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MATH FOR ECONOMISTS

PURPOSE OF COURSE

Math for Economists will help you assemble a toolkit of skills and techniques to solve fundamental problems in both macroeconomics and microeconomics. The material covers both pre-calculus and calculus concepts and should help you identify the best option to solve problems. Advanced courses in economics typically utilize mathematical techniques beyond basic calculus; so, gaining practice in fundamental skills can serve as a good basis for further study. Of note, this course applies pre-calculus and calculus; this is different from “applied math,” which economists typically use to refer to probability and statistics.

This course begins with a survey of basic optimization tools and then applies them to solve problems over several periods in time. The role of risk in financial decisions is explored in relation to individual choices and macroeconomic processes. The equitable distribution of resources is then considered. A specific application, game theory, is presented as one of the major recent advances in economic theory. The final topic returns to microeconomic problems such as taxes, elasticity, and specific types of supply and demand curves.

The materials in the course are a framework to apply quantitative skills. Students should liberally use Saylor course materials ( MA001 , MA003 , MA005 , ECON103/MA101 ,

MA102 , and ECON105/MA103 ) to refresh their general techniques. This course fits into the major as a bridge between quantitative theory and specific applications to problems in economics. Completing this course can greatly help students successfully build a toolkit for the intermediate courses, ECON201 and ECON202 .

COURSE REQUIREMENTS

In order to take this course, you must:

 Have access to a computer.



Have continuous broadband Internet access.



Have the ability/permission to install plug-ins or software (e.g., Adobe Reader or

Flash).



Have the ability to download and save files and documents to a computer.

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Have the ability to open Microsoft files and documents (.doc, .ppt, .xls, etc.).



Have competency in the English language.

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Have complete dthe following courses: ECON103/MA101 , MA102 (suggested), and

ECON105/MA103 .

COURSE INFORMATION

Welcome to ECON001: Math for Economists. General information about this course and its requirements can be found below.

Course Designer: {Optional}

Primary Resources:

Yoram Bauman’s

Quantum Microeconomics with Calculus,

Version 4.02

Requirements for Completion:

Time Commitment:

Tips/Suggestions:

LEARNING OUTCOMES

Upon successful completion of this course, students will be able to:

 develop a toolkit of quantitative methods to solve problems in macroeconomics and microeconomics;

 apply specific quantitative tools to solve problems in macroeconomics and microeconomics;

 analyze specific problems in microeconomics and macroeconomics from a quantitative perspective and interpret results;

 synthesize problem solutions by combining various elements of the quantitative toolkit; and

 evaluate the results of quantitative solutions within the broader context of economic theory and practical applications.

CONTENT OUTLINE

UNIT 1: BASIC TOOLS OF OPTIMIZATION IN ECONOMICS

Time Advisory: This unit should take you approximately {X} hours to complete.

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Subunit 1.1: {X} hours

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Economics can be described as the study of how individuals, firms, and governments make the best choice among competing alternatives. Individuals seek to maximize utility or happiness, firms want to maximize profits, and governments attempt to maximize social well-being. Sometimes the possible and expected consequences of a choice can be quantified to help rank and determine the best, or optimal, selection. Economists use a variety of quantitative tools to evaluate alternatives, and this unit will build a foundation of basic optimization techniques.

Decision trees map a formal process of possible expected outcomes of a decision.

Their main value is identifying secondary and tertiary consequences of a choice. The additional, or marginal, benefits and costs at each step in the decision process can be clearly assigned. As a measure of risk, probabilities for possible outcomes can be calculated to help the decision maker identify the most likely scenarios. Remember that decision trees can be used to model almost any rational choice process.

Calculus can be used to model most marginal decisions because derivatives measure the instantaneous rate of change. In other words, calculus describes what happens when we change “just a little more” of some variable in a process. Economists can then examine the whole range of choices—considering just a little bit more and more—until all options have been exhausted to determine the optimal one. Using advanced calculus, economists can isolate a single variable or input so that decisions can be broken down into their constituent components.

Learning Outcomes

Upon successful completion of this unit, you will be able to:

 develop a toolkit of quantitative methods to solve optimization problems in macroeconomics and microeconomics;

 apply specific quantitative tools (decision trees, algebraic equations, derivatives, integrals) to solve problems in macroeconomics and microeconomics;

 analyze specific problems in microeconomics and macroeconomics from an algebraic and calculus perspective and interpret the similarities and differences among tools.

1.1

Decision Theory

1.1.1

Individual Choice Optimization

1.1.2

Firm Choice Optimization

1.2

Decision Trees

1.2.1

Definition and Structure of Simple Decision Trees

1.2.2

Example of a Monopoly’s Decision Tree

1.3

Optimization and Calculus

1.3.1

Derivatives

1.3.1.1

Definition

1.3.1.2

Assumptions

1.3.2

Partial Derivatives

1.3.2.1

Definition

1.3.2.2

Assumptions

1.3.3

Application of Derivatives

1.3.3.1

Simple Techniques

1.3.3.2

Advanced Techniques

1.3.3.2.1

Addition

1.3.3.2.2

Multiplication

1.3.3.2.3

Exponentiation

1.3.3.2.4

Subtraction

1.3.3.2.5

Division

1.3.3.2.6

Euler and Natural Logs

1.3.3.2.7

Partial Derivative Techniques

1.3.4

Integration

1.3.4.1

The Fundamental Theorem of Calculus

1.3.4.2

Antiderivatives

UNIT 2: TEMPORAL OPTIMIZATION

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Humans perceive their environment in four dimensions. The first three—length, width, and height—define our physical universe. We can pinpoint any location using those three coordinates. The fourth dimension, time, is perceived as moving linearly in one direction. Humans can observe how things change over time, and economists try to explain and predict these changes using quantitative tools.

The relative value of an asset will change over time. Some assets, like gold, remain fairly unchanged while others decay, like a wood house. Even when assets don’t decay, their relative value to goods and services may change. For example, an ounce gold coin purchased in 2006 was worth about $600 and more than doubled in value in 2010.

Present value calculations are used to figure out how much less (or more) an asset will be worth at a future time. These are important calculations for anyone saving their income.

How assets change over time has broader implications for whole societies. Economic growth models assume that investments in capital depreciate over time, and a sustainable growth pattern requires the correct rate of capital replacement. While simple models may explain growth over two periods, more dynamic ones explain changes over generations or even whole empires. Remember that economists try to assist people to make optimal choices not only today but also well into the future.

Learning Outcomes


 develop a toolkit of quantitative methods to solve optimization problems over time in macroeconimics and microeconomics;

 apply specific quantitative tools (present value calculations, organic growth qquations, derivatives) to solve problems in macroeconomics and microeconomics; and

 analyze returns to capital over time, economic growth, and period models from an algebraic and calculus perspective and interpret the differences between tools.

2.1 Present Value

2.1.1 Definition

2.1.2 Decision Tree

2.2 Distributions

2.2.1 Lump Sums

2.2.2 Annuities

2.2.3 Perpetuities

2.3 Capital Theory

2.3.1 Definition

2.3.2 Growth over Time

2.3.3 Initial Endowments

2.4 Present Value and Budget Constraints

2.4.1 Simple Compounding

2.4.2 Continuous Compounding

2.5 Applications of Concepts—Organic Growth Model

2.5.1 Explicit Costs

2.5.2 Opportunity Costs

2.5.3 A Two-Period Model

2.5.4 An Infinite-Period Model

2.5.5 Mathematical Applications to Solve Organic Growth Models

UNIT 3: FINANCIAL THEORY, RISK, AND APPLICATIONS

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Financial theory is predicated on the concept of inflation or the general rise in prices over time. Inflation has the power to erode a person’s savings or erase their debts. The rate of inflation varies over time, and its effects must be factored into a variety of quantitative economic models. In this unit, we will calculate the real value of a currency two different ways and select which is best for various models.

Risk is a measurable phenomenon. Every time a decision is made, there is a risk that the expected outcome will not be the actual result. In this respect, risk is ever-present because our environment is continually changing. The fairness of a system can be characterized by a risk and reward calculation, and models of risk tolerance can quantify an individual’s or firm’s appetite for risk.

Arbitrage is a typical example of a risk application. Generally defined as exploiting price differentials by buying a financial asset in one market and selling it in another, arbitrage involves calculating risk and reward. Economists will evaluate the similarities and differences in assets and markets over time to arrive at an optimal and potentially risk-free transaction.

Learning Outcomes


 develop a toolkit of quantitative methods to solve problems in financial theory and risk;

 apply specific quantitative tools (expected value, probabilities, and algebraic equations) to solve problems in financial theory and risk;

 analyze returns to capital over time with inflation risk, diversification, and probabilities from an algebraic perspective; and

 choose between using real and nominal interest rates when building an economic model.

3.1 Inflation

3.1.1 Nominal and Real Interest Rates

3.1.2 Inflation

3.1.2.1 Definition

3.1.2.2 Historical Trends

3.1.2.3 Significance of Problem

3.1.3 Mathematical Techniques

3.1.3.1 Using Nominal Rates

3.1.3.2 Using Real Rates

3.2 Risk

3.2.1 The Risk Process

3.2.1.1 Decision Trees

3.2.1.2 Expected Value

3.2.1.3 Fair and Unfair Bets

3.2.2 Risk Reduction

3.2.2.1 Diversification

3.2.2.2 Law of Large Numbers

3.2.3 Indifference Curves

3.2.3.1 Definition

3.2.3.2 Mathematical Application

3.3 Arbitrage

3.3.1 Definition

3.3.2 Classifications

3.3.2.1 No Arbitrage

3.3.2.2 Rush-Hour Arbitrage

3.3.2.3 Financial Arbitrage

3.3.2.3.1

3.3.2.3.2

Comparable Investments

Risk Tolerance

UNIT 4: EQUITABLE DIVISION OF RESOURCES AND CONSUMPTION

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{Etc…}

Economics is sometimes classified as the science of scarcity. Everything in our universe is scarce. There is only so much fresh water, food, and energy on the planet.

Because we are mortal, our time is scarce, and even intangible things like good ideas are not in infinite supply. Economists can quantify this scarcity to help individuals, companies, and governments make the optimal choices about how to divide our resources.

Exactly how to divide up resources has been a constant debate since ancient times.

Economists can view the problem on a very personal level, like how best to divide up a pizza pie among a group of children, or on a global scale, like assessing fishing rights in international waters. There is no one right way to measure fairness, and context must be considered. Nevertheless, the essential fairness of a decision can be measured using objective mathematical tools.

Vilfredo Federico Damaso Pareto was an Italian economist who spent much of his life measuring efficiency and fairness. He successfully quantified trade theories and distribution patterns of wealth in nations. Jeremy Bentham proposed a calculation system for pleasure and suffering to calculate “the greatest good for the greatest number.” Individuals can use a cost-benefit analysis to make better decisions about the allocation decisions they face.

Learning Outcomes


 develop a toolkit of quantitative methods to solve problems in resource allocation and product consumption;

 apply specific quantitative tools (cost-benefit analysis) to solve problems in choice theory;

 analyze differences in equity and fairness using economic definitions and tools; and

 evaluate the optimal equity solution within the broader context of economic theory and practical applications.

4.1 Introduction to Game Theory vis-à-vis Equity and Fairness

4.1.1 Five Essential Components

4.1.2 Applications and Choices

4.2 Cake-Cutting: The Problem of Fair Division

4.2.1 Definitions of Fairness

4.2.1.1 Proportionality

4.2.1.2 Envy-Free

4.2.1.3 Equitability

4.2.1.4 Pareto Efficiency

4.2.2 Questions to Guide Optimal Fairness Choices

4.2.3 International Trade

4.2.3.1 Pareto Efficient and Inefficient Choices

4.2.3.2 Coase Theorem

4.3. Economics and Social Welfare

4.3.1 Cost-Benefit Analysis

4.3.2 Pareto Efficiency Improvements

4.3.3 Examples

4.3.3.1 Taxes

4.3.3.2 Monopoly Pricing and Price Discrimination

UNIT 5: GAME THEORY


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{Etc…}

Game theory was first applied to economic problems by the Nobel-Prize-winning economist John Nash. His discoveries were dramatized in the movie A Beautiful Mind .

In essence, game theory describes and predicts how participants will make decisions based on their expectations of consequences. In the classic game, the prisoner’s dilemma, two criminals either choose to confess or maintain innocence. There are four possible outcomes and associated benefits and costs with each. The choices can be quantified, and a “Nash equilibrium” can be calculated. Although abstract, this kind of game can describe behavior in oligopolies.

Games can become substantially more complex as additional moves and choices are incorporated into models. Decision trees can be used to describe sequential move games, and a payoff matrix approximates simultaneous move games like the prisoner’s dilemma. Algorithms will provide some solutions to Nash equilibriums while partial derivatives yield answers to other games. The right tool must be matched to each kind of game.

As a practical example, game theory can be used to describe some kinds of auctions.

This kind of market has gained increasing popularity with the advent of online auctions at eBay. When calculating a price, the bidder must analyze available data, but some information is held back. The introduction of uncertainty into the process makes the optimal calculations extremely complex.

Learning Outcomes


 develop a toolkit of quantitative methods to solve problems in game theory;

 apply specific quantitative tools (decision trees, Nash equilibria) to solve problems in game theory;

 analyze differences in outcomes of auctions using game theory; and

 evaluate the optimal strategy within the broader context of economic theory and practical applications for games.

5.1 Sequential Move Games

5.1.1 Definition and Characteristics

5.1.2 Backward Induction

5.2 Simultaneous Move Games

5.2.1 Definition and Characteristics

5.2.2 Strictly Dominant Strategies

5.2.3 The Prisoners’ Dilemma

5.3 Iterated Dominance and Nash Equilibrium

5.3.1 Iterated Dominance

5.3.1.1 Definition and Characteristics

5.3.1.2 Example: Player Elimination

5.3.2 Nash Equilibrium


5.3.2.2 Algorithms for Finding Nash Equilibria

5.3.3 Infinitely Repeated Games


5.3.3.2 Trigger Strategy

5.3.4 Mixed Strategies


5.3.4.2 Finding Mixed Strategy Nash Equilibria

5.4 Auctions

5.4.1 Kinds of Auctions

5.4.1.1 Ascending Price Open Auction

5.4.1.2 Descending Price Open Auction

5.4.1.3 First-Price Sealed Bid Auction

5.4.1.4 Second-Price Sealed Bid Auction

5.4.2 Bid-Shading and Truth-Revelation

5.4.3 Auction Equivalences

5.4.4 Auctions Potpourri

5.4.4.1 All-Pay Auctions

5.4.4.2 The Winner’s Curse

5.4.4.3 Multi-Unit Auctions

UNIT 6: MICROECONOMIC THEORY AND APPLICATIONS

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Economists make many assumptions about the nature of supply and demand. Curves are smooth continuous functions and are nicely behaved in Euclidean space. A basic graphic argument can be constructed and then extrapolated to algebra and then calculus.

In the real world, these assumptions and mathematical extrapolations often need modification, but it’s a good beginning point to understand how supply and demand behave in a perfect theoretical world.

Joseph Louis Lagrange was an important mathematician in the eighteenth century; he developed a mathematical technique to calculate minimum and maximum points of a function. Originally applied in physics, his work has been used in a variety of fields from quantum mechanics to the general theory of relativity. Because economists seek to maximize (or minimize) outcomes, Lagrangian techniques are a mainstay of any economic toolkit.

Calculus is the primary tool to calculate the instantaneous slope of a function at a given point. The slope reveals the rate of change and gives insight into marginal choices a consumer or firm makes. The double derivative describes the rate of change of the rate of change. While this may sound abstract, the information indicates whether the rate of change is increasing, decreasing, or remaining the same intensity. This has applications in consumer choice and profit maximization.

Learning Outcomes


 develop a toolkit of quantitative methods to solve problems in microeconomics;

 apply specific quantitative tools (algebraic techniques, derivatives, partial derivatives, Lagrangian techniques) to solve problems in game theory;

 analyze differences in supply curves, demand curves, and utility functions using quantitative techniques; and

 synthesize problem solutions in taxation, elasticity, and optimization by combining various elements of the quantitative toolkit as presented throughout the course.

6.1 Supply and Demand

6.1.1 Movement along a Curve

6.1.2 Shift in a Curve

6.1.3 Expressing Supply and Demand Mathematically

6.2 Taxes

6.2.1 Per-Unit Taxes on the Sellers

6.2.2 Per-Unit Taxes on the Buyers

6.2.2.1 Sales Taxes

6.2.2.2 Subsidies

6.2.3 Tax Equivalence

6.2.4 Tax Incidence

6.2.5 Expressing Taxes Mathematically

6.3 Elasticities

6.3.1 The Price Elasticity of Demand

6.3.1.1 Elastic Demand

6.3.1.2 Inelastic Demand

6.3.1.3 Unitary Elastic Demand

6.3.1.4 Calculating the Price Elasticity of Demand

6.3.2 Perfectly Elastic and Inelastic Supply

6.3.3 Elasticity Examples

6.3.3.1 Total Revenue and Elasticity

6.3.3.2 Uniform Monopoly Pricing

6.3.3.3 Tax Incidence

6.3.4 Calculus Techniques for Price Elasticity

6.3.4.1 Applying Derivatives

6.3.4.2 Applying Partial Derivatives

6.4 Individual and Market Demand Curves

6.4.1 The Algebra of Markets

6.4.2 The Shape of the Demand Curve

6.4.2.1 Substitution Effect

6.4.2.2 Income Effect

6.4.3 The Shape of the Supply Curve

6.4.4 Integrating Supply and Demand

6.5 Deriving Supply and Demand Curves

6.5.1 Cost Minimization

6.5.1.1 Utility Functions and Indifference Curves

6.5.1.1.1

6.5.1.1.2

Cobb-Douglas Utility Function

Marginal Rate of Substitution

6.5.1.2 Production Functions and Isoquants

6.5.1.2.1 Cobb-Douglas Production Function

6.5.1.2.2 Marginal Rate of Technical Substitution

6.5.1.3 The Cost Function

6.5.1.4 Solving the Individual’s Cost-Minimization Problem

6.5.1.5 Solving the Firm’s Cost-Minimization Problem

6.5.1.6 The Necessary First-Order Condition

6.5.1.6.1 The Last Dollar Rule

6.5.1.6.2

6.5.1.6.3

Graphical Explanation

Lagrangian Techniques

6.5.2 Supply Curves

6.5.2.1 Choosing Output to Maximize Profits

6.5.2.2 Choosing Inputs to Maximize Profits

6.5.3 Demand Curves

6.5.3.1 Marshallian (Money Held Constant) Demand Curves

6.5.3.2 Hicksian (Utility Held Constant) Demand Curves

6.5.3.3 Comparison of Marshall and Hicks

6.5.4 Market Structures

6.5.4.1 Monopolies

6.5.4.1.1

6.5.4.1.2

Simple Monopolies

Monopolies with Price Discrimination

6.5.4.2 Duopolies

6.5.4.2.1

6.5.4.2.2

Collusion

Cournot (Quantity) Competition

6.5.4.2.3 Stackleberg (Leader-Follower) Competition

6.5.4.3 Perfect Competition Adjustments

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