UGC NET MATHEMATICAL ECONOMICS MATERIAL
INPUT-OUTPUT MODEL
The Input-Output Model was given by Leontief in 1951. Before this, Quesney Tableau was used
to analyze the interdependence between industries.
Assumptions:
1. Constant Returns of Scale 2. Fixed input coefficient or requirements
3. No externalities 4. No economies and diseconomies of scale
5. No technical change or technical progress 6. Homogenous good
7. Factor supplies are given 8. Input prices are given
9. Demand is given 10. Perfect competition in the production
11. No input is unutilised or underutilised
Demand for any product should be large enough so that it can be used as inputs. The theory
is based on general equilibrium and also involves empirical investigation. The input-output
model is concerned only with production.
Types of Input-Output Model
1. Closed Input-Output Model: It does not consider any external demand but only internal
demand which can be satisfied by the industries. There is no final demand sector.
2. Open Input-Output Model: It involves inter-industry demand. Final demand is by the
households. The primary input in this model is the Labour.
3. Static Input-Output Model: In this, investment is exogenous in nature and includes both inter
industry and final demand.
4. Dynamic Input-Output Model: It is given by Dorfman, Samuelson, Solow. It is an extension of
static model but treats Investment as an endogenous variable. It highlights the capital
requirements of different sectors. Part of the output is kept to add to the capital stock. However,
the output should be large enough so that it covers Current Production, Current consumption,
and Addition to the capital stock
Capital in the beginning of any period should be large enough to be able to produce the
output in the current period.
Hawkins-Simon Conditions
These conditions are used to maintain the feasibility and viability of the system. The conditions
are as follows:
1. Determinant of (I-A) should be positive.
2. Diagonal elements of (I-A) should be positive.
3. All principal minors in the matrix should be positive.
4. Sum of input coefficient should be less than one. This condition is also known as Solow’s
condition. Mathematically it is expressed as: ⅀aij < 1
5. Sum of all elements in each column of the matrix including the labour coefficient should be
equal to one, that is, ⅀aij = 1 including labour. If we do not include labour, then ⅀aij < 1.
Conditions which must be satisfied:
1. Viability Condition: It means that no element of technology coefficient matrix can be less than
zero.
2. Feasibility Condition: Sum of elements in each column of Inverse coefficient matrix must be
equal to one.
LINEAR PROGRAMMING PROBLEM
The concept of Linear Programming Problem was given by Dantzig. It is used to maximize
or minimize the objective subject to certain constraints in the form of inequalities.
Linear programming is an optimizing technique which is aimed at maximizing or minimizing
an objective function subject to a number of constraints in the form of inequalities.
Objective function is also known as the criterion function. It is the function required to be
maximized or minimized.
Structural Constraints: These are the limitations within which optimization has to be
accomplished. They are the bounds that are imposed on the solution and are expressed in the
form of inequalities.
Feasible Solution: All possible solutions that satisfy the constraints.
Infeasible Solution: It is the situation when the feasible region is empty.
Feasible Region: It is a set of all possible, feasible solutions.
Optimal Solution: It is the feasible solution with the largest objective function.
Unbounded region: It is the region where the feasible region is continuing endlessly.
Assumptions of Linear Programming Problem
1. Linearity/Proportionality: It simply means that the relations between the variables can be
depicted in the form of straight lines, revealing the constant returns to scale.
2. Divisibility: All decision variables can take non-negative fractional values, that is, they are
continuous quantities and need not necessarily be complete units.
3. Additivity: Total value of objective function equals the sum of contributions of each variable to
the objective function.
4. Certainty: The parameters are known with certainty and the optimum solution that is derived is
predicted on perfect knowledge of all parameters.
5. Constant prices: It implies perfect competitive situation. Input-Output prices remain constant
and therefore, a perfectly competitive approach is followed.
6. Finiteness: A finite number of activities and constraints are considered in any problem.
7. Homogeneity: All units of the same resource are identical.
NON-LINEAR PROGRAMMING
Kuhn-Tucker Conditions
Maximization: Minimization:
• F’(X) ≤ 0 • F’(X) ≥ 0
• X ≥ 0 • X ≥ 0
• F’(X)*X = 0 • F’(X)*X = 0
Kuhn-Tucker Sufficiency Theorem
Max y = f(X)
Subject to: gi ≤ rj, where X ≥ 0
1. Objective function f(X) must be differentiable and concave (f’(X) ≤ 0)
2. Each constraint must be differentiable and it should be convex.
3. Kuhn-Tucker conditions must satisfy the conditions for maximization.
Game Theory : Game theory is the formal study of conflict and cooperation
Game is a strategic situation given by Neumann-Morgenstein in Theory of Games and
Economic Behaviour in 1944. It is the study of decision making where several players must make
choices that potentially affects the interest of other players.
Players, here, are the decision making agents. Moves are the actions undertaken by the players
and are thus also known as strategies.
Game: It is a situation in which two or more participants or players confront one another in
pursuit of achieving some objectives.
The game should have the following features:
1. Finite number of players
2. Finite number of strategies available to each player.
3. Each player should know the rules governing the choice of each action
Pay-Off: Utility that a player gets given a certain outcome of the game.
Pay-Off Matrix: Shows players, their actions and their pay-offs in a game.
Pure Strategy: The player will surely follow a particular action. It provides a complete definition of
how a player plays a game. It determines the move a player will make for any situation he or she
could face. The probability of a pure strategy is equal to one.
Mixed Strategy: An assignment of probability to each player in which he will randomly select a
pure strategy. So a course of action will be selected according to the probability distribution.
Dominant Strategy: In a game if one or more strategies of a player are inferior to atleast one of
the remaining strategies, then it is known as a dominant strategy.
Optimal Strategy: Optimal strategy is the course of plan which puts the player into the most
favourable situation, irrespective of what are the strategies followed by other player.
Two Person Game or N-People Game: The game which has two players is called a Two-person
game, while the game having more than two players, is known as N-Person game.
Zero-Sum Game: When the gain of one competitor is the loss of the other, then it is a zero sum
game. It is also known as a matrix or rectangular game.
Constant Sum Game: Sum of shares of two players add upto the same amount.
Minimax: Out of maximum losses, minimum value is chosen. In simple words, it is the minimum
loss out of the maximum losses. It is a decision rule in game theory for minimising the maximum
losses and is called the lower value of the game.
Maximin: It is the maximum out of the minimum and the concept is used in case of profits. It is
also known as the upper value of the game.
Fair Game: It is the point where minimax and maximin are equal to zero.
Nash Equilibrium: Each player believes that it is doing the best it can, given the strategy of the
other player. No player can improve upon the strategy unilaterally.
Co-operative Game: Game in which participants can negotiate binding contracts that allow them
to plan joint strategies. Example: Bargain between buyers and sellers.
Non-cooperative Game: Game in which negotiations and enforcements of binding contracts are
not possible.
Repeated Game: Game in which actions are taken and payoffs received over and over again.
Tit-for-Tat Strategy: Repeated game strategy in which a player responds in kind to an
opponent’s previous play, co-operating with co-operative opponents and retaliating against un-
cooperative ones.Differentiation in Economics Application I
Total Costs = TC = FC + VC
Total Revenue = TR = P * Q
Profit = TR –TC
Break Even: TR = TC
Profit Maximization: MR = MC
Application I: Marginal Functions (Revenue, Costs and Profit)
Calculating Marginal Functions
MR = d(TR) / dQ
MC= d(TC) / dQ
Applications II
Elasticity of Demand: how does demand change with a change in price……
ed= Proportional change in Demand/Proportional change in Price
ed is negative for a downward sloping demand curve
– Inelastic demand if | ed |<1
– Unit elastic demand if | ed |=1
– Elastic demand if | ed |>1
Optimization in Economics
Maximum and minimum values
For Minimization
๐๐ฆ/๐๐ฅ=0 and ๐2๐ฆ/๐๐ฅ2>0 convex from below
For maximization
๐๐ฆ/๐๐ฅ=0 and d2๐ฆ/๐๐ฅ2<0 concave from below
QUESTIONS FOR CLARIFICATION
1.An input-output model which has endogenous final demand vector is known as
A. Open input-output Model
B. Closed Input-output Model
C. Static input-output model
D. Dynamic input-output Model
2. In Linear programming problem involving two variables, multiple solutions are obtained when
one of the constraints are
A. The objective function should be parallel to a constraint that forms boundary of the feasible
region
B. The objective function should be perpendicular to a constraint that forms the boundary of the
feasible region
C. Neither a nor b
D.Two constraints should be parallel to each other
3.Given production function Q = ALK; a,b>0,
Increasing returns to scale requires that
A. A+B=1 B. A+B=0
C. A+B>0 D. A+B>1
4. Which of the following statements is true concerning the optimal solution of linear program
with two decision variables
A. There is only one solution to a linear program
B. The optimal solution is either an extreme point or is on a line connecting extreme points
C. All resources must be used up by an optimal solution
D. All of the above
5. The production function is input-out analysis propounded by w.w. Leontief has implied the
value of elasticity of substitution between input is
A. One B. Zero
C. Constant D. Infinite
6. For a Demand function p=16-q-o.5q the price elasticity of demand at q=4 is
A. 0.5
B. 0.2
C. 0.7
D. 0.3
7. Let the two regression lines be given as=3x=10+5y and 4y=5+15x.then the correlation
coefficient between x and y is
A. -0.40
B. 0.40
C. 0.89
D. 1.05
8. Simon-Hawkins conditions relate to which of the following
A. Static Leontief model
B. Dynamic Leontief Model
C. These are necessary and sufficient conditions for the solution of the model
Choose the correct answer from the codes given below
A. A and B B. B and C
C. A, B and C D. A and C
Answers:-
1) B 2) A 3) D 4) B 5) A 6) A 7) C 8) C
INTEGRATION
It is opposite process of differentiation. It is opposite process of differentiation. If we wanted
to find out function from derivative we use integration
Important formulas of Integration:
1. ∫ ๐๐ฅ = ๐ฅ + ๐
2. ∫ ๐ฅ๐๐๐ฅ =๐ฅ๐+1๐+1+๐
3. ∫ ๐๐ฅ๐๐ฅ = ๐๐ฅ + ๐
4. ∫1/๐ฅ๐๐ฅ=๐๐๐๐ฅ+๐
Some Properties of Integration
.∫[๐(๐ฅ) ± ๐ (๐ฅ) ]๐๐ฅ = ∫๐ (๐ฅ) ๐๐ฅ ± ๐ (๐ฅ) ๐๐ฅ + ๐
∫๐๐ (๐ฅ) ๐๐ฅ = ๐ ๐ (๐ฅ) ๐๐ฅ + ๐ where a is constant
Consumer Surplus
Consumer surplus is the monetary gain obtained by consumers because they are able to
purchase a product for a price that is less than the highest price that they would be willing to
pay
Area can easily be found using definite integral
๐ช๐๐๐๐๐๐๐ ๐บ๐๐๐๐๐๐ = ∫๐๐
๐–๐๐
Problem:-
Given demand function is P=15−2๐−๐²
๐น๐๐๐=2, P=15−2x2−2²
=15−4−4=7
Therefore, ๐ถ๐ = ∫ 15−2๐−๐²๐๐ฅ−7x2
15X- 2 X²/2 – X³/3 – 14
15*2- 2*4/2 – 8/3 – 14 = 28/3
Producer Surplus
Producer surplus or producers' surplus is the amount that producers benefit by selling at a
market price that is higher than the least that they would be willing to sell for.
Producer’s Surplus= px - ∫๐๐
๐
TO FIND OUT TC from MC:
If it is given to find out Total Cost from Marginal Cost, then we use Integration.
∫ ๐๐ถ ๐๐ฅ = ๐๐๐ถ + ๐ = ๐๐ถ
Example: The marginal cost function for some product is 1+2๐ฅ+6๐ฅ² where ๐ฅis output. Find
the total cost function if fixed cost is Rs.100 when output is zero.
๐๐ถ=1+2๐ฅ+6๐ฅ²
๐๐ถ = ∫ 1+2๐ฅ+6๐ฅ²๐๐ฅ
=๐ฅ+2๐ฅ²/2+6๐ฅ³/3+๐
Since fixed cost is given to be 100, ๐๐ถ=๐ฅ+๐ฅ2+2๐ฅ3+100
Similarly we can find Total Revenue from Marginal Revenue function.
IS LM Model (With Numerical Problems)
Problem -1 The following equations describe an economy: C = 10 + 0.5 Y (Consumption
function)I = 190-20i (Investment function) Derive the equations for IS curve
Problem 2: Given the following data about the monetary sector of the economy : Md= 0.4 Y
–80iMs= 1200 crores. where Md is demand for money, Y is level of income, Ms is rate of
interest and M is the supply of money. Derive the equation for LM function
Solution:
For money market to be in equilibrium:
Md= Ms
0.4 Y –80 i= 1200
80 i= 0.4 Y –1200
i= (0.4Y/80) –(1200/80)
i= (1/200) Y –15 ….. (i)
Thus we get the following LM function: i= (1/200) Y –15
Alternatively, LM equation or function can also be stated as:
Y = 200i + 3000 …(ii)
Harrod Domar Model
The growth rate of GDP can be calculated very simply.
The ICOR is defined as the growth in the capital stock divided by the growth in GDP Since
Investment ( is defined as the growth in the capital stock, the
ICOR is equal to Investment divided by the growth of GDP Investment will be equal to savings
and Savings is equal to the APS times GDP
If we divide both sides by the ICOR and we divide both sides of the equation by GDP we have
the result that the growth rate of GDP will equal the Average Propensity to Save ( by the
Incremental Capital Output Ratio
Thus if the APS is 12 and the ICOR is 3 the growth rate of GDP, G(Y), would be 4
Problem: In a Harrodian economy, ICOR is 4.5 : 1, population growth is 2% per annum, and
the investment rate is 27%. Hence the annual growth of per capita will be :
๐ฎ๐๐๐๐๐ ๐๐๐๐ ๐๐ ๐ฎ๐ซ๐ท ๐๐๐๐๐๐๐๐๐ =๐จ๐ท๐บ/๐ฐ๐ช๐ถ๐น−๐ฎ(๐ท)
=๐๐๐.๐๐−๐=๐−๐=๐
Ans: 4%
Using Harrod Domar growth equation, what is the rate of growth of a closed economy with
C/Y=3/4 and ฮ๐ฒ/ฮ๐=3? Where C=consumption, Y=output and K=Capital.
(a) 8.66% (b) 8.50% (c) 8.33% (d) 7.50%
๐บ๐๐๐๐ ๐ช/๐=๐/๐, ๐๐๐
๐ช/๐=๐๐๐, ๐๐ฅ๐ฌ๐จ ๐ฐ๐ ๐ค๐ง๐จ๐ฐ ๐ญ๐ก๐๐ญ ๐๐๐+๐๐๐=๐
Therefore APS = ๐/๐ ๐๐ง๐ ๐ฐ๐ช๐ถ๐น=ฮ๐ฒ/ฮ๐=๐
๐น๐๐๐ ๐๐ ๐ฎ๐๐๐๐๐=๐จ๐ท๐บ/๐ฐ๐ช๐ถ๐น=๐/๐/๐=๐/๐๐=๐.๐๐๐๐ ๐.๐.๐.๐๐%
Game Theory and Economics
Game theory is the study of how people behave in strategic situations.
Strategic decisions are those in which each person, in deciding what actions to take, must
consider how others might respond to that action.
Game Theory was given by John Von Neumann and Oskar Morgenstern.
Game Theory is “the study of Mathematical Models of conflict and cooperation between
intelligent rational decision makers.”
If the market is composed by a small number of firms, each firm must act strategically
Each firm affects the market price changing the quantity produced
Suppose 2 firms are
producing 100 units
If one of the firms decides to increase the production by 10 units
The market supply will increase from 200 to 210 and the price has to drop to reach an equilibrium
Therefore, it also affects the profits of other firms
Each firm knows that its
profit depends not only on how much it produced but also on how much the other firms
produce.
A game is a situation where the participants’ payoffs depend not only on their decisions, but
also on their rivals’ decisions. This is called Strategic Interactions:
My optimal decisions will depend on what others do in the game.
Four elements to describe a game:
1. players;
2. rules: when each player moves, what are the possible moves, outcomes of the moves;
3. outcomes of the moves;
4. payoffs of each possible outcome: how much money each player receive for any specific outcome
Matching Coins
Each player selects one side of a coin if the coins match player 1 wins and gets 1 dollar
from player 2 if the coins don’t match player 2 wins and gets 1 dollar from player 1.
Solutions of the Games
To predict what will be the solution/outcome of the game we need some tools:
1. dominated and dominant strategies;
2. Nash equilibrium.
Dominated and Dominant Strategy
Dominant Strategy:
A strategy that gives higher payoffs no matter what the opponent does.
“ I will do whatever I want to do, no matter what you do.”
Dominated Strategy:
A strategy is dominated if there exists another strategy that is dominant.
So far we have only assumed that each player is rational to determine the outcome of the game.
No Dominant Strategies
In most games there are no dominant strategies for all players.
We cannot use this method to predict the outcome of the game.
Nash Equilibrium
John Nash developed a criterion for mutual consistency of players known as Nash Equilibrium
The decisions of the players are a Nash Equilibrium if no individual prefers a different choice.
In other words, each player is choosing the best strategy, given the strategies chosen by the
other players.
I will do whatever I want to, provided what you do.”
It is a solution concept of non-cooperative game theory involving two or more players in
which each player is assumed to know the equilibrium strategies of the other player and No
player has anything to gain by changing only their strategy.
Saddle Point : It is the equilibrium point by Nash Analysis.
QUESTIONS FOR CLARIFICATION
1. Suppose that a unit tax of $2 is imposed on producers and that the initial equilibrium price of
the good is $10. With a vertical demand curve and an upward-sloping supply curve, we can
predict that
A. the price faced by consumers is 12 after the tax.
B. the price faced by consumers is 8 after the tax.
C. the price faced by consumers is 10 after the tax.
D. the price faced by consumers is 11 after the tax.
2. if the elasticity of supply is 2, this means that if
A. The price rises by one dollar, the quantity supplied will rise by two dollars
B. The price falls by one dollar, quantity supplied will rise by two percent
C. The price rise by one percent ,the quantity. Supplied will fall by two percent
D. The price rises by two percent ,the quantity supplied will rise by one percent
3. Which of the following measures the degree of monopoly power.
A. AR-MR/AR B. AR/AR-MR
C. MR/AR-MR D. AR-MR/MR
4. Given the production function Q=2.K1/3.L2/3.find the output level when 8 units of capital and
27 units of labour are used
A. 36 B. 54 C. 18 D. 24
5. Given the demand function as P=1/4q-1/2,consumer surplus at q=25 is
A. 1.12 B. 1.25 C. 1 D. 0.5
6. Rate of return to cover a risk of investment and decrease in purchasing power, as a result of
inflation is known as
1. Nominal rate of return
2. Accrual accounting rate of return
3. Real rate of return
4. Required rate of return
7. Which of the following is the conditions of economic viability and technological feasibility of
Leontief static model
A. Perron–Frobinius root
B. Value of the objective function of the primal should equal the value of the objective function of
the dual of the primal
C. Kuhn-Tucker conditions
D. Hawkins-Simon conditions
8. The demand and supply function are given as: Pa=30-5x and px3x-10 respectively .The
consumer surplus
A. 12.5 B. -62.5 C. -125 D. 62.5
Answers:-
1) A 2) C 3) C 4) B 5) B 6) A 7) C 8) D
