Welcome to

Microeconomics 3

(advanced undergraduate microeconomics)

at the

Economics Program, University of Copenhagen.

Fall semester 2006

Lecture 1

Today’s programme

² Welcome. Practical information.

² Thematic overview of the course.

² Theme 1: Preference and demand. (1a) utility representations.

² Lectures:

Martin Ljunge.

Tel: 35 32 44 11. E-mail: martin.ljunge@econ.ku.dk

O¢ce: Studiestraede 6 (4th ‡oor)

Homepage: http://www.econ.ku.dk/okomlj/

² Classes:

Frederik Roose Øvlisen

E-mail: frederik.oevlisen@econ.ku.dk

NB: Classes are starting on Friday 15/9 (i.e. Friday this week).

Problem sets can be downloaded from webpage (…rst problem set is alreadyavailable)

http://www.econ.ku.dk/oevlisen/micro3.htm

1 - English Frederik Roose Øvlisen HO 1 Monday 15-17

2 - Danish Frederik Roose Øvlisen HO 1 Wednesday 14-16

3 - Danish Frederik Roose Øvlisen HO 1 Friday 12-14

² The course webpage (time & place, lecture plan, reading list, problem sets,slides, midterm project stu¤, news, links, etc.):

http://www.econ.ku.dk/okomlj/teaching/micro3f6/micro3.htm

² Link to ISISk page :

http://isis.ku.dk/kurser/index.aspx?kursusid=24927&xslt=default

² Some practicalities :

- Please, sign up for the course at ISISk (in order to be added to the mailinglist).

- Remember to check your mail regularly (for example by forwarding thee-mail to your ordinary e-mail address).

- If you have any questions to or technical problems with this, please noticeme (since otherwise I will be assuming that everybody is receiving my e-mails).

² About lectures:

² 2h on Tuesday and Thursday on odd weeknumbers.

² Lecture slides will be posted on the course homepages in pdf-format. (dur-ing the afternoon, the day before the lecture).

² Lectures will aim to:

-motivate and explain new concepts (+ refresh "old" concepts).

-explain in detail di¢cult passages in the book (sometimes Varian is a bitsketchy) .

-provide a guide to what is important and what is less important, in general,as well as for the exams (hopefully you will …nd that there is a closeconnection)

-re‡ect on the perspectives (Is it useful? And for what?)

-give examples of recent research (generalizations, variations, applications)

-NOT go through the material "line-by-line"

² You are invited to ask questions during or after lectures.

² About classes:

² 2h/ week, every week

² Important complement to lectures.

² Exercises will be posted on the course homepages (latest immediately afterthe Thursday lecture).

² Active participation.

² Some hints and recommendations:

² Preparation for lectures:

-Print (and bring) handout for lectures, if available. Add my additions +your own comments and notes to the handout during the lecture.

-Read the course material....(needless to say).

-....and remember to re‡ect on what you learn. Always be sceptical.

² Preparation for classes:

- print (and bring) problem set to classes.

-Try always to solve the problems on your own, before attending classes.

² Create small study groups (for discussing the material, discussing problemsets, for the mid-term project, etc.)

² Aim of the course:

To provide the microeconomic foundations for theoretical and appliedcourses at the masters level.

² Which courses? Here are some examples:

"Micro theory" courses

– Advanced Micro

– Game Theory

– Mechanism Design

"Applied micro" courses

– CGE models

– International Trade and Investment

– Public economics / policy

– Environmental Economics

– Health Economics

– Internet Economics

– Behavioral/Experimental Economics

– Auctions

– Labor Economics

– Governance and Decision Making

and micro theory also plays an important role in, e.g.:

– Growth theory

– Development economics

– Advanced Microeconometrics

etc.

"Financial economics" courses

– Theory of Finance

– Corporate …nance

– Asset pricing theory and portfolio analysis

– Macro …nance

Literature

(Varian) Hal A Varian: Microeconomic Analysis, W.W. Norton & Company,Third edition, 1992 (116 pages).

(DDL) David Dreyer Lassen: Aggregate demand (7 pages).

(TWP) Toke Ward Petersen: An introduction to CGE-modelling and an illus-trative application to Eastern European Integration with the EU, Working paper1998-04 (22 pages) (or Toke Ward Petersen: Introduktion til CGE-modeller.Nationaløkonomisk Tidskrift 135 (1997) pp. 113-134 (22 pages)).

(PNS) Peter Norman Sørensen: Externalities in the Koopmans Diagram (15pages). (also available in Danish)

(GJ-PR) Chapter 6 in Geo¤rey A. Jehle and Philip J. Reny: Advanced Micro-economic Theory, Addison-Wesley, Second edition, 2000 (26 pages)

Total number of pages: 186.

The book by Varian can be purchased in the book store (AKADEMISK BOGHAN-DEL, Studiestræde 3).

Papers by DDL, TWP (English version), and PNS (both English and Danishversion) are available from the course homepage.

TWP (Danish version) and GJ-PR are available from the Study O¢ce.

Lecture plan

1. Introduction to the course. Preferences and demand (2 lectures).

(a) Utility representations. Varian section 7.1 (5 pages).

(b) Revealed preferences. Varian sections 8.7-8.8 (5 pages).

(c) Aggregate demand. DDL (7 pages).

2. General equilibrium (3 lectures).

(a) Equilibrium and welfare in exchange economies. Varian ch. 17 (25pages).

(b) Production economies. Varian sections 18.1-18.8 (16 pages).

(c) Computable General Equilibrium models. TWP (22 pages). Guestlecture by Poul Schou.

3. Imperfections (4 lectures)

(a) Public goods. Varian ch. 23 (18 pages).

(b) Externalities. Varian ch. 24 (8 pages).

(c) Externalities in production economies. PNS (15 pages).

4. Project (1 week).

5. General equilibrium over time and asset markets (2 lectures).

(a) Time. Varian ch. 19. (10 pages).

(b) Asset Markets. Varian ch. 20 (19 pages).

6. Welfare economics (1 lecture).

(a) National income test. Varian section 22.1 (6 pages).

(b) Welfare functions. Varian section 22.2 (2 pages).

7. Social choice (1 lecture).

(a) The problem of social choice and Arrow’s impossibility theorem. GJ-PRsections 6.1-6.2 (11 pages).

(b) Measurability and interpersonal comparability. GJ-PR section 6.3 (8pages).

(c) Justice. GJ-PR sections 6.4 (3 pages)

Assessment

Midterm: 1 week project in the beginning of November (Wednesday 1/11 -Wednesday 8/11)

(NB: no lectures during mid-term project, but classes will continue).

Fall Break: Lectures and classes are cancelled in week 42.

Two-hour closed book exam: Date to be announced.

Theme 1: Preferences and Demand

Focus: A single consumer (today we disregard all questions and problemsrelated to aggregation, interpersonal comparisons etc.)

How should we model consumer preference?

We can choose between:

1. A preference relation

2. A utility function

3. A demand function

Q:

- Does it matter which model we choose "as a starting point"?

- What characterizes a rational consumer?

These are some of the most fundamental - and yet di¢cult - questions ofeconomic theory.

There has been a lot of research in this area, and still new results come upadressing all sorts of variations of these questions.

Our aim: to understand the basic connnections between 1.,2.,& 3.

The link between 1) & 2)

Remember: ONE consumer.

Consumption set X ½ Rk+. A consumption bundle x is a tuple (x1, ..., xk) 2

X.

NB: X is the set of possible consumption bundles. If x 2X then the consumermay, or may not, be able to actually choose x. Given income and prices, thebudget set is the set of a¤ordable consumption bundles.

Usually, we just assume X = Rk+. (And sometimes even X = Rk).

º is a binary relation on X.

x º y is interpreted as "x is at least as good as as y".

Sometimes economists say that "x is weakly preferred to y" which has thesame meaning.

Sometimes the symbol % is used instead of º (usually just a matter of notation)

º is complete (or "total") if for all x, y 2X we have x º y or y º x.

interpretation: any two bundles can be compared.

º is transitive if x º y and y º z implies x º z.

this imposes a certain rationality on º.

NB: Sometimes economists say that º is a "preference relation" if º is acomplete and transitive binary relation. Warning: terminology may di¤er fromauthor to author.

The function u : X ! R is called a utility function.

We say that u represents º if:

u(x) ¸ u(y), x º y,

for all x, y 2 X.

Theorem: If X is …nite, then the following is equivalent:

1) º is complete and transitive.

2) º can be represented by a utility function.

Remarks:

This theorem is not mentioned in the book.

The assumption that X is …nite is crucial: The theorem does not hold forX = Rk

+ (some additional assumption are required).

Important counterexample: Lexicographic preferences (to be discussed inclasses).

From º we de…ne two other binary relations  and » .

x  y if [x º y and not y º x].

x » y if [x º y and y º x].

Two "technical" remarks to Varian Section 7.1.

Varian’s always assumes that º is complete and transitive. This has twoimplications:

1) First, it allows him to state the de…nition of  in the following simpli…edway:

x  y if not y º x.

2) Second, it allows him to give the following alternative (but equivalent!)de…nition of a utility representation:

u represents º if:

u(x) > u(y), x  y.

(Try to think about why this de…nition is equivalent in this case)

General remark:

In a context of utility functions / preference relations, it is always a goodidea to be very explicit about your assumptions (what is meant by a utilityrepresentation? is it assumed that º is complete? transitive? etc.)

Bonus Problem:

If the relation º is complete and transitive, what can we say about the relations and » and (complete? transitive?).

In the following, suppose that º is complete and transitive.

Continuity: For all y 2 X the sets

fx j x º yg, called the "at least as good as" setfx j y º xg, called the "no better than" set

are closed.

How about the "indi¤erence" set fx j x » yg then. Is it closed?

Yes, it is closed. The "indi¤erence" set is the intersection of the "at least asgood as" and the "no better than" sets, which are closed and the intersectionof two closed sets is closed.

The continuity axiom rules out sudden preference reversals.

Weak monotonicity: If x ¸ y then x º y

Strong monotonicity: If x ¸ y and x 6= y then x  y.

Let X = Rk+.

Theorem: Suppose that º is complete, transitive, continuous, and stronglymonotonic. Then there exist a continuous utility function u which representsº.

Proof for k=2:

De…ne

u(x1, x2) = fx j ((x1, x2) » (x, x)g.

By continuity and strong monotonicity such x exists for any pair x1, x2, and itis unique (why?)

x1

x2

45 degrees

(x1,x2) Better

Worse

(x,x)

Figure 1:

Hence u is a well-de…ned function u : X ! R

Does u really represent º ?

Yes!, since (z1, z2) º (y1, y2) if and only if u(z1, z2) ¸ u(y1, y2) (try toillustrate in the …gure).

In fact, u is also continuous (but we won’t prove that. You can …nd the proofin other advanced microeconomics textbooks, e.g. MasColell et al. 1995).

Convexity assumptions on º are often used to guarantee convenient propertiesof the associated utility function (or the associated demand function for thatmatter).

Convexity: For every y 2 X, the upper contour set fx j x º yg is convex.

Strict convexity: For every y 2 X, the upper contour set fx j x º yg isstrictly convex.

x1

u(x1)

x1

u(x1)

NB: The vertical axis should be labelled x2.

Why is strict convexity nice?

Because given a linear budget constraint, the optimal bundle is uniquely de-termined.

(and if it is an interior point on the budget line then ¡MRS = ¡p1p2

.)

x1

u(x1)

x1

u(x1)

Budget lines

NB: The vertical axis should be labelled x2.

Suppose that º is complete and transitive.

If u represents º, then:

1. º is weakly monotone if and only if u is nondecreasing.

2. º is strictly monotone if and only if u is strictly increasing.

"weakly monotonone" and "strictly monotone" are ordinal properties (why?)

Note: Properties of a utility function that are invariant for any strictly increas-ing transformation are called ordinal.

º is convex (= u representing º concave.

but

º is convex ; u representing º concave.

"concave", and "strictly concave", are not ordinal properties (why?)

But the following holds (NB: this is actually not mentioned in Varian’s book):

If u represents º, then:

1. º is convex if and only if u is quasiconcave.

2. º is strictly convex if and only if u is strictly quasiconcave.

"quasiconcave", and "strictly quasiconcave" are ordinal properties (why?)

What does it mean that a function u is quasi-concave?

Def: u quasi-concave if

u(tx+ (1¡ t)y) ¸ minfu(x), u(y)g,for all x,y, t.

If u is concave then it is also quasi-concave.

But the converse implication does not necessarily hold.

x1

u(x1)

x1

u(x1)

Figure 2:

Final remark

Sometimesº (a complete and transitive binary relation) represents preferencesover probability distributions (i.e. lotteries).

In this case, some special assumptions on º ("axioms") have appeal, whichgives us not only existence of a utility function representing º , but - moreinterestingly - a utility function which has a special form.

In fact,it allows us to write up the utility function as an expected utility.

In Problem set 1, you are asked to recall the details of this expected utilitymodel.

Next week

Problem: In real life, your are not likely to observe directly the preference rela-tion º or utility function u (or the demand function x(p, m) for that matter.)

What you might observe is a …nite number of actual choices (with varyingbudget constraints)..

Question: Consider the situation where we have such data. Can we then …nda utility function such that consumer’s choice is perfectly consistent with thisutility function?

We examing this question (and other questions) in the next lecture.