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Economics Asked by a SnapAndSolve user · about 2 hours ago

1 What is mathematical economics? Outline the 6 points that summarize the concept of mathematical...

1.1 What is mathematical economics? Outline the 6 points that summarize the concept of mathematical economics. b) State 5 noticeable criticisms against and 3 defenses in favor of the mathematical economics.
1.2 Trace historically the development of mathematical economics and the contributions of authors like W.S. Jevons; Augustin Cournot Léon Walras; Tâtonnement; Francis Ysidro Edgeworth; Vilfredo Pareto; Paul Samuelson.
2.1(a) What is the increment of a function? State the increment of the function Z=f(X,Y).
(b) State a function of two variables and outline the three ways the increments of the function can arise
(c) State the formulae for the increment of the above function. If Y=f(X), highlight the steps involved in finding the differential of Y.
2.2(a) Define the total differentiation of a function
(b) Suppose we have a function (i) Find the differential of Y if Y= (2X-5). If X increases from 5 to 5.01, find Y before and after the change. (ii) Suppose Y=9X² + 10X + 8, calculate the differential of the total value of Y, when if x is to change from 10 to 10.02.
2.3(a) state and define the seven rules of total differentiation
(b) Find the total differentials for the following functions. (i) Z=3X²+Xy-2y³ (ii) U=2X₁+5X₂+X₃

Answer

  1. Mathematical Economics: Mathematical economics involves the application of mathematical methods to represent economic theories and analyze problems. The six points summarizing the concept include:

    1. Use of mathematical symbols and equations to represent economic models.
    2. Application of calculus and algebra to solve economic problems.
    3. Development of theoretical models to predict economic behavior.
    4. Quantitative analysis of economic data.
    5. Optimization of economic functions.
    6. Simulation of economic scenarios.

    Criticisms:

    1. Over-reliance on assumptions that may not hold in reality.
    2. Complexity that may obscure understanding.
    3. Potential for mathematical errors leading to incorrect conclusions.
    4. Limited applicability to real-world problems.
    5. Neglect of qualitative factors in economic analysis.

    Defenses:

    1. Provides clarity and precision in economic analysis.
    2. Facilitates the derivation of testable predictions.
    3. Enhances the ability to handle complex economic systems.

  2. Historical Development: The development of mathematical economics can be traced through contributions by:

    • W.S. Jevons: Introduced the use of calculus in economics.
    • Augustin Cournot: Developed models of market competition.
    • Léon Walras: Formulated the general equilibrium theory.
    • Francis Ysidro Edgeworth: Advanced the theory of utility and indifference curves.
    • Vilfredo Pareto: Known for Pareto efficiency and welfare economics.
    • Paul Samuelson: Integrated mathematics into economic theory, contributing to modern economic analysis.

  3. Increment of a Function: The increment of a function refers to the change in the function's value as its variables change. For Z=f(X,Y), the increment is ΔZ = f(X+ΔX, Y+ΔY) - f(X,Y).

  4. Function of Two Variables: For a function Z=f(X,Y), increments can arise through changes in X, changes in Y, or simultaneous changes in both X and Y.

  5. Formulae for Increment: For Y=f(X), the differential of Y is given by dY = f'(X)dX. Steps involve finding the derivative f'(X) and multiplying by the change in X, dX.

  6. Total Differentiation: Total differentiation involves finding the total change in a function with respect to changes in its variables. It is expressed as dZ = (∂Z/∂X)dX + (∂Z/∂Y)dY for a function Z=f(X,Y).

  7. Differential of Y:

    1. For Y = (2X-5), if X increases from 5 to 5.01, Y changes from 5 to 5.02.
    2. For Y = 9X² + 10X + 8, if X changes from 10 to 10.02, dY = (18X + 10)dX = (18*10 + 10)(0.02) = 3.8.

  8. Rules of Total Differentiation: The seven rules include:

    1. Linearity
    2. Product Rule
    3. Quotient Rule
    4. Chain Rule
    5. Implicit Differentiation
    6. Higher-Order Derivatives
    7. Partial Differentiation

  9. Total Differentials:

    1. For Z=3X²+Xy-2y³, dZ = (6X + y)dX + (X - 6y²)dY.
    2. For U=2X₁+5X₂+X₃, dU = 2dX₁ + 5dX₂ + dX₃.

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