Mathematical Economics: Theory and Applications

Author: Zhen Xianshui
Publisher:
Publish Date: 1999-07-01
Features:
This book is divided into 6 chapters. Chapters 1 and 2 focus on microeconomics; Chapter 3 discusses five fundamental issues of mathematical economics; Chapters 4 and 5 discuss computable general equilibrium analysis and balanced growth in linear multi-sector economic systems; Chapter 6 discusses balanced growth and optimal growth of macroeconomic systems. At the end of the book, there is an appendix on the mathematical equation and block diagram representations of economic systems, as well as exercises and their reference answers. This book can be used as a textbook for undergraduate and graduate students in economics and management, as well as for economic workers for study and reference.
Excerpt:
Chapter 1 Utility Functions and Demand Functions
This chapter starts from the basic definitions of economics and explores how to quantify social and individual goals. First, it provides the mathematical expressions of several commonly used utility functions in practical applications, and then, based on the utility maximization principle, it gives the corresponding mathematical expressions of several commonly used demand functions.
1.1 Assumptions and Mathematical Expressions of Utility Functions
What is economics? The definition of economics is: using limited resources to arrange production reasonably, and then reasonably distributing the produced goods among consumers to achieve the greatest satisfaction for humans in the present and future. The greatest satisfaction of humans is the goal of the economic system. The primary task of quantitative economics is to provide a quantitative description of human satisfaction or the goal of the economic system. Some people believe that "human satisfaction" cannot be quantified, but this view is incorrect. We often hear reports such as: "China's overall economic situation in 1997 was better than in 1996." If U（97）represents the overall economic goal value of China in 1997, then this statement means: U（97）＞U（96). Quantitatively describing the overall economic goal value can indicate the specific numerical value of the goal, such as whether U（97）is 50 or 30, or it can only indicate the order of the goal values corresponding to different states. Therefore, when we say, "China's economic situation in 1997 was better than in 1996," we have actually made a quantitative conclusion about the economic goal value. In our daily lives, when reading newspapers or listening to the news, we encounter many similar quantitative conclusions about economic goals. When the government makes decisions on major economic issues, it also cannot do without a quantitative description of the economic goal value. For example, with industrial production and people's consumption levels rising, environmental pollution is becoming increasingly severe. Now, if there is 100 million yuan available for either developing production or treating environmental pollution, the government has an infinite number of strategies to choose from: Strategy 1: Use 50 million yuan to treat environmental pollution and the remaining 95 million yuan for production; Strategy 2: Use 100 million yuan to treat environmental pollution and the remaining 90 million yuan for production. Without a doubt, the government must choose one strategy from the infinite number of options. If Strategy 2 is chosen, it means: U（2）＞U（1). That is to say, Strategy 2 is the best, as it can bring the greatest satisfaction to the people in this region in the present and future. So, how can we be sure that Strategy 2 will achieve the greatest economic goal value? We cannot make various quantitative conclusions based solely on experience and feelings. Therefore, the primary task of quantitative economics is to provide a quantitative description of the economic system goal value. Quantitatively describing the economic system goal value can be done in many different ways. Since human satisfaction is related to the satisfaction of all individual members, the economic system goal value should be a function of individual satisfaction levels U1, …, Um: U＝U（U1，…，um）（1.1）
where Ui is the satisfaction level of the ith individual. The specific mathematical expression of the human happiness function in Equation (1.1) will be further discussed in Section 3.3.