Welcome!Econ A494

Math Econ & Advanced Micro Theory

Spring 2013

Prof. Jim Murphy

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Course Overview

• Introduce you to the mathematical tools behind microeconomic theory

• This is not a pure math econ course• Rather, my goal is to show you how to use and apply the tools of math

econ to microeconomics.• Success in the class requires you to apply the math and interpret the

results• It is not sufficient to just be able to do the math, you also need to interpret it.

• Purpose of the course• Prepare you for grad school• Provide an intellectual challenge• Understand the theoretical underpinnings of microeconomics

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Stuff I assume you already know

• Intermediate Microeconomics• College algebra

• including how to solve systems of equations

• Calculus• Differentiation with single and multiple variables

• Make sure you are comfortable with the rules for finding derivatives!

• Unconstrained and constrained optimization

• The pre-requisites require at least a B in Intermediate Micro and in Calculus. Success in this class requires that you are already confident in your understanding of that material.

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Expectations

• This course is designed to be rigorous and challenging.• For those of you thinking about grad school, my goal is to give

you a taste of what to expect.• Problem sets are important!

• OK to work in groups.• Must be neat, clearly shows your work, proper notation.

• Office hours• Drop by if you have questions—except right before class• At least try to solve the problem before coming by.

• My first question will always be “what have you tried?”

• Do not wait until the last minute!• Do not “punt” by turning in an assignment with notes like “I wasn’t sure

how to do this one…”

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Agenda

• Own your own…• Review Chiang Ch 2 and Hoy Ch 2

• In Hoy, pay particular attention to the notation

• Readings• Silb Ch 1. • Handout #1 – Paradigm of Economics

• Calculus Practice Quiz• Overview – structure of economic theory• Problem set #1 is due Mon Jan 28 or Wed Jan 30

• Calculus review – can start now• Due date depends upon how much review we will need• Will announce due date next class

The Paradigm of Economics

Silb ch 1Handout 1

Introduction

• How would you define microeconomic theory?• Microeconomics addresses the allocation of scarce resources among

competing uses. • Theory of choice• Decisions, or choices, are the consequence of scarce resources• Neoclassical economics:

• Changes in behavior due to changes in constraints faced by rational, self-interested economic actors• Firms• Individuals

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Theories and Models

• Microeconomic Analysis• Theories are used to explain observed phenomena in terms of a set of

basic rules and assumptions.• For example

• The Theory of the Firm • The Theory of Consumer Behavior

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Theories and Models

• Microeconomic Analysis• Models

• a mathematical representation of a theory used to make a prediction.

• Validating a Theory• The validity of a theory is determined by the quality of its prediction, given the

assumptions.

• Evolving the Theory• Testing and refining theories is central to the development of the science of

economics.

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What is a theory?

• A theory is a set of explanations or predictions about various objects in the real world

• Theories consist of 3 parts:1. Assertions, or postulates

2. Assumptions, or test conditions

3. Events, or predictions

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Structure of economic theories

• 1-2. Assertions (postulates) & assumptions• Webster’s definition of postulate:

• “to assume without proof to be true, real, or necessary, especially as a basis for argument.”

• These give structure to the model

• Silb. distinguishes between assertions and assumptions.• Assertions: not observable, but can use to observe implications

• e.g., “Firms maximize profits”• Do not have to be realistic as long as they predict well

• Assumptions: observable. Must be realistic• “a relative price declined”; “consumer income increased”

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Structure of economic theories

• 2 types of initial postulates or assertions• Behavioral postulates

• Suppositions about how agents behave• “Firms maximize profits”• “Consumers maximize utility”

• Market postulates or assertions• Institutional setting for economic agents to perform

• “Firm operates in a perfectly competitive market”

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Structure of economic theories

• 3. Events or predictions• Theory says that the behavioral assertions imply that if the assumptions

are valid, then certain events will occur.• Events are logical implications of postulates and assumptions• Both the test conditions and events must be observable• Meaningful theory yields refutable predictions (which can be tested)

• Example• ASSERTION: consumers maximize utility subject to budget constraint• ASSUMPTION: Consumer income increased• EVENT: Consumption of good increased

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Economics: A theory of choice

• Economics studies individual choice in the face of constraints• “You can’t always get what you want”

• Mick Jagger – Economics degree from London School of Economics

• Choice can be thought of as the interaction of 2 phenomena:• Opportunities or constraints () – observable• Tastes or preferences () – not observable

• Individual’s decision () depends on both and

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Types of variables

• Exogenous variables, or parameters • These are beyond the individual’s control• Agents take these as given• Examples:

• Competitive firm: prices, government policy• Consumers: prices, income, tastes & preferences

• Endogenous, or choice, variables • Variables determined by the model• Choices by individual agents• We will solve for these in our models• Examples:

• Competitive firm: how much output to produce to maximize profits• Consumers: how much of a good to consume to maximize utility

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Example

• Jim is hungry and has $1.00 .• The candy machine has the following:

• Small Snickers $0.70 • Large Snickers $1.10 • Granola bar $0.80

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Exogenous variables, or parameters Jim’s income Candy prices

Endogenous variables, or choice variables Jim’s decision about what candy to buy

The function quantifies Jim’s candy consumption , given prices & income

Paradigm of economics

• Although it may not be possible to predict the original choices made by individuals, it may still be possible to predict how those choices change when opportunities/constraints change .

• Mathematically, can we predict a sign for ?• What if the price of a Granola bar increased?

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Example

• Assertion• Firm is a profit-maximizing monopolist facing a per-unit tax,

• Assumption• The tax rate faced by the firm increased

• Event• A profit-maximizing firm should decrease output when the tax rate

increases.

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Profit maximization

• Goals:• Build a model that describes the decision-making process by which the

firm will choose the level of output to produce.• Set up the objective function

• Use this model to formulate refutable hypotheses about the firm’s behavior when exogenous parameters (constraints) change.• How does the firm react to things they can not control? comparative statics

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Refutable hypotheses

• A refutable hypothesis has the possibility that it will be wrong.• “It will rain tomorrow”

• is refutable because it might not rain tomorrow.

• “It may or may not rain tomorrow” • is NOT refutable because this statement is consistent with all possible weather

outcomes.

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Refutable hypotheses

• Postulates are refutable only through:• making logically valid predictions about real observable events based on

the postulates, • then discovering that the predictions are false.

• Example: Assertion of profit maximization• Certain behavior should be observed under certain assumptions

• Firms will increase output when the price increases.• If this is what we observe, the theory will be confirmed, but not proven.

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Mathematical formulation

• Profit-maximizing firm faces a per-unit tax• Some definitions. Let:

• firm’s total output (endogenous – firm chooses )• total revenue function• total cost function• per unit tax (exogenous, or parameter)

• Set up the firm’s objective function:

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( ; ) ( ) ( ) yMax y t R y C y t y

Typo corrected

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Implications of model

• Some testable hypotheses logically follow from the assertion of profit maximization.

• Later we will show that one prediction of this model is:• If the government increases the tax rate, then the firm will reduce its

output. • Mathematically:

( ; ) ( ) ( ) yMax y t R y C y t y

Typo corrected