per unit of output (Y) rather than per unit of input
(X), because there is usually more than one variable cost item involved in the production of
agricultural commodities.

VC= g(Y) , function of out put

Since fixed costs (FC) do not vary with output, FC are equal to some constant dollar value k. i.e

FC= k

Total cost (TC)= VC +FC

TC= g(Y) + FC

AVC= variable cost per unit of output

AVC= VC/Y= g(y)/Y

AFC= FC/Y= k/Y average fixed cost

ATC= AC= TC/Y average total cost

ATC= AVC + AFC

= VC/Y + FC/Y

MC= marginal cost change in total cost (VC) resulting from an incremental change in output.

MC= dTC/dY= dVC/dY

The production function increased initially at an increasing rate until the inflection point was
reached, then it increased at a decreasing rate. T he cost function first increases at a decreasing rate
until the inflection point is reached, then increases at an increasing rate.

Cost function on the output side

Birr TC

VC

FC

17
 Y

Birr

MC
AC

AVC

AFC

Profit Maximization from the Output Side:

Profit (P) is maximized when MC= MR Marginal cost= Marginal revenue

P= TR-TC

dP/dy= dTR/dy – dTC/dy=0

MR-MC= 0

MR= MC

Under the assumption of pure completion price is constant.

Hence, dTR/dy= Po dy/dy= Po= MR

1st derivative corresponds to cost minimization

2nd derivative corresponds to profit maximization

dMR/dy-dMC/dy= + or - ?. Negative sign indicates a maximum and positive sign a minimum. Slope of
MC must be greater than the slope of MR for profits to be maximized.

Cost function and profit function

Birr
TR

TC
VC

18
Birr Profit

Y
MC

A B MR
AC

AVC

MR and MC intersects at two points (A & B).

At point A slope of MC is negative, point of minimum profit. The minimum point on profit
function represent the maximum loss for the farmer.

At point B slope of MC is positive, point of maximum profit. Maximum gain for the farmer

Cost data for maize production Price= 4

Yield
(qt) TVC FC TC AVC AFC AC MC MR Profit
40 90 75 165 2.3 1.9 4.13 4 -5
50 110 75 185 2.2 1.5 3.7 2 4 15
60 130 75 205 2.2 1.3 3.42 2 4 35
70 140 75 215 2.0 1.1 3.07 1 4 65
80 155 75 230 1.9 0.9 2.88 1.5 4 90
90 175 75 250 1.9 0.8 2.78 2 4 110
100 200 75 275 2.0 0.8 2.75 2.5 4 125
110 230 75 305 2.1 0.7 2.77 3 4 135
120 270 75 345 2.3 0.6 2.88 4 4 135
130 320 75 395 2.5 0.6 3.04 5 4 125
140 380 75 455 2.7 0.5 3.25 6 4 105

To know the exact point of profit maximization, it requires exact mathematical function.

19
The Duality of Cost and Production:

The shape of the total variable cost function is closely linked to the shape of the production
function that underlies it.

Assume X= input, Y= output, vo= unit cost of input, TVC= variable cost

Then APP= Y/X,

TVC= X*vo,

From APP, X/Y= 1/APP

AVC= TVC/ Y = X* vo /Y= vo /APP

AVC= vo /APP

MC= dTC/dY= vo dX/dY= vo /MPP

MC= vo /MPP

Given price of input vo= 0.15

Maize
Nitrogen/ MC= AVC=
yield/ MPP 1/MPP Vo APP
ha Vo/MPP Vo/APP
ha
0 0
20 16.5 0.825 1.212 0.15 0.18 0.83 0.18
40 35.2 0.935 1.070 0.15 0.16 0.88 0.17
60 55.1 0.995 1.005 0.15 0.15 0.92 0.16
80 75 0.995 1.005 0.15 0.15 0.94 0.16
100 94 0.95 1.053 0.15 0.16 0.94 0.16
120 110.7 0.835 1.198 0.15 0.18 0.92 0.16
140 124.2 0.675 1.481 0.15 0.22 0.89 0.17
160 133.3 0.455 2.198 0.15 0.33 0.83 0.18
180 136.9 0.18 5.556 0.15 0.83 0.76 0.20
20
 -
200 134 -6.897 0.15
0.145 -1.03 0.67 0.22
220 123.4 -0.53 -1.887 0.15 -0.28 0.56 0.27
-
240 103.9 -1.026 0.15
0.975 -0.15 0.43 0.35

The Inverse of Production Function:

Any production function has an underlying dual cost function or correspondence. PF has input
on the horizontal axis, and out put on the vertical axis. The corresponding cost function
expressed in physical terms is the production function with the axis reversed. The result is the
inverse PF, or cost function expressed in physical terms. The cost function is dual to the PF.

If PF is increasing at an increasing rate, the inverse PF increases at a decreasing rate. if PF increases at a

decreasing rate, the inverse PF increases at increasing rate.

Production
function Inverse PF
X= Y/2=
Y= 2x 0.5Y
Y= bx X= b/y
Y= x0.5 X= y2
Y= x2 X= y0.5
Y= axb X= (y/a)1/b

If the cost function is known, it is frequently possible to determine the underlining PF.

The total cost for the input expressed in terms of units of output is obtained by multiplying the
inverse function times the input price.

Given input price Vo

Inverse AVC=
PF APP MPP function Vo/APP MC= Vo/MPP

Y=X0.5 1/x0.5 0.5/x0.5 X= y2 Vox0.5 Vo x0.5/0.5= 2Vo x0.5

Ep= MPP/APP

Ep= AVC/MC

1/EP= MC/AVC

Stage I. MC<AVC, EP>1, MPP>APP

Stage II. MC>AVC, 0<Ep<1, MPP<APP

Stage III. MC<AVC, Ep<0, MPP<APP

21
Supply Function for the Firm:

The MC curve that lies above the AVC curve will be the supply curve for the farm. Each point
on the MC curve above the AVC curve consists of a point of profit maximization if the output
sells for the price associated with the point. The supply function for the farm will consists the
series of profit maximizing points with respect to price of product.

PF, Inverse function (cost function), derive supply function

Exercise:

Given Y= 0.4x+0.09x2 - Pi=

0.003x3 2
APP=0.4+0.09x-0.003x2
MPP= 0.4+0.18x-0.009x2
AVC= MC= EP=
X Y APP MPP Pi/APP Pi/MPP MPP/APP 1/EP

0 0 0.4 0.40 5.00 5.00 1.00 1.00

1 0.487 0.487 0.57 4.11 3.50 1.17 0.85

2 1.136 0.568 0.72 3.52 2.76 1.27 0.78

3 1.929 0.643 0.86 3.11 2.33 1.34 0.75

4 2.848 0.712 0.98 2.81 2.05 1.37 0.73

5 3.875 0.775 1.08 2.58 1.86 1.39 0.72

6 4.992 0.832 1.16 2.40 1.73 1.39 0.72

7 6.181 0.883 1.22 2.27 1.64 1.38 0.72

8 7.424 0.928 1.26 2.16 1.58 1.36 0.73

9 8.703 0.967 1.29 2.07 1.55 1.34 0.75

10 10 1 1.30 2.00 1.54 1.30 0.77

Unit 3. Production with two variable inputs

Introduction

This chapter underlines a production function indicating the technical relationships in which two
inputs are used in the production of a single output, also called the factor- factor model.

Two inputs are allowed to vary. The resulting production function is;

Y= f(X1, X2)
22
If there are more than two or n different inputs, the PF might be written as

Y= f(x1, x2/ x3, x4…xn)

The inputs x3… xn will be treated as fixed and given, with only the first two inputs allowed to
vary

3.1.An Isoquant and the Marginal Rate of Substitution

In the two input case, there may be many different combinations of inputs that produce exactly
the same amount of output.

A line can be drawn that connects all points representing the same output. This line is called
Isoquant.

As one moves along an isoquant (isos mean equal, quant mean quantity), the proportion of the
two inputs vary but output remains constant/ the same. Every point on the Isoquant represent the
same yield/ output level, but each point on the line also represent a different combination of the
two inputs.

Diminishing marginal rate of substitution

X2

Y2

Y1

Yo

X1

Yo, Y1, Y2 are isoquant

The slope of Isoquant is referred as the Marginal Rate of Substitution (MRS) or (Rate of
technical substitution (RTS) or Marginal rate of technical substitution (MRTS).

MRS is a measurement of how well one input substitutes for another as one moves along a given
isoquant.

MRSx1,x2 mean increase of x1 and decrease in x2 to produce the same output. X 1 is replacing
input and X2 is the input to be replaced, moving down along isoquant.

23
Along the isoquant representing constant output, each incremental unit X 1 i.e X1(x1) replaces less
and less X2 i.e.x2 (x2).

The diminishing marginal rate of substitution between inputs accounts for the usual shape of an
isoquant bowed inwards, or convex to the origin.

The slope of an isoquant can also be defined as

MRSx1x2= x2 /x1

The diminishing MRS is normally a direct result of the diminishing marginal product of each
input.

Depending on the relative productivity of the two inputs isoquants might be positioned nearer to
or farther from one of the two axes.

Isoquant Patterns:

 if dX2/dX1 <0 and d2X2/d2X1>0, the isoquant are negatively sloped. Convex to the origin .
the factors are imperfect substitute.

X2

X1

 if dX2/dX1 <0 and d2X2/d2X1=0, the isoquants are straight lines. The factors are perfect
substitute. In this instance, one factors of production substitutes for the other in a fixed
proportion. Has constant MRS.

X2

X1
 when the inputs are combined in fixed proportions, there is no factor substitutability. The
marginal product of inputs at the vertex are zero (e.g. a tractor and operator/ driver). This

24
 can occur when two inputs must be used in fixed proportion with each other. The first
and second derivatives of the function are not every where defined.

X2

Y1
Yo

X1

Isoquant and ridge lines:

Ridge line is a line that connects all points of zero slope on the isoquant. Marks the division
between stage II and III for x1 or x2. Connect points where MRS for x1 is zero and infinite for
x2. For a ridge line to be drawn isoquant must assume zero or infinite slope. Between ridge line
production is feasible.

Ridge line and a family of PF for Input x1

X2
x2*

x2** Economic region

Ridge
line
for x2
x2***
Ridge
line
for
x1

X1
Y Y=f(x1/x2*)
Y= f(x1/x2**)

Y=f(x1/x2***)

X1

25
3.2Marginal rate of substitution and Marginal product
The total change in output resulting from a given change in the use of two inputs is the
change in each input multiplied by its respective MPP.

Y= MPPx1 x1 + MPPx2 x2

Along an isoquant Y is exactly equal to zero. Hence;

Y= 0= MPPx1 x1 + MPPx2 x2

MPPx1 x1 + MPPx2 x2 = 0

MPPx1 x1 = -MPPx2 x2

MPPx1 = - MPPx2 x2 x1

MPPx1 MPPx2= - x2 x1

The MRS between two inputs is equal to the negative ratio of the MPPs.
MRSx1x2= -MPPx1 MPPx2
3.3.Partial and Total Derivatives and the Marginal Rate of Substitution
Consider PF Y= f(x1,x2)
MPPx1=
df/dx1 x2

MPPx2= df/dx2 x1
Suppose that the PF is Y= x10.5 x20.5

MPPx1= 0.5x1-0.5 x20.5

MPPx2= 0.5x10.5 x2-0.5

MRSx1x2= -MPPx1/ MPPx2

x -0.5 x20.5
0.5 1

0.5x10.5 x2-0.5
MRS1,2= -X2/X1

Similarly
Total change inMRS2,1
dY/dx1= canX1be
the MPP for canfound?
be obtained by dividing the total differential of the PF by
dX1. The result is;
dMRS/dx1=

dY/dx2=

dMRS/dx2= 26
 The total change in output as a result of a change in the use of x 1 is the sum of two effects.
The direct effect ( measures the direct impact of the change in the use of x1 on
output. The indirect effect measures the impact of a change on the use of x1 on the use of x2
which in turn affect Y through . Indirect effect because if out put is to
remain constant on the isoquant, an increase in x1 must be compensated with a decrease in x2.

Factor Interdependence:

Case I. Y= f(X1,X2)

If f’1,2>0 and f21,2>0 factors are technically complementary

Case II.

If f’1,2= 0 that means the marginal productivity of one input is not affected by the
change in the level of other input. The factors are technically independent.

Case III.

If F’1,2<0 the factors are technically competitive. Increasing the level of one input
decreases the marginal productivity of the other.

Elasticity of scale:

3.4.Maximization in the two- input case

Isoquant connects all points producing the same quantity of output.

The maximum of a function:

Suppose PF Y= f(x1,x2)

1st order or necessary conditions for the maximization of output are;

point where the slope of PF is

zero

point where the slope of PF is zero

27
The 2nd order condition for the maximization of output require that the partial derivative be
obtained from the 1st order conditions. There are four possible 2nd derivatives obtained
2
( y/ x1)/ x1= y/ x21= f11
2
( y/ x1)/ x2= y/ x21x22= f12
2
( y/ x2)/ x1= y/ x2x1= f21
2
( y/ x2)/ x2= y/ x22= f22

Young’s theorem states that the order of the partial differentiation makes no difference and that
f12 = f21

The 2nd order condition for a maximum require that;

f11 < 0

F22 < 0

f11f22 > f12f21

taken together the 1st and 2nd conditions provide necessary and sufficient condition for the
maximization of the two input PF that has one maximum.

given PF y= 10x1 + 10x2 – x21 –x22

1st order- f1= 10-2x1=0

X1= 5

F2= 10-2x2= 0

X2= 5

The critical values for a function is a point where the slope of the function is equal to zero.
Critical values are x1= 5 x2=5. Along the ridge line. This point could be a maximum, minimum or
saddle point.

For a maximum, the 2nd order conditions require that f11 <0 and f11f22 > f12f21

Maximizing a profit function with two inputs:

Consider PF y= f(x1,x2)

TVP= p*y

Where

P= price

Y= output

28
TVP= total value product

TFC= v1x1 + v2x2

Where

TFC= total factor cost

V1= price of x1

V2= price of x2

Profit (P)= TVP – TFC

= p*y –v1x1 –v2x2

= p*f(x1,x2) –v1x1-v2x2

P/ x1= p*f1 –v1=0

P/ x2= p*f2 –v2=0

VMP= MFC= P/ x1= p*f1= v1

VMP= MFC= P/ x2= p*f2 =v2

VMP= value of marginal product

MFC= marginal factor cost

P*f1/v1=p*f2/v2= 1

P*f1/ p*f2 = v1/v2

f1/ f2 = v1/v2

but f1/ f2 = MPPx1/MPPx2= MRSx1x2

MRSx1x2= v1/v2

Assuming fixed input price, the 2nd order condition holds;

P*f11 < 0 VMP functions are downsloping

P*f11 p*f22 - pf12 pf21 > 0

With fixed input price, the input cost function will have a constant slope or the slope of MFC
will be zero.

The Budget Constraint and the Isoquant Map:

If the farmer faces a budget constraint in the purchase of input x1 and input x2, and as a result is
unable to globally maximize profit, the next best alternative is to select least- cost combination
29
where the budget constraint faced by the farmer comes just tangent to the corresponding
isoquant.

Co/v2

X2
Expansion
path
Y3

Y2
Y1

Yo
Co/v1
X1

Iso-outlay line /budget line/ and isoquant map

Co = v1 x1 + v2 x2

C0= fixed budget

V1= price of x1, v2 price of x2

MRS1,2= -dx2/dx1- slope of isoquant

Slope of budget line is –v1/v2

At point of tangency the two slopes are equal.

-dx2/dx1= -v1/v2

dx2/dx1= v1/v2

MPP1/ MPP2= v1/v2

Given price of output po

VMPx1/v1= VMPx2/v2= k

The ratio indicates least cost combination of inputs with the level of budget. Accordingly, k
imply the outcome of the last dollar spent on input.

Equimarginal return principle- ensures that if a farmer is not at the point of

profit .maximization at least costs are being minimized for the level of output that can be
produced. In another way, maximum output is being produced for a given budget outlay. As one
moves outward along the expansion path the value of k declines.

3.5.Constrained maximization and minimization

Constrained revenue maximization:

30
 If the farmer is unable to globally maximize profits (at the edge of the two ridges) the next
best alternative is to find a point of least-cost combination. The least cost combination of
inputs represent a point of revenue maximization subject to the constraint imposed by the
availability of cash for the purchase of inputs.

Any problem involving maximization or minimization subject to constraint can be termed a

constrained optimization problem. The constrained optimization problem consists of two
component parts:

1. The objective function to be maximized or minimized and

2. The function representing the constraint on objective function.

Suppose objective function is to maximize revenue

R= P*Y or

R= p*f(x1,x2)

Constraint to purchase input x1 and x2 represented by;

C= v1x1 + v2x2

C- v1x1 - v2x2 = 0

Here Lagrange function is used to solve problem involving optimization. Lagrange multiplier is
employed.

The general expression for Lagrange function L is

L= (Objective Function to be maximized or minimized) + (constraint on the

objective function)

L= p*y + ( C- v1x1 - v2x2)

L= p*f(x1,x2) + ( C- v1x1 - v2x2)

The necessary condition for the optimization of objective function subject to

constraints are first order conditions. This condition requires that the first derivatives of
L with respect to x1, x2 and be found and be set equal to zero.

Necessary condition

dL/dx1= p*f1 - v1= 0

dL/dx2= p*f2 - v2 = 0

dL/d = C- v1x1 - v2x2=0

Where f1= MPPx1, the marginal product of X1 holding X2 constant

31
f2= MPPx2, the marginal product of X1 holding x1 constant

p*f1 and p*f2 can be interpreted as the VMP of x1 and x2 respectively

v1 and v2 are input prices or the marginal factor costs (MFC) for x1 and x2

the third equation indicates that when the objective function has been maximized, all cash
available for the purchase of x1 and x2 will have been spent.

Equation 1 & 2 can be rearranged as below

p*f1/v1=

P*f2/v2=

P*f1/p*f2= v1/v2

Price is constant

MPPx1/MPPx2= v1/v2

point of tangency between isoquant and budget

line

- Shadow price

- Imputed or implicit value of last cash spent on an input

- The worth of the incremental cash spent on inputs to the firm if the
inputs are allocated according to the expansion path conditions.

- The shadow price may or may not the same as the input price.

- p*f1/v1= , VMP/v1=

- The value of the incremental cash spent is VMP on the purchase of the
input

To ensure that revenue is maximum rather than minimum the 2nd order condition are needed;

32
 solve using determinant and if the sign is –ve
the function is maximized

If VMPx1/v1=VMPx2/v2= = -2.8 If –ve imply the farmer is operating beyond the point of
output maximization for both inputs.

If VMPx1/v1=VMPx2/v2= = 0 a farmer operates precisely at the global point of output

maximization, ridge line

If VMPx1/v1=VMPx2/v2= = 1global point of profit maximization, where the two ridge lines
connect.

The Lagrangean Method:

To find the solution of the problem

Max (min) f(x,y) subj. g(x,y)=c

Proceed as follows:

1. Write down the Lagrangean function

L (x,y)= f(x,y)- λ (g(x,y)-C)

2. Differentiate L with respect to x, y and equate the partial to zero

Find the only possible solution to the problem max x2y3z s.t x+y+z=12 ----(1)
3. The two equations in 2 together with the constraint yield the following three
x,y,z #0
equations.
L= x2y3z –λ(x+y+z-12) -------( 2)
f’1(x,y)= λ g1’(x,y)
Text. is;
Soln. the Lagrangean

dL/dx= 2xy3z-λ= 0
Sydsaeter knut & Hammond peter J. 2009. Mathematics for
2 2
dL/dy= 3x y z-λ= 0---------------------- (3)
economic analysis. Pearson edition.
dL/dz= x2y3-λ= 0

dL/dλ= x+y+z-12= 0

from the two first equation in (3) we have 2xy3z = 3x2y2z so y= 3x/2. The 1st & 3rd equation in (3)33
likewise gives z= x/2. Inserting the value of y & z into the constraint yields x+3x/2+x/2= 12 so x= 4.
Y= 6 & z= 2 thus the only possible solution is (x,y,z)= (4,6,2).
Cost minimization subject to a revenue constraint:

The problem of finding a point on the expansion path representing the least- cost combination of
inputs can also be constructed as a problem of minimizing cost subject to a revenue function.

Obj. function Min. cost C= v1x1 + v2x2

Sub. Revenue constraint R= py= pf(x1,x2)

L= v1x1 + v2x2 +

1st order condition equal to zero

2nd order condition less than zero

3.6. Returns to Scale, Homogenous Production Functions, and Euler’s Theorem

Economies and diseconomies of size

Economies of size- a situation in which as the farm expands output the cost per unit of output
decreases. reason;

 The farm may spread its fixed cost over a large amount of output

 Possible to do more field work with the same level of machinery and equipment

 Lower depreciation cost per unit of output

 Increase in the supply of critical inputs (land, fertilizer, seed…)

 Take advantage of pecuniary economies. As the size of operation increases the farmer
may pay less per unit of variable input because inputs can be bought in larger quantity.

34
Diseconomies of size- increase in the per unit cost of production arising from an increase in
output. Reason;

 As output increase the manager’s skill must be spread over the larger farm.

 The farm may become so large that the assumptions of the purely competitive model are
no longer met. Input price and output price might significantly vary.

Economies and Diseconomies of scale-

Scale imply proportionate increase in the amount of inputs. Both fixed and variable inputs are
subjected to vary in the production function.

If inputs double, as a result output doubles neither economies nor diseconomies of scale take
place.

Economies of scale exists when an increase in all categories of input result into more amount of
output (greater than percentage of input increase). If output doesn’t double it is diseconomies of
scale.

For economies or diseconomies of size to take place all that is required is that the output level
change. All inputs need no change proportionately. However, if economies or diseconomies of
scale are to take place, not only output change but each of the inputs must change.

Scale also linked with the length of time involved. The long run average cost curve represents
the possible units associated with each possible scale of operation.

Homogenous production function:

Help to define economies or diseconomies of scale.

A production function is said to be homogenous of degree n’ if when each input is multiplied by

some number t’ output increases by the factor t’.

Assumes time period is sufficiently long and all inputs treated as variable and included in the PF
n’.

The degree of homogeneity refers to the returns to scale. Represent a variety of transformation
between inputs and products.

A Production Function (PF) of homogenous degree 1 said to have constant returns to scale or
neither economies or diseconomies scale.

A Production Function with homogenous of a degree greater than 1 is said to have increasing
returns to scale or economies of scale.

A Production Function is homogenous of degree less than 1 is said to have diminishing returns
to scale or diseconomies of scale.

While there are many different PF only certain kinds of PF are homogenous. In general they are
multiplicative rather than additive also a few exceptions exist.
35
 1. the production function

Y= ax10.5 x20.5

Multiply x1 and x2 by t’ to get

a (tx1)0.5 (tx2)0.5

a tx10.5 x20.5

t1y constant returns to scale, homogenous of degree 1

2. the production function

ax10.5 x20.8

a(tx1)0.5 (tx2)0.8

t1.3y increasing return to scale, homogenous of 1.3

3. the production function

ax10.5 x20.3

a(tx1)0.5 (tx2)0.3

t0.8y decreasing return to scale, diseconomies of scale, homogenous of 0.8

For multiplicative function of the general form ax1 x2 the degree of homogeneity can be
determined by summing the parameters and .

Returns to scale and individual production elasticity:

Consider PF y= ax1 x2

If the PF is homogenous of degree n, and all inputs are represented in the production function,
the parameters representing the returns to scale is the degree of homogeneity. For a
multiplicative power production function with g inputs, the degree of homogeneity and the
returns to scale is determined by summing the g respective parameters ( ) which are the
elastic ties of production for the individual inputs.

Duality of production and cost for the input bundle:

Euler’s Theorem

The theorem is mathematical relationship that applies to any homogenous function. The theorem
states that if a function is homogenous of degree n’ the following relationship holds;
36
Where n degree of homogeneity. If the function is a PF then

MPP1 *X1 + MPP2 *X2= ny or

MPP1*X1 + MPP2*X2= Ey

Where E’ is the returns to scale parameter or function coefficient.

If the equation is multiplied by price of output p’ the result is

pMPP1*X1 +pMPP2*X2= E*p*Y

VMP1*X1 + VMP2*X2= E*TR

TR= total revenue

VMP1= value of marginal product for input X1

VMP2= value of marginal product for input X2

VMP= contribution to revenue

VMP*X1= return to each unit of input multiplied by the quantity that is used.

The input that contribute little to TR in terms of VMP would be paid less and
impoverished than that contributed more.

Unit 4. Types of production function

PF is a technical and mathematical relationship describing the manner and extent to which a
particular product depends on the quantity of inputs and services of inputs used. PF is of two
types;

i. Continuous PF- dozens of inputs and outputs can be split into smaller units. Eg
Fertilizer can be applied to hectare of land ranging from kilogram to any quantity.

ii. Discontinous PF- obtained for inputs which are used in whole numbers such as one
plough, two plough. One can shift only from one point to other.

These PF may be related to short run or long run period.

i. Short run PF- planning period during which one or more resources remain fixed.
During this period output can be changed only by intensive use of fixed resources.
The input output relationship that prevails during this period is called short-run PF.

ii. Long run PF- planning period during which all resources vary in quantity. The
supply of the product can be fully adjusted according to demand.

Choice of the function- in deciding which function to use, it is required to plot the observed data
on an arithmetic paper to determine whether trend is linear or non-linear. If trend is not linear
37
and slope of curve is downward and concave upward or upward and concave downward, the data
are plotted on semi logarithmic paper. In this manner an examination of nature of curve will
guide in deciding the type of function to be fitted.

4.1. The Linear production function

This is also known as the first degree polynomial function and is the simplest form.

Y= ao +a1x1+a2x2, ao intercept a1 slope

Y= ao+a1x1+a2x2

intercept

X1

Not widely used. Some of its limitation are

 second order derivatives (not less than zero) miss sufficient

condition

 returns to scale are not decreasing

iso-quant is linear, as X1 increase more x2 is given up in the production process. Input x 1 and x2
are perfect substitute. Such r/ship b/n input is rare. When ever such relationship is found the two
inputs are added together to treat as a single input.

X2

38
 X2=(y-ao-a1x1)/a2

X1=(y-ao-a2x2)/a1
X1

4.2. Quadratic production function

Since square of any number is always positive, y takes positive value in this PF.

2nd degree polynomial function can be expressed as

Y= ax2

Y= a+bx-cx2

The –ve sign before c denotes diminishing marginal return but not both. A declining and
negative marginal productivity. Thus this function explains stage II and III but not stage I of PF.

The maximum physical product can be obtained

X= b/2c

X= 0.5b/c

The shape of TPP. MPP and APP is presented below

TPP

39
Y

APP

MPP X

Elasticity of production is not constant. EP declines with the level of input applied.

Ep= MPP/APP

4.3. The Cobb-Douglas Production Function

Y= Ax1 x2

Where x1= labour

X2= capital

The function has three characteristics;

1. homogenous of degree n,

2. exibit diminishing marginal return to capital or labor, when the other held constant so the
law of variable proportion held. Parameter A represent technology of the society.

3. Easy to estimate by log10 or natural log (e)

Lny= lnA + lnx1 + 1- ln x2

Assumption is constant returns to scale

Let bo= lnA

Lny= bo + lnx1 + 1- ln x2

Where A= ebo

b1=

b2=

= regression error term

Linear in the parameter or coefficient

40
 Fixed inputs can be contained in parameter A.

The function needs to be expanded in terms of inputs.

Cobb-Douglas type of function , more than two inputs or factors of production
Y= Ax1 x2 x3 x4

As the number of inputs treated as variable expanded, the sum of parameters on the variable
inputs should also increase assuming each variable input has a positive marginal product.

Some characteristics of the Cobb-Douglass Type of Function:

 Homogenous of degree

 The returns to scale parameter or function coefficient is equal to the sum of the B values
on the individual inputs, assuming all inputs are treated explicitly as variable.

 The B values represent elasticity of production with respect to the corresponding input
and are constant.

 The partial elastic ties of production for each input are simply the B parameters for the
input.

Some of limitation of cob-Douglas PF are;

 Ep for an input is constant, irrespective of the input used. This is unlikely in the
neoclassical three stage PF.

 MPP and APP for each input never intersect, exception is when Ep =1 for one of the
input. If Ep is 1, imply MPP and APP are equal everywhere irrespective of input used. If
Ep<0, MPP always less than APP vice versa.

 All inputs must be used for the output to be produced. Since PF is multiplicative, absence
of one input makes TPP=0 even if other inputs are readily available.

 There is no finite output maximum at a finite level of input use. The function increases up
the expansion path at a rate that corresponds to the value of the function coefficient. If
b=1 the function increases at a constant rate, b<1 function increases at decreasing rate,
b>1 increase at increasing rate.

 For a given set of parameters, the function can represent only one stage of production for
each input, and ridge lines do not exist. If Ep for each input is less than 1, the function
will depict stage II everywhere. If the Ep (function coefficient) is less than 1, there will
normally be a point of global profit maximization at a finite level of input use.

4.4. The Spillman Production Function

Spillman primarily interested in determining whether or not the law of diminishing returns had
empirical support within some rather basic agricultural production process.

Spillman Function is;

41
 Y= A(1-R1x1)(1-R2x2)

Where A, R1, R2 are parameters to be estimated. The value of R 1, R2 normally expected to fall
between zero & 1. R1 + R2 would normally be less than or equal to 1.

The relationship between yield and inputs is non-linear.

4.5. The Transcedental Production Function

Mid 1950 the limitation of cob-douglas PF recognized. Though parameters are estimated the
function didn’t very well represent the neoclassical three stage production function. Fixed
production elasticity was major concern.

Researchers sought to make modifications in the Cobb-Douglass to allow for the three stages of
production and variable production elastic ties, yet at the same time retain a function that was
clearly related to the cob-douglass and was easy to estimate from agricultural data. A PF
introduced in 1957 looked like a slightly modified version of Cobb-Douglas. The base of natural
logarithm e’ was added and raised to a power that was a function of the amount of input that was
used.

The two input function was;

Y= AX1a1X2a2 e(b1x1 + b2x2)

The corresponding single input function is

Y= AXa ebx

dy/dx=MPP= aAxa-1 ebx + bebxAxa

MPP= Y(a/x + b)

Since APP= Y/X;

Ep= MPP/APP= y(a/x + b) = a+ bx changing partial elasticity

Y/x

Properties of single input Transcendental under varying assumptions

with respect to parameters a & b (a+bx)

Value of a Value of b What happens to Y and E

y increases at a decreasing rate until x= -a/b, then decreases as x

0<a<=1 <0 increases, Ep is declining.

the neoclassical case: Y increases at increasing rate until x=(-a+ sqrt

a)/b, increases at a decreasing rate untile x= -a/b, then decrease as x
>1 <0 increase Ep is declining.
42
 Value of a Value of b What happens to Y and E

0<a<1 0 y increases at a decreasing rat, Ep is constant equal to a

y increase at a constant rate Ep= 1 MPP and APP are the same every
1 0 where

>1 0 y increase at an increasing rate, Ep is constant equal to a

y increases at a decreasing rate until x= (-a+ sqrt a)/b, then increase at

0<a<1 >0 increasing rate, Ep is increasing .

>=1 >0 y increases at an increasing rate, Ep is increasing

4.5. de Janvry Modifications

Y= x1g(x1,x2) x2b(x1,x2)ej(x1,x2)

Ep allowed to vary with input variation.

Modification to cob-Douglas PF

4.6. Polynomial forms

The PF described so far require that a positive amount of each input be presented for output to be
produced. Isoquant come asymptotic to , but don’t intersect the axes. When isoquants intersect
an axis, output is possible even in the absence of the input represented by the other axis. A
polynomial form is inherently additive rather than multiplicative.

Y= A+bx1 + cx12 + dx2 + ex22

a,b,c,d,e are constant parameters

MPP1= b+2cx1= 0

MPP2= d+2ex2= 0

Y= A+bx1 + cx12 + dx2 + ex22 +fx1x2

MPP1= b+2cx1+fx2= 0

MPP2= d+ 2ex2 + fx1= 0

Eg. Y= x1 + x12 – 0.05X13+ x2 + x22 – 0.05x23 + 0.4x1x2

1st order condition;

MPP1= 1 + 2x1 – 0.15X12+ 0.4x2 =0

MPP2 = 1+ 2x2 – 0.15x22 + 0.4x1= 0

2nd order condition;

43
dMPP1 & dMPP2 < 0

4.6. Square root production function

allow diminishing total product and also marginal product to decline at a declining rate.

0.5
if y= a-bx+ cx

-0.5 =
MPP= -b+ 0.5cx 0

Applicable in the case of biological nature of production and where MPP may be greater at lower
level of input and TPP declines after attaining certain maximum.

Ep increase with increase in input use and output levels.

The SQRT function with two variable input can be expressed as;

Y= a-b1x1-b2x2+b3x10.5+b4x20.5 + b5x1 0.5x20.5

Main properties of SQRT production function:

 MPP is declining

 TPP reaches maximum

 Elasticity of production is declining

 Isoquant intersect the axis indicating inputs are perfect substitute

 Recommended to estimate fertilizer response at lower level

4.7. The elasticity of substitution

Elasticity of substitution is the percentage change in the use of x 2 divided by the percentage
change in the use of x1.

esb= (dx2/dx1) (x1/x2)

esb= elasticity of substitution

esp= MRS1,2 (x1/x2)

MRS1,2= marginal rate of substitution x1 for x2

f11= slope of MPP1

f22= slope of MPP2

44
f12= f21 (by Young’s theorem) the change in the slope (MPP1) with respect to a change in the use
of x2 or vice versa.

Formula for calculating the elasticity of substitution is;

es= [f1 f2 (f1x1 + f2x2)/ (x1x2(2f12f1f2 – f12f22-f22f11)]

es= elasticity of substitution x1 for x2 at a particular point on the isoquant for any production

4.8. Constant Elasticity Substitution Production Function (CES)

Had two principal features;

1. the es b/n the two inputs could be any number between zero and infinity

2. for a given set of parameters the es was the same on any point along the isoquant,
regardless of the ratio of input use at the point. Hence the name CES.

CES production function is;

Y= A [bx1- + (1-b)x2- ]-1/

= correlation coefficient between inputs

b+ (1-b)= 1 constant returns to scale

4.9. Derivation of cost and supply function from production function

Production function is used to derive the corresponding cost and supply function. Both short run
and long run cost and supply function can be derived from a given production function.

If input prices are given and constant, all information about short run (SR) and long run
(LR) can be derived from PF.

Given the market price and assuming perfect competition SR supply function can be
obtained from the derived short run cost function and long run supply function from the
derived long run cost function.

a. Derivation of SR cost function:

Given

The PF y= f(x1) and price of input, px1

Required

Short run cost function (TC) in terms of output y, that is

TC= K + h(y)

45
 Where TC is total cost, k is fixed cost and h(y) is total variable cost

Assumption

Input price px is constant

Cost function and production function by nature are inversely related. Knowledge of one
employs knowledge of the other given input prices.

Steps to get TC

Suppose the PF is y= x0.5, px= 2 Birr

Step 1. Find the inverse of the PF or x= f-1(y), means write x in terms of y

The inverse function is x= y2

Step 2. Set the total cost equation as TC= k+px* x and insert inverse function in place of
x

TC= k + Pxy2

TC= k + 2y2

Short run average cost (AC)= TC/y= k/y + 2y

MC= dTC/dy= 4y

AVC= TVC/y= 2y

MPP= 0.5x-0.5 = Px/ MC

APP= px/AVC= x-0.5

Ep= MPP/APP= 0.5x

b. Derivation of short run supply function

Given

The production function, y= f(x), px, py

Required

Short run supply function (output y) in terms of output price (py)

Y= h(py)

Assumption

Sufficient capital

Perfect competition (input price px, and py are constant)

46
 Profit maximizing farmer (that is decision is based on MR= py= MC

Under these assumption the supply function will be the series of profit maximizing points
or MC curve above AVC.

Steps to get short run supply:

Step 1. Derive total cost function from production function (as above)

PF, y= x0.5

X= y2

TC= k + 2 y2

Step 3. Drive MC= 4y

Step 4. Set MC= py and solve for y(output)

Py= 4y

Y= py/4 short run supply function

Once the short run supply function is derived, we an get elasticity of a farm w.r.t product
price as;

c. Derivation of Long run cost and supply function

a. Derivation of LR cost function

Given

The PF, y= f(x1,x2) and p1 and p2

Required

LR cost function or Total cost (TC) in terms of output (y) i.e TC= h(y)

There are two ways of deriving long run cost function- Minimum cost approach and
expansion path approach.

i. Minimum cost approach

Assumptions

 Price of x1 and x2 are constantPerfect competitive market

 Long run cost for the product of output y is the minimum

Required

Suppose the PF is y= x10.3x20.6, p1= 1Birr and p2= 3 Birr

47
Step1. Find the inverse function for each input holding the other input as constant

y= x10.3x20.6 inverse function for inputs are;

x1= y1/0.3x2-2 inverse function for x1

x2= y1/0.6x2-0.5 inverse function for x2

step 2. set the long run total cost function as TC= p 1X1 + p2X2 and substitute the inverse function
for x1 in the equation in place of x1.

TC= y1/0.3x2-2 + 3X2

Step 3. Take the derivative of total cost function in terms of X 2 and y obtained in stept 2 w.r.t x2.
Set it equal to zero and solve for x 2. This gives the level of X 2 that minimize cost for a given
level of output y.

For y= x10.3x20.6 first find dTC/dX2 as

dTC/dx2= -2y1/0.3x2-3 + 3= 0 then by solving for X2 we find

= x2-3 = -3/-2y1/0.3

x2= (1.5)-1/0.3

y-1/0.9

x2= (1.5)-1/.3 y1/0.9

x2= (1.5)-1/.3 y10/9

step 4. Determine the level of X 1 which minimize cost for a given output by substituting the
corresponding X2 (obtained in step 3) directly in the inverse function for x1 obtained in
x1=y1/0.3x2-2 inverse function for x1. This gives the minimum cost level of x1 as;

x1= (1.5)2/3 y10/9

step 5. Substitute the minimum cost level equations for x 1 and x2 in the total cost equation given
in step 2. This gives the long run cost function as TC= h(y)

for y= y= x10.3x20.6 substitute the vale of X 1 and X2 in TC= 1X1 +3X2 which gives the long run
cost function as

TC= [(1.5)2/3 + 3(1.5)-1/3] y10/9 = 9/2(1.5)-1/3 y10/9= 3.93y10/9

The Long Run TC function is 3.93y10/9

The LR AC= TC/y= 3.93y10/9

The LR MC= dTC/dy= 4.37y1/9

ii. Expansion path approach

48
Assumptions:

 Price of inputs constant

 Perfect competitive market

 Farmers decision is based on least-cost combination of inputs

 The farmer seeking to produce a given level of output at minimum cost would always use
input x1 and x2 in combinations indicated along the expansion path

 The farmer simultaneously change all inputs along the expansion path

 The LR cost function represents a series of minimum cost points of input combinations
for a given total output

Requires

Production function, p1, p2

PF= y= x10.3 x2 0.6 p1= 1, p2= 3

Step 1. Determine the equation for expansion path

 Set MPP1/MPP2= p1/p2 and solve for x1

MPP1/MPP2= 0.5x2/x1= 1/3

X1= 1.5 x2

 Substitute the value of X1= 1.5 x2 in y= x10.3 x2 0.6 and solve it for X2.

Y= (1.5x2)0.3 x2 0.6

= 1.50.3x20.3 x2 0.6

= 1.50.3x20.9

X2= 1.5-1/3 y10/9

Step 3. Set the LR TC equation TC= 1 P1 + 3P2 and insert the value of x 1 from expansion path
equation in place of x1 , this gives total cost in terms of only x2

TC= X1 + 3X2 and insert the value of x1, x1= 1.5x2

TC= 1.5X2 + 3X2

TC = 4.5 X2

Step 4. Substitute the value of X2 obtained above in the TC

TC= 4.5 [1.5-1/3 y10/9]

TC= 3.93 y10/9 the final long run cost in terms of output y.
49
 b. Derivation of long run supply function

Given

Pf y= f(x1,x2), p1,p2 and price of output py

Required

Long run supply function or output (Y) in terms of product price (py), that is Y=
g(py)

Assumptions

 P1, p2, py are constant

 Perfect competitive market

 Sufficient capital to purchase more input

 Farmer decision based on least cost combination of inputs

Step 1. Use the already derived cost function

TC= 3.93 y10/9 the final long run cost in terms of output y.

Step 2. Derive LR MC

dTC/dY= LMC= 10/9*3.93y1/9

LMC= 4.37y1/9

Step 3. Set LMC equal to product price as LMC= 4.37y1/9 = py

Y= 1.73[py]9

Advantages of cost function over production function:

 The cost function can be specified from factor input prices for a given level of
output.

 To estimate supply and demand for agricultural commodities and agricultural

inputs

 To present the r/n ship b/n quantity of output and input in monetary relationships.

9.10. The Demand for Input to the Production Process

The demand for input to the agricultural production is a derived demand. That is, the input
demand function is derived from the demand by buyers of the output from the farm. In general
the dd for factors of production depends on;

i. The price of output or output being produced.

ii. The price of the input

50
 iii. Price of other inputs that substitute for or complement the input

iv. the parameters of the PF that describe the technical transformation of input into output

v. availability of cash to purchase input

 Price of input is assumed to vary

Birr

MFC,v’

MFC, v’’
AVP

MFC, v’’’

MFC, v’’’’

Demand
VMP for input
Demand function for input x

Price of input is assumed to vary

Profit maximized at P.MPPx= VMPx= v

VMPPx= value of marginal product

P= price of out put

V= input price

 Intersection b/n VMP and V(MFC) at any point represent the demand for
input

 Increase in py shift VMP curve up ward to the right, increase dd for input.

 As the productivity of the underlying PF increases, the MPPx will increase.

This in turn will increase the dd by farmers for inpu x.

Assume PF is y= Axb

A= positive parameter

0<b<1
51
MPPx= bAXb-1

X= demand for input

1st order for profit maximize

VMPPx= v=MFC

Py* B*A Xb-1= v

Xb-1= v/ py*B*A

X= v1/b-1 [py*.b*A)1/b-1
X= v1/b-1 py-1/b-1.(b*A)-1/b-1 here the dd for input x is a function only of the price of the input (v), the price of the product (py), and the coefficient or

parameter of the underlying PF (b).

Assume A= 1, b= 0.5

Demand for units of input x under various

assumptions about output price, py
Price of Price of y (dollar)
x(v)dollar 2 4 6 8
1 1 4 9 16
2 0.25 1 2.25 4
3 0.11 0.44 1.00 1.78
4 0.0625 0.25 0.5625 1
5 0.04 0.16 0.36 0.64

X= 0.25v-2 py2 = 0.25p2/v2

As Px increase dd for x decrease. As py increase dd for input x increase. An increase in the py

causes a shift upward in the entire dd schedule or function.

The elasticity of input demand:

The own- price elasticity of demand for an input is defined as the percentage change in the
quantity of the input taken from the market divided by the percentage change in the price of
input.

(dx/dv)* (v/x)

dx= change in qty dd of x

x= average/ mid value

dv= change in price of x

v= average/ mid value

52
Given the input function X= v1/b-1 [py.bA)1/b-1

dx/dv= (1/b-1) v-1 [v1/b-1 [py.bA)1/b-1] but [v1/b-1 [py.bA)1/b-1] = x

= (1/b-1) * x/v

dx/dv*v/x= (1/b-1) *( x/v ) * v/x

edx= (1/b-1) by how much demand for input increases as price of input increases

elasticity of demand with respect to output price (ed py)= -1/(b-1) an increase in py
will increase the demand for x.

Technical Complements (Tcml), Competitiveness (TCmp) and Independence (TIdp):

Technical Complement- an input x2 can be defined as a technical complement for another input
(X1) if an increase in the use of input X2 causes the MPP of X1 to increase. Most inputs are
technical complements of each other. Notice that inputs can be technical complements and still
substitute for each other along a down ward sloping isoquant. Eg the presence of phosphate may
make the productivity of nitrogen fertilizer greater.

Tcml can be defined by d(MPPx1)/dx2 > 0

Technical Independence- an input is said to be technically independent of another input when

the use of x2 is increased the MPPx1 doesn’t change. This requires

TIdp can be defined by d(MPPx1)/dx2 = 0

Eg. y= ax1 + bx12 +cx2 + dx22

dy/dx1= MPP1= 2bx1

dMPP1/dx2= 0 TIdp

For additive production function with out interaction terms, inputs are technically independent.

Technically competitive- an input x2 is said to be technically competitive with another input

(x1) if when the use of x2 is increased, the X1 (MPP1) decreases. This requires that

Tcmp can be defined by d(MPPx1)/dx2 < 0 PF of additive function with

negative interaction term qualify this situation.

Y= ax1- bx1x2+cx2

dy/dx1= a –x2= MPP1

dMPP1/dx2= -1 the two inputs are competitive

inputs that are technically substitute for each other would include inputs that are
very similar to each other.

53
Unit 5. Production of more than one product
A production possibility curve represents the amount of each output that can be produced given
that the available resources of inputs are taken as fixed and given. Considers alternative outputs,
not inputs.

Production possibilities curve

Pulse for resource bundle

Classic Prod. Possibilities

curve Cereal

A society couldn’t operate on a point outside its production possibilities curve in that this would
require more resources than are available to the society.

A production possibilities curve thus represent the possible alternative efficient sets of output
from a given set of resources.

Simple equation;

X= g (y1, y2) product transformation function

where X= fixed qty of resources available to the society

y1= qty of pulse that is produced

y2= qty of cereal that is produced

dx= (dg/dy1)dy1 + (dg/dy2) dy2

since input is fixed, dx= 0

hence; (dg/dy1)dy1 + (dg/dy2) dy2 = 0

(1/Mpp1)dy1= -(1/MPP2)dy2

-MPP2/ MPP1= dy2/dy1

y / y = RPTy1,y2= slope of production possibility curve called the rate of product transformation
d 2 d 1

(RPT). The RPT is the slope of the product transformation function and indicate the rate at which
one output can be substituted for or transformed to the production of the other output as the input
bundle is reallocated.
54
y / y -ratproduct transformation of y1 for y2. y1 is substituting and y2 is being substituted product.
d 2 d 1

5.1. Competitive, Supplementary, Complementary, and Joint Products

Competitive output- product transformation function is downward sloping. RPTy 1,y2 will be
negative.

dy2/dy1 < 0 implies competitive products

Supplementary output- positive level of output y1 is possible without any reduction in the
output of y2. RPTy1,y2 is either zero or infinite depending on which output appears on the
horizontal axis.

dy2/dy1= 0 or dy2/dy1= infinity

eg. chicken farm, did not reduce input from other activities

Complementary output- if production of y1 causes the output of y2 to increase. The RPTy1,y2 is

positive at least for certain combination of y1 and y2. Eg. Crop rotation, intercropping

dy2/dy1 > 0 for certain production level for y1 & y2.

Joint product- that must be produced in a fixed ratio to each other. As a result the product
transformation function will either be a single point or a right angle. Elasticity of product
substitution between y1 and y2 is zero. Eg. beef and hide

Y2 Y2

Supl
range

Competitive Y1 Supplementary Y1

Y2 Y2

Compl
range

55
 Complementary Y1 Joint Y1

Given y1= 2xy1

Y2= 3xy2

Xy1 + xy2= x

Y1,y2 alternative output

Xy1, xy2 qty of x used in the production of y1 and y2. The sum of these quantities must equal X.

Xy1= y1/2

Xy2= y2/3

Therefore

y1/2 + y2/3= X

dx= 1/2dy1 + 1/3 dy2

since x is fixed, dx= o

1/2dy1 = -1/3 dy2

dy2/dy1= -3/2 RPTy1, y2

RPTy1, y2 = -1.5 constant downward slope .

The slope arises directly from the fact that the underlying single- input function exhibit constant
marginal returns to the input bundle x.

Given

Y1= xy10.5

Y2= xy20.33

Xy1 + xy2= x

Soln.

xy1= Y1 2

xy2=Y21/0..33

X= Y1 2 + Y23

56
Total differential of the equation;

dx= 2y1 dy1 + 3y22 dy2 =0

2y1 dy1= -3y22 dy2

-dy2/dy1= -2y1 3y22 RPTy1,y2 slope is negative and changing

5.2. Product transformation and the output elasticity of substitution:

It is the percentage change in the output ratio divided by the percentage change in the rate of
product transformation. Provides clue on the shape of product transformation function.

Eps= percentage change in the output ratio (y2/y1) divided by the percentage change in the rate
of product transformation.

= [ d(y2/y1)y2/y1]/ ( dRPTy1y2/RPTy1y2)

5.3.Maximization in a two-output setting:

If the amount of both output are at a global maximum, an additional unit of input bundle will
produce no additional output of either y1 or y2. Accordingly,M PPxy1 and MPPxy2 is zero.

The Isorevenue line: the revenue function (R) for two products is ;

R= p1y1 + p2y2

The slope of isorevenue line is –P1/P2 a constant ratio of the two output prices.

At any point on the isorevenue line, total revenue is the same, but if total revenue is allowed to
vary a new isorevenue line is drawn.

Y2 Prod.
Trans.
function Output expansion path
Isorevenue
line

Product transf. function,

isorevenue lines, & the output
expansion path Y1
57
Profit is maximized from combination of y1 and y2 that result in to equality of the absolute value
of slope of isorevenue line and RPT. p1/p2= RPTy1,y2.

5.4.Simple mathematics of constrained revenue maximization

Max. p1y1 + p2y2

Subj. Xo= g(y1,y2)

Xo is fixed

The Lagrangian 1st order or necessary conditions are

L= p1y1 + p2y2 + λ (Xo- g(y1,y2)

dL/dy1= p1 - λ dg /dy1= 0

dL /dy2= p2 - λ dg /dy2= 0

dL/ dλ= Xo- g(y1,y2)= 0

-MPPxy2/ MPPxy1= -p1/p2

RPTy1y2= -p1/p2

1st order condition find the point where the slope of isorevenue line is the same as the slope of
the product transformation function. Both are downward sloping.

The equimarginal return principle from the output side is;

p1 - λdg/dy1= 0

p2- λdg/dy2= 0

p1/dg/dy1= λ

p2/dg/dy2= λ

P1*MPPy1= λ

P2 * MPPy2= λ

VMPxy1= VMPxy2= λ the farmer should use the input bundle such that the last physical unit of the
bundle returns the same VMP for both enterprises. The analysis assumes that the resource or
input bundle is already owned by the farmer and therefore the decision to produce will cost no
more than the decision not to produce.

5.5. Two outputs and two inputs

The general equimarginal return rule requires that;
58
 P1MPPx1y1/v1= P2MPPx1y2/v2= P1MPPx2y1/v2= P2MPPx2y1/v2= k

The VMP of each input in the production of each output will be the same and equal to some
number k. the number k is actually a Lagrangian multiplier or an imputed value of an additional cash
available in the case of for the purchase of input to be used in output1 or output2 production.

Two inputs in the production of two outputs

py1= 4 py2=8 v1=10 v2=10
Yi
Yiel VM Yiel VM el VM Yiel VM Vmpp Vmpp Vmpp Vmpp
Inp p
dx1y Px1y dx2y Px2y d Px2y dx2y Px2y x1y1/ x1y2/ x2y12/ x2y2/
ut1 1
1 1 2 2 x2 1 2 2 v v v v
y1
0 70 4 30 80 20
1 90 4 80 35 40 95 60 30 80 8 4 6 8
11
2 105 4 60 40 40 60 38 64 6 4 6 6.4
0
12
3 115 4 40 43 24 40 44 48 4 2.4 4 4.8
0
12
4 120 4 20 45 16 20 47 24 2 1.6 2 2.4
5
12
5 122 4 8 47 16 12 48 8 0.8 1.6 1.2 0.8
8
13
6 122 4 0 49 16 8 48 0 0 1.6 0.8 0
0
13
7 120 4 -8 50 8 4 47 -8 -0.8 0.8 0.4 -0.8
1
13
8 118 4 -8 49 -8 0 45 -16 -0.8 -0.8 0 -1.6
1
13
9 114 4 16 47 -16 -4 42 -24 1.6 -1.6 -0.4 -2.4
0
12
10 109 4 20 44 -24 -8 38 -32 2 -2.4 -0.8 -3.2
8
The general profit maximization relationship requires that;

P1MPPx1y1/v1= P2MPPx1y2/v2= P1MPPx2y1/v1= P2MPPx2y1/v2= 1

On the input side; MRSx1x2= v1/v2

On the output side; RPTy1y2= p1/p2

Unit 6. Enterprise Budgeting and Marginal analysis

Budgeting is a planning device.

 help to determine what crops to grow and livestock to be raised.

 Determine how inputs should be allocated between enterprises

Hypothetical enterprise budget for maize per ha

Description Unit Unit Quantity Total
59
 cost
(Birr)
(Birr)
Gross revenue
Sale of grain Qt 400 20 8000
Total revenue 8000
Variable cost of
production
Seed Qt 0.5 800 400
Fertilizer Qt 1 870 870
Pesticide Kg 20 3 60
fuel Litre 50 16.5 825
Labour Mdys 12 80 960
Oxen power Ox.days 15 60 900
Total variable cost 4015
Fixed cost of
production
Equipment
% 5 70
depreciation
Store depreciation % 5 80
Interest on input % 300
Tax on land Birr 200
Total fixed cost 650
Total fixed and
4665
variable cost
Net returns overall
cost (Return to 3335
owner)

Assumption is made;

 Composition and magnitude inputs

 Price of output and inputs

 Expected yield

 Labour and oxen days to be used

 Wage rate for labour

 Opportunity cost of resources use/ imputed costs

 Depreciation of equipment and implements and stores to account for costs associated
with the wearing out of the inputs that are used in the enterprise for more than one season

 Interest charge on borrowed money or the opportunity cost of the farmers own money
invested in the farm.

60
6.1. The level of output to be produced
Level of output:

The first question is level of output to be produced and represented in the budget.

The farm enterprise budget is usually developed on a per hectare or per animal basis.

 If an entrepreneur is interested in maximizing profit, the level of output to be chosen is

that output level where MC= MR= Product price

 Farmers usually produce more than one product and have limitation on the availability of
fund to purchase inputs for each enterprise. The product- product model provides better
set of decision making rules for determining the allocation of fund for the purchase of
inputs when the farmer produce many different outputs and faces constraints. The basic
decision rule is;

VMPxy1/v= … = VMPxyi/v= … = VMPxym/v= λ Equimarginal return principle on

the output side. The rule implies that the farmer would attempt to make equal the
ratio of returns to costs for all enterprises.

The Variable input levels:

Variable inputs are factors of production which the farmer planed to control or alter during the
upcoming season. Are readily allocable to specific enterprise.

VMP= MFC

Constrained maximization indicates that inputs should be allocated to each output in such a way
that the last birr spent for each input returns the same amount for each enterprise.

VMPxy1/v1= … = VMPxyi/v1= … = VMPxym/vn= λ

The Fixed input allocation:

A farm budget is frequently used as a planning device for the coming production season. Fixed
inputs would thus include only those factors of production the farmer did not intend to change or
control over the coming season. In many instances, not readily be allocated to an enterprise.

Method of determination of cost of inputs and services available at farm:

There are many inputs and services which are furnished by own family members. While
calculating the cost of crops and livestock it is necessary to calculate the value of such own
inputs and services. The value of such inputs are computed in the following manner.

i. Own input and services

1 Family labour on the basis of wage paid to permanent labor in the locality
2 Owned bullock labor On the basis of maintenance costs of bullock
3 owned seed At the price prevalent in the village during sowing season
61
 4 Farmland manure at the rate prevalent in the village
5 Rent of own land Rent prevalent for some kind of land in the village or locality
On the values of permanent assets maintained by farmers
Interest on investment in such as farm building, irrigation structures, tube wells,
6
own fixed asses machineries, equipment, implement, livestock etc. interest are
charged at the rate which is payable at long term loans.
the interest on working capital is charged for half of the
Interest on working capital
7 period crop occupies the field. The interest is charged at the
or operational cost
rate to be paid on short term loans.
payment made in kind should be evaluated on the basis of
Payment made in kind and
8 prices prevalent in the village while value of perquisite on the
perquisites’ to farm labor
basis of market price.
There is no generally accepted basis for this. However, 10%
9 Management allowance
of farm income can be given under this head.

ii. Implements and machinery

a. Implement drawn by bullock in proportion of bullock power days used in various

crops

b. For other implements, the depreciation is allocated in proportion of human labor

used in each crop

c. The implements used in specific crops, the entire depreciation of that implement
should be allocated in the account of that crop.

d. The depreciation on irrigation structure should be allocated in proportion of

irrigated days used for each crop.

iii. Rent paid for leased in land: actual amount paid should be taken into account.

iv. Rent of own land: in proportion of area grown under crop out of self cultivated land.

Allocation of joint fodder cost among animals:

It is allocated in the proportion of standard animal unit. Standard units are determined as follows;

1. Bullock, cow, buffalo above 2 years of age 1animal unit

2. Camel above 2 years 2 animal units

3. Sheep and goats 0.2 animal units

4. Young stocks of 1 to 2 years 0.5 animal units

5. Young stocks of below 1 year 0.25 animal units

Hypothetical examples of cost of production (wheat),Birr/ha

62
 Amount
Description Remark
(Birr)
A Operational cost
Family labor 1090
Hired labor 2018
Bullock labor 500
Machine power 4778
B Material cost
Seed 1801
Manure & fertilizer 3329
Plant protection & chemicals 879
Irrigation 1606
C Total working capital (A+B) 16001
D Other cost
Interest on working capital 329
Rental value of land 7500
Depreciation of value of farm assets 721
Depreciation of value of fixed farm assets 138
Sub total 8688
E Total cost (C+D) 24689
Grand total Cost (total cost + 10% of total
F cost as managerial allowance and cost of 27157.9
risk factors)
G Yield of main product (Q/ha) 40
H Yield of byproducts (Q/ha) 38
I Selling price of main product (Birr/q) 999
J Selling price of by product (Birr/q) 125
Gross return (return from main product +
K 44710
return from byproducts)
L Net return over total cost 20021 K-E
M Net return over grand total cost 17552.1 K-F
Cost of production of main products at total
N 617.225 E/G
cost (Birr/Q)
Cost of production of main products at
O 678.9475 F/G
Grand total cost (Birr/Q)
Input-output ratio at total cost (Gross
P 1.81 K/E
return/Total cost)
Input-output ratio at Grand total cost
Q 1.65 K/F
(Gross return/Grand total cost)

Economics of Milk production on a Dairy Farm:

Amount Remark
Description (Birr)
A. Fixed costs:
1. Interest on investment in animals @10%
2. Depreciation on investment on animals @10%

63
3. Interest on the value of investment on fixed asset @10%
4. Depreciation on the value of investment on fixed assets
@10%
5. Amount of insurance (if animals are insured)
B. Variable costs:
1. Cost of green and dry fodder
2. Cost of concentrate
3. Cost of veterinary medicine and vaccines
4. Cost of human labour
5. Cost of rope/ chains
6. Electricity charges
7. Interest on working capital @10% for 6months (interest
would be calculated on cost of items 1 to 6)
C. Total cost: TFC + TVC
D. Returns:
a. Milk production in liter (lactation period has been
assumed 250 dys
b. Returns from milk (production of milk X average price
per liter)
c. Value of manure
d. Value of calf
e. Total return
f. Net return over variable cost
g. Net return Over total cost
h. Output-input ratio: Gross return/ Total cost

Economics of Egg production on a poultry Farm:

Amount Remark
Description (Birr)
A. Fixed costs:
1. Depreciation on initial construction cost of farm building
(2% is recommended)
2. Interest on the value of investment in farm building
3. Depreciation on equipment
4. Interest on investment in equipments
Total fixed cost
B. Variable costs:
1. Value of one day layer chicks
2. Cost of feed
3. Cost of veterinary medicine and vaccines
4. Human labor charges
5. Electricity charges
6. Other miscellaneous expense

64
 Interest would be
calculated for 9months
treating production cycle
of 18 months (i.e 50% of
7. Interest on working capital production period)
8. Opportunity cost of land
Total variable cost (1 to 8)
C. Total cost: TFC + TVC
D. Returns:
Value of annual sell of eggs (average yearly egg production per
hen: 264*no of egg lying hen)
Poultry manure
Sale of bird
Sell of empty gunny bags
Total returns
Net Return= TR-TC
Cost of egg production= Total cost/ no. of eggs produced
Benefit cost ratio= Total returns/ Total cost

Unit 7. Decision Making in an Environment of Risk and Uncertainty

Farmers face situations in which outcomes are uncertain. Nature has impact on farming. Farming
is inherently linked to the path of nature. Market affect farmers, prices are high when less
production and less when more production. Prices are largely determined by external forces
outside the control of farmers. Farming takes place in an environment characterized by risk and
uncertainty.

In uncertain environment possible outcomes and their respective probabilities of occurrence are
not known.

In a risky environment both the outcomes and probabilities are known.

To deal with risk all that is needed is insurance policy. The insurer can discover the outcome and
the probabilities of their occurrence and write a policy with a premium sufficient to cover the
risk and net a profit to the insurer.

Uncertainty: It is state of mind in which one can only guess the outcomes to a particular event
and is completely in dark about the probability of occurring the outcome. The term uncertainty
used to include all circumstances in which decision is required to be made without perfect
knowledge of future events. There is no way to predict the probability distribution of outcome
and only anticipation of the future can be made. Is not insurable. It can’t be included into the cost
like risk. It is entirely subjective in nature and vary from individual to individual.

Risk:- is uncertainty that affect individual’s welfare and is associated with adversity and
economic loss. It involves the probability of loss of income and resources of farmers, harm to
their health and welfare. It refers to a situation in which one is not sure of outcome but
probabilities of alternative outcomes can be established and variability of outcome can be

65
quantified. Major sources of risk are drought, flood, insect pests, plant disease, frost, hail, cattle
epidemic, sudden fall in farm prices (input, output). Minor sources of risk are a). risk due to
technical causes such as breakdown of machinery, defective seeds, delay in the transport of
perishable farm products. B). risk due to special hazards such as theft, robbery, fire, labor strike
etc.

Types of risk:

1. Yiled risk

2. Price risk

3. Institutional risk- tenure condition, mode of operation of credit and other farm input
supply agencies, government policies and regulations, opportunistic behavior of parties in
contract farming.

Probabilities and outcomes not

Probabilities & outcomes are known known

Risky events Uncertain events

Risk- uncertainty Continuum

Uncertainty cannot be dealt with as easily.

Basic differences between Risk and Uncertainty

Risk Uncertainty
Parameters of probability
Variability of income is
distribution of outcomes can not be
measurable
established in quantitative manner

Subjective in nature/ vary from

Objective in nature
individual to individual
Can be incorporated as Can't be entered as component of
component of cost cost
Is insurable Not insurable

7.1. Farmer attitude towards risk and uncertainty:

No one would normally enter an environment characterized by risk and uncertainty without
expectations of gains greater than would be the case in the absence of risk and uncertainty.
66
That individuals vary markedly in their willingness to take on risk and uncertainty can be
illustrated with a simple class game as follow.

Alternative income generating strategies

Income (outcome) Probability of each
Strategy
(Birr) outcome
A 1,000,000 0.3
(500,000) 0.2
0 0.5
B 100,000 0.3
50,000 0.4
0 0.2
(20,000) 0.1
C 50,000 0.7
30,000 0.2
0 0.1
D 30,000 0.4
25,000 0.4
15,000 0.2
Suppose a person is confronted with four d/ft strategies. Each strategy produce various level of
income and probabilities attached to it.

The probability attached to each strategy represent the expected proportion of times the specified
income is expected to occur relative to the total times the particular strategy is pursued.

One way to determine which strategy to pursue would be to calculate the expected income
occurring as the result of each strategy.

The expected income is the income resulting from the strategy weighed by its probability of occurrence.

Expected income
Strategy A= (1,000,000*0.3) +(-500,000*0.2) + (0*0.5)= 200,000.00
Strategy B= (100,000*0.3) +(50,000*0.4) + (0*0.2)= 48,000.00
(-20,000*0.1)+(50,000*0.7)+(30,000*0.2)+
Strategy C= (0*0.1)= 41,000.00
Strategy D= (30,000*0.4)+(25,000*0.4)+(15,000*0.2)= 25,000.00

So based on Expected income, strategy ‘A’ would always be pursued, despite the fact that
strategy ‘A’ also allows for the greatest potential loss.

The strategy that is pursued depends in part on the person’s particular financial situation. By
pursuing strategy A’ if a positive income was not achieved, the farmer would lack funds
necessary to meet the basic needs of life and would starve. Such a person would be reluctant to
pursue any strategy other than D’ . but a person with 1mill. Already in the bank would probably
choose strategy A’. the worst that person could do is lose half of what he/she already had.

7.2. Actions, States of nature, Probabilities and Consequences

To make decision a farmers should have alternatives.
67
Actions/ strategies are alternatives available to the farmer.

State of nature represent the best guess by a decision maker with regards to the possible events
that might occur. Determine outcome in combination with Actions.

Probabilities- Can be attached to each outcome. Represent the manager’s guess as to the number
of occurrence of particular outcome relative to the total number of possible outcomes resulting
from particular strategy.

Consequences- represent outcomes that are produced by the interaction of the manager’s action and
the states of nature. Represent what could happen to the manager.

State of nature
High
Action yields Low yield
Grow soybeans 20,000 3,000
Grow Wheat 15,000 10,000
0.6 0.4
Probabilities

Expected income
Grow soybeans= (20000*0.6)+(3000*0.4) 13,200
Grow Wheat= (15000*0.6)+(10000*0.4) 13,000

If a farmer is interested in maximizing income, he/ she would be better of to grow Soybean than
Wheat. However, the farmer might also be concerned with income variability.

7.3.Risk Preference and Utility

The farmer’s willingness to take risk is in large linked to his or her psychic makeup. The
maximization of utility subject to constraints imposed by the availability of income is the
ultimate goal of the farmer.

Utility function links utility or satisfaction to the amount of one or more goods that are available.

The possible function linking Utility to income

Utility Utility

B
A

Risk averter Income Risk- neutral Income

68
 Utility

Risk-Preferrer Income

Assuming that the farmer can achieve greater income only at the expense of taking on greater
risk or uncertainty

Graph A- risk averter will have a utility function that increases at a decreasing rate as income
rises.

Graph B- utility function for risk- neutral farmer will have a constant slope. Increase in
satisfaction is equivalent to increase in income.

Graph C- for risk preferrer utility/ satisfaction increases as income increases. Utility increases at
increasing rate.

Quadratic utility function; U= z + bz2

Where z= some variable of concern that generates utility for the manager, such as income.
Suppose that there exists uncertainty with regards to the income level, so that z is replaced by the
expected z or E(z). therefore expected utility is;

E(U)= E(z) + b* E(z2)

Expected value of squared variable is equal to the variance of the variable plus the square of the
expected value. Therefore;

E(z2)= σ2 + [E(x)]2. Hence

E(U)= E(z) + b [E(x)]2 + b σ2

Thus utility is a function not only of expected income, but also its variance.

Indifference curve linking the variance of income with the expected income
Expected Expected
income Indiff. curve income

Indiff. curve
B
A

Risk averter Income Risk- neutral Income

69
 variance variance
Expected
income

Indiff.
curve
C

Income
Risk-Preferrer variance

Indifference curve yield the same amount of utility and shows possible combination of income
and its variance. Assuming U equal Uo and taking the total differential of the utility function;

dU= dUo/dz = 1+ 2b [dE(x)] + b [dσ2]= 0

therefore;1+ 2b [dE(x)] = -b [dσ2]

-b [dσ2] 1+ 2b [dE(x)]= 1

To solve for utility function differentiate expected income with respect to income
variance, then;

dE/dσ2 =

the denominator [1+2bE(X)] will always be positive. The shape of indifference curve will
depend on the value of b. if b is zero, the farmer neither desire nor dislike risk. The farmer is
risk neutral. If b is positive the farmer loves risk, and indifference curve will have a negative
slope. If b is negative the farmer is risk averse and will have indifference curve sloping upward
to the right.

7.4. Risk, Uncertainty, and Marginal Analysis

So far the assumption was input prices, output prices and outputs were known with certainty.

There exist several ways of incorporating risk and uncertainty in to the models, while relying on
marginal analysis. One method is to use expected price and expected yield in to the model.

Predicted prices are obtained from various sources but also need to attach subjective probabilities
with respect to its accuracy.

Current and past prices are helpful to formulate price expectation.

Fore example, if a farmer experienced maize yield of

 130 qts per ha last year,

 114 qts per ha the year before, and

70
  122 qts per ha the year before that.

Method 1. A simple way of formulating a yield expectation might be to average the yield
over the past three years. Thus 122 qts per ha can be taken as expected yield.

Method 2. To weigh more heavily data from the recent past relative to earlier data.
Expected output becomes a distributed lag of past output levels. Fore example a farmer
may place a weight of 0.6 on last year’s data, 0.3 on the year before and 0.1 on the year
previous to that. Accordingly, the expected yield is Y= 0.6(130) + 0.3(114) + 0.1(122)=
124.4 qts per ha.

Once price and output expectations have been formulated, they could be inserted directly
into the model. The marginal conditions would then be interpreted based on expected
rather than actual prices.

To over come limitations associated to expected price additional constraint should be

added in to the model. If y 1 and y2 are produced with a given resource bundle. Due to
price and output instability, there is income variability associated with both y 1 and y2. The
income variability associated with y1 is y 1σ2 and income variability associated with y 2 is
y2 σ2. The income variability associated with the first commodity may partially offset or
add to the income variability from the second commodity. An interaction term or
covariance term is needed. This term is 2y1y2 σ12

The total income variability is µ= y1σ2 + y2σ2 +2y1y2 σ12

The farmer is interested to maximize revenue subject to the constraint that income
variability not exceed a specified level µo and the constraint imposed by the availability
of cash for the purchase of the input bundle x. so the Lagrangian is.

L= p1y1+ p2y2 + λ(µ- y1σ2 - y2σ2 -2y1y2 σ12) + ó(vxo- vg(y1,y2)

Compute 1st order condition w.r.t y1, y2, λ, ó equate to zero.

7.5. Strategies for Dealing with Risk and Uncertainty

Insure against risk:

If an insurance policy is available, income variability due to that source of risk can be reduced by
purchasing the policy. People purchase fire insurance not because they expect their house to burn
down, but because the cost of the insurance is low relative to the potential loss that could occur
should the house burn.

Insurance policies work if the probabilities attached to the event is low. In other word, insurance
should be used in situations where there is a low probability of a large loss.

Crop insurance plans have the effect of making the farmers income from one year to the next
more even, despite the fact that the farmer may pay in the form of premiums some what more
than is returned in the form of claims over a 10-year period.

Contracts:
71
The future market can be thought of as a device which allows farmers to contract for the sell of
specified commodity at specified price for delivery at some future point in time. Reduce or
eliminate price uncertainty.

Flexible facilities and equipments:

If a farmer is to adjust to changing relative product and input prices, it must be possible to adapt
buildings and equipments lasting more than one production season to alternative uses as input
and output price ratios change.

Diversification:

Strategy long used by farmers for dealing with price and output uncertainty. The strategy is to let
profit from one type of livestock or crop enterprise more than offset losses in another enterprise.
May also make effective use of labor and other resources throughout the year, thus increasing
income in both good year and bad year. may lose pecuniary and other internal economies open
to the specialized counterpart.

Government programs:

Programs designed to support farm incomes when output levels are low. Eg. food security, safety
net, pastoral development, drought resistant variety. Research, moisture stress zones. Etc

Price support programs

Central store and buffer stocks.

Marketing, information

Unit 8. Time and agricultural production process

Time is an inherent part of all agricultural finance issues. Marginal analysis can be used as a
basis for making decisions within a time frame encompassing several production periods.

8.1. Alternative goals of a farm manager over many seasons

Long run profit maximization:

A farmer interested in long run profit maximization over a number of years has long run
planning horizon and will make investment decisions with expected payoff some years away
consistent with the longrun goal. Eg. coffee, apple, fruit plantation etc

A goal of long run profit maximization will frequently require short-run profit and income
sacrifices during the early years of planning horizon, with the hope or expectation of making
greater profit during the latter years. However, a dollar today is worth more than a dollar
obtained a year or more from now.

Interest rate, inflation, season to season variability of income/ profit, affect decision to invest
limited cash.

Accumulation of wealth:

72
A farmer interested in the accumulation of wealth might more likely use part of last year’s profit
as a down payment on additional land, and the profitability of the crop for the coming season
may be reduced to a degree.

Other goals:

Nearly every farmer wants to be recognized by his neighbor as a good farmer. To own larger
size of land for home construction. To raise more livestock compared to other farmers. To
construct additional building. To own more machinery. Seek for prestige and recognition, by
accumulating much resources.

8.2. Time as an input to the production process

Labour, machinery etc provide stream of services over time to generate profit to the farm.

Time suitable for performing planting, tillage and harvest activities is limited by weather
conditions and field time used to perform one operation can not be reused to perform another
field operation.

8.3. Discounting revenue and costs

Discounting is used to determine what specific amount of revenue obtained at some future point
in time would be worth today or to determine the current amount of a cost incurred at some
future point in time.

The present value (PV) of a dollar:

A general rule for determining the PV of a dollar earned at the end of each of n’ years is;

PV= ∑(1/1+i)n

PV= present value

n= number of years

i= Market interest rate

accordingly PV of revenue and costs are calculated. Hypothetical example.

i= 10%
Revenue
Year (R) Cost © DF PV R PV C Profit
0 0 500 1 0 500 -500
1 1000 1200 1.1 909.0909 1090.909 -181.818
2 1500 1400 1.21 1239.669 1157.025 82.64463
3 2000 1600 1.331 1502.63 1202.104 300.5259
4 2500 1800 1.4641 1707.534 1229.424 478.1094
5 3000 2000 1.6105 1862.764 1241.843 620.9213
800.3831

73
Unit 9. Linear Programming and Marginal Analysis
Classical optimization methods involve the maximization or minimization of a function subject
to one or more constraints. To do this new variable called Lagrangian multiplier is added for
each constraint, and the maximization or minimization entails setting the partial derivatives of
Lagrangian function with respect to each variable, including the Lagrangian multipliers equal to
zero.

The Lagrangian multiplier can be interpreted as the increase (decrease) in the function to be
maximized (minimized) associated with a relaxation of constraint by 1 unit.

9.1. Classical optimization and Linear Programing

Mathematical programming is divided into two major subcategories;

i. Non linear programming eg. constrained optimization using Lagrangian function

ii. Linear programming

Y= Ax1aX2b non linear function

C= v1x1 + v2x2 linear function

Linear programming- involves optimization of a linear function subject to linear constraint.

Unlike classical optimization problems, in which atleast one of the functions was nonlinear, with
LP every function is linear.

Classical optimizations require at least one of the functions to be nonlinear.

9.2.Assumptions of Linear Programing

Five basic assumptions underlie LP;

i. Linearity- objective function and the constraints in LP problem are linear.

ii. Additivity- suppose one unit of Y1 require 2x1 and 3x2. Hence 100unit of y1 require
200x1 and 300x2. Constant returns to scale.

iii. Divisibility- if 1 unit of Y1 can be produced by 2x1 and 4x2 then ½ unit of Y1 is
produced with 1x1 and 2x2

iv. Non-negativity- the solution should not require that negative quantities of an input be
used and output be produced.

v. Single-valued expectations- LP assume that coefficients such as input requirements

and prices are known a priori with certainty.

A. Simple Constrained Maximization Problem

Maximize. 4y1 + 5y2

s.t constrains

74
 Input x1: 2y1 + y2 ≤12

Input x2: y1 + 2y2 ≤16

There are 12 units of input x 1 and 16 units of input x 2 available. Units of y1 each require 2units of
x1 and 1 unit of x2. Unit of y2 require 1 unit x1 and 2 unit x2.

y1 y2
2 1 x1
1 2 x2

The column of matrix referred as activities.

The raw of matrix referred to as resource constraints.

Graphical method solution

Y2
12

Constraint 1 (amount
of x1 available)
8

Feasible
region Constraint 2
(amount of x2
20/3 available

8/3
6 16 Y1

Optimal solution, Y1= 8/3, Y2= 20/3

Simplex method solution

Maximize. 4y1 + 5y2

75
s.t constrains

Input x1: 2y1 + y2 ≤12

Input x2: y1 + 2y2 ≤16

Y1, y2 ≥0

1st. introduce two variables called slack variables (s1 and s2). Slack variables used to convert
inequalities into equalities. 12 and 16 called the right hand side (availability of inputs).

The problem can be written as;

column
Row y1 y2 s1 s2 RHS
x1 2 1 1 0 12
x2 1 2 0 1 16
Objective 4 5 0 0 0

step

 Start with units of output with largest price or coefficient. Here y 2 selected, named pivot
column. Since the objective function is maximization, the most limiting input must be
determined

(x2 is 16, 16/2= 8) and

x1 is 12, 12/1= 12).

Hence 8<12, imply x2 is the most limiting input.

column
Row y1 y2 s1 s2 RHS
x1 2 1 1 0 12
New, ½= 0/2= 16/2=
nx2 0.5 2/2= 1 0 ½= 0.5 8
Objective 4 5 0 0 0

 The nx1 is obtained by subtracting from the old x1 the product of nx2.

column
Row y1 y2 s1 s2 RHS
nx1 1.5 0 1 -0.5 4
nx2 0.5 1 0 0.5 8
Objective 4 5 0 0 0

 Similar approach is used for objective function row. The new element for the intersection of the
objective row the RHS is 0-8*5= -40

76
 column
Row y1 y2 s1 s2 RHS
nx1 1.5 0 1 -0.5 4
nx2 0.5 1 0 0.5 8
5- 0- 0- 0-
4-0.5*5= 1*5= 0*5= 0.5*5= 8*5=
nObjective 1.5 0 0 -2.5 -40

 If all numbers appearing in the column representing outputs are 0 or negative, the optimal
solution has been found. In this example , the value at the intersection of the y 1 column
and the new objective row is positive (1.5) indicating that production of y1 will further
increase profits. The entering row is y1

column
Row y1 y2 s1 s2 RHS
nnx1 1 0 0.67 -0.33 2.67
n nx2 0 1 -0.33 0.67 6.67
nnObjective 0 0 -1 -2 -44

The optimal solution has been found that maximizes revenue from the sale of y1 and y2
subject to the two constraints. The 2.67 and 6.67 represent the output y 1 and y2,
respectively. The -40 is the negative of the objective value. The solution produce 44
revenue.

In the objective function row, s1 and s2 are the negative of imputed values of additional
unit of x1 and x2.

If one additional unit of x1 were available it would contribute one additional dollar to
revenue. Additional unit of x2 contribute 2 dollar to the revenue. These are shadow prices
for x1 and x2.shadow prices indicate the maximum amount that the farmer would be
willing to pay for the next unit of x1 and x2.

The shadow prices obtained from LP can be interpreted in exactly the same manner as the
Lagrangian multipliers obtained using classical optimization methods. In both cases they
represent the change in the obj. function associated with a relaxation of the corresponding
constraint by 1 unit. The shadow price really applies only to the next unit of the input.
Shadow price usually decline as the availability of input is increased.

If LP solution doesn’t use all available units of an input, its shadow price or implicit
worth will be zero. Additional units of an input already in excess have an imputed value
of zero and are worth nothing to the farmer.

Duality:

Any LP problem can be converted to its corresponding dual. If the primal is a constrained
maximization problem, the dual will be constrained minimization problem.

77
 The dual of the maximization problem is;

Primal :

Maximize. 4y1 + 5y2

s.t constrains

Input x1: 2y1 + y2 ≤12

Input x2: y1 + 2y2 ≤16

Y1, y2 ≥0

Dual

Maximize. 12x1 + 16x2

s.t constrains

2x1 + x2 ≥ 4

x1 + 2x2 ≥ 5

x1, x2 ≥0 imputed costs of inputs

notice: the rows of the primal are the columns of the dual and vice versa.

Slack variable have negative sign when they appear, since constraints are greater or equal to.

Row would be selected on the basis of largest ratio of the RHS.

Unit 10. Frontiers in Agricultural Production Economics

The farmer performs three functions;

i. Selects the amount of each output and mix of output to be produced in the production
process

ii. Determine the proper quantity of each input to be used and allocate inputs among the
various outputs

iii. Bear the risk associated with the production and marketing of the products.

10.1. Alternative approaches to management

Management is treated the same way as seed and fertilizer. This approach yield a production
function such as;

Y= A x1a x2b Mc

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Where; y= output, x1 and x2 are two variable inputs M is management with an elasticity of
production of c.

Production function with variable elasticity of production Y= A x1a(M) x2b (M)

Where a & b are individual production elasticity which are each a function of the ‘level’
of management M. this model suggest that a given quantity of input will somehow
produce greater output on the farm of a skilled manager than on the farm of a farm
manager who lacks skill.

A final possibility is that manager’s skill are embodied in the coefficients or parameter ’A’. the
parameter ‘A’ in a Cobb-Douglass type of production function is a sort of garbage dump,
embodying the collective influences of everything that t he researcher didn’t wish to treat as an
explicit input in the production function. One possible equation for A’ is;

A= M cz

Where Mc= management with an elasticity of production c’

Z= parameter with management variable excluded

10.2. Management and profit maximization

Inputs are categorized into Land, Labor, Capital and Management. Each input category receives
a payment. Land receives rent, labor receives wage, capital receives interest and management
receives profit.

10.3.New technology and the agricultural production function

New technology usually comes in the form of an improvement in one or more of the inputs used
in the production process.

Possible impacts of new technology;

 Raise MPP on inputs (MPP= dy/dx)

 Increase elasticity of production (Ep= MMP/APP)

 Make slope of new PF greater than the old PF for a given level of input (MPPnew >
MPPold)

 VMP of the input raises, result into increase in profit maximizing level of use

 Shift intercept but not the slope of the PF. Output with the new PF is increased relative to
the old PF, but the MPP of the inputs is unaffected.

 Lower per unit cost of production (TC/Y). one or more of input prices are reduced.

Ex1. Suppose the PF

Y= a + bx + cx2

Where a,b> 0 c<0 diminishing marginal return

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 If the new technology shift the new PF without any change in MPP of x, the parameter
‘a’ (intercept) will have increased. An increase in the MPP of X could occur as a result of
parameter b’ becoming larger or as a result of parameter c’ less negative.

Ex2. Y= a + bx1 + cx12 + dx2 + ex22 new technology that affect x 1, will not change the MPP of
x2, as there is no interaction.

Y= a + bx1 + cx12 + dx2 + ex22 + gx1x2 since there is interaction between X 1 and X2, new
technology that affect X1 will probably change the parameters b, c, and g. if g is positive, the
new technology would increase the MPP of x2 as well.

Ex3. Cob-Douglas PF, y= Ax1ax2b one explanation for parameter A’ of Cobb-Douglass type of
PF is that it represent the current state of the production technology at any point in time. A
change in the parameter A’ will change the slope of the PF and the individual MPP for both
inputs. Parameter A’ appears multiplicative in each MPP. If as a result of the new technology the
price of one of the input declines, there will normally be an increase in the use of the input that
experienced the price decrease. The use of the other inputs may increase, decrease or stay the
same depending on whether the other inputs are technical complement, competitive or
Independent.

10.4. Time and Technology

For those dealing with a problem in a static, timeless environment, the impact of new technology
are little important. A PF estimated from a single period cross-sectional data has as an underlying
assumption the state of the technology that existed at the time for which the data are available.

However, if a pF is to be estimated from data over several production periods, technology does
become of importance. Moreover, it is often difficult to find direct measure of technology over
time. To overcome this problem, time variable is incorporated into a production function.

If Cobb-Douglass PF is considered y= Ax1ax2b the effect of time might be captured by parameter

A’

A= α + β

Where α is the parameter with the impacts of technology (time) ignored, β the
parameter associated with change in technology.

If technology is assumed to affect elasticity of in put x1, the parameter a’ of x1 is

made a function of the measure of technology in this case time ‘T’.

A= ó + λT

Where ó is the base production elasticity, λ the change in Ep with respect to a

change in technology per unit of time.

More complicated production function could be easily developed that would allow for variable
rate of change in the technology. The PF becomes a Cobb-Douglass type with variable
production elasticity.

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 y= Ax1ax2b eλT

T= measurement thought to represent technology such as time.

Another approach could be to estimate separate PF for each year in the data series. This help to
get separate estimates of every parameter every period. When the technology affect all input and
difficult to separate for each input.

A similar approach is to use a Transcendental function y= Ax1a1x2a2 eλ1Tx1 + λ2Tx2

Where T is the technology measure. The values for λ1, λ2 would indicate the extent to which the
new technology favor input x1 or input x2.

10.5.Conceptual issues in estimating agricultural production functions

Estimation of PF from survey data of farmers is common. A common approach might be survey
sample farmers of certain size, with regards to quantity of seed, fertilizer, chemical, and other
inputs used and attempt to estimate a single production function as individual observation. Then
find the point where VMP= MFC point of profit/ revenue maximizing.

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