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Sanathanan Velluva

Demand Function
Demand is a multivariate relationship, that is , it is determined by many factors
...
Demand function is
expressed as
Q = f(p,Y,Ps, Pc, Ta, A etc)
Where,
Q is the quantity demand of good X
P is the price of X
Y is the income of the consumer
Ps is the price of substitute
Pc is the price of complementary goods
Ta is the taste and preference of the consumer
A is the level of advertisement
The simplest of form of the demand function is written is Q =f(P) - ceteris paribus
Linear and Non –linear demand functions
The demand function Q =f(P) can be modeled by the simple linear equation
P = a-bQ, where a and b are constants
...
(example Qd = 12-2p)
A typical non-linear demand equation takes the following form Qd = α/P eg Qd = 8/p

P

P
a

vertical intercept
P = a - bQ
Q= α/P
Slope is -b

Horizontal intercept
a/b
Q
Linear demand curve

Q
Non linear demand curve

(Practice drawing demand curves based on Lecture notes)

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Supply function
There are several variables that influence the supply of good X
...
Mathematically, supply function takes the
following form
Qs = f(P,C,Po,Te,N,O)
Where Qs is the quantity supplied of good X
P is the price of the good X
C is the cost of production
Po is Price of other goods
Te is the technology
N is the number of other producers in the market
O is other factors
The simplest of form of the supply function is written is Qs =f(P) - ceteris paribus
...
Eg, Qs = 20p
A supply function is shown graphically
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Slope = d

Law of Demand : The law of demand states that there is a negative, or inverse,
relationship between price and the quantity of a good demanded and its price
...

The law of supply states that there is a positive relationship between price and
quantity of a good supplied
...

Equilibrium Price
Equilibrium price is the price determined by the forced of demand and supply
...
Qd = Qs

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( Given Qd= 2- 0
...
2 +0
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A Price Elasticity of Demand: PED measures the responsiveness (sensitivity) of quantity demanded to
changes in the price
...

percentage change in quantity demanded
Ed = -----------------------------------------------Percentage change in price

% ∆Qd
----------% ∆P

For a demand function Q = f(P) ed = | dQ/dP x P/Q|
1
For a demand function P = f (Q) ed = ------ x P/Q
dP/dQ
Marginal function
Or Ed = -------------------------Average Function
Types of Price Elasticity of Demand
1
...
Price
elasticity of demand is less than one
...
Elastic Demand: Quantity demanded responds strongly to changes in price
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3
...

4
...

5
...

(Draw graph for each case and explain)

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Along a linear demand curve of the type P = a – bQ elasticity varies from -∞ to zero

-∞
>1
=1
<1
= 0

Constant Elasticity Curve: For demand function of the type QPn = C, Ed will be equal to the power of
P
...
Try these questions
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]
Relationship between Elasticity of Demand and Revenue
(a) Relationship between Elasticity of demand and TR
The total revenue test is a method of estimating the price elasticity of demand by observing the
change in total revenue that results from a price change (when all other influences on the quantity sold
remain the same)
...

▪ If a price cut decreases total revenue, demand is inelastic
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(a) Relationship between Elasticity of demand and MR and TR
Marginal revenue is defined as the change in total revenue as the number of units sold changes
...
So, there is a
relationship between elasticity and marginal revenue
...

If Price falls and demand is inelastic we know TR falls and so MR is negative
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(c) Marginal Revenue and Ed are related MR = AR (1 – 1/Ed) or P (1 – 1/Ed)
(d) Ed = AR/AR – MR or P/P - MR
( For the above Give proof from class room notes)
B Income Elasticity of Demand
Income elasticity of demand measures how much the quantity demanded of a good responds to a
change in consumers’ income
...

percentage change in quantity demanded
Ey = ------------------------------------------------

% ∆Qd
-----------

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Percentage change in Income

% ∆Y

Ey = dQ/dY x Y/Q
If the income elasticity of demand is greater than 1, demand is income elastic and
the good is a normal good
...

If the income elasticity of demand is less than zero (negative), the good is an
inferior good
...


C Cross elasticity of demand
CED measures how much the quantity demanded of one good responds to a change in the price of
another good
...

The cross elasticity of demand for
a substitute
The cross elasticity of demand for a complement is negative
...
Price elasticity of supply is the percentage
change in quantity supplied resulting from a percentage change in price
...

Supply is unit elastic if the supply curve is linear and passes through the origin
...
)

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Supply is perfectly elastic if the supply curve is horizontal and the elasticity of supply is infinite
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or U = f(q1,q2,q3, …… qn)
...
An IC gives all combinations of x and y for which utility, U, has the same value
...

Total Utility: The amount of satisfaction obtained by consuming specified amounts of a product per
period of time
...
Assigning numerical values
to the amount of satisfaction
Ordinal Utility: Not assigning numerical values to the amount of satisfaction but
indicating the order of preferences
...

If U = f(x), then dU/dx = MUx = U’
If U = f(x,y), then we have MU of x and MU of y
∂U/∂x = MUx is the marginal utility w
...
t to good X
∂U/∂y = MUy is the marginal utility w
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t to good Y
As the quantity consumed of a good increases, the marginal utility from consuming it decreases
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Qty of goods
Production Function
A production function illustrates the relationship between input and output
...
The production function shows
the maximum quantity of output that can be produced as a function of the quantities of inputs used in
the production process
...
For most firms,
the capital, is fixed in the short run
...
( law of variable
proportion)
The Long Run The long run is a time frame in which the quantities of all resources
can be varied
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Average Product
Average product is defined as the ratio between total product and number of units of variable
factor
...

For a production function Q = f(L)
MPL = Change in Total Product / Change in Variable Factor
OR
MPL = ∆ TP/ ∆ L or dQ/dL
For a production function Q = f(L,K)
MPL = ∂Q/∂L and MPK = ∂Q/∂K
Relationship between AP and MP ( Prove that when AP got a maxima AP=MP or MP always crosses
at the maxima of AP

For a production function Q = f(L), APL = Q/L
d(APL) = d(Q/L) = L x dQ/dL) – Q) x 1/L2= 0
Rearranging this we have
dQ/dL = QL or MP of labour = AP of labour ( hence proved)

Note: The Law of Variable Proportion
The law of variable proportions examines input-output relationship when output is
increased by varying the quantity of one factor (variable input) while other factors are
kept constant
...
(explain three
stages and its importance)

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RETURNS TO SCALE - long run production
The laws of returns to scale explain the behavior of output in response to a proportional and
simultaneous change in inputs
...

When a firm increases both the inputs proportionately, there are three possibilities Total output may
increase more than proportionately Total output may increase proportionately Total output may
increase less than proportionately Accordingly, there are three kinds of return to scale Increasing
returns to scale Constant returns To Scale and Decreasing returns to scale
Let a production function Q = f(L,K)
If we increase both the factors by λ (Lambda) times , we have Q = f(Lλ,Kλ)
...
Assume the scalar λ got a power n
...

⚫ If doubling all inputs yields less than a doubling of output, the production function is said to exhibit
decreasing returns to scale
...


Graph given below examine these three cases (Explain)

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A typical example of Linearly homogeneous production function is Cobb-Douglas Production function
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Mathematically, C = C(y) where y is the optimal output level
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Symbolically a cost function is given as
C = f(Q,T,Pt)
C= cost, Q = output, T = technology, Pt = price of factors
Cost function can be modeled as C = f(q) + b , where q = f(x1, x2)
( Where f(q) is variable cost and b is fixed cost)
Cost Concepts
Total Costs – TC of producing any output is defined as the minimum cost that must be incurred to
produce that output
...

– Fixed costs are those costs that do not vary with the quantity of output produced
...

TC = TFC + TVC or C = f(q) + b

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Total fixed cost is the same at each output level
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Total
cost, which is the sum of TFC and TVC also increases as output increases
...

Average Costs: Average costs can be determined by dividing the firm’s costs by the quantity of output
it produces
...
Average total cost (ATC) is
total cost per unit of output
...
AFC = b/q
Average variable cost (AVC) is total variable cost per unit of output
...
It
is the slope of the TC curve
...


MC =

(change in total cost) TC
=
(change in quantity)
Q
MC = dc/dq = f ‘(q)

The shape of the MC curve is U shaped (draw)

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Relationship between different Costs
The AFC curve shows that average fixed cost falls as output increases
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As output increases, average variable cost falls to a minimum and then increases
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The MC curve is very special
...
Where AVC is rising, MC is above AVC
...

Similarly, where ATC is falling, MC is below ATC
...

At the minimum ATC, MC equals ATC
...
MC curve always passes through the
minimum of AC
...
When MC is less than
AC, average cost must be falling
...

If MC < AC, then AC is falling
If MC = AC, then AC is at its minimum
If MC > AC, then AC is rising
Mathematical Proof
We know AC = C/q
Therefore, the critical points in AC can be obtained by taking the first derivative
...
A Economics 2020(private circulation only)
By Dr
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Ec = MC/AC = (dC/dq) / C/q or dC/dq x q/C
Revenue function
A firm receives revenue when it sells output
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TR = PQ
Marginal Revenue: the marginal revenue is the rate of change in total revenue per unit
increase in output
MR = d(TR)/dq or dR/dQ
Average Revenue : AR is defined as the revenue per unit of output sold
...
It implies that AR function and demand function are
same
The relationship between AR and MR (define and AR and MR)
(For a demand function p= a-bq show that slope of MR is twice as that of the slope of AR?)

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AR and MR curves have well defined graphical and mathematical relationship
...
See graph below
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Similarly setting MR function equal to zero we can see output at zero MR
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In the graph, it is seen that OA is half of OB
Revenue curves under Perfect Competition: Since the price is same for all the sellers, in a
perfectly competitive market, AR=MR=P is a horizontal line parallel to output axis
...
Therefore, MR curve passes through the mid of AR and Y axis
...


P = a-bq
AR=MR=P
MR=a-2bq

AR and MR under PC

AR and MR under Monopoly
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Profit function
Profit is the difference between Revenue and Cost
...
Cost is given as C(q)
Economic profits () are the difference between total revenue and total costs
 = R(q) – C(q) = P(q)q – C(q)
The necessary condition for choosing the level of q that maximizes profits can be found by
setting the derivative of the  function with respect to q equal to zero ( Necessary condition)

d
dTR dTC
=  ' (q) =
−
=0
dq
dq
dq

or

dTR dTC
=
dq
dq

MR=MC

To maximize economic profits, the firm should choose the output for which marginal revenue
is equal to marginal cost
...
Symbolically, C = f(Y)
...
Hence consumption
is dependent on income
...
C = a + bYd
Where C = current consumption
a = autonomous consumption
b = marginal propensity to consume
Yd = disposable income
b = MPC = ΔC/ΔYd is the slope of the consumption function
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The y intercept of the consumption function is a (autonomous consumption)
An example of linear consumption function is C = 100 + 0
...
Symbolically,
S = f(Y)
...
S = -a + (1- b)Yd
Where S = current saving
-a = autonomous saving
1-b = marginal propensity to save
Yd = disposable income
1-b = MPS = ΔS/ΔYd is the slope of the saving function
The y intercept of the saving function is - a (autonomous consumption)
An example of linear saving function is S = -100 + 0
...


In this analysis Keynes has used two technical attributes or properties of consumption
function and two attributes to saving function
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A Economics 2020(private circulation only)
By Dr
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Symbolically: APC = C/Y
Where: C : Consumption, Y : Income
Average Propensity to Save (APS) :It is the ratio of total saving to total income
...
That is APS = S/Y
Since APS is a counterpart of APC, both together constitute total income
...
APC falls as income increases because the proportion of income spent on
consumption decreases but APS (Average propensity to save) increases
...

Symbolically: MPC = ∆C/∆Y or dC/dY or b
Where, ∆C = change in consumption and ∆Y = change in income
For a linear consumption function C = a +bYd, MPC is given by dC/dY = b
An interesting feature of MPC is that it obeys a double restriction, 0 < MPC < 1
Marginal Propensity to Save (MPS): It is the ratio of incremental (changing) saving to
incremental (changing) income
...

Therefore, it is expressed as: MPC + MPS = 1 or MPS = 1 –MPC
We know, Y = C+S
A change in income leads to change in both C and S, hence we have
∆Y = ∆C+∆S
Dividing through by ∆Y we have, ∆Y/∆Y = ∆C/∆Y +∆S/∆Y = 1 = MPC + MPS
Graph given below shoes MPC and MPS

C

Y= C+S

C,S

Y = C+S

C = a+bY
S = -a + (1-b)Y

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O

O
Y

Y

Features of MPC:
(1)The value of MPC is greater that zero but less than one (0 < MPC < 1)
(2) MPC cannot be negative (always positive)
(3) As income increases MPC may fall
...

(2) Ordinarily, APC and MPC both declines as income increases but MPC declines at a faster
than decline in APC
...


Major inferences of Keynes
...


“the fundamental law, upon which we are entitled to depend with great confidence,
……is that men (women) are disposed, as a rule and on the average, to increase their
consumption as their income increases, but not by as much as the increase in their
income
...
It implies that as income
increase, consumption also increases but the increase in consumption is less than
proportionate to increase in income
...
Consumption is a stable function of real disposable income
...
The consumption income relationship is reversible
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In the long run a smaller proportion of income will be consumed as income increases

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ISOQUANT (Equal Product Curve Or Iso Product Curve or production indifference curve)
An isoquant shows the different combinations of two factors (say Labour and Capital) which
produce same level of output
...


K

Qconst = f(L,K)

L

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Along the isoquant the maximum possible output is constant
...
In an isoquant map output increases as we move to higher from isoquants
...


MRTS is defined as the rate at which one factor is substituted for another factor without altering the
level of output
...

The slope of the isoquant at any point measures the marginal rate of technical substitution—the
ability of the firm to replace capital with labor while maintaining the same level of output
...

Also there, exist a relationship between MRTS and the marginal product of two factors
...
That is dq = 0
...

( students are advised to draw graph from the above table to examine MRTS)

CHARACTERISTICS OF ISOQUANTS ( Properties): The isoquants have the same general
characteristics of ICs
...

Isoquants are negatively slopped: This is because increase in the quantity of one factor is
followed by decreased in the quantity of other factor
...
This is given by the Marginal Rate of Technical Substitution (MRTS)
...
That is as we move down
along an isoquant, the absolute value of its slope or MRTS declines and the isoquant is convex
...
That is the quantity of
labour increases and quantity of capital reduces
...
Hence we need only less and less capital for compensating the loss of labour
...
( draw graph)
( that is d2K/dL2> 0 Second derivative is more than zero)
Two isoquants will never interest
Higher isoquants shows higher levels of output
In between two isoquants there are large number of isoquants
Isoquant specify cardinal measure of output
Isoquants never touch either X or Y Axis
Elasticity of Substitution σ
A better measure for factor substitution is elasticity of substitution
...
The elasticity of substitution is defined as the percentage change in the capital labour raio
divided by the percentage change in MRTS
...


Homogeneous Production Function: A production function is said to be homogeneous of degree n
if multiplying each inputs by a constant λ changes the value of the function by the multiple λ
...
Now if we take λ outside we have λ f(L,K)
...

then the function is homogeneous of degree n, then
f(L λ,K λ) = λ nf(L,K) = λ nQ
If n=1 we have production function with first degree and CRS, n<1 DRS and n> 1 we have IRS
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Linearly Homogeneous Production Function: Production functions homogeneous of the first
degree are commonly referred to as linearly homogeneous production functions
...
The best known linearly homogeneous production function is cob-Douglas production
function
...

Definition Q = AL α K 1- α

or

Q = AL α Kβ

(Where α +β = 1)

Q = Output, A = Efficiency parameter, L and K are Labour and Capital, α and 1- α are elasticity
coefficient of labour and capital respectively
Properties ( for proof see lecture notes)
It is a linearly Homogeneous Production Function ( A typical case of constant returns to scale)
Marginal Product of Labour MPL= α
...
APK or 1-α Q/K
MRTSLK = α/1- α X Q/L
Elasticity of substitution = σ, which is equal to one (1)
Labour Share is equal to α
Capital Share is equal to 1- α
It satisfies Euler’s Theorem ( Product Exhaustion Theorem), that is if factors are paid
according to Marginal Product the product will be exhausted
That is Q = MPL L + MPK K
Cobb Douglas Production Function And Returns To Scale
In the case of Cobb Douglas Production Function if α + (1- α) ( or α +β = 1) there is constant
returns to scale ( give proof from notes)
If α + (1- α) <1 , there is diminishing returns to scale or α +β < 1
α + (1- α) > 1 there is increasing returns to scale or α +β > 1
ISOCOST LINE
An isocost line includes all possible
combinations of labor and capital that can be
purchased for a given total cost
...

The slope of isocost is given by the ratio of
prices of two factors, that is, w/r
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LEAST COST INPUT COMBINATION – PRODUCER’S EQUILIBRIUM
(mathematical derivation – use lecture notes)
1
...
We use
isoquants and isocost to examine the producers equilibrium
...
Give Graph
...
It is the point where the isoqaunt curve just touches the isocost line
...
Slope of the isoqaunt curve and isocost line are the same at this point
...
Minimisation of cost for a given level oof output
...
In this case we have one isoquant and several isocost
...
In other words LP covers problems in which a linear function is
maximized or minimized subject to a system of linear inequalities
...

The linear model consists of the following components:
– A set of decision variables
...

– A set of constraints
...
The objective function should be expressed as a linear function of decision
variables
...
If the primal of the objective
function is to maximize output, then its dual will be the minimization of cost
...
As far as a
consumer is concerned his budget is a constraint
...

a11X1 + a12X2 ≤ b1
a21X1 + a22X2 ≤ b2

Constraints

In the above example the maximum quantity of b1 and b2 available is the constraint
...
That is X1, X2, …Xn ≥ 0
Feasible Region: The set of all points that
satisfy all the constraints of the model is
called a feasible region
Optimal solution is a feasible solution that
results in the largest possible objective
function value when maximizing (or smallest
when minimizing)
...
That is the problem
to be anlysed through linear programming is
expressed entirely in terms of linear
equations
...


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Mathematical Formulation of LP Problem
A
...

Zmax = X1P1 + X2P2 + X3P3 + X4P4 +
...
a1nXn ≤ b1
a21X1 + a22X2 + a23X3 +
...
a3nXn ≤ b3

...
amnXn ≤ bm
Non negative constraint is given as
X1, X2, …Xn ≥ 0

Constraints

Non-negativity constraint

We can also write down the general LP problem in the abbreviated form with the help of
summation sign and matrix algebra
...
m)
j=1
Xj ≥ 0 ( j= 1,2,3,
...
a1n
X1
b1
a21 a22 a23
...
a3n
X3 ≤ b3

...

am1 am2 am3
...
Minimisation Case ( m variables and n constraints)
The dual of the above maximization problem is summarized below
C = b1y1 + b2 y2 +
...
m
Subject to the constraints,
a11y1 + a21 y2 + a31y3 +
...
+ am2ym ≥ c2

...

a1ny1 + a2n y2 + a3ny3 +
...
am1
a21 a22 a32
...


...

a1n a2n a3n
...
But, actually, corresponding to
every minimisation problem (programme) there always exist a counterpart maximisation problem
...
The original problem
is usually referred to as the primal problem, and its counterpart is known as the dual problem
...
Suppose the primal problem is a maximization problem
...

Maximise, Z = C1 X1 + C2 X2
Subject to the Constraints,
a11X1 + a12X2 ≤ b1
a21X1 + a22X2 ≤ b2
Non Negativity constraint

X1, X2  0

In matrix notation this can be written as,
Objective function is [C1 C2]
X1
X2
Constraints are given as
a11 a12
a21 a22

X1
X2

b1
b2
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Dual Minimisation Problem
From the above a primal problem its dual may be formulated as followed
Minimise, C= b1y1 + b2 y2
Subject to the Constraints,
a11y1 + a21 y2 ≥ c1
a12y1 + a22 y2 ≥ c2
Non Negativity constraint
In matrix notation this can be written as,
Objective function is [b1 b2]
y1
y2
Constraints are given as
a11 a21
a12 a22

Y1, Y2  0

y1
y2

c1
c2

(Students interested to write ‘n’ variable ‘m’ constraint for presenting primal dual can use the
section from Mathematical formulation of LP given above)
An Example is given below
Primal:
Maximise,
Zmax =
14x1 + 12x2 + 18x3
Subject to
2x1 + x2 + x3 ≤ 2
x1 + x2 + 3x3 ≤ 4
x1 ≥ 0 x2 ≥ 0 x3 ≥ 0

Dual:

Minimise,
Zmin
Subject to,

2y1 + 4y2

2y1 + 1y2
1y1 + 1y2
1y1 + 3y2
y1, y2≥0

≤ 14
≤12
≤ 18

Another example of the primal-Dual Relationship

Solution to LP
1
...
Graphic
Approach (see Class Note)
2
...

1
...
Transportation problems: To decide the routes, number of units, the choice of factories so that
the cost of operation is a minimum
...
Manufacturing Problems: To find the number of items of each type that should be made so as to
maximise the profits
4
...

5
...
Job assigning problems : to assign jobs to workers for maximum
results subject to restrictions of wages and other costs
...
Military problems: Military applications include the problem of selecting an air weapons system
against the enemy so as to keep them pinned down and at the same time minimise the amount of
aviation gasoline used
...
The linear progrmming problem assumes the linearity of the objective function and
constraints
...
Such
problems cannot be solved by LP technique
...
There is no guarantee that the solution by LP technique will give integer valued solution
...
In such cases, LP cannot be used
...
LP model does not take into consideration the effect of time and uncertainty
4
...
But in real life situation,
they are frequently neither known nor are they constants
5
...
When above situations arise, other techniques such as integer
programming can be utilised
...
Sometimes large scale problems cannot be solved with LP technique even when assistance
of computer is available
...

Slack variable: A variable used to convert a less than or equal to a constraint into an equality
constraint by subtracting it from the left hand side of the constraint
A Feasible solution: It is a solution which satisfies all the constraints including the non negativity
constraints
Feasible region : It is the collection of all feasible solutions
...
The dual variables are the variables of the dual LPP

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Input-Output Analysis
Input Output analysis examines the interdependence of the economy as a whole
and studies the pattern of movements of intermediate products from one industry to
other industries and the consumers, given the resources and the state of technology
...
His model appeared in the "Structure of the American Economy" published
in 1936
...
The economy is divided into 'n' different interdependent sectors or industries
2
...

3
...
That that is, Xij = aijXj or aij = Xij/Xj,
4
...
It shows that if each of the Xij is multiplied by a constant, the corresponding
Xj is multiplied by the same constant
...
No technical progress: This assumption ensures a constant aij
...
One single activity for each output:
7
...
An industry may use some of its own product
...

8
...
Xn be the output levels of 'n' sectors so that the input output transaction table is
given as:
Input Output Transaction Table
Producing Total Output Input Requirement of Producing Sectors Final
Sectors
Demand
1
2
3
N
1
X1
X11
X12
X13 ……
X1n
C1
2
X2
X21
X22
X23 ……
...

Xnn
Cn
Since the total output of an industry is fully used up either in industries as input or in meeting
final demand we can write this in the form of the following equations, called balance
equations
...
+ X1n + C1
X2 = X21 + X22 +
...


...


...
+ Xnn + Cn
Here Xij is the output of the ith sector which is used as input in the jth sector and Ci is the final
demand of the ith sector which includes private demand, government demand, exports over
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imports stock of capital formation,etc
...

Xi = Xi1 + Xi2 +
...
X1n
X21 X22
...

Xn1 Xn2
...
For instance, Column 1 indicates the
amounts of inputs used by the first industry from all industries including itself
...

The Technology Matrix
We define aij = amount of Xi needed to produce one unit of Xj, and consider these quantities
as the entries of a matrix A = (aij)
...
That is, the
amount of ith commodity required to produce a unit of jth commodity
...

Input Output Transaction Table
(Structural form)
Producing Total Output
Sectors
1
X1
2
X2
3
X3

n

Xn

Input Requirement of Producing Sectors
Final
Demand
1
2
3
N
a11x1 a12x2 a13x3 …… a1nxn
C1
a21x1 a22x2 a23x3 …… a2nxn
C2
a31x1 a32x2 a33x3 …… a3nxn
C3

an1x1

an2x2

an3x3

……

annxn

Cn

X1 = a11X1 + a12X2 +
...
+ a2nXn + C2
……………………………
...


...

Xn = an1Xn + an2Xn +
...
For instance, the first column of A shows the inputs
required from all sectors for producing one unit of output in the first sector
...

Equation (1) can be rearranged as
X - AX = C, and using the fact that IX = X, where I is the identity matrix,
(I - A) X = C
(2)
Thus, provided the matrix (I – A) is invertible, there is a unique solution
X = (I - A) -1 C (3)
Here (I-A) -1 is called the Leontief inverse
...
Similarly, (3) is used to obtain the total output when A and C are
given
...
Leontief
...
The transaction table is given as
Transaction Matrix
Industries Inputs to
Inputs to Final
Total
Agriculture Industry Demand Output
1
2
Labour

X11
X21
X01

X12
X22
X02

C1
C2
0

X1
X2
X0

Then the leontief production function takes the form, X1 = F1 (X11, X21, X01) and X2 = F2
(X12, X22, X02), Where X1 and X2 are total outputs
...

Adding across the rows, we get the balance equations,
X11 + X12 + C1 = X1 (1)
X21 + X22 + C2 = X2 (2)
X01 + X02 + 0 = X0 (3)
35

Using the proportionality assumption we get the following input-output coefficients aij =
Xij/Xj
...


Technological Coefficient Matrix
Industries
1
2
labour

Inputs to
a11
a21
a01

Inputs to
a12
a22
a02

Final
C1
C2
-

Total
X1
X2
X0

Using the proportionality assumption, our structural equation take the form:
a11X1 + a12X2 + C1 = X1 (4)
a21X1 + a22X2 + C2 = X2 (5)
a01X1 + a02X2 + 0 = X0 (6)
SOLVING AN INPUT-OUTPUT SYSTEM
In a two sector economy, we have the structural equations,
a11X1 + a12X2 + C1 = X1
a21X1 + a22X2 + C2 = X2
In matrix format,
a11 a12
a21 a22

X1
X2

In matrix symbols,

C1
C2

+

=

X1
X2

AX + C = X (1)
C = X - AX
(2)
C = (I-A) X
X = (I-A) )-1 C
(3)

Therefore,

Hawkin-Simon Conditions and Economic viability
In a two industry example we have,
1 - a11

-a12

-a212

1-122

(I-A) =

Then the H-S condition states that (a) all diagonal elements are strictly positive, that is, 1-a11
>0 and 1-a22 >0, (b) all principal minors are positive, that is,
│1-a11 │ > 0

and

1 - a11 -a12
-a21 1-122

>0

That is : │1-A │ > 0
Applications or uses of I-O Analysis
36

1
...
Similarly, we may use the
method in planning, company transactions, social accounting, military mobilisation,
transport, location, trade, regional economics, etc
...
It is useful in the study of effects generated in the economy as a result of changes
made in one or more sectors of the economy
3
...
In a situation of less than full employment resulting from the deficiency of aggregate
demand, suitable changes in final demand in the I-O table can suggest the level of
economic activity necessary to achieve full employment
...

5
...
Structural disequilibrium
arises when there is inefficient, improper use of available resources, shortage of
entrepreneurs, unemployment, scarcity of capital and foreign exchange, etc
...
production, consumption and balance of payments, etc
...
It does not allow for substitution among inputs
...

2
...
In practice, these assumptions hold only in a stationary
economy
...
Model does not tell us anything as to how technical coefficients would change with
changed conditions
...
The assumption of linear equations, which relates output of one industry to inputs of
others, appears to be unrealistic since factors are mostly indivisible, increase in
outputs do not always
5
...

6
...

7
...


37



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