# Applications of Mathematics in Economics

Edited by Warren Page
Series: MAA Notes
Volume: 82
Edition: 1
Pages: 153
https://www.jstor.org/stable/10.4169/j.ctt5hh861

1. Front Matter
(pp. i-vi)
(pp. vii-viii)
3. Preface
(pp. ix-x)
4. Notes on the Sections
(pp. xi-xvi)
5. Problems and Subject Areas
(pp. xvii-xviii)
6. 1 Microeconomics
(pp. 1-16)
Mary H. Lesser and Warren Page

Microeconomics is the study of decisions made by individual economic units. It is often called “price theory” because the emphasis is on how prices bring decisions made by individuals (consumers or producers) into balance. Marginal analysis is key in microeconomics and decisions are “made at the margin,” meaning the decision often is reduced to “should one more unit be consumed, or produced?” When the benefit from one more unit is less than its cost then the answer is “no” and an optimum has been achieved.

Most standard textbooks at the intermediate level approach microeconomic analysis using geometry and algebra, with...

7. 2 Scenarios Involving Marginal Analysis
(pp. 17-26)
Julie Glass, Lynn Paringer and Jane Lopus

Many economic decisions involve investigating the effects of small changes on various outcomes. Analyzing the effects of small or incremental changes allows us to investigate factors affecting cost, revenue, and profit in microeconomics. Economists use the term “marginal” to connote the change due to a one unit change in something. For example, to an economist, marginal cost (M C) means the change in total cost from producing one more unit ofq, and marginal revenue (M R) is the change in total revenue from selling one more unit ofq. Since the rate of change at a value is represented...

8. 3 Intermediate Macroeconomic Theory
(pp. 27-44)
Michael K. Salemi

The organizational chart for course work leading to an undergraduate major in economics has the shape of an inverted pyramid. All students begin by completing a one- or two-semester principles of economics course. Majors next complete three intermediate theory courses including intermediate microeconomics, intermediate macroeconomics, and statistics for economics. In many universities, all three intermediate theory courses are prerequisites for field courses so that students master a common set of tools, vocabulary, and problem solving techniques before specializing.

There is a good bit of variety in the course content of the intermediate macroeconomics theory (IMAC) course. The mainstream course teaches...

9. 4 Closed Linear Economies
(pp. 45-54)
Warren Page and Alan Parks

Alinear economyis governed by linear (matrix) equations. Aclosed economyis self-contained in some way – for instance, all goods produced are consumed, or expenditures equal income. Wassily Leontief’s fundamental work [2], for which he received the Nobel Prize in Economics in 1973, explains how production interacts with consumption, labor, and prices. We present three models of a closed linear economy and show how each can attain equilibrium (state of stability). In Section 2, the Production Adjustment Model explains how production adjusts to satisfy demand, and the Price Adjustment Model finds prices that balance income and expenditures. Relating...

10. 5 Mathematics in Behavioral Economics
(pp. 55-74)
Michael Murray

Economists and psychologists who have explored the borderland between their two disciplines have created the field of behavioral economics. Economists have long understood that people don’t always maximize their satisfaction or profits in the exact fashion that standard economic theory depicts. Behavioral economics has drawn economists’ attention to numeroussystematicdeviations of human behavior that conflict with the predictions of standard economic theory². These systematic deviations can often be accounted for by psychological insights into human behavior. Social psychology and cognitive psychology have contributed particularly rich insights to behavioral economics.

This chapter introduces four topic areas in which behavioral economics...

11. 6 Econometrics
(pp. 75-90)
Rae Jean B. Goodman

Ragnar Frisch, 1969 recipient of the Nobel Prize in Economic Sciences, stated that econometrics is “the unification of economic theory, statistics and mathematics.” [2] Economic theory starts with a set of assumptions and produces a series of qualitative statements or hypotheses. Economists use data that are different from the data used in most mathematical statistics courses. These data are not usually generated by a controlled experiment. For example, economists are not able to change a tax rate to see how people respond, and then make policy recommendations. Instead they use past responses to changes in tax rates to estimate what...

12. 7 The Portfolio Problem
(pp. 91-98)
Kevin J. Hastings

“Buy low, sell high” may sound like wise advice, but it is certainly not easy to follow. If we open the daily paper to the stock quotations and see that currently a share of IBM stock sells for$103\frac{3}{4}$, down from$105\frac{1}{2}$the day before, is the stock currently “low” and ready for an advance, or is it “high” and on the way down? Even when a stock is behaving in a relatively stable way over a time period, its daily volatility might make us nervous. We all want a decent return on our investments, but not at the...

13. 8 Topics in Modern Finance
(pp. 99-114)
Frank Wang

Two most important concepts in investing areriskandreturn. Much of the economic study of these concepts falls within the sphere of modern finance, which has three major building blocks: Modern Portfolio Theory, the Capital Asset Pricing Model (CAPM, pronounced “cap-em”), and the Black-Scholes formula. In 1952, Harry Markowitz pioneered the Modern Portfolio Theory. His seminal paper [7] established the framework for selecting investments to reduce risk. Extending Markowitz’s theory, William Sharpe devised the CAPM, that describes the relationship between risk and expected return [8]. Fisher Black and Myron Scholes, assisted by Robert Merton, found an analytic formula for...

14. 9 Maximizing Profit with Production Constraints
(pp. 115-130)
Jennifer Wilson

Mathematical applications to economics are rarely introduced in Calculus II or III. This is a missed opportunity since so many important concepts in second and third semester calculus courses can be discussed in terms of production, profit, utility, and social welfare functions, which are central to microeconomics. In this paper, we focus on mathematical techniques for optimizing profit functions with and without constraints. We illustrate these techniques with examples, and provide additional problems at the end of each section for student use.

Section 2 (Production Functions) introduces production functions and discusses several of their key properties.

Section 3 (Unconstrained Optimization)...