Diksha Mission Economics No.38. Mathematical Economics

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Diksha Mission Economics No.38 Mathematical Economics

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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

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 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

𝑑𝑦/𝑑𝑥=0 and 𝑑2𝑦/𝑑𝑥2>0  convex from below

 For maximization

𝑑𝑦/𝑑𝑥=0 and 𝑑2𝑦/𝑑𝑥2<0 concave from below

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QUESTIONS FOR THE DAY

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4.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

8. 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

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22.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

21. 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

25. 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

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29. 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

34. 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

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42. 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