Math in Economics

Susheng Wang

• Publisher: WSPC
• Year: 2015
• ISBN:
  9789814663830
• Edition: 2
• Topic: Business
• Subtopic: Business Mathematics
• Index: Business

This textbook concisely covers math knowledge and tools useful for business and economics studies, including matrix analysis, basic math concepts, general optimization, dynamic optimization, and ordinary differential equations. Basic math tools, particularly optimization tools, are essential for students in a business school, especially for students in economics, accounting, finance, management, and marketing. It is a standard practice nowadays that a graduate program in a business school requires a short and intense course in math just before or immediately after the students enter the program. Math in Economics aims to be the main textbook for such a crash course.

The 1st edition was published by People's University Publisher, China. This new edition contains an added chapter on Probability Theory along with changes and improvements throughout.

Request Inspection Copy

This textbook concisely covers math knowledge and tools useful for business and economics studies, including matrix analysis, basic math concepts, general optimization, dynamic optimization, and ordinary differential equations. Basic math tools, particularly optimization tools, are essential for students in a business school, especially for students in economics, accounting, finance, management, and marketing. It is a standard practice nowadays that a graduate program in a business school requires a short and intense course in math just before or immediately after the students enter the program. Math in Economics aims to be the main textbook for such a crash course.

The 1st edition was published by People's University Publisher, China. This new edition contains an added chapter on Probability Theory along with changes and improvements throughout.

Request Inspection Copy

Readership: Students ranging from final undergraduate year, to master's and PhD level in economics, accounting, finance, management, and marketing courses.
Key Features:

• It concisely covers main math knowledge and tools useful for business and economics studies, which includes matrix analysis, basic math concepts, general optimization, dynamic optimization, ordinary differential equations, and probability theory
• Basic math tools, particularly optimization tools, are essential for students in business schools, and especially for students in economics, accounting, finance, management, and marketing

Chapter 1

Linear Algebra

We will focus on the part of linear algebra that is most useful for business studies — matrices. Basic knowledge of linear algebra is assumed.

1. Vector

A scalar is either a real number a ∈ ℝ or a complex number z ∈ ℂ, where ℝ is the set of all real numbers and ℂ is the set of all complex numbers. A vector is an ordered sequence of numbers:

The numbers x_i are called entries, coefficients or elements. The entries can generally be complex numbers. However, except when eigenvalues are involved, we assume real entries and denote an n-dimensional vector by x ∈ ℝⁿ. For convenience, we sometimes write it as (x_i) or (x_i)_n. In business studies, for example, a vector can represent a financial asset, with its entries representing features of the financial asset.

A special vector is the zero vector: 0 ∈ ℝⁿ. Given a vector in (1.1), we can define its transpose and denote it by x′ or x ^T, with

x′ = x^T ≡ (x₁, x₂, . . . , x_n).

We typically write a vector as a vertical/column vector; its horizontal/row version is x′. For a vector x ∈ ℝⁿ, we can define its length as ||x||, where

and call it the norm. Then we can define the distance between any two points x and y in ℝⁿ by the norm ||x – y||.

Given two vectors a = (a_i)_n and b = (b_i)_n, we can define their summation a + b, subtraction a − b, and multiplication a′b, 〈a, b〉 or a · b respectively by

We can also multiply a vector a ∈ ℝⁿ by a number λ ∈ ℝ:

These operations are intuitively shown in Figure 1.1, and, if θ is the angle between vectors a and b, we have

a · b = ||a|| · ||b|| · cos(θ).

Proposition 1.1.For vectors a, b, c ∈ ℝⁿ, we have

(a) The associative law of summation: (a + b) + c = a + (b + c).

(b) The commutative law of summation: a + b = b + a.

(c) The commutative law of multiplication: a · b = b · a.

(d) The distributive law: a · (b + c) = a · b + a · c. ◼

A vector β ∈ ℝⁿ is a linear combination of vectors α₁, . . . , α_m ∈ ℝⁿ, if there exist λ₁, . . . , λ_m ∈ ℝ such that

β = λ₁α₁ + · · · + λ_mα_m.

Figure 1.1.Graphical Illustration of Vector Operations

For example, a mutual fund is a linear combination of some basic ass...