Experiment-2

RETURN TO SCALE, FACTOR SHARE, ELASTICITY OF PRODUCTION

Returns to scale: Returns to scale in microeconomics describe a production situation that
occurs in the long run when the scale of production increases when all inputs used are
variable, which affects the output level. A proportionate change in output results from a
proportionate change in input. Returns to scale explain what happens to total output when all
production inputs increase, assuming that technology is constant and the market is perfectly
competitive.
Returns to scale is a term in economics that refers to a rate at which a change in output leads
to a change in input. It is a long-run theory of production
In the short run, the firm cannot build a new factory to increase its returns to scale because it
takes time and money to build one. It can, however, hire additional workers to increase its
short-run returns, but only up to a certain point due to the law of diminishing marginal
returns. However after the factory is built, the firm can hire more workers. These changes in
the firm's inputs (labour, land, etc.) can help increase the firm's output, known as returns to
scale.

Statement of the Laws of Returns to Scale

When a producing unit increases all its inputs proportionately, technically, there are three
possibilities, i.e., the total output may increase proportionately, more than proportionately, or
less than proportionately. Accordingly, we have three laws of returns to scale which are as
follows:
1) Constant returns to scale: If a producing unit increases all its inputs in a given quantity
(say X %) and the total output also increases in the same proportion (X %). then it implies the
existence of Constant Returns to Scale. To take an example, suppose all inputs are doubles,
and then if total output is also doubled, this would have been possible only when there exist
constant returns to scale. In short, if increase in the total output is proportional to the increase
in inputs, it means a situation of constant returns to scale exists.
2) Increasing returns to scale: If a producing unit increases its inputs say by X% and the total
output increases by more than X%, then it implies the existence of Increasing Returns to
Scale. To take an example, suppose all inputs are doubled, and then if total output is more
than doubled, this would have been possible only when there exist increasing returns to scale.
In short, if increase in the total output is greater than the proportional increase in the inputs, it
means that a situation of increasing returns to scale exists.
3) Diminishing returns to scale: If a producing unit increases all its inputs say by X% and the
total output increases by less than X%, then it implies the existence of Diminishing Returns
to scale. To take an example, suppose all inputs are doubled, and then if total output is less
than doubled, this would have been possible only when there exists diminishing returns to
scale. In short, if increase in output is less than proportionate to the increase in inputs, it
means that a situation of diminishing returns to scale exists.
Production Function and Returns to Scale
The laws of Returns to Scale can be explained more precisely through the Production
Function which has already been introduced in Unit 8. Let us take a production model
involving only two variable inputs, capital (K) and labour(L) and one commodity X. The
Production Function can then be expressed as: Qx = f(K,L)
Where Qx denotes the quantity of commodity X, K stands for capital and L for labour
employed. Let us further assume that both the inputs K and L are increased in the same
proportion say p. It is quite likely that if all the inputs are increased in proportion p, the total
output may not increase in p proportion. Suppose we represent the proportion by which
output rises by h then the Production Function may be expressed as hQx = f(pK, PL)
where, h denotes the h-time increase is Qx as a result of p-time increase in inputs, K and L.
The proportion h may be equal to, greater than, or less than p, accordingly, it brings out the
three laws of returns to scales.
i) If h= p, the production function reveals constant returns to scale.
ii) ii) If h > p, the production function reveals increasing returns to scale.
iii) iii) If h < p, the production function reveals diminishing returns to scale. Let us
take a numerical example to explain three laws of returns to scale. Suppose a
producing unit employs 5 labourers and one machine which gives it an output of
say 100 units of a commodity. Assume that labour and machine are both variable
inputs such that we are considering a situation of the long run.
Let us take a numerical example to explain three laws of returns to scale. Suppose a
producing unit employs 5 labourers and one machine which gives it an output of say 100
units of a commodity. Assume that labour and machine are both variable inputs such that
we are considering a situation of the long run.
Now, suppose the producing unit doubles the labour and capital employing 10 labourers
and two machines and if the total output increases to 200 units, it means we have the
existence of Constant Returns to Scale since 100% increase in labour and capital leads to
100% increase in output.
Further suppose that the employment of 10 labourers and two machines raises total output
to 300 units, then 100% increase in labour and capital leads to 200% increase in output.
This is, accordingly, termed as Increasing Returns to Scale. Similarly, suppose that the
employment of 10 labourers and two machines raises total output to 150 units, and then
100% increase in labour and capital lead to 50% increase in output. This is a situation of
Diminishing Returns to Scale.
ISOQUANTS AND LAWS OF RETURNS TO SCALE
The concept of Isoquants can be used to express the returns to scale. Look at Figure
below where returns to scale are represented. Four Isoquants have been drawn showing
level of output 100, 200, 300, and 400 units. Line OS has been drawn passing through the
origin O. As we move along OS, the inputs of labour and capital vary. But since the OS
line passes through the origin, the ratio between labour and capital remains the same
throughout, though absolute amounts of labour and capital keep rising. So, increase in
labour and capital along line OS represents the increase in the scale. Even OT line
represents increase in the scale.
 Constant Returns to Scale Returns to scale are constant if output increases in the same
proportion as the increase in all factors. Constant returns to Scale are shown with the help
of Isoquants

Diminishing Returns to Scale

When output increases in a smaller proportion than the increase in all inputs, diminishing
returns to scale are said to prevail.

FACTOR SHARE
Factor shares and earner shares in agriculture
Any new farm technology has the income distribution effect and new rice technology
developed in 1960 is not an exception. The major criticism of the new rice technology is that
it is labour saving and reduced the returns to labor and increased the returns to other factors.
Therefore, it is important to analyse the nature of income distribution and factor shares. There
are two methods to estimate the distribution of farm earnings one is accounting method and
another is production function method. In accounting method, the observed distribution of
farm earnings among the direct participants and among the factors of production is estimated.
In production function analysis, the production elasticities of different inputs are estimated.
Further in accounting method, we can know which participants have actually benefited and
how the benefits are related to ownership of factors of production; in the production function
analysis, we get an idea of the distribution of output among inputs that would prevail if each
input was paid based on its marginal product. The concepts and the empirical examples are
discussed below:
Accounting Method
Distribution of farm earnings
Consider a tenant farm operator whose landlord shares some of the production costs; then the
share of output accruing to the three main classes involved can be calculated as:
i. payment to landlord is the value of output given as rent on land minus cost of production
given by the landlord.
ii. payment to hired labor is the sum of the wage rates times the days of all crop operations.
plus the value of output given in kind such as harvesters
iii. payment to operator is the value of output minus the sum of payments to landlord, hired
labour, and current inputs.
The output share of each earner class is arrived by dividing each payment by the total value
of output. Dividing each payment by relevant price index gives the real income going to each
earner class.
Factor shares
Factor shares are the ratio of costs of factor inputs used in the production process to the total
value of output. Factor shares can be calculated as:
 payment to current input(C) is the sum of expenses on fertilizer, and other chemicals
plus irrigation cost and rent on power sources such as tractors.
 payment to land(A) is the payment to landlord plus imputed rent of owned land.
 payment to labor(L) is the payment to hired labor plus imputed value of family labor.
 payment to capital(K) is the imputed value of the service of capital equipment.
 operator's profit is the value of output minus(i + ii + iii + iv)

The imputed costs of family labor, owned land and capital can be calculated as: the
imputed wage rate of family labor is calculated using market wage rate; the imputed cost
of land is calculated using average rental rate; the imputed cost of capital is the
depreciation, repairs and interest which is the opportunity cost of the capital. The shares
of land, labour and capital can then be measured by dividing, ii, iii, vi by their sum. The
operators residual, calculated in the event of non-availability of capital data is the value of
output minus (i+iiiii). This residual will reflect the returns to operator's capital and
management. All variables are defined in terms of flow. If the firm purchases inputs and
sells output at constant unit prices (p,i,w,1 and p. respectively), factor shares of the firm's
input are: Factor share of current input are;
Factor share of current input = PC/PQ
Factor share of capital = iK/PQ
Factor share of labor = WL/PQ
Factor share of land =rA/PQ
where, C, K, L, and A are the physical quantities of each input factor used in production,
and Q is the physical quantity of output produced.
Effects of any change, which affects an existing production function, and a relative price
structure of outputs and inputs can be observed through associated changes in factor
shares. Factor share is a powerful concept in analyses when dealing with economic
growth, technological changes, and income distribution.
Factor shares can be estimated for an economy as a whole, for an industry, for all farms in
a country or region,, or for an individual farm. It is sometime meaningful to estimate
factor shares by farm category; by crop rice or vegetables, by land condition - irrigated or
rainfed; by size small or large; or by tenure status owner operator, leaseholder, or share
tenant. The type and level of aggregation used for factor share estimation depends on the
purpose of the research. Factor and income shares can be reported on per hectare basis.
Factor payments can be presented in many ways. For crop farming, they can be expressed
per hectare or per farm. To simplify comparison, it is sometimes convenient to present
them per unit of land. It is also convenient to discuss factor payments for inter region,
inter-country and overtime comparisons in terms of relative input and output prices.
Production function method

A farm's production process is generally expressed by a production function that gives the
quantity of output as a function of the quantities of its inputs: Q=f(C,K,L,A)
The farm seeks to maximize profit in the production process. Profit (n) is the difference
between total revenue and total costs:
Π= PQ-(Pc+ iK + wL+ rA)
Substituting = Π= Pf(C, K, L,A) - (Pc+ iK + wL+ rA)
Profit is a function of inputs and is maximized with respect to these variables. Standard
 economic theory states that profit is maximized when each input is utilized to the point
where its value of marginal productivity equals its market price.
Mathematically, it is expressed as: P.f1=p, P.f2 =i, , P.f3= w, , P.f4= 4
Where fj ( j=1,2,….4) is the partial derivative of the production function with respect to the
jth input, i.e, the marginal productivity (MPP) of jth input. It is assumed that the
production function of equation 1 satisfies all conventional requirements for a production
function.
The value of MPP of an input (P.fj) is the rate at which the farm's revenue would increase
when one unit of the jth input is added to the production process, if other input levels are
held. constant. The conditions of profit maximization, equation shows the farm can
increase profit as long as an addition to the total revenue earned from using an additional
input unit exceeds its input cost.
Substituting equation for factor share equations, the factor share of labor is written as:
Factor share of labor =wL/PQ=f3.L/Q or rewriting,

f3. L/Q=Q/L. L/Q=MPP/APP

where APP is the average productivity of labor. If profit maximization is satisfied, the
factor share of an input equals its production.elasticity the proportionate rate of change of
output Q with respect to input. The factor share, which is equivalent to the production
elasticity of an input, can be expressed as a ratio of the marginal and average
productivities of the input at profit maximizing level.
ELASTICITY OF PRODUCTION

The elasticity of production, also called the output elasticity, is the percentage change in
production divided the percentage change in the quantity of an input used for that production.
For example, if a firm increases the number of workers by 10%, and the number of products
produced per month increases by 20%, the elasticity of production will be:

Elasticity of Production = 20% / 10% = 2

It is also called the partial output elasticity, because it refers to the change in the output
when only one output change (that is, it’s the partial derivate of the production function, as
opposed to the total derivative). If the production function has only one input, the elasticity of
production measures the degree of returns to scale, but usually, production function has more
than one input.

Elasticity of production measures how responsive the output of a firm is to changes in the
input of a factor of production, such as labor or capital. It is a useful concept for
understanding how firms optimize their costs and production functions, and how they react to
changes in market conditions.The elasticity of production is a measure of the responsiveness
of the production function to the change in one input.

The quantity supplied depends on several factors. Some of the more important factors are the
price of the good or service, the cost of the input and the technology of production
EXPERIMENT-

RISK ANALYSIS THROUGH LINEAR PROGRAMMING

Linear programming can be used to analyze risk through sensitivity analysis. In this method,
one objective function coefficient value is varied within the range of optimality while all
others are held constant.

Linear programming can be used in the banking sector to determine the optimal allocation of
risk among different assets. This can include loans, investments, and derivatives. The
objective could be to minimize the overall risk exposure or to achieve a target risk level while
maximizing profitability

Linear programming can also be used for financial decision-making. This can
include: Optimizing resource allocation and efficiency, Reducing costs and risks, Increasing
revenue and profit, Balancing objectives and constraints, and Exploring different scenarios
and sensitivities

Linear programming models are widely used in finance to solve various problems related
to: resource allocation, portfolio optimization, risk management, and production planning.

Linear programming is a technique for determining an optimum schedule of interdependent

activities in view of the available resources. Programming is just another word for 'planning'
and refers to the process of determining a particular plan of action from amongst several
alternatives. Linear programming applies to optimization models in which objective and
constraint functions are strictly linear. The technique is used in a wide range of applications,
including agriculture, industry, transportation, economics, health systems, behavioral and
social sciences and the military. It also boasts efficient computational algorithms for problems
with thousands of constraints and variables. Indeed, because of its tremendous computational
efficiency, linear programming forms the backbone of the solution algorithms for other
operative research models, including integer, stochastic and non-linear programming.

General algebraic simplex method.

 It also gives concrete ideas for the development and interpretation of sensitivity
analysis in linear programming. Linear programming is a major innovation since
World War II in the field of business decision making, particularly under conditions of
certainty.
 The word 'linear' means the relationships handled are those represented by straight
lines, i.e the relationships are of the form y = a + bx and the word 'programming'
means taking decisions systematically. Thus, linear programming is a decisionmaking
technique under given constraints on the assumption that the relationships amongst
the variables representing different phenomena happen to be linear.
 A linear programming problem consists of three parts. First, there objective function
which is to be either maximized or minimized. Second, there is a set of linear
constraints which contains the technical specifications of the problems in relation to
the given resources or requirements.
 Third, there is a set of non negativity constraints - since negative production has no
physical counterpart.
AIM 1. To find and know more about the importance and uses of 'linear programming'.

2. To formulate a linear programming problem and solve in simplex method and dual
problem

DATA COLLECTION IN LINEAR PROGRAMMING is a versatile mathematical

technique in operations research and a plan of action to solve a given problem involving
linearly related variables in order to achieve the laid down objective in the form of
minimizing or maximizing the objective function under a given set of constraints.

CHARACTERISTICS

• Objectives can be expressed in a standard form viz. maximize/minimize z = f(x) where z is

called the objective function.

• Constraints are capable of being expressed in the form of equality or inequality viz. f(x) =
or ≤ or ≥ k, where k = constant and x ≥ 0.

• Resources to be optimized are capable of being quantified in numerical terms.

• The variables are linearly related to each other.

• More than one solution exist, the objectives being to select the optimum solution.

• The linear programming technique is based on simultaneous solutions of linear equations.

USES There are many uses of L.P. It is not possible to list them all here. However L.P is very
useful to find out the following:

Optimum product mix to maximize the profit.

• Optimum schedule of orders to minimize the total cost.

• Optimum media-mix to get maximum advertisement effect.

• Optimum schedule of supplies from warehouses to minimize transportation costs. •

Optimum line balancing to have minimum idling time.

• Optimum allocation of capital to obtain maximum R.O.I

• Optimum allocation of jobs between machines for maximum utilization of machines.

• Optimum assignments of jobs between workers to have maximum labor productivity.

• Optimum staffing in hotels, police stations and hospitals to maximize efficiency.

• Optimum number of crew in buses and trains to have minimum operating costs.

• Optimum facilities in telephone exchange to have minimum break downs. The above list is
not an exhaustive one; only an illustration.

ADVANTAGES
• Provide the best allocation of available resources.

• meet overall objectives of the management.

• Assist management to take proper decisions.

• Provide clarity of thought and etter appreciation of problem.

• Improve objectivity of assessment of the situation.

DEFINITION OF TERMS : There are instances where number of unknowns (p) are more
than the number of linear equations (q) available. In such cases we assign zero values to all
surplus unknowns. There will be (p-q) such unknowns. With these values we solve 'q'
equations and get values of 'q' unknowns. Such solutions are called Basic Solutions.
 The variables whose value is obtained from the basic solution is called basic
variables.
 The variables whose value are assumed as zero in basic solutions are called non-basic
variables. :
 A solution to a L.P.P is the set of values of the variables which satisfies the set of
constraints for the problem. : A feasible solution to a L.P.P the set of values of
variables which satisfies the set of constraints as well as the non-negative constraints
of theproblem. :
 A feasible solution to a L.P.P in which the vectors associated with the non-zero
variables are linearly independent is called basic feasible solution Linearly
independent: variables x1 , x2 , x3......... are said to be linearly independent if k 1
x1+k 2 x2+.........+k n x n=0, implying k 1=0, k 2=0,........
A Feasible solution of a L.P.P is said to be the optimum solution if it also optimizes the
objective function of the problem.
 Linear equations are solved through equality form of equations. Normally,
constraints are given in the "less than or equal" (≤) form. In such cases, we add
appropriate variables to make it an "equality" (=) equation. These variables added
to the constraints to make it an equality equation in L.P.P is called stack variables
and is often denoted by the letter 'S'. Eg: 2x1 + 3x2 ≤ 500 2x1 + 3x2 + S1 = 500,
where S1 is the slack variable
 Sometime, constraints are given in the "more than or equal" (≥) form. In such
cases we subtract an approximate variable to make it into "equality" (=) form.
Hence variables subtracted from the constraints to make it an equality equation in
L.P.P is called surplus variables and often denoted by the letter 's'. Eg: 3x1 + 4x2 ≥
100 3x1 + 4x2 - S2 = 0, where S2 is the surplus variable.
 Artificial variables are fictitious variables. These are introduced to help
computation and solution of equations in L.P.P. There are used when constraints
are given in (≥) "greater than equal" form. As discussed, surplus variables are
Graphical L P solution The graphical procedure includes tw steps :
1. Determination of the solution space that defines all feasible solutions of the
model. 2. Determination of the optimum solution from among all the feasible
points in the solution space. The proper definition of the decision variables is an
essential first step in the development of the model. Once done, the task of
constructing the objective function and the constraints is more straight forward.
2. FORMATION OF MATHEMATICAL MODEL OF L.P.P There are three forms : •
General form of L.P.P • Canonical form of L.P.P • Standard form of L.P.P These
 are written in 'statement form' or in 'matrix' form as explained in subsequent
paragraphs
Linear programming, a mathematical method used to optimize the allocation of
resources, utilizes several key terminologies. Here's a brief overview of some
commonly used terms:

1.Decision Variables: These represent the quantities or levels of the decision to be

made in the problem. They are typically denoted by ( x1, x2, ..., xn ).

2.Objective Function: The objective function defines the quantity to be maximized or

minimized in the optimization problem.
3.Constraints: Constraints are restrictions or limitations placed on the decision
variables. They define the feasible region of solutions. Constraints can be of different
types:
- Equality Constraints:These constraints enforce specific equality relationships
among the decision variables.
-Inequality Constraints:These constraints enforce restrictions on the decision
variables in the form of inequalities.
- Non-negativity Constraints: These constraints specify that decision variables must
be non-negative

4.Feasible Region:The feasible region is the set of all possible solutions that satisfy all
the constraints in the linear programming problem.

5. Optimal Solution:The optimal solution is the solution that maximizes or minimizes

the objective function while satisfying all the constraints. In graphical terms, it's the
point within the feasible region that either maximizes or minimizes the objective
function.

6. Optimal Value:The optimal value is the value of the objective function

corresponding to the optimal solution.

7. Shadow Prices (Dual Prices):In linear programming with constraints, shadow prices
represent the rate of change in the optimal value of the objective function per unit
change in the right-hand side of a constraint. These prices indicate the marginal value
of additional resources or changes in constraints.

8. Binding Constraint: A constraint is binding if its associated decision variable is at

its upper or lower limit in the optimal solution. These constraints determine the shape
of the feasible region.

9. Unbounded Solution: An unbounded solution occurs when the feasible region

extends indefinitely, and the objective function can increase or decrease without limit.

These terminologies are fundamental in understanding and formulating linear

programming problems and interpreting their solutions.
EXPERIMENT-
TECHNOLOGY, INPUT USE AND FACTOR SHARES
Technology is nothing but the application of improved knowledge on production
relationships and thus technology has the effect of raising the production function. More
output per unit of input is possible with the new technology.
This indicates that production can be Increased with improved technology through the same
amount of inputs that were used with traditional technology or the production level with
traditional technology can be reached with fewer inputs with improved technology.
Consider the Figure, where curve AB refers the traditional technology production function
and curve AC refers the improved technology production function. Traditional technology
needs X1 units of input to produce Y1 units of output and X2 units of inputs to produce Y2units
of output.
However, with improved technology. Y2 units of output can be produced with X1 units of
input. The difference between X1, and X2 units input is the saving in input due to improved
technology, which can be used in additional areas to increase the production. Thus improved
technology helps in the efficient use of inputs. particularly when the inputs are scarce.

Improved technology also reduces cost of production. For the same output different
technologies have different cost. As cost curves are mirror images of product curves, the MC
and AVC are lower due to higher MPP and APP under improved technology. The lowest cost,
under improved technology is arrived when MCi intersects AVCi, and the lowest cost(c1)
under traditional technology is arrived when MC, intersects AVC. However, c, is below c, and
the difference indicates the effect of improved technology on cost of production.
New technology shifts the optimum input level. It is evident from Figure that with the new
technology, the allocative (price) efficiency is improved, as the profit maximizing level of
input occurs at a higher level (X2) with new technology than with existing technology (X1).
For example, if the total yield increases, then the MPP also increases. So the higher MPP with
the given input-output prices will move the economic optimum to a higher level of input use.
Thus new technology encourages higher Level of input use which futher helps to achieve
both technical and allocative efficiencies.
increases the MPP and APP, which in turn causes MC and AVC to decrease. The efficiency
level a of these technologies is: improved technology, α= 1.0; intermediate technology, α= 0.7
and traditional technology, α = 0.5. As shown in Figure with improved technology the
rational production stage starts when the output per unit of input is Y1units, with intermediate
technology. the rational stage starts when output per unit of input is Y2units and with
improved technology. the rational stage starts when output per unit of input is Y3 units. The
effect of these technologies on production can be explained as:

Given the production function, Y=a+b1X+b2X2+b3X3

When efficiency level is incorporated, the production function becomes,

Y=a+b1 αX+b2 (αX )2 + b3 (αX )3

The efficiency incorporated profit equation is

π =Py(a+b1 αX+b2 (αX )2 + b3 (αX )3 -px.X

Profit maximum level of input use is,

d π /dX=Py(b1 α + 2b2 α2X + 3b3 α3X2 -pX = 0

Using the estimated production function,

Y=44.00+33.69X+66.40X2 - 4.42X3

he optimum number of irrigations with varying technologies are:

Traditional technology (α=0.5), X=19.84

Intermediate technology (α =0.7), X=12.56

Improved technology (α =1.0), X=10.10

This indicates that when the input-output prices are constant, to get the same level of profit
with improved technology, the traditional and intermediate technologies use more water, thus
wasting the available water supplies. This has serious implications for water allocation
decisions. For example if every farmer is given a water supply based on economic optimum
level of 10.10 irrigations per hectare, then intermediate and traditional technologies will
reduce the yield correspondingly as indicated below:

Given the economic optimum level of input use(X=10.10), the yield under different
technologies is arrived using the estimated production function and efficiency parameters:
Given the estimated, efficiency incorporated production function (8.6) the yield under
different technologies are:

Improved technology: Y=2603.80kgs.

Intermediate technology: V=2039.18kg .

Traditional technology: Y=1338.26kgs.