Econ 340: Introduction to Mathematical Economics (2017)

General informationThursday, 4:30-7:10 PM, Buchanan Hall (formerly, Mason Hall), D005, Fairfax campus.

Econ 306: Intermediate Microeconomics

General informationEnnio E. Piano, Department of Economics and F.A. Hayek Program, George Mason University,Buchanan Hall (formerly Mason Hall), D137-7.epiano@gmu.eduOffice hours: Thursday, 9:00-10:30 AM or by appointment. Buchanan Hall, F.A. Hayek program office, D137-7.Course objectivesThe course is aimed at familiarizing the students with the basic tools of economic analysis, including the theory of consumer behavior, the theory of seller behavior, partial and general equilibrium theory, and game theory. The focus will be on the application of these tools to the analysis of human behavior in all its forms.PrerequisitesPrinciples of Microeconomics (ECON 103), Principles of Macroeconomics (ECON 104), and one semester of calculus (MATH 108 or 133). I expect you to be familiar with the use of algebra to solve simple sets of simultaneous equations, graphing, and to understand the concept of a derivative. ReadingsRequired textbooksHirshleifer, J., Glaezer, D., and Hirshleifer, D. 2005. Price theory and applications: Decisions, markets, and information (7th edition). Cambridge University Press.ArticlesAllen, D. W. "Property rights, transaction costs, and Coase: One more time." (https://drive.google.com/file/d/0B3CY-zqfWBQCRTlLSDltcVJ2Vzg/view?usp=sharing)Buchanan, J. M. "Cost and choice." (https://drive.google.com/file/d/0B3CY-zqfWBQCdnNXU3hOeEZzTTQ/view?usp=sharing)Cowen, T. "Public goods." (https://drive.google.com/file/d/0B3CY-zqfWBQCZjJJdWpXREc0VVk/view?usp=sharing)Shughart III, W. "Public Choice." (https://drive.google.com/file/d/0B3CY-zqfWBQCRlZpTG9DSjN1Q1U/view?usp=sharing)Staten M. and Umbeck J. "Economic inefficiency." (https://drive.google.com/file/d/0B3CY-zqfWBQCQjlzUERzQnpWRlE/view?usp=sharing)Varian, H. "How to build an economic model in your spare time." (https://drive.google.com/file/d/0B3CY-zqfWBQCMFY2dVlkYTBnMW8/view?usp=sharing)Recommended readingsAlchian, A. A., and Allen, W. R. 1969. Exchange and production: Theory in use. Wadsworth.David Friedman. Price Theory: An Intermediate Text. 1985. (Available free online: http://www.daviddfriedman.com/Academic/Price_Theory/PThy_ToC.html)Scheduled outlineAugust 28th-30thClass overview and introduction. Hirshleifer, chapter 1.September 4th-6thPreferences and choice. Hirshleifer, chapter 3; Buchanan, "Cost and choice."September 11th-13thConsumption and the derivation of the laws of demand. Hirshleifer, chapter 4. Quiz 1.September 18th-20thDemand theory, extensions. Hirshleifer, chapter 5. Quiz 2.September 25ht-27thPure exchange. Hirshleifer, chapters 2 and 14. Quiz 3.October 2nd-4thProperty rights, transaction costs, and the Coase theorem. Hirshleifer, chapter 14; Allen, D. W. "Property rights, transaction costs, and Coase: One more time." Quiz 4.October 10th-11thThe economics of time. Hirshleifer, chapter 15. Quiz 5.October 16thMidterm review.October 18thMidterm.October 23rd-25thThe theory of the firm and introduction to production theory. Hirshleifer, chapter 6.October 30th-November 1stThe theory of the competitive firm. Hirshleifer, chapter 7. Quiz 6.November 6th-8thThe theory of the monopolistic firm. Hirshleifer, chapter 8. Quiz 7.November 13th-15thIntroduction to game theory and the theory of oligopoly. Hirshleifer, chapter 10. Quiz 8.November 20th-27thWelfare economics. Hirshleifer, chapter 16; Cowen, "Public goods;" Staten and Umbeck, "Economic inefficiency." Quiz 9.November 29th-December 4thPublic choice theory and "How to build an economic model in your spare time!" Hirshleifer, chapter 17; Shughart, "Public choice." Quiz 10.December 6thFinal review.Other important datesSeptember 5th, last day to add and drop classes without penalty.September 19th, last day to drop with a 33% tuition penalty.September 29th, last day to drop with a 66% tuition penalty.November 22nd-26th, Thanksgiving recess.December 9th, last day of class.December 13th-20th, exam period.GradingYour grade in this course consists of ten weekly quizzes (30%), a midterm exam (35%), and a final exam (35%).Quizzes: Quiz questions are based on the assigned readings and lectures. This means you cannot and will not do well in this class unless you do the assigned readings and attend the lectures. The quizzes will always be taken on Wednesday. I will drop the lowest quiz grade to allow for unexpected events. No make-up quizzes are available. No exceptions. Midterm: The midterm date is October 16th. You cannot make-up the midterm under any circumstance. If you know ahead of time of some extenuating circumstance that will prevent you from taking the midterm, you must contact me ahead of time (i.e. at least 48 hours prior to the exam). In the event that you are excused from taking the midterm upon contacting me, I will shift your midterm grade towards your final (so your midterm will be worth 0% and your final 60%). If you do not contact me ahead of time and do not take the midterm, you will receive a zero. No exceptions. Final: The final exam date is to be announced. The final exam is comprehensive and you must take it on this date. If you know ahead of time of some extenuating circumstance that will prevent you from taking the final on the scheduled date, you must contact me ahead of time (i.e. at least one week prior to the exam). In the event that you are excused from taking the final on the scheduled date upon contacting me, we will arrange an alternate date. If you do not contact me ahead of time and do not take the final on the scheduled date, you will receive a zero. No exceptions.The grading scale is as follows:A+: 97-100%; A: 92-96%; A-: 88-91%; B+: 84-87%; B: 80-83%; B-: 76-79%; C: 70-75%; F:<70%.Academic integrityGeorge Mason University’s Honor Code requires all community members to maintain the highest standards of academic honesty and integrity. Cheating, plagiarism, lying, and stealing are all prohibited. Honor Code violations will be reported to the Honor Committee. Plagiarism is not accepted (statements from Macon web site: http://mason.gmu.edu/montecin/plagiarism/htm#plagiarism). The use of electronic devices is prohibited during an exam or a quiz; failure to comply with this will result in your failure of the assignment and potentially the failure of the class. Make sure to familiarize yourself with the GMU Honor Code, which is stated in the George Mason University Undergraduate Catalog.

Homework 1:Macro

ExercisesSolve the following differential equations:1) \(\dot{y}+y=4\)\(y\left(0\right)=0\)\(\dot{y}+y=0\)\(\dot{y}=-y\)\(\frac{dy}{dt}=-y\)\(dy=-ydt\)\(-\frac{1}{y}dy=dt\)\(-\int\frac{1}{y}dy=\int dt\)\(-ln(y) +c_a = t + c_b\) With \(c_b - c_a = c\)\(e^{ln(y)} = e^{-t} e^c\) With \(A=e^{c}\)\(y=Ae^{-t}\)\(\dot{y}=-Ae^{-t}\)Now we substitute using the conditions:\(-Ae^{-0}+0=4\)\(A=-4\)\(y=4-\dot{y}\)\(y=4+Ae^{-t}\)\(y=4+Ae^{-t}\)\(y=4-4e^{-t}\)2) \(\dot{y} =23\)\(y(0)=1\)\(\frac{dy}{dt} =23\)\(dy=23dt\)\(\int dy = 23\int dt\)

Econ 340: Week 2

# Miscellaneous

## Summation notation

- Take the sum \(x_1+x_2+x_3+...+x_n\)
- We can express this sum using the summation symbol \(\Sigma\): \(\sum_{i=1}^n x_i=x_1+x_2+x_3+...+x_n\)
- What does the notation mean? What does the
**superscript**mean? The**subscript**? - IMPORTANT: Whenever I do not specify superscript and subscript, always assume \(i=1, n=n\)
- How do we interpret the following? \(\Sigma_{i=30}^{35}N_i\)
- Do example 1, p56

## Rules of sums & Newton's binomial formula

**Additivity property**: \(\Sigma(a_i+b_i)=\Sigma a_i + \Sigma b_i\)**Homogeneity property**: \(\Sigma (c\cdot a_i)=c\Sigma a_i\), where \(c\) is a constant- From the homogeneity property, it follows that \(\Sigma c=n\cdot c\). Why?
- Do example 2 p60: Derive the fact that the sum of the difference between \(x_i\) and its
**arithmetic mean \(\mu_x\)**is equal to zero: \(\Sigma (x_i-\mu_x)=0\)

### Useful formulas

**Gauss' formula**: \(\sum i= \frac{1}{2} n(n+1)\)- Proof:

- \(x=1+2+...+(n-1)+n\)
- We can rewrite this as \(x=n+(n-1)+...+2+1\)
- From these we can write \(2x=[n+(n-1)+...+2+1]+[1+2+...+(n-1)+n]\)
- Rearranging we get \(2x=(n+1)+(n-1+2)+...+(2+n-1)+(1+n)=(n+1)+(n+1)+...+(n+1)+(n+1)\)
- Finally, we get \(2x=n\left(n+1\right)\). Solving for \(x\): \(x=\frac{1}{2}n(n+1)\). QED

- Two other useful formulas are \(\sum i^2=\frac{1}{6}n(n+1)(2n+1)\) and \(\sum i^3=[\sum i]^2\)
**Newton's binomial formula**: \((a+b)^m= a^m+\left( \begin{array}{c} m \\ 1 \end{array} \right)a^{m-1}b+...+\left( \begin{array}{c} m \\ m-1 \end{array} \right)ab^{m-1}+\left( \begin{array}{c} m \\ m \end{array} \right)b^m\) where the**binomial coefficients \(\left( \begin{array}{c} m \\ k \end{array} \right)=\frac{m(m-1)...(m-k+1)}{k!}\)**are defined for \(m=1, 2, ...\) and \(k=0,1,2,...,m\)- Application of Newton's formula:

- \((a+b)^3=a^3+\left( \begin{array}{c} 3 \\ 1 \end{array} \right)a^{2}b+\left( \begin{array}{c} 3 \\ 2 \end{array} \right)ab^{2}+b^3\). We can use the formula from above to write \((a+b)^3=a^3+\frac{3}{1}a^{2}b+ \frac{3\cdot 2}{1\cdot 2}ab^{2}+b^3=a^3+3a^2b+3ab^2+b^3\)

## Double sums

- It's possible to calculate the sum of sums by using the following formula: \(\sum_{i=1}^m \sum_{j=1}^na_{ij}\)
- Do example 1 p65: \(\sum_{i=1}^3\sum_{j=1}^4(i+2j)=\sum_{i=1}^3[(i+2)+(i+4)+(i+6)+(i+8)]=\sum_{i=1}^3(4i+20)=(4+20)+(8+20)+(12+20)=84\)

## A few aspects of logic

- Do example 1, p66
**Propositions**: assertions that are either true or false. When an assertion contains one or more variables for which it can be true or false we say it is an**open proposition**- The
**implication arrow**: \(\Longrightarrow\) - \(P\Longrightarrow Q\) you read as "p implies q" or "if p then q"
- The
**logical equivalence arrow**: \( \Longleftrightarrow\) - \(P \Longleftrightarrow Q\) you read as "p if and only if q"
- Do example 2 p67
**Necessary and sufficient conditions**:

- If \(P \Longrightarrow Q\) we say that p is
**sufficient condition**for q - If \(P\Longrightarrow Q\) we say that q is
**necessary condition**for p

- Explain

## Solving equations

- Do examples 3-4 pp68-69

## Mathematical proofs

- Every mathematical theorem can be formulated as an implication. Indeed, all proofs of mathematical theorems rely on the establishment of implications between premises (or assumptions) and conclusions.
- Examples of proof methods:

**Direct proof**: We start from the premises and we keep deriving their implications until we get to the conclusions.**Indirect proof**: We deny the conclusions and show that the premises must also be false. The indirect proof relies on the fact that \(P\Longrightarrow Q\) is equivalent to \(\backsim Q \Longrightarrow \backsim P\) (non-q implies non-p)

- Do example 1 p72

## Essentials of set theory

- We partially covered this last week
- You should be familiar with the following notions/notations
**Set**and**elements**: \(S= \{e_1, e_2, e_3, ..., e_n\}\)- Two sets,
**A**and**B**, are said to be equal if every element in**A**is also an element in**B** - Do examples

### Property of a set

- We can use the following notation to specify the property of a set: \(S=\{e: p\}\), where e=typical elements and p= defining properties
- Example: the budget set. \(B=\{(x, y): px+py\leq m, x\geq 0, y\geq 0\}\). Explain.

### Set membership

- We covered this in class last week. Remember the meaning of the following symbols:

- \(\subset\)
- \(\subseteq\)
- \(\in\)
- \(\notin\)

### Set operations

**Union**: \(\cup \). All the elements that belong to__at least one__of the sets.**Intersection**: \(\cap\). All the elements that belong to__both__sets.**Minus**: \(\setminus\) .All the elements that belong to one set but not the other.

- Do example 1 p76
- Other important notions:

**Disjoint set**: the empty set. For example, the intersection between two sets that do not share any members**The universal set**, \(\Omega\): The set containing all potential subsets of a family of sets.**Complement set**: \(A^c=\Omega\setminus A\). It contains all elements of the universal set not contained in A.

Demand for insurance

Let's \(EU=X^\alpha\) be an individual's VNM expected utility function, where \(\alpha<1\), that is to say, the individual is risk averse. The individual's income is \(W_a\) with probability \(p\) and \(W_b\) with probability \((1-p)\). The individual has access to the insurance market, where he buys \(I\) units of insurance at the price \(s\) per dollar of insurance. The individual's expected utility maximization problem is

Max \(EU=p(W_a-sI)^{\alpha}+(1-p)[W_b+(1-s)I]^{\alpha}\) with respect to \(I\)

The first order condition yields

\(-\alpha s p (W_a-sI)^{\alpha-1}+\alpha (1-p)(1-s)[W_b+(1-s)I]^{\alpha-1}=0\)

Rearranging yields

\(sp(W_a-sI)^{\alpha-1}=(1-p)(1-s)[W_b+(1-s)I]^{\alpha-1}\)

Exponentiating both sides of the equation by \(\frac{1}{\alpha-1}\) yields

\((sp)^{1/\alpha-1}W_a-[(1-p)(1-s)]^{1/\alpha-1}W_b=[(sp)^{1/\alpha-1}+[(1-p)(1-s)]^{1/\alpha-1}]I\)

Solving for the optimal value of \(I^*\) gives us the individual's demand for insurance

\(I^*=\frac{(sp)^{1/\alpha-1}W_a-[(1-p)(1-s)]^{1/\alpha-1}W_b}{[(sp)^{1/\alpha-1}+[(1-p)(1-s)]^{1/\alpha-1}]}\)

Note that the demand for insurance is increasing in \(W_a\) and decreasing in \(W_b\), as well as increasing with respect to the probability of the event against which the individual is insuring (this is a bit roundabout to derive, but it follows from the fact that \(\alpha<1\).

Problem Set #1

Introduction to Differential Equations.

Solve the following differential Equations:

1) \(\frac{dy}{dt}+y\left(t\right)\ =\ 4\ \ ;\ \ y\left(0\right)=0\)

Homogeneous

\(\frac{dy}{dt}+y\left(t\right)\ =\ 0\)

\(\frac{dy}{dt}=\ -y\left(t\right)\)

\(dy\ =\ -y\left(t\right)dt\)

\(\int dy=\int -y\left(t\right)dt\)-

\(y\left(t\right)=e^{-t+c}\)

\(y(t)=e^{-t}e^c\)

\(y(t) = Ae^{-t}\)

Particular

\(0+y(t) = 4\)

\(y\left(t\right)=4\)

General Solution

\(y\left(t\right)=Ae^{-t}+4\)

Specific Solution

\(y\left(0\right)=Ae^{-t}+4=0\)

\(y\left(0\right)=Ae^{-\left(0\right)}=-4\)

\(y\left(0\right)=A\left(1\right)=-4\)

\(A=-4\)

\(y\left(t\right)=-4e^{-t}+4\)

2) \(\frac{dy}{dt}=\ 23\ \ ;\ \ y\left(0\right)=1\)

\(dy\ =\ 23dt\)

\(\int dy\ =\ \int 23dt\)

\(y\left(t\right)\ =\ 23t+c\)

specific solution

Econ 365: Week 1

# The facts of the social sciences

## Who is F.A. Hayek?

- Hayek was a 20th century Austrian economist
- He work on a variety of issues, from the theory of the trade cycle to monetary theory, from epistemology and philosophy of economics to legal theory, from political philosophy to the history of ideas
- He was awarded the nobel prize in 1974 for these contributions

## Why did he write this paper and who were the targets?

- Starting in 1937 (Economics and Knowledge) Hayek started noticing that economics was gravitating more and more towards methodological formalism. He though this was due to the uncritical adoption of the (supposed) methodological principles of the physical sciences
- Thus Hayek's criticism of behaviorism, the belief that human beings respond to external stimuli in completely passive and therefore exactly predictable ways
- Hayek adopted a sophisticated formulation of methodological dualism (later he rejected this notion thanks to the influence and Popper and shifted towards his own formulation of methodological universalism based on the notion of explanation of the principle)

## The methodology of the social sciences

- What do we mean by "social sciences"?
- Demography vs economics, law, sociology, linguistic etc.
- The origins of scientific knowledge in the social sciences
- The role of introspection: We know more about human behavior thanks to the fact that we intuitively understand our fellow men than we would if we were to merely observe them as if they were atoms
- Example: The Marxian and the train station
- One of the most important problems in the social sciences is that of classification
- Individual things, facts, and events do not exist out there
- In the social sciences, the things are what people think they are
- It's not physical properties that matter, but the relationship between objects (even non existing objects) and the mind of individual agents. As social scientists, we must impute these meanings in order to make sense of our observations
- Take money: the notion of money (universally/generally accepted medium of exchange, store of value) must exist in the mind of the social scientist before he can ever identify it
- Of course, the human mind is fallible, thus we can make mistake
- The monology assumption: the LOGICAL structure of the human mind is the same across cultures, ethnicities, etc.
- Human action implies the universal notion of purposefulness (means-ends, causation, time, etc.)
- Praxeology as "only a kind of logic." We can derive all of the fundamental laws of economics starting from the notion of choice.
- Derive the law of demand.
- Once we have classified human behavior and the emergent results of human behavior in society (exchange, conflict) we can start to look outside the window and make sense of it
- This is where timology comes in (interpretation of human events through the imputation of meaning onto the minds of the agents)
- What is a battle? What is a revolution? Can we observe a government?
- What we refer to as "historical facts" are really just theories
- Social theory is EPISTEMOLOGICALLY and LOGICALLY prior to historical experience and therefore analysis
- There is no such thing as historical analysis without theory. There is only untested and tested (logically and empirically) theories
- A theory that has been so derived can never be falso (as long as it does not contain logical inconsistencies) but merely irrelevant for the purpose at hand

# The economic approach

## Who is Gary Becker?

- Arguably the greatest social scientist of the second half of the 20th century
- Like Hayek, wrote on a variety of issues that were thought to be outside of the scope of economics proper (racial discrimination, the family, crime, addictive behavior, fertility, politics, and so forth)
- He was awarded the nobel prize in 1992
- Champion of "economic imperialism" and of the economic approach to human behavior

## What is the economic approach?

- How to define economics?
- By its scope (Coase, Knight, Buchanan?): 1) Allocation of material goods; 2) the market
- By its method (Mises, Becker, Tullock)
- Beyond homo economicus

### The three fundamental assumptions of the economic method

- Maximizing behavior
- Market equilibrium
- Fixed (given) preferences

- What economics does not assume: perfect information, superhuman rationality, perfect foresight

### Maximizing behavior

- This is the same as purposeful behavior from above. People strive to achieve goals. In doing so the have to make choices.
- Maximization does not require rationality. But as a first approximation, we should not attribute peculiar forms of behavior to irrationality
- Problem: what do we need by rationality. Instrumental vs. epistemic rationality.
- Instead, when we do observe peculiar behavior we should investigate further. Most likely, we ignored or failed to identify some major cost associated with the course of action we initially predicted
- In fact, animals maximize too. Are animals rational?
- This means the same as that people respond to incentive. Is the opposite true?
- Mayweather vs Mcgregor

### Market equilibrium

- Extending the notion of markets
- What do we mean by equilibrium
- Prices, shadow prices, and costs

### Fixed (given) preferences

- Defining preferences
- Not the common meaning of the word
- "Underlying preferences ... over fundamental aspects of life": Health, prestige, sex, etc.
- Why don't individual maximize happiness?
- Do they maximize on "life"? Is all death suicide, in some sense?
- The importance of tradeoffs

## Implications

- The implications of these three principles are far reaching
- The law of demand
- The equimarginal principle
- The law of supply
- Important: economics deals with human action/choice. Everything that is outside of this realm does not belong to economics (is not to be explained by economics). It can picture into the context of economic analysis and influence the equilibrium outcome.
- Are the three assumptions redundant?
- \(\)

Econ 340: Week 1

# Introduction

## General information

- Instructor: Ennio E. Piano,
*Department of Economics*&*F.A. Hayek Advanced Program in Politics Philosophy and Economics**, George Mason University* - Email: epiano@gmu.edu
- Class schedule: Thursday, 4:30-7:10 PM, Buchanan Hall D005
- Office hours: Thursday, 9:00-10:30 AM, Buchanan Hall, D137-7 (this is in the Hayek Program Office)

## Course structure and objectives

- 13 lectures over three months and a half
- Main goal: familiarizing yourself with the basic tools of mathematics as they are used by economists.
- Focus on the economics, not the mathematics. The mathematic is instrumental to the economic argument one wishes to communicate.
- Most of the class will cover basic calculus. If we have time we will also cover integration (unlikely). Basic calculus is what you need to know to understand most papers in economics.
- We will mainly look at
*microeconomic applications*. Why? Contemporary macro is basically microeconomics "addressed in a loud voice" (Richard Wagner) - Textbook: Sydsaeter et al.
*Essential Mathematics for Economic Analysis*(any edition after the second would work, available at the bookshop on campus) - Notes: The last two weeks we will review some important applications of the math learned throughout the semester, as they will be prominently features on the final exam. Plus: a good exercise to review the content of the class. I will provide my own notes with the derivation of important models.

## Tentative outline

- See syllabus

## Grading

- 10%: Homework (two sets: #1 to be distributed on September 21st and due October 5th; #2 to be distributed on November 16th and due on November 30th).
**Bring a physical copy to class. Do not send me an electronic copy. I do not accept late work under ANY CIRCUMSTANCES. Also: no make up work.** - 40%: Midterm (3-5 problems depending on difficulty)
- 40%: Final (3-5 problems depending on difficulty, cumulative)
- 10%: Class participation (class participation is
for a class like this one. Mathematics is hard to grasp and master and requires continuous involvement (George Polya). You will be asked to come to the whiteboard and attempt to do an exercise. Everyone in the class will have to do it.*extremely important***You will not be graded based on whether you know the answer, but on whether you participate in the collective effort to find one.** - For this reason, class attendance is strongly recommended although it is not required.

# Why mathematical economics?

## Why economics?

- "There is only one social science..." (Gary Becker)
- Simple tools to explain the social world
- From purposefulness to utility maximization to the law of demand
- The theory of everything

## Why mathematics?

- "The objective of mathematicians is to discover and to communicate certain truths." (Daniel Solow)
- The "clarity argument" (e.g., Paul Romer)

Good mathematical theory is valuable because it encourages clear writing and thereby produces clearer thoughts in the mind of the author. It encourages clear writing by limiting the vocabulary that the author uses. The mathematical equations depend on a limited set of symbols. In good theory, each symbol is tightly bound to specific, precisely defined word or phrase from the vocabulary. The combination of the equations and the words used in the theory give the words and symbols precise meaning. The prose in good theory does not pull in vaguely defined terms that are not in this vocabulary.

## Example (also from Romer)

- Say we want to investigate the relationship between two economic variables: human capital and knowledge
- We start with an abstract conception of the relationship:

\(H\rightarrow A \rightarrow H\)